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chapter V Quantum Mechanical Scattering Theory
CHAPTER
v
Quantum Mechanical Scattering Theory
I n the preceding four chapters we have given the basic mathematical tools needed for a mathematically rigorous study of conventional nonrelativistic quantum mechanics. T h e actual applications have been limited either to obtaining very general, albeit fundamental, results of quantum mechanics or to computing the bound states of specific systems (e.g., the hydrogen atom). Within the context of nonrelativistic quantum mechanics, scattering theory deals with collision phenomena of particles moving at velocities small by comparison with the speed of light. Since scattering theory is concerned with completely or partially free particles, it is of utmost interest from the experimental point of view. From the theoretical point of view, scattering theory is still the object of intensive research for the physicist as well as the mathematician. We shall give in this chapter the most basic ideas of nonrelativistic quantum scattering theory. I n illustrating these ideas, we shall limit ourselves to two-particle scattering, which is the only case which has received a thorough treatment until now. T h e scattering of n particles, usually referred to as the n-body problem, is a field of active current research, which deserves at least one entire volume for a minimally adequate treatment. A very short introduction to this general case will be given in the last section of the present chapter. 1.
Basic Concepts in Scattering Theory of Two Particles
THEORY AND 1 .l. SCATTERING
THE
INITIAL-VALUE PROBLEM
I n studying in Chapter I1 n-particle systems in quantum mechanics, we were concerned exclusively with problems involving bound states. 391
392
V. Quantum Mechanical Scattering Theory
Such states are related to the point spectrum of the internal energy operator of the system, as described in $5, Chapter 11. We shall now turn our attention to two-particle states, which are not bound states. T h e solution of the general initial-value problem for an n-particle system is in essence contained in Axiom S3 of Chapter IV, §3-namely, if we could prepare by some means a state Yo of the system at the instant t o , and if the Hamiltonian H were given, then, in principle, we could compute the state* Y(t)= e x p [ - i H ( t
-
to)]Yo
at all later instants t . However, such a computational job, besides being in most practical cases unmanageable due to its extremely great computational complexity, is not warranted by the needs of the experimentalist who carries out experiments at the present level of technical sophistication. T h e experiments carried out at present in molecular, atomic, nuclear, and elementary particle physics which involve, from the theoretical point of view, an initial value problem are exclusively scattering experiments. Such experiments are characterized by the fact that the system undergoes a preparatory measurement at some instant t o , a determinative measurement at a later instant t , , and that for most of the time span t , - to of the experiment there is no interaction (or practically negligeable interaction) between the different constituent parts of the system. As a matter of fact, these parts interact with one another only for a very small part of the duration t , - to of the experiment. For the rest of the time they are spatially too far apart to have the very shortrange forces make themselves felt. This and some other features of scattering experiments, which will be mentioned later, make it unnecessary to solve completely the initial-value problem in order to relate the theoretical predictions to the available experimental data. It is clear from the above remarks that for the greater part of a scattering experiment, the different constituent parts of the physical system under observation move independently from one another, i.e., they are “almost free.” I n order to elucidate this intuitive concept of “almost free” motion in a mathematically clear and concise form, it is desirable to consider this concept first in the framework of classical mechanics, which readily appeals to intuition.
* In this chapter we adopt a system of units in which Fz
=
1.
1 . Basic Concepts in Scattering Theory of Two Particles
393
1.2. ASYMPTOTIC STATESI N CLASSICAL MECHANICS Let us consider a two-particle classical system. I n the Newtonian formalism of classical mechanics, the state of such a system is given by a vector-valued function
T h e trajectory of the kth particle, k = 1, 2, can be described by a threedimensional vector-valued function rk(t),which geometrically is a curve in three dimensions. If the two particles of the system undergo the kind of motion which was depicted earlier as typical in a scattering experiment, then the trajectories rk(t),k = 1, 2, should have as characteristic the main qualitative features of the curves in Fig. 4: at the beginning t w to
FIG. 4. Asymptotic states in classical mechanics.
of the experiment and for a relatively long time afterwards the motion of each particle should be “almost free,” i.e., “almost” along a straight line rF(t) at a uniform speed vf, k = 1, 2; then, during a relatively short period, while the particles are interacting with one another, the motion is, in general, very complex; finally, at the end, t w t, , of the experiment and for some time before that, the motion of the particles should be again along straight lines rBut(t)and at “uniform speed” viUt. Now, in classical mechanics a particle is said to be in free motion if its state is of the form r(t) = ro
+ vt,
so that r ( t ) = 0. Hence, at times t w t o , we expect that
V. Quantum Mechanical Scattering Theory
394
and similarly for t (1.3)
M
t,
rk(t)m
ryt + vFtt,
k
=
1,2.
The vectors r? and v p characterize a free state of the kth particle, k = 1, 2, which is called the incoming asymptotic state. Similarly, the free
state characterized by riutand vPt is called the outgoing asymptotic state of the kth particle. It is not a priori clear in what precise sense the approximation signs in (1.2) and (1.3) should be understood. One obvious choice is to take them to mean that (1.4)
I rk(t)- (rp + vpt)~w 0,
I i.k(t)
- vf
I
w 0,
t w to, t
M to,
where I * I above denotes the length of three-dimensional vectors. Since one can reasonably expect that the situation does not intrinsically change if we increase t , - t o , and, moreover, the above approximations might hold with increasing accuracy when to --+ - 00, it seems natural to give to (1.4) the following precise meaning:
lim
t-1-m
1 r k ( t )- Vkin I = 0.
Similar considerations lead to the following corresponding requirements:
The conditions (1.5) and (1.6) define in a precise way the asymptotic states of a single particle. Naturally, the asymptotic states xin(t) and xout(t) of the entire system is the aggregate of the asymptotic states of its two constituent particles. Incoming asymptotic states defined by (1.5) can be thought of as being states in which the particles of the system would be at all times if there were no interaction between the two particles. Similarly, the outgoing states would correspond to the fictitious case of particles which have interacted in the “infinite past,” and afterwards move independently.
395
1. Basic Concepts in Scattering Theory of Two Particles
We note that according to the definitions in (1.5) and (1.6), if asymptotic states exist at all for a given classical physical process, then they are unique for each state, since vf and viUtare uniquely determined by the second of the respective relations in (1.5) and in (1.6), and then rin and riutare uniquely determined by the first relations in (1.5) and (1.6), respectively. It is, however, natural to ask whether asymptotic states in the above sense exist for every classical physical process of importance. The answer to the above question is negative. I n the case of longrange forces, i.e., forces which are of significant intensity even at arbitrarily large separations between particles, asymptotic states in the above sense do not exist. A notable example of such a force is the Coulomb force (see Exercises 1.1 and 1.2). The intuitive physical explanation of this phenomenon is that long-range forces never sufficiently loosen their grip on the particles to allow them to travel in a free manner when they are at large distances from one another. A generalization of the above concept of asymptotic state is obviously necessary to deal with this situation. However, the already introduced concept is of sufficient generality to cope with most of the practically important cases. Even the Coulomb potential is in practice most often “screened,” i.e., instead of having to deal with the Coulomb potential V ( r ) l / r , we are faced with a potential which at large distances decreases much faster than l/r, and for which asymptotic states, in the above sense, exist.
-
STATESAND SCATTERING STATES 1.3. ASYMPTOTIC IN THE SCHROEDINGER PICTURE OF QUANTUM MECHANICS It is straightforward to adapt the concept of asymptotic states defined in the classical two-particle case by (1.5) and (1.6) to the case of quantum mechanics. We can achieve this by stating that in the Schroedinger picture, the “free states” Yin(t) and Yout(t) are, respectively, the asymptotic incoming and outgoing states of a two-particle system in state Y(t)if and only if lim 11 Y(t)- Y’”(t)lI= 0,
t+-m
In order to define precisely the meaning of the concept of a “free state” we must assume that in addition to the “total” Hamiltonian W 2 ) of the two-particle quantum mechanical system we have at our disposal a “free” Hamiltonian Hi2),which would describe a system consisting of two particles with the same physical characteristics (mass, intrinsic
396
V. Quantum Mechanical Scattering Theory
spin, etc.), but which do not interact among themselves. For instance, in the case where Hc2)is given essentially” by
then Hi2) is taken to be essentially 1 2m1
A -1
1 2%
- -A2
3
where the domains of definition of the above differential operators are adequately defined,t as was done in 97 of Chapter IV. However, if both particles move in an external force field, given by a potential Vext(rl , r2 , t), then instead of (1.8) we have
and instead of (1.9) we have
This will be the case when, for example, the system under consideration consists of two spinless charged particles moving in the force field generated by a much heavier particle (such as the heavy nucleus of some atom). Since the heavier particle is practically unaffected by the motion of the lighter particles, such a problem can be treated fairly accurately as a two-body problem in an external field rather than a three-body problem. Consider now the general case when the “free” Hamiltonian Hh2) and the total Hamiltonian W 2 )are given by the self-adjoint operators associated with the system. We say acting in the Hilbert space 3‘(2) that a nonzero vector-valued function Y(f)(t),t E R1, is a free state in the Schroedinger picture if and only if it satisfies the relation Y q t ) = exp( -iH(y’t)
Y‘f’(O), t E R1,
and if Y(r)(O)is orthogonal to the closed linear subspace 3‘;;)of bound of the system with Hamiltonian Hi’). In the absence of states in i?F2)
* See the discussion in $7 of Chapter IV on the essential self-adjointness of the Schroedinger operator. t We note that in the preceding two operators, as well as throughout this chapter, we have adopted a system of units in which fi = 1.
397
1. Basic Concepts in Scattering Theory of Two Particles
external forces both Hi" and its internal energy part Hit), obtained after subtraction from Hi2) of the center of mass motion, are operators = (0). For example, with a pure continuous spectrum so that 2;;) this is the case when Hi2)is given by (1.9), and therefore Hit) is essentially -(1/2m0)A acting on wave functions +(R, r), where A is the Laplacian containing derivatives with respect to the relative position coordinates of r and m, = m,m,(m, mz)-l. I n reality, the particles interact, and any Schroedinger picture state Y(t)in which they can be found, which will be called occasionally an interacting state, has to satisfy the relation
+
Y(t)= exp(--iHW) Y(O),
t E R1.
Due to the unitarity of the operator exp(-2P2)t) we have (1.10) 11 ~ ( t-) ~ " " ( t ) l l= 11 exp(--iH(2)t) Y(O) - exp(--i@)t) Ul""(0)ll
11 Y(O)- exp(iH(%) exp(--i@)t) Y ~ " ( O ) ~ I = 11 exp(i@t) exp(--iH(2)t) Y(O) - Y""(O)\~, =
where "ex" is a symbol which will be frequently used in the future, and which stands for "in" and i'out." In view of the preceding notation, we see that (1.7) implies that
or, equivalently, (1.12)
lim
t+Tm
11 ~ ~ " (0 exp(iH?)t) ) exp(--iH(2)t) Y ( O )= I~ 0,
where the interpretation of the above notation is that one should let t --f -co when "ex" stands for "in," and t -+ +co when "ex" stands for "out." Since, conversely, (1.11) or (1.12) implies that (1.7) is true, we can state the following theorem.
Theorem 1.1. T h e state Y(t)= exp(-iH(%) coming and outgoing asymptotic states Yex(t):
Y(0) possesses in-
(a) if and only if the strong limits (1.13)
Yex= s-lim exp(iHt)t) exp(--iH%) Y(o), t+Tm
exist and belong to
0;)&:;
398
V. Quantum Mechanical Scattering Theory
E 2P @ (b) if and only if there are vectors Yex(0) such that
%At),ex = in, out,
Y(O) = s-lim exp(iH(2)t)exp( - i ~ p ’ t )~ “ ( 0 ) .
(1.14)
t+Tm
An interacting state Y(t)which has an incoming as well as an outgoing asymptotic state is called a scattering state. We recall from relation (3.43) of Chapter IV that
represents in the interaction picture the same state which is represented in the Schroedinger picture by Y(t)= exp(--iH(2)t) Y(0).Thus, pex, ex = in, out, can be thought of as being the asymptotic states of p(t) in the interaction picture.
1.4.
WAVE
MBLLER
OPERATORS
Denote by RY) and R!? the set of all vectors which are the limits (1.14) for some Yex(0) E J f ( 2@ ) Jfi2)when t -+ - 00 and t .--t 00, respectively, i.e.,
+
Rf’ (1.15)
= {f+ : f+ =
~ ‘ 2= ) {f-
~<~i~Q‘”(t)f,f~S OXP’},
:j- = s-lim d 2 ) ( t ) f f , E t++m
s 0~ f ) } ,
~ ( ~ ’ (=t e) x p ( i ~ ( ~ exp(-iHp)t). )t)
We see from (1.14) that if a nonzero vector g belongs to either RY) or R?), then exp(--iH%)g does not represent a bound state, since it is intuitively obvious that a bound state cannot be asymptotically free. Hence, it seems natural to assume that in any realistic quantum mechanical two-particle theory
Rt) = RL2) = %?)A, where 2i2’ is the space of bound states. We shall see later that the conclusion of this heuristic argument is confirmed when dealing with potentials which usually occur in practice.
Lemma 1.1.
If A(t) is a uniformly bounded family of operators,
It A(t)tl < c,
--co
< t < +a,
1. Basic Concepts in Scattering Theory of Two Particles
399
then the set M of all vectors f E & for which s-limA(t)f, t+tO
-a < to < +a,
exists is a closed linear subspace of
&.
Proof. Since the existence of the strong limits of A(t)f and A(t)g for t -+ to implies the existence of the strong limits of A(t)(af bg) = aA(t)f + b A ( t ) g ,it follows that M is a linear subspace of x. Let f be a vector in &? which is the strong limit of a sequence f, ,f , ,... of vectors from M. If
+
gn = s-lim A ( t ) f n , t+to
then we can write
for all t sufficiently close” to to ,we see by glancing at the right-hand side of (1.16) that
II A(t1)f
- A(t2)fII
<
f
for all such t , , t , which are sufficiently close to t o . Hence, the strong limit of A ( t ) f for t -+ to exists, and thereforefE M. Q.E.D. The operator function L P ( t ) defined in (1.15) assumes as value a unitary operator for every value o f t E W, and consequently it is uniformly bounded: -a < t < +a. 11 rn(2’(t)lI = 1,
* If to =
&a,sufficiently “close” to to means sufficiently large.
400
V. Quantum Mechanical Scattering Theory
Thus, we can apply Lemma 1.1 to conclude that the families MT’ of ) Afb? for which the strong limits Q c z ) ( t ) f for all vectors f~ X c z0 t -+ -a and t -+ +a,respectively, exist are closed linear subspaces 02;;). Hence, the family of Mf)
=
MF) n MF)
of all vectors f for which both limits, s-lim W ) ( t ) f = j+ , t+Tm
exist is a closed linear subspace of X ( 2 ) . Let EM?’be the projectors onto MY’, (1.17)
E M y % = MF’.
Then, according to the above definition of MY), the limits
exist for all vectors f E X , because E $ ’ ~ EMF). Then the operators (1.18)
exist and are defined on the entire Hilbert space X ( z T ) .h e operator L??) is called the incoming M0ller wave operator and QF) is called the outgoing M0ller wave operator. We shall refer to these two operators as the ‘Lin’’ and “out” wave operators. 1.5. THESCATTERING OPERATOR T h e scattering operator (or Soperator) S for the problem at hand is defined in terms of the wave operators
Qz),
Qz)
Since the wave operators are strong limits of uniformly bounded operators Qcz’(t), /I Q(z’(t)\\ 1, they are bounded. Furthermore, we have seen that Q, are defined everywhere in 2. Hence, the adjoints QF)*exist and are bounded (see Chapter 111, Theorem 2.7). Thus, S2) is a bounded linear operator, defined on the entire Hilbert space iFZ).
<
1. Basic Concepts in Scattering Theory of Two Particles
401
The importance of the scattering operator lies in the fact that many quantities which are readily measurable by scattering experiments are easy to compute if we know S ( z ) . T o provide an illustration, assume that a scattering state Y-(t) of the system has been prepared at the beginning of the scattering experiment, and that at the instant to of the preparatory measurement Y(t)was essentially a free state YF(t),
Furthermore, assume that a determinative measurement at an instant t , , long after the two particles of the system have interacted, can determine the value of the observable Etl of the form
which is represented by the projector on the scattering state Y-(t) at t = t , . Then, if Y+(t) is normalized, themean value of Etl is given by the expression (1.20)
PY++Y-
=
W+WI Etl Y+(t1)>= IV+(t1) I y--(h))IZ*
We recall from the discussion at the end of $1 in Chapter IV that the above expression is usually referred to as the transition probability from Y+(t)to Y-(t),due to the usual assumption that if the outcome of the determinative measurement of Etl is X = 1, the system is left afterwards in the state Y-(t). We note that the expression in (1.20) is actually independent of t , , (1.21)
due to the fact that the operator exp(-zW2)t) is unitary. Let Yout(t) be the outgoing asymptotic state of Y-(t): lim
t++m
11 ~ - ( t) ~ ? ~ ~ (= t >0,l l
Since according to (1.14)
Y-out( t ) = e x p ( - - i ~ t ’ t )YP(o>.
402
V. Quantum Mechanical Scattering Theory
we arrive at the following expression for the transition probability:
This expression is very convenient and helpful in practice, since it involves free states which in the typical scattering experiment are computationally easy to derive from the available experimental data.
SCATTERING CROSSSECTION 1.6. THEDIFFERENTIAL In practice, the typical two-particle scattering experiment is of the kind depicted in Figure 5 : a beam of particles 6, impinges on a target ector
INCIDENT B E A M COLLIMATING
'* Yo
FIG. 5. Scattering of a beam of particles on a target.
consisting of particles 6,. Assuming that the particles in the beam do not interact with one another, and that each particle 6, in the beam interacts with only one particle 6, in the target, the experiment can be viewed as consisting of a large ensemble of independent scattering experiments of two-particle systems" 6 = @I 9 6 2 > .
* The two mentioned conditions can be satisfied by taking a "weak" beam of particles (i.e., a beam with few particles per unit volume), and taking for the target a slab of material which is sufficiently thin to eliminate the possibility of a particle 6,scattering from more than one particle in the target.
403
I . Basic Concepts in Scattering Theory of Two Particles
The above experimental setup is ideally suited to determine the percentage of the particles in the beam which can be found in a given volume of space, occupied by a detector, after they have been scattered by the target. I n any scattering experiment the detector is placed as depicted in Fig. 5, i.e., sufficiently far from the target so that it never interferes with the scattering process. Hence, the experiment obviously provides information about the probability that a particle of the beam, coming towards the target from the direction 0 <40
wo = (+0,4)),
< 277,
0
< 80 < -,
will be scattered within the solid angle dw around the direction = (4, el,
(1.23)
o < + G 2-,
o G e < T.
If N denotes the number of two-particle scatterings per unit time, and dE] of incident if the solid angle dw and the energy range [ E , E particles are sufficiently small, then this probability can be written in the form N n ( E , w,, , w ) dE dw, since it is obviously proportional to N as long as the two earlier mentioned conditions on beam and target are satisfied. It can be expected that T(E,wo , w ) is dependent on the energies E and E’ of the particles of the system 6 = (6,, S,} before and after collision. However, since energy is conserved, we must have E = El, so that the dependence on E‘ does not have to be displayed. I t has become customary to orient the frame of reference so that y50 = Oo = 0, and to call the quantities
+
(1.24)
o(E, w )
=
( N / J oT(E, ) 0,w ) ,
a(E) =
1
a(E, w ) dw
QS
(where Jo is the incident JEux, i.e., the number of incident particles per unit time and unit beam cross section) the dzjferential cross section and the total cross section, respectively. The notation du/dw instead of a(E, w ) is often encountered in literature. 1.7. THETRANSITION OPERATOR In potential scattering W 2 and ) Hi2’ are given by (I .8) and (1.9), and one can easily relate o(E, w ) to the S operator. T o establish this relation, it is computationally advantageous to express S(2)in terms of a new operator, (1.25)
S‘2’
=
1 - 2Ti77‘2’
where the operator F 2 )is referred to in physical literature as the transition operator or T operator. I t owes its name to the fact that in the case when
404
V. Quantum Mechanical Scattering Theory
there is no interaction present, the system stays in the same free state (i.e., there are no “transitions”), and the operator F 2 )is zero. We easily establish that T ( 2 )= 0 when there is no interaction by noting that the “absence of interaction” means that = Hi2) and therefore LF2)(t)= 1 , i.e., Qf)
~
fp= s(2)= 1.
We have seen already in 97 of Chapter I1 that in the absence of external forces the Schroedinger operator Hi2’ of a two-particle system can be written in the form H p
=
H p
+z p
where HA2)is the center-of-mass energy operator and Hi2)is the internal energy operator. T h e Hilbert space A?(2)= L2([w6)of wave functions #(R, r) is obviously the tensor product ( 1.26)
x(2) =
#t’@
#(I)
of the Hilbert spaces A?‘,“ = L2([w3)and &(I) = L2([w3)of wavefunctions $,(R) and #(r), respectively, where R and r are the center of mass variables given in (7.2) of Chapter 11. It becomes evident from looking at the differential operators (1.27)
-
and (7.3) in Chapter I1 that HA’) = HA’) @ 1 and H12) = 1 @Hi1), where HL1) and Hi1) act in XL1)and &?(I), respectively. It is natural to require that, in general and when no external forces are present, the free and total Hamiltonians can be written in the form
Since the operators HL2’ and HAY’, as well as HA2) and Hj2’, obviously commute, we can use Theorem 2.4 of Chapter IV to infer that exp( -i~,’2’t) = exp(--iH,(2)t) exp( - i ~ $ ) t ) , exp(iH‘”t) = e x p ( i ~ f ) t )exp(iHi2)t).
1. Basic Concepts in Scattering Theory of Two Particles
405
The above two relations are derivable directly from (2.13) in Chapter I V by noting that in the above case all operators are bounded and defined on the entire Hilbert space A?(2). From (1.28) we get (1.29)
d ” ( t ) = exp(iH(2)t)exp( -iHf’t)
Inserting the above expression for L P ( t ) in (1.18) and (1.19), we easily arrive at the following result.
Theorem 1.2. If the free and total Hamiltonians Hi2) and H ( 2 )of a two-body system can be decomposed into sums (1.28) of commuting Hamiltonians for center of mass and internal motion, then
where QF) and S1) are operators in the internal-motion Hilbert space of the tensor product A?(2)= 26)8 A?(1). The above theorem shows that in case of two-body potential scattering in which no external forces are involved, we can work in the Hilbert space 2 ( l ) which is related to the internal motion of the system. If we work in momentum space, we can introduce the relative momentum variables p, related to the relative position vector r. We might hope that operator could be written as an integral operator in which, the S1) in case of spinless particles, would be of the form (1.30)
However, we shall see in the $2 that S1) commutes with the free Hamiltonian Hi;). Lemma 1.2 shows that, on account of this fact, the representation (1.30) is not feasible.
Lemma 1.2,
Let K be a bounded integral operator on L2(R3)
406
V. Quantum Mechanical Scattering Theory
with a kernel K(p, p’) which is Borel measurable in Rs. If K commutes with the self-adjoint operator Hi:),
then K is the zero operator.
Proof. If f(p) is a function of compact support A , then E p ( A ) f =
f E gH;;) , where Ep(d) is the spectral measure obtained as the product
(1.13) in Chapter IV of the spectralmeasuresof the three momentum component observables: (EP(4g)- (P)
= XA(P)g’(P)
T h e operator HA:)Ep(d)is obviously bounded, and commutes with K ,
=0
(almost everywhere).
Since the above expression has to vanish almost everywhere for all f(p) of compact support, we have =0
(almost everywhere in R3 x R3 = Re).
Now, the set of points in R6 for which I p I = I p’ I (called the energy shell) is of Lebesgue measure zero. Hence, we deduce that K(p,p’) vanishes almost everywhere in R6. Iff (p) and g(p), f,g E L ~ ( R ~are ) , bounded functions of compact support, then g*(p) K(p, p’)f (p’) is certainly integrable on R6 since K(p, p’) is Borel measurable and vanishes almost everywhere. From the relation
we get by applying Fubini’s theorem
(g I K f )
=
j
R
dPL?*(P)
j
R
dP’K(P9 P’MP’) = 03
for all f,g E WE(R3), Since W 3 W ) is dense in L2((a3),we get Kf = 0 for all f E VE(R3). Thus, K is a bounded linear operator which is zero on a dense subset V@3) of L2((a3),and therefore K = 0. Q.E.D.
407
1. Basic Concepts in Scattering Theory of Two Particles
0
Let
us
introduce
spherical
< +GO,0 < 8 < T , 0 < 4
coordinates
(p, w)
=
( p , 8, C$),
< 27r, in the space R3 of the variables
p. We shall see later in this chapter that the T ( l )operator can be written as an integral operator with respect to the variable w, (1.31)
over the set Q,, defined in (7.10) of Chapter 11, which is essentially isomorphic to the unit sphere in three dimensions. T h e function T ( l ) ( p ;w, w’) is uniquely determined by the operator T ( l ) (see Exercise 1.6) if we restrict ourselves to continuous kernels for the integral operator in (1.3 1). 1.8. THET-MATRIXFORMULA FOR THE DIFFERENTIAL CROSSSECTION T h e differential cross section o(E, w) can be computed from the p ; w, w’) which occurs in the integral representation (1.31) function F)( of the T operator. I n order to arrive at an expression relating o(E, w ) and T ( l ) ( p w, ; w ’ ) , we have to express in theoretical terms those essential features of a scattering experiment which enter in the definition of the differential cross section. I n any such scattering experiment, a beam of particles scatters from the particles of a target. T h e experimental setup is such that the relative momentum p of the incoming particles is well defined. Let us introduce spherical coordinates ( p , 8,+) in the p space, and suppose that the incoming particles are prepared in a scattering state Y(t),which is such that p lies within the set (1.32)
Denote by Yin(t) the incoming asymptotic state of Y(t).Suppose that Yin(0)is represented by the function (see Exercise 1.5) (1.33)
$in(p,P) = 4@) $dPL
of the center of mass momentum P and relative momentum p. Let us denote by E‘P’(B)the spectral measure (1.34)
(~(pYw)(P, P)
= XB(P)
$w,PI-
408
V. Quantum Mechanical Scattering Theory
It can be shown (see Exercise 1.8) that for any Bore1 set B
E
g3
Since the support of the function +(P, p; t ) representing Y(t)is within R3 x C(O),we deduce from the above relation that
Hence, $o(p) vanishes almost everywhere outside C(O). Denote by Yout(t)the outgoing asymptotic state of Y(t).It is easy to show (see Exercise 1.4) that ( I .36)
Yout(o)= S'2'Yi"(0) = ( 1 x S'")Yin(o).
Hence, by applying the same type of argument which led to (1.34) to Y o u t ( t ) , we obtain
Consequently, the probability P(lmo+ Iu0,)of finding that the relative momentum p of the system, a long time t after the scattering had taken place, is within the cone
c(+) = {P: (0, c) EL,^+,
L ~=,[e,', eo' + 4~x
[+ot,
+or
+4
1
is given by (see also Exercise 1.12) (1.37)
l(S(1'#o)(P)12dP.
We shall see in the next section that S1) commutes with EHA?(B) (see Theorem 2.3), where (E
(B )#)-
(PI
=
X{pa,2m,EB)(P) $(PI.
409
1 . Basic Concepts in Scattering Theory of Two Particles
It follows that for the complement B, E 39, of the set {p2/2m:p we have EHdt’(Bl) s(l’#o = S‘l’EHd?(B,) #o
=
E
I,,},
0
since E%’(B1) t,!Jo = 0 because $,(p) vanishes outside e(O).This means vanishes when p2/2m$ B, , i.e., when 1 p I 4 Ipo. Therethat (,S1)t,!J0)(p) fore, if the integral in (1.37) is expressed in spherical coordinates, the integration in p extends in effect only over the set Ipo. Thus, we can write
Let us consider the case when the direction of relative motion of the two particles in the system has changed as a result of intersection, i.e., n IUo, = 0 . If we use the definition (1.25) of when wo # wo’ and IUo the T operator to write S“’#, = #o - 27TiT‘1’#,
and recall that result
go(p,w ) vanishes when w is outside Iwo, we arrive at the
Hence, we get for the probability density P(wo‘)(per unit solid angle) of scattering in the direction wo’
provided that ( T ( l ) # o ) ( pw, ’ ) is continuous in w’ in some neighborhood its integral of wet. Substituting in the above relation for the operator F1) representation (1.31), we obtain
410
V. Quantum Mechanical Scattering Theory
where the integrations in o1and w, extend over the unit sphere in three dimensions. The request that ( T(ljh0)(9, 0’) be continuous can be met by demanding that T ‘ l ) ( p ;w’, w) be continuous in w‘. At this stage one must remember that the expression (1.41) is the probability density of scattering of two particles in the relative direction wo‘ with respect to one another. We have already indicated, however, that in a scattering experiment a beam of particles is being scattered on a target consisting of a large number of particles. Thus, in a scattering experiment, a great many two-body scatterings are observed simultaneously. These scatterings between particles in the beam and particles in the target are taking place at different locations with respect to the laboratory frame of reference. Hence, in order to arrive at an expression for the differential cross section, an averaging of P(wo’) over all these two-body scatterings at different ‘locations has to be carried out, If a particle 6,” in the target is at a location ro in relation to another particle G,’, while 6,’ is in position r with respect to a particle G1 in is at the location r ro in relation to the incident the beam, then 6,” particle 6,. Since, with the exception of location, all the other variables are the same, we conclude that if of 6,’ and 6,”
+
*m represents Yin(0)for {G1, G,’}, the wave function
represents Yin(0) for { G I , G,”}. By taking Fourier transforms, for {GI, G2”}we easily get &,(PI
J~(P),
= ex~(--~ro)
where J0(p) refers to {G1, G,’}, and JrJp) refers to {GI, G,”}. Hence, the probability Pro(wo’) for {Gl, S,”} is (1.42) Pro(w,,’) = 477’
J”
J”
d p p 2 dwl d o z T*(p;w,,’, 1%
wl)
instead of (1.41). We shall now make the assumption that the scatterers in the target are uniformly distributed, so that No scatterers interact with the incident particles per unit target volume and unit time. Then the average number
1. Basic Concepts in Scattering Theory of Two Particles
411
of particles scattered* per unit time and target volume in the direction is
w,'
(1.43)
4-%7= No J V P,(W,')
dro
1,dr, 1dppa 1
= 4?raN,
dw,
x Tc')(p;WO',
w8)
dw, exp[iro(p, - p,)] F1)* ( p ; w',
$O*(Pl) $O(o(pBI)IP1-P~-o
wl)
9
assuming that the scatterers in the target are sufficiently dense that we may, with sufficient accuracy, replace the sum over the individual scatterers with an integral over the volume V of the target. We shall evaluate the integral (1.43)under the assumption that the target is a slab of thickness 6 , which extends to infinity in the directions x, and yo (see Fig. 5). Setting k = pa - p1 and reversing the orders of integration in (1.43), we obtain (1.44)
Y(W0))
= 47r2No
I dw, I dr, j d p dw8p2eikroF(w,';p,
w , ,ma),
where we have introduced the function (1.45)
WJ;; P , w1 ,w2) =
T"'*(p; wgl,
4T ' Y p ; w;, 4$ O * ( P , 4 JO(P'
WZ).
Since I p1 I = I p2 I = p, only two of the components of k are independent. We can substitute for w 2 the variable w which represents the spherical coordinates 0 and of p2 in a coordinate system with its z axis in the direction of p1 . An easy computation yields
+
By substituting for o
p2 dp dw
= p 2 sin ll
=
k,
=p
sin ll cos+,
k,
=p
sin ll sin 4.
(8,+) the variables k, and
k, , we obtain
dp dll d+ = p2 sin 0 a@, a(e2 ' +') dp dk, dk, k,) 9
=
1 dp dk, dk, . cos e
* In practice, the scatterers in the target are approximately at rest with respect to the laboratory frame, so that the relative scattering direction w i computed in the center-ofma88 frame is very often almost equal to the scattering angle measured in the laboratory reference frame.
412
V. Quantum Mechanical Scattering Theory
Using the properties of Fourier transforms applied to the variables
k, , k, and their conjugates xo and yo , we obtain dxo dY0 = 4n2
J dk, dk, eirokjdP , B1F ( w o ’ ; P ,
j F b o ’ ;P ,
w12
w2)
cos 0
1.0
I
k,=ky=O
w1
, w2)
P2 dP.
Since obviously dp dw, = dp dw, the above relation can be used in = computing the integral in (1.44). When k, = k, = 0, we have and O1 = 8, or = and 8, = 7~ - 0 2 . Since F(wo’;p , wl, w,) vanishes when w, or w2 are outside Iwo, it follows that when Iwois sufficiently small only the first alternative = and = 8, yields a nonzero result. This means that k, = 0, 8 = 0, and the integration in zo in (1.44) can be immediately performed:
+, +,
+, +,
+,
&C
We note that No6 = N is the total number of scatterings per unit time and target area, i.e., the incident flux Jo . Let us choose dp, , Ad,, and A+o very small, and the direction of the incident beam along the z axis, so that 8, = +o = 0. Then, since
we get by the mean-value theorem of the integral calculus
We note that the application of the mean-value theorem is warranted ; w ) is continuous on account of the implict assumption that T ( l ) ( p w’, in some neighborhood of w = (0,O). Since v(wo’)is the number of scatterings in the direction wo’ per unit time, we have” according to the definition of a(E, wo) v(-w0‘) a(E, wo’) = ~.
* The minus sign is due to the choice 8,
Jo
0. Since the origin of the coordinate system is at the target, this choice means that the beam travels in the -e, direction. However, in physics the convention is that in the formula for o(E, w ) angles should be measured from the direction of the incident beam, i.e., in a coordinate system with its zoaxis pointing in the -e, direction (see Fig. 5). = #o =
1. Basic Concepts in Scattering Theory of Two Particles
413
Thus, we have obtained the expression (1.47)
-ur, 0)(2
for the differential scattering cross section under the assumption that T ( l ) ( p w; l ' , wl) is continuous in wl' and w1 when wl' is in a neighborhood of w' and w1 in a neighborhood of (0,O).
EXERCISES
+
1.1. Let rEx vIXt,R = 1, 2, be the asymptotic states for the state rk(t),k = 1 , 2, of a classical two-particle system. In the center of mass coordinates (7.2) of Chapter I1 the state of the system is given by R(t) and r(t), where R(t) = R, V,t describes the uniform motion of the 1 - TiX, center of mass of the system. Show that if rex = rex vex = vFx - vgx, then I r(t) - (rex vex t)l ---t 0, (l/t) r(t) -+vex and I r(t)l - t I vex I rex- vex/Ivex I when t -+ ~ C Qrespectively. ,
+
+
--f
1.2. Prove that a system of two particles interacting via a potential V ( r ) = l / r (gravitational or Coulomb potential) does not possess any asymptotic states satisfying (1.5) and (1.6).
1.3, Show that the adjoint K* of a bounded integral operator K ,
defined on L2(Rn), is an integral operator of the form
1.4. Prove that if Yex(t)E M,(ex = in, out) are the asymptotic states of Y(t),and S is the scattering operator, then !Pout(0)= S!Pin(0).
1.5. Show that the formula (1.47) for the scattering cross section stays valid in the general case when Yin(0)is represented by an'arbitrary wave function p ( P , p ) , which is not necessarily in the form of the product occurring in (13 ) . 1.6, Prove that if T ( l ) ( p w ; , w ' ) and 7'A1)(p;w , w ' ) are two kernels satisfying (1.31) for a given operator P ) ,then T f ) ( p ;w , w ' ) = T ( l ) ( pw; , w ' ) almost everywhere with respect to the measure p!') x p p x p p on R1 x sZp x 12,.
414
V, Quantum Mechanical Scattering Theory
1.7. Suppose Y(t) and Y,,(t),t E W,are vector-valued functions assuming values in the Hilbert space %, and that there is a constant C such that I/ Y(t)lI C and 11 Y,,(t)ll C for all t E W.Let A be a bounded 11 Y(t)- Y,,(t)ll= 0 implies that operator on 2. Show that liml+FFm
<
<
1.8.
Use the result of Exercise 1.7 to prove that ‘t
lim (Y(t)I E(p)(B)Y(t)>= (Yex(0) I E(D)(B)Yex(O)), w
+
where (HA2)$)“(P, p) = (P2/2M0 p2/2m,) $(P, p), with M,, and mo denoting, respectively, the total and the reduced mass of the system. 1.9.
Consider the case of one-particle wave mechanics, when
H, 2 (1/2m)A. Using the well-known relation (exp(-iH,t)+)“(k)
=
exp[ -i(k2/2m)t] $(k)
derive that, for functions $(r) which are Lebesgue integrable as well as square integrable, (exp(--iH,,t)+)(r)
=
J
( m / 2 ~ i t ) ~ / ~exp[--i(m/2t)(r
-
r’)2]+(r’)dr‘
R3
almost everywhere in R3. 1.10.
Verify that the operators Uil) and U\2’, defined for real t # 0
by ( ut(l)+)(r) = (m/it)3/2 exp[-im(r2/2t)l
(U,(2)+)(r)= exp[-im(r2/2t)l
$(rnr/t),
+(TI,
are unitary. Use the result of the preceding exercise to show that for H , 2 -(1/2m)h p f O t -
-
u, u, . (1)
(2)
1.11. Use the result of Exercise 1.10 to prove that the operator families eciHol and U f ’ defined there satisfy Iim ll(e-iHOt - U,(1))+ 11
t+Fm
for all,$ EL~(W).
=
o
415
2. General Time-Dependent Two-Body Scattering Theory
1.12.
Consider the cones
c(+) = {(r, e,+): y 2 o , ~ G , e G 0,
+ ~ 84, , G + G + A+) +o
and C(-) = {r: -r E C+}in R3. Use the results of Exercises 1.7 and 1.1 1 to prove that for H, 2 -(1/2m)A
Remark. This result indicates that asymptotically in the future the probability of finding the particle spactially located in the cone C(+) is equal to the probability of finding its momentum within F+).
2. General Time-Dependent Two-Body Scattering Theory 2.1.
THEINTERTWINING PROPERTY OF WAVEOPERATORS
In $1 we defined the Moller wave operators SZ, by means of strong limits of SZ(t), Q* =
(2.11
:diz Q(t)
~ ( t=) eiHt
EM+
e-iHot
,
H
=
H*,
Ho = Ho*,
in the time parameter t . In this definition, the closed linear subspaces M, of X are spanned by all vectorsf which are such that eeiHot f is a free state and the respective strong limits s-limt-,TmSZ(t)f exist." T h e version of scattering theory based on this procedure of computing the wave operators, and eventually the scattering operator (2.2)
s = Q-*Q+,
is called time-dependent scattering theory. In this section we shall derive some of the basic properties of wave and scattering operators, which follow from the general definitions (2.1) and (2.2), and which do not require the specification of Ho and H within
* A more common way of defining Q+ features EH(SL)instead of EMi in (2.1),where S,", is the absolutely continuous spectrum of H (in practice, Sg = S,", though in general S,",C S,"). Since, in general, the strong limit of Q(t)EH(S)E for t co might not --f
exist, this alternative definition gives rise to the question of existence of wave operators. In our version, the wave operators always exist, but one must check in each particular case whether M i is large enough to warrant a physical meaning to Q+ and S.
416
V. Quantum Mechanical Scattering Theory
the confines of some particular model. It can be easily seen by comparing (2. I), (2.2) with (1.1 8), ( I .28), and (1.29) that the derived theorems apply and S 2 as ) well as to their counterparts Qg)and to the operators in the “internal motion” Hilbert space %’(l). Naturally, in the first case one should take H = and H , = Hi2),while in the second case one should take H , = Hi1)and H = H(l).
Qz)
Theorem 2.1. properties
T h e wave operators 62, have the intertwining
(2-3)
Q*pHO
=
eitHQ+ ,
t
E
R1,
Proof. If fl belongs to the respective subspaces M, , then according to the definition of M, , the respective strong limits Q ( t ) f l , t -+ co, exist. Thus, we can write
11 exp(iHt) Q,fl =
lim
t14m
= lim
tl-’Fm
=
lim
tl+FW
- s-lim[exp(iHt,) exp( -iHOtl)] exp(iH,t) fl I/ tlpTm
(1 exp(iHt) Q*fl
- (exp(iHt,) exp(--iHotl)) exp(iH,t)fl
I/ exp(iHt) Q+fl - exp(iHt) exp[iH(t, 11 Q+fl - exp[iH(t,
-
t)] exp[-iH,(tl
Hence, exp(iHot)fl also belongs to M,
- t)] exp[-iHo(tl -
- t)]fl
(1
t)]fl j j = 0.
, and since
s-lim exp(iHtl) exp( -iH0t,) exp(iHot)fl tl+kC
(1
= Q+ exp(iHot)fl
,
we have proved that
for all t E R?. Now, if f2 E M i fact, for allf, E M,
=
S 0 M, , exp(iH,,t)f, also belongs to M i . I n
= (fi I exp(--iHot)f,)
= 0,
2. General Time-Dependent Two-Body Scattering Theory
417
since exp(iHot)f, E M, according to the preceding discussion. Thus, due to definition (2.1) of Q, , we have Q, fi = 0 and Q, exp(iHot) fi = 0. Consequently (2.6)
Q*
exp(iHot)fz
=
fi E M i .
exp(iHt) 0,fi = 0,
Since every f E 2 can be decomposed in the form
we immediately obtain (2.3) from (2.5) and (2.6). The relation (2.4) can be easily derived from (2.3) by methods frequently employed in Chapter I V (see Exercise 2.1). Finally, by making use of the resdlt (3.6) in Theorem 3.1, Chapter IV, and of the fact that 52, are bounded and therefore continuous, we easily derive that for f E aH0 Q*Hof = Q* s-lim t+O
= s-lim t+O
1 zt
1
zt (exp(iH,t)
1
= s-lim t-0
zt
1
at
1)f
(Q* exp(iHot)- 521) f
= s-lim 7-(exp(iHt) 0, t+O
-
-Q ),
(exp(iHt) - 1) Q*f
f =
HQ*f. Q.E.D.
2.2. THEPARTIALISOMETRYOF WAVEOPERATORS We recall that in the last section M, were defined to be the sets of all for which the respective limits vectors f E 2 02$)
f*
= s-lim Q ( t )f t+im
exist, while R, are the sets of all vectorsf, , According to Lemma 1.1, the sets M, are a closed linear subspace of 2.We shall prove the same statement for R, , by showing that Q, are partially isometric operators.
Definition 2,1, A bounded linear operator U defined on the Hilbert space lf is said to be partially isometric with initial domain M and final domain N if U maps M isometrically onto N and maps M I = lf 0 M into the zero vector, i.e., if 11 Ufll = llfll f o r f E M and 11 Ufll = 0 f o r f e Z 0 M.
418
V. Quantum Mechanical Scattering Theory
Theorem 2.2. The sets R, are closed linear subspaces of H ;the wave operators SZ, are partially isometric with initial domains M, and final domains R, , respectively. Proof. According to the definition of R, , any vector f, E R, is the strong limit f + =A ;T Q(t)f = Q+f
. Moreover, since D(t) is unitary,
for some vector f E M,
II Q+f II
(2.7)
II Q(4f II
= t-1-m lim
= llfll
iff E M, . Hence, the restriction 0, of SZ, to M, has an inverse 0;' , which is bounded on account of (2.7). T o prove that R, is closed, let h E X be the strong limit of a sequence h, , h, ,... E R, . By (2.7), f, = 6$z1 , f , = ,... is a Cauchy sequence of elements from M, . Since M, is closed; fi ,f, ,... has a limit f E M, . Therefore, h = 0,f E R, , which proves that R, is a closed set. The linearity of R, is an obvious consequence of the linearity of 0,. Thus, R, is a closed linear subspace of X,and 0,maps M, isometrically onto R, . Moreover, if g _L M, , then Q+g = s-lim Q(t) En+g = 0. t+-m
Thus, 0, is partially isometric with initial domain M, and final domain R, . The corresponding result for R- is established in a similar manner. Q.E.D.
Lemma 2.1, If U is a partially isometric operator on # with initial domain M and final domain N, then U* is a partial isometric operator with initial domain N and final domain M, and the following relations are true: (2.8)
U*U
= EM,
(2.9)
UU*
=
Proof. If g 1M, then Ug Hence, iff E M, then (g
=
EN.
0 and consequently U*Ug = 0.
I u * w =
for all g E MI, and therefore U* UfE M. Since M is linear, we also have f - U*Uf E M for any f E M. On the other hand, (h
If-
U * U f ) = ( h I f ) - (Uh I U f ) = 0
2. General Time-Dependent Two-Body Scattering Theory
419
for all h, f E M (see Exercise 2.2), which implies that f = U*Uf. Thus, U*Uf = f for all f E M and U*Ug = 0 for all g 1M, i.e., U*U = EM and (2.8) is established. We shall prove now that U* is partially isometric. If h E N, there is a unique f E M such that h = Uf, 11 f I( = 11 h 11. Consequently, U*h = U*Uf = f by (2.8). On the other hand, if h, 1N, then (U*hl I fl) = (h, I Ufl) = 0 for all fi E %, since Ufl E N always. Consequently, U*hl = 0 whenever h, 1N. Thus, U* is partially isometric with initial domain N and final domain M. By reversing the roles of U and U* in (2.8) we show that (2.9) is true. Q.E.D.
If we apply this lemma to the operators Q, relations :
, we arrive at the following
(2.10) (2.1 1)
In particular, if M, = M- = M, = Z , we would have Q+*Q* = 1. We shall see in later sections that in problems of physical interest Mi’) coincides with the subspace #(2) 0Xh;) of all vectors which are = ZL1) 0&?hi), where [see (1.26)] orthogonal to the subspace C Z(l)is the space spanned by the eigenvectors of the “internal” free Hamiltonian Hit) given in (1.28). Ordinarily, Hit) has a pure = (0) and Mi2) = X ( z ) . continuous spectrum, in which case
%A2)
2.3. PROPERTIES OF THE S OPERATOR We turn our attention now to the S operator, giving special notice to the practically very important property of its “unitarity.”
Theorem 2.3. The S operator and the T operator commute with the free Hamiltonian Ho . Proof. Let EHo(B)be the spectral measure of Ho . Using the intertwining property (2.2) and its equivalent (2.12)
Qi*EH(B)= EHo(B)Qi*, B E B’,
derived from (2.2) by taking adjoints, we obtain E ~ O ( B )=S E ~ O ( B SZ-*Q+ ) = Q - * E ~ ( BQ+ ) = Q - * Q + E ~ ~ (= B )S E ~ O ( B ) .
Thus, S commutes with Ho . I t is evident from (1.25) that T also commutes with H , . Q.E.D.
420
V. Quantum Mechanical Scattering Theory
We recall that Theorem 2.3 has been used in proving that (1.30) is not feasible. I n Theorems 2.4 and 2.5 we show that S*S = EM, if M, = M, .Thus, the condition that S # O coincides with the condition that M, # {0}, i.e., that there are asymptotic states and that the wave operators do not vanish.
Theorem 2.4. The S operator maps the subspace M, into M-, and maps its orthogonal complement Z 0 M, into zero; moreover it maps M, isometrically into M- (i.e., I/ Sfll = 11 f 11 for all M,), if and only ifQ R, C R- . Proof. According to Lemma 2.1, the final domain of Q-* is identical to the initial domain M- of Q- . Therefore, Sf = Q-*(Q+f) E M- for a l l f e 2.However, iff 1M- , then Q,j = 0 and consequently Sf = 0, which establishes the first statement in the theorem. According to Theorem 2.2, Q, maps M, isometrically onto R,, i.e., 11 Q+gII = I / g11 for g E M, , By Lemma 2.1, Q-* maps R- isometrically onto M- . Thus, we have 11 Q-*Q+g I/ = /I Q+g/I = I( g 11 for all g E M, if R, C R- . However, if R, 0(R- n R,) # {0}, then the elements of R, 0(R- n R,) are mapped by Q-* into zero, and S = Q-*Q+ is not isometric on M, . Q.E.D.
The above theorem indicates that the S operator might be unitary if M, = M- = M, and S is restricted to M,. Theorem 2.5 gives an important necessary and sufficient condition for this case to occur.
Theorem 2.5. The S operator is partially isometric with initial domain M, and final domain M- and satisfies the relations (2.13)
S*S
=
EM+,
SS*
=
EM_
if and only if R, = R- . Proof. Assume that R,
= R - . Using ER&Qi= Q,
and the identity (2.1 I), we obtain that (2.13) holds: S*S
= Q+*Q-Q-*Q+ = Q+*ER-Q+ = Q+*ER,Q+
SS* = Q-*Q,Q+*Q-= Q-*ER +Q- = Q-*ER-Q-
= Q+*Q+ =
EM+,
= Q-*Q- = E M - .
* In theories which are invariant under time reversal M, = M- = M, and there is a one-to-one antilinear mapping of R- 0 R, onto R, 0 R - . Hence, in such theories either R, E R- or R, is not a subset of R- , and S is not isometric on M, .
42 1
2. General Time-Dependent Two-Body Scattering Theory
Now, by Theorem 2.4, when R- = R, , the S operator is partially isometric, with initial domain M, and final domain M- equal to its range 9Zs . Since, according to the same theorem, 9Zs C M- , we obtain by using (2.13) and Lemma 2.1 that gs= M- . Conversely, assume that the relations (2.13) are true. We get by expressing in (2.13) S*, S, and EM* in terms of the wave operators Q, :
Q-*Q+Q+*Q= Q-*Q-.
Q+*Q-Q-*Q+ = Q,*Q,
Multiplying the above two relations from the left by Q+ and from the right by Q+*, respectively, we obtain, after using (2.10) and (2.11),
Thus, we have for any f E 2
11 E R + f l / 2
=
(f 1 E R + f )
= (ER+fl
=
(f 1 E R + E R - E R + f )
ER-ER+f)
= 11 E R - E R + f 1 / 2 *
The above equality implies (see Exercise 2.3) that ER+f = ER ER+f for a l l f e 2,i.e., ER-ER+ = ER+ = El+ = ER+E,- . Inserting the above relation in (2.15), we get ER-
= ER_ER+ER-= Ei-ER+ = ER-ER+ = ER+ .
Q.E.D.
I n the case of two-body potential scattering, i.e., scattering theory in which H and H , are essentially of the form (1.8) and (1.9), we will be able to prove that indeed M, = M- = M, and R, = R- when the potential V(r) fulfills certain conditions. However, it is still interesting to investigate the physical consequences of a theory in which R, # R- , i.e., when the S operator is not a partial isometry from M, to M- . Let us assume that go is a vector which belongs to R, but does not belong to R- . According to the definition of R, , the fact that go E R, implies that there is a vector gin E M, such that go = Q+gi, , i.e.,
Hence, the interacting state
e-iHt
go has an incoming asymptotic free
422
V. Quantum Mechanical Scattering Theory
state. This means that e-iHlg, is not a bound state. However, since go 4 R- F lim eiHt
t++m
-Mot
e
go
does not exist. Hence, by Theorem 1.1 e--iHtgo has no outgoing asymptotic state, Thus, e--iHtgois a “quasi-bound” state, which is approximately a free state in the distant past, but which does not become a free state again at any time in the future. On the other hand, if there is a vector g, E R- which does not belong to R,, then the theory possesses an interacting state e-{Htg, which becomes a free state in the distant future, but wich has never been a free state in the past. Thus, the condition of “unitarity” of the S operator, which is usually imposed on scattering theories, is equivalent to the requirement of not having any quasi-bound states in the theory.
2.4. PROPERTIES OF R,
AND
M,
Thus far in this section we have never used the basic assumption that a state eciHotg is called a free state only if it is not a bound state of the Hamiltonian Ho . As a matter of fact, all the theorems of this section stay valid if we define M, to be the sets of all vectors f E Y? for which the respective strong limits of Q ( t ) f , t ---f oc), exist. If M, are defined this way and we choose H identical to H , , we would have Q(t) = 1 and M, = M, = A?,always. However, if Ho has bound states, then not all e-i’foif, f E &, are free states, and M, defined in $1 is a nontrivial subspace of X . Let MI = EH(S : ) Z denote the closed subspace of Y? onto which the spectral measure E H ( Sf) belonging to the continuous spectrum of H projects.
Theorem 2.6. T h e closed subspaces R, of Y? are also subspaces E of M, = EH(Sf)A?, and are left invariant by EH(B), i.e., E H ( B ) ~ R, for all B E 98, whenever f~ R, .
Proof. Since R, are the ranges of Q, , we can find for any g , vectors f* , such that g , = Q,f, . Hence, using (2.4), we get
E
R,
which establishes that R, are left invariant by H. We shall prove that R, C M, by showing that Q J , is orthogonal to all the eigenvectors of H .
423
2. General Time-Dependent Two-Body Scattering Theory
Let g be an eigenvector of H with the eigenvalue A,, EH({Ao})g= g. Given any f, E M, , we can write (g
so that
I Qifl) = <-w{AoHg I Qih)= (Q**EH({Ao))gIfl) I fi>
=
= (g I Q*@%AO))fl)
= 0,
where we have made use of (2.12) to derive the third equality; the last f, = 0, since no f, E M, is an inner product vanishes because EHo({Ao}) eigenvector of Ho . Since Q,fZ = 0 whenever f2 1M, , we derive, by setting f =fl f 2 , f l E Mo , f 2 IM* 9
+
( g I Q * f > = ( g I Q*fl>
for any f E X . Hence, R, g o f H . Q.E.D.
+( g I
Qif2>
=0
are orthogonal to any eigenvectors
We will be able to prove in later sections that in potential scattering with certain classes of potentials, we have R, = R- = M, . I n order to arrive at results of this kind, one usually starts by proving first that R, = R- . This establishes immediately that the S operator is unitary. Now we shall prove a theorem which states a sufficient condition for the identity of R, and R- . This theorem applies to operators with a simple continuous spectrum. The concept of one operator A with a simple spectrum was introduced in $5 of Chapter IV: the self-adjoint operator A has a simple spectrum if the one-element set {A}is a complete set of operators. According to the theorems of $5 in Chapter IV it is equivalent to say that A has a simple spectrum if there is a vector fo which is cyclic with respect to A. If SA, and S;t denote the continuous and point spectrum of A, then the closed subspaces #A, = EA(SA,)X and 8:= EA(S;t)X onto which E A () :S and E A ( Sf:)project, respectively, are left invariant by A. T o see that, note that X = X t 0Xf:and that
(f I A g )
=
I+" d ( f I E$g> -m
A
=0
iff E $?A, a n d g E S;t n g A or , iff E X;t a n d g E #A, n BA. The restrictions A, and A, of A to Scand X,., respectively, are obviously self-adjoint operators with spectral resolutions (2.16) (2.17)
A
.-I AdE?,
A,
-
=
j h dEp,
EAc(B) = EA(B n Sf), EAp(B) = EA(Bn S,"),
424
V. Quantum Mechanical Scattering Theory
respectively. When A , or A, are operators with a simple spectrum, we say that A has a simple continuous spectrum or a simple point spectrum, respectively. I t is easy to establish that A has a simple spectrum if and only if both its continuous spectrum and its point spectrum are simple (see Exercise 2.4).
*Lemma 2.2. Let A be a self-adjoint operator with a simple spectrum, and let M, , k = 1, 2, be invariant subspaces of A , i.e., for any f E M, A a,,,we have Af E M, , k = 1 , 2. If the restrictions A , , k = 1, 2, of A to M, have identical spectra, then M, = M, . Proof. Since M, is left invariant by A, the operators A and EM, commute (see Exercise 2.5). Hence, by Theorem 5.6 of Chapter IV, the projector EMk is a function of A, i.e., EMk= Fk(A). From the properties EM,= E* and E;, = EM,, by using (2.26) of Chapter IV Mk we prove that Fk(h) is a real function satisfying
and therefore F,(h) can assume only the values 0 and 1 for X E SA. Thus, if we set F,(h) = 0 for h $ S A , we show F,(h) is the characteristic function of a Bore1 subset B, of SA. We obviously have Al, = AEM,
= AF,(A).
Hence, the spectrum S A k of A, coincides with B , . Consequently, Q.E.D. S A 1 = S A a implies that F l ( A ) = F,(A), i.e., EM, = E M , . The following theorem can be very useful in establishing that S is unitary.
Theorem 2.7. R, = R - .
If the continuous spectrum of H is simple, then
Proof. Let H , be the restriction of H to the closed subspace M, = E H ( S F ) X . T h e operator H , is obviously a self-adjoint operator with a pure continuous spectrum which is identical to the continuous spectrum of H . Hence, according to the assumption, the spectrum S H 1 of Hl is simple. According to Theorem 2.6, R, are closed subspaces of M, which H , leaves invariant. Since the spectrum of H, is simple, we can apply Lemma 2.2 and infer that R, = R- . Q.E.D.
2. General Time-Dependent Two-Body Scattering Theory
425
2.5. DYSON’S PERTURBATION THEORY We saw in 91 that the knowledge of the S operator is sufficient for the calculation of the differential cross section and of transition probabilities. This shows that a complete solution of the dynamical problem, which would consist in computing the evolution operator U(t, to) introduced in 93 of Chapter IV, is not necessary for the theoretical description of a scattering experiment. Since the computation of Y(t)= U(t, to)Yo for given Yo can be a difficult and intricate computational task, it is desirable to develop methods for computing the S operator in a direct manner. Dyson’s perturbation theory provides one such method. I t follows from (2.1) that for anyf, g E 2 (2.18)
(f 1 Q-*g>
=
( Q - f I g>
=
!jE = i$
Since Q*(t) is unitary, we have 11 Q*(t)g 11 = 11 g 1 . On the other hand, if g E R- , it follows from the fact that Q-* is a partially isometric operator with initial domain R- (see Theorem 2.2 and Lemma 2.1) that 11 Q-*g (1 = 11 g (I, so that I/ Q*(t)g /I = 11 Q-*g 11. According to Lemma 6.2 of Chapter IV, this result in conjunction with (2.18) implies that Q-*g = s-lim E,-Q*(t)g,
g
t++m
S
R-
.
= R- , we have
Hence, if S is partially isometric, so that R, (2.19)
E
=
EM_ s-lim Q*(t)(s-lim Q(t,) EM+)
=
EM- s-lim(s-lim Q*(t)Q(to)) EM,,
t++m t++m
to+m
to-l-m
where the operator (2.20)
V(t, 2,)
=
Q*(qQ(to)= eiHot
e--iH(t--to)
e-iHoto
is obviously unitary for any t, to E R1. Comparison of
*
- eiHot
QO-
e
-iHt
with (3.43) of Chapter IV shows that Q*(t) is the operator which determines the time dependence of the state p(t)in the interaction picture. Hence, we have (see Chapter IV, Exercise 3.4)
(2.22)
H,(t) = eiHOt(H- H,)
for any f E g Hn B H o .
e-iHot
426
V. Quantum Mechanical Scattering Theory
We have seen in 97 of Chapter IV that in potential scattering, under , very reasonable assumptions on the potential, g Hcoincides with gH0 When this is the case, and the vector f belongs to g H= g H 0we , have exp(--iHoto) f E gH0and exp(iHt,) exp( -iHoto) f E g H (see Exercise 2.6), so that Q(to)f E g H= BHn gH0 and (2.23)
av, at
~
to+
=
s-lim V(t At-0
+4
At
v(t,to) f = - -iHI(t)v,t 0 ) f .
--i@
I Hdt)
to) -
Thus, for any vector h E 2? (2.24)
(a/at)(h
I
w,t o m
=
v,
t0)f).
If the function ( h I H I ( t ) V ( t ,to)f) is integrable in t, we obtain from (2.24), after having observed that V(to, to) = 1, the following relation: (2.25)
(h I
v(t,t o ) f ) = ( h If)
-
-i
f( h I HI(~I)
If the integral --i
t
Jto
v(t1 t01.f)
t0
( h I HI(t1) V
l
3
t o m dtl
is defined for any h E X , it determines for each fixed f E gHo a continuous linear functional in h, and therefore by Riesz's Theorem (Theorem 2.3 of Chapter 111), it defines a mapping f -+ Wl(t,t o )f. I t is easy to see that Wl(t,t o ) is a linear operator defined on BH0. We write, symbolically,
I n this notation (2.25) is equivalent to the statement that (2.27)
V ( t , to)f = (1 - -i
St Hdtd $0
v(t1 , 10) 4)f,
f E
If we treat the above integral equation recursively in V(t,to),we obtain after n steps
v(t,to)f
=
+ vi(t,t o ) + + vn-i(t, + W d t , to)
"'
tO)lf,
f E gHo
Y
where Vk(t,t o ) and W J t , to) are defined by the relations
(2.29) W,(t, to) = (-iy
1 dt, St'dt, t
to
to
r-' to
dt, HI(tl)... HI(t,-l) V(t, , to),
427
2. General Time-Dependent Two-Body Scattering Theory
whose precise meanings are established in the same manner as in the case of W,(t, to). It has become customary to introduce a chronological ordering operation T by setting (2.30)
T[HI(~,)HI(tz)
H~(t,)l= Hdt,,) HdtiJ
* a *
til 2 tia >, * * * 2 tt, .
HdtiJ,
This operation orders the time-dependent operators in the product H,(t,) ... HI(tk)in a chronological sequence of increasing values of the time variable from right to left, so that the integrand in (2.28) becomes symmetric under permutations of t, ,..., t, when T is set in front of it. Since there are n! such permutations, it is easy to check that
Suppose that
exists for every k
=
(2.33)
1, 2, ..., and that W,f
=
w-lim(w-lim W,(t, t o ) f ) t-fm
to+-m
also exists and converges (at least in a weak sense) to the zero vector Under these assumptions we obtain from (2.19), when n -+ +GO. (2.20), (2.30), and (2.31) the following expansion: S
+ + vs + * . . ) E ~ l + f *
EFI_(~ vi
The above expansion is known as a perturbation expansion of the S operator. In the zero order for the case M, = M,, it becomes S = E;, = EM,,i.e., it describes the case when there is no scattering. The computation to higher orders is facilitated in practice by an explicit knowledge of the commutator [HI(tl), H,(t,)] = D(tl , t z ) . This makes possible the computation of the integrand of V , by successive use of the relation T[HI(tl) HI(tZ)l = HI(t1) W,)-
wz
-
4) Wl > tz).
V. Quantum Mechanical Scattering Theory
428
Computations of this nature occur very frequently in quantum field theory, and techniques have been developed for coping with them in a systematic manner. OF ASYMPTOTIC STATES 2.6. THEEXISTENCE
There are cases even in potential scattering in which M, = {0}, i.e., when there are no asymptotic states in the sense of (1.7). One notable example is the case of Coulomb scattering, when we are dealing with the Schroedinger operator H with a Coulomb potential. Namely, it can be shown that quantum Coulomb scattering shares with the classical Coulomb scattering the feature of not having asymptotic states in the conventional sense (1.7), though asymptotic states can still be introduced in a somewhat different sense of the word." This particular example indicates that it is desirable to produce some criteria for the existence of asymptotic states. T h e rest of this section deals with the derivation of Theorem2.8,first proved by Jauch andZinnes [1959], which will be found very useful in establishing the existence of asymptotic states in potential scattering (see Exercise 2.8 and $7).
*Theorem 2.8. Suppose 9,, 9,C g Hn g H ois, a linear submanifold of 2 which is such that e-'iHot f E g Hn gH0 for all f E 3, . Then the closure 6, of 9, belongs to the initial domains M, of the wave operators 52, if and only if when t, ---t - 00 and t, ---t co the respective strong limits of
+
Il(tl , t 2 ) f =
(2.34)
t2
eiHtH,epiH0tfdt,
Hl
=H
-
H, ,
tl
exist for every f E 9, . A sufficient condition for the existence of these limits is that for some t , , t, E [wl
sf", II
ffl
exp(--iHot)fII &
< fa,
/:p
II Hl exp(--iHot)fIl dt < f a ,
respectively. T o understand the above theorem, we must comprehend the meaning of the integral appearing in (2.34). This integral is defined by the requirement that
* For details, see Dollard [1964].
429
2. General Time-Dependent Two-Body Scattering Theory
determines a continuous linear functional in g. I n fact, if the integral in (2.35) exists, then, by Riesz' theorem (Chapter 111, Theorem 2.3), there is a unique vector f, = Il(tl, t z )f such that (fi I g ) = +r(g;t , , tz). Thus, in order to know that Il(tl,te) is well defined we only have to establish that the Riemann integral in (2.35) really exists. For this task we need Lemma 2.3. For any two vectors f E gH0 and g E g Hthe function
Lemma 2.3.
(2.36) F ( t ) = (exp( -iHot)f I H exp( -iHt)g) - ( H , exp( -iHot)f I exp( -iHt)g) is continuous in t E R1, and (2.37)
1(
i
(
-
I exp(--iHt)g)) dt
<[-Q(t,)- -Q(W I g).
Proof. T o establish the continuity of the first term in (2.36), we note that for any T E R1 (2.38)
+
+
I(exp[--iH0(t .)If I H exp[-iH(t T ) ] g) - (=p( -iHot)f I H exp(--iHt)g)l d I(exp[--iHo(t .)If1 H(exp[--iH(t T ) ] - exp(-iHt))g)I I((exp[--iHo(t T ) ] - exp(--iH,t))f I H exp(--iHt)g)l d llfll I1 H(exp[--iH(t .)I - .xp(--iHt))g I1 ll(exp[-iHo(t .)I - exp(--iH,t))fI/ I/ Hexp(--iHt)gl/.
+
+ +
+ + +
+
We recall now that by Theorem 2.7 of Chapter IV, the operator H(exp[-iH(t T ) ] - exp(-iHt) is a function of the self-adjoint operator H , and consequently
+
H(exp[-iH(t
+
T)]
- exp(-iHt)) g = (exp[-iH(t
+
T)] -
exp(-iHt)) Hg.
Since exp(-iHt) and exp(--iHot) are strongly continuous operatorvalued in t (see Chapter IV, §6.3), the expression on the right-hand side of (2.38) converges to zero when T -+ 0. T h e continuity in t of the second term in (2.36) is even easier to prove by following the same line of reasoning, since we have
(Ho exp[-iHo(t f .)If[ exp[--iH(t f T ) ] g) - (Ho exp(-iHot)fI exp(-iHt)k')l d I/ Ho(exp[--iHo(t 41 - exp(--iHot))fI/ II g /I /I Ho exp(--iHot)fII Il(exp[--iH(t dl - exp(--iW)g I/.
+
+
+
430
V. Quantum Mechanical Scattering Theory
The cqntinuity of F ( t ) in the variable t implies the existence of the integral in (2.37) for any given t , , t, E R1. Thus, the first part of the lemma is established. T o establish the second part, we recall from Theorem 3.1 of Chapter IV that
s-l.l-n ;(exp[-iA(t + 2
T)]
- exp( -iAt))g = A exp(-iAt)
g
for any A = A* in general, and consequently for A = H and A = H , in particular. I n view of this fact, a glimpse at (2.36) suffices to show that iF(t) = (d/dt)(e-iHOtf 1 f?-"""g>.
Q.E.D.
Hence, (2.37) immediately follows.
Let us return now to the proof of Theorem 2.8. When
f Egl then exp(--iH,t)fE
gH
gHo >
g1C g H so , that
1 H exp( -iHt)g)
=
(exp(iHt) H exp( -iHot)f
1 g),
and F ( t ) can be written in the form F ( t ) = (exp(iHt) Hl exp( -iHot)f ] g).
Thus, according to (2.37), we have (2.39) ([Q(t,) - C!(tl)]f I g)
=i
fa
(exp(iHt) Hl exp( -iHot)f
I g) dt
tl
for any f E g1and g E g H Since . g His dense in 2,the integral on the right-hand side of the above relation can be said to determine a unique linear functional in g E Z . Thus, we can write symbolically (2.40)
[L?(t2)- Q(t,)]f
=i
1 exp(iHt) Hl exp( -iHot) f dt. t2
tl
It follows from the above relation that in the case s-lim Q ( t ) f exists for t -+ -co or t -+ +a,the respective strong limits of the integral 00 also exist. in (2.34) for t , -+ - co or t , -+ Conversely, let us assume that
+
s-lim t,-+-m
~11
e x p ( i ~ t H~ ) exp(--iH,,t)fdt
43 1
2. General Time-Dependent Two-Body Scattering Theory
exist for all f E 9, . Then, it follows from (2.40) that s-lim Q(t,)f also exists when t , -+- co and f E 9,. However, by Lemma 1.1, the set of all vectors f for which this last limit exists is closed, and consequently the closure G, of 9,must be contained in it. Thus, the theorem is established. For all practical purposes, the investigation of the existence of the strong limits in t, and t, of the integral in (2.34) reduces to the investigation of the existence of integrals (2.41)
for some t, E W, and of
for some t , E R1. I n fact, by using the Schwarz-Cauchy inequality we get (2.43)
Hence, the existence of (2.41) implies, in conjunction with (2.39), that l l f l l = \I S+fII, w-lim Q(t,)f exists when t , -+ -a.Since / / Q ( t l ) f i l we infer by using Lemma 6.2 of Chapter IV that s-lim Q(t,)f also exists for t1 -+ -a. T h e argument for t, -+ + co is completely analogous.
<
2.7. THEPHYSICAL ASYMPTOTIC CONDITION T h e condition which the Schroedinger-picture asymptotic states Y e x ( t ) (ex = in, out) of the interacting state Y(t)have to satisfy, (2.44)
lim /I Y(t)- Y e x ( t ) ( I = 0,
t+*m
has no direct physical meaning, since the above vector norm is not a measurable quantity. We recall from $41 and 2 of Chapter IV that the directly measurable quantities are the probabilities
432
V. Quantum Mechanical Scattering Theory
for any set of compatible fundamental" observables. Hence, we expect asymptotic states to satisfy some kind of physical asymptotic condition which would specify that the probabilities (2.45) for the interacting state Y(t)approximate arbitrarily closely the corresponding probabilities for the respective asymptotic states Yinpout(t),when t -+ fCO. That this is actually the case is stated in Theorem 2.9. Let the Hamiltonian H and a set 0,of additional fundamental observables be such that any other observables of the interacting system or the free system is a function of compatible observables from 0, = { H } u 0, or OFx = {H,} u 0,,respectively. Furthermore, assume that whenever { H , A,', ..., An'}, A,', ..., A,' E 8,, is a set of compatible observables, then H , , A,', ..., A,' are also compatible. Under these assumptions, the asymptotic states Y e x ( t )(ex = in, out) of the scattering state Y ( t ) satisfy the physical asymptotic conditions
*Theorem 2.9.
(2.46) (2.47)
for any sets {A, ,..., A,) C 0,, {H, A,', ..., An'}C 0, of compatible observables. Proof. I t is straightforward (see Exercise 1.7) to prove that (2.46) holds, i.e., lim /(Y(t)I EA1s...BAm(B) Y ( t ) )- ( Y e x ( t ) I
t+*m
( B )Y e x ( t ) ) I
EA1s...*Am
=0
when (2.44) is satisfied. Since A,', ..., A,' commute with H as well as with H, , we infer that for arbitrary B E 9Yn eiHt
e-iHotEAl'
,...,A , , ' ( B , )
= EA''
....,an'(^,) e i H t e --iHot .
By applying the operators on both sides of the above equation to vectors f E M, and letting t -+ f03, we obtain immediately EAl',...,An'(B,) f E M, and (2.48)
Q&Ai'..
.. an'(^,)f = *A''..
...An' (B,) Q*f
*
* See the discussion in the beginning of 92 in Chapter IV concerning fundamental observables.
433
2. General Time-Dependent Two-Body Scattering Theory
IM, , then (EA”*-.’A”’(B2)g If)
Furthermore, if g
( g I EA”....*A”’(B2)f) = 0
=
for all f E M, , and therefore EA1’,...,A,‘g M, . Consequently, (2.48) holds for arbitraryf E 2.Combining this relation with (2.4))we conclude that (2.49)
QiEHO(B,) EA1’....*An’(B2) = EH(B,)EA1’,,,,*A*‘(fj2) Qi
for any B , E B1 and B , E gn. As a consequence of the basic properties of spectral measures (see Chapter 111, the proof of Theorem 5 4 , we can write for any B E gn+l EHo.AI‘s...*An‘(B) = inf
1 1EHO(BP))EAL’-....An‘(BF)): Bc k
u BP) x
B$)/,
k
where the projector infimum is taken only over finite unions of sets Bik) x BLk’) with Bik)E 491) BLk)E gn. Hence, EHo3*1*...,*-(B)is the strong limit of some sequence of finite sums
1EHo(BP))
EAi’
.....An‘(Bp))
k
corresponding to a sequence of sets Uk(Bik)x BLk)),with B as an intersection. This result in conjunction with (2.49) implies that Q*EHO’Al‘
(2.50)
I....
A”@)
= EH.A”
.....A”’@)
Qk
for any B E gn+l. We recall that (2.51)
where ex
Y(0) = Q*YeX(O), =
in for 52, and ex
=
out for 52-
. Hence,
after noting that
434
V. Quantum Mechanical Scattering Theory
where the last step follows from the fact that Q,*Q,?Pex(0) = Yex(0), since Yex(0)E M, . Hence, (2.47) is established. Q.E.D. All nonrelativistic quantum mechanical theories satisfy the conditions of the above theorem. For example, in wave mechanics for a single particle interacting with a potential V(r), one can take
W C -(1/2m)A
+ V(r)
and 0,consisting of L2,L,(n E R3, I n I = l), X , and P,(K = 1,2,3); here, L2 denotes the total angular momentum operator (encountered in 97 of Chapter 11), L, = n * L is the angular momentum projection in the direction n, and X , , P , are the position and momentum components
with respect to the Kth axis of a Cartesian inertial reference frame. From the physical point of view, it is sufficient if asymptotic states satisfy the physical asymptotic conditions (2.46) and (2.47), rather than (2.44). As a matter of fact, albeit that for long-range potentials (such as the Coulomb potential or V ( r ) r - ~ 0, < cy. < 1) there are no asymptotic states satisfying (2.44), there are free states satisfying (2.46) and (2.47). One could call such states physical asymptotic states, and develop" a scattering theory based on the existence of these states. As a matter of fact, in this case one could define the wave operator Q, to be the operator which maps an incoming asymptotic state into the corresponding interacting state, i.e., as the operator satisfying (2.51); Q- could be defined in a similar manner. As soon as the wave operators are defined, one can proceed in the usual manner and introduce the scattering operator by means of (2.2) and the transition operator by the old formula
-
(2.53)
T
z=
(1
-
S)/2ni.
For the rest of the quantities of importance in scattering theory (transition probabilities, differential scattering cross section, etc.), one can also retain the old definitions.
EXERCISES 2.1. Assume that C is a bounded operator in A?, and A, , A, are self-adjoint operators in H with spectral measures EA1(B)and EA2(B), respectively. Prove that the relation eiAltC= CeiAat is satisfied for all t E R1 if and only if EA1(B)C = CEA2(B)for all B E 9P.
* For details see the article by PrugoveEki [1971a], which also contains further references.
435
3. General Time-Independent Two-Body Scattering Theory
2.2, Show that if U is a partially isometric operator with initial domain M, then ( U f I Ug) = ( f I g ) for all f,g E M. 2,3. Show that if E is a projector and )I Ef then f = Ef.
11
=
llfll
for some f
E
2,
2.4. Show that f o is a cyclic vector with respect to A if and only if EA():S f o is a cyclic vector of A , and EA( S;) f o is a cyclic vector of A, . 2.5, Prove that if a self-adjoint operator A leaves the closed linear , M ]= 0 for all B E 9 I l . subspace M invariant, then [ E A ( B ) E 2.6. Show that if A is self-adjoint and f all t E R1.
E
g A ,then
eiAi
f
E
g Afor
2.7. Suppose that the potential V(r) is Lebesgue square integrable Apply the result of Exercise 1.9 to prove on R3 so that J) V ) )< +CQ. that if $(r) is Lebesgue integrable as well as square integrable on R3, then
almost everywhere in R3, and consequently
where ( Ve+Hof $)(r)
=
V(r)(e-iHOt$)(r).
If the potential V(r) is Lebesgue square integrable on R3, then = L2(R3)in potential scattering for V(r). Explain how this statement follows from Theorem 2.8 and Exercise 2.7. 2.8.
M,
=
M-
3. General Time-Independent
Two-Body Scattering Theory
3.1.
THERELATIONOF TO THE
THE
TIME-INDEPENDENT
TIME-DEPENDENT APPROACH
I t is possible to construct the Mdler wave operators Q, by taking limits with respect to parameters other than the time parameter t . Scattering theory based on such time-independent procedures for computing the basic operators Q, , S, and T is called time independent. The relation of the time-dependent to the time-independent approach is best exhibited by showing that Q, can be obtained as strong ,, for E -+ +0, where the parameter E , limits of operator functions Q
436
0
V. Quantum Mechanical Scattering Theory
+
CO, is not related to the time parameter t and has no physical meaning, and the operators SZ,, satisfy the equations
Let us show that the relations (3.1) can play the role of definitions of ,, . linear operators Q Since e i H t and are strongly continuous functions of t (see $7 of Chapter IV), Q(t) = e i H t is also strongly continuous (see Chapter 111, Exercise 5.6). Hence, for anyf, g E A?,the function ( g I Q ( t ) f > = ( g I eiHt i?iH~"f>, t
E
UP,
is continuous, as well as bounded in t :
l ( g I e i H t e-iHotf >I
< It g II /I eiHte-iHOtfII = l l f
I/ It g II.
Consequently, the first improper Riemann integral in (3.I), (3.4
converges absolutely for any f,g E A?:
Thus, the functional ( g If) is defined on A? x 2. It is straightforward to verify that ( g If ), f,g E Z, is a bilinear form. Furthermore, we see from ( 3 . 3 ) that (. I *) is bounded. Hence, there is a unique bounded linear operator (see Chapter IV, Exercise 2.5), which we denote by Q-, , such that (3.4)
(g I Q J >
m
=
( g ~ f =) E
0
e-Et(g I eiHte- i H ~ft > d t
for allf, g E 2,We write the above relation in the symbolic form
(3.5)
Q-, = E
1 m
0
eCt eiHtePiHQtd t ,
E
> 0,
3. General Time-Independent Two-Body Scattering Theory
437
under the agreement that its real meaning is given by (3.4). I n fact, (3.5) is an example of a Bochner integral (see Theorem 3.4 in Appendix 3.7 to this section). It can be shown in a completely analogous manner that for any E > 0 there is a unique bounded linear operator Q,, which satisfies
T h e above relation is written in the symbolic form for all f, g E 8. 0 Q ,
=
J"
ert e i H t
--m
e-iH t dt,
E
> 0.
We are interested in the existence of the strong limits of Q,,f E -+
$0.
as
From (3.3) we immediately infer that
By applying Lemma 1.1 we deduce that the sets N, and N- of all vectors f E %' for which the respective limits of Q,,f exist, (3.8)
N*
= {f:s-lim Q*, E++O
f E if]
are closed linear subspaces of 3. We shall prove now that M, C N, and M- C N- .
Theorem 3,l. If for some f E 2, s-limt-,+m52(t)f exists, then dim6++,, 52-, f also exists, and (3.9)
s-lim Q(t)f t++m
= s-lim Q-, f. f+fO
Similarly, if for some g E S, s-limt-,.-m Q(t) g exists, then s-limE++,,52+Fg also exists, and (3.10)
s-lim Q(t)g t+-m
=
s-lim .R+,g. f+,O
Proof. Let us write f-
=
s-lim Q(t)f t++m
438
V. Quantum Mechanical Scattering Theory
when the above limit exists. Then, for any g E % Kg
I
Q-€f
-f-h
I < €1
m
=
E
0
e-ct(g I Q ( t ) f ) dt
-E
/
m
0
e-€$(g If-)
dtl
m
0
= /I g
II
e-€tl(gISZ(t)f-f-)ldt<€IIgl/ J m e - € t ~ ~ ~ ( t ) f - f - ~ ~ d t
J
0
m
e-u I/ Q(u/.)f
-f- /I du.
We set out to prove that the above integral is smaller than any given q > 0 for all sufficiently small c > 0. Choose some a > 0 and split the interval of integration:
s," 1," j: =
+
*
For the first integral we obtain the following upper bound:
Now, take some
(Y.
satisfying o
and then choose
E
1 rl 2 ltfll
+ Ilf-I1 '
> 0 so small that
(3.11)
for all t 2 a / € . For any such
j
m
e-" II Q(u/.>f
E,
we compute that
-f- I/ du
Consequently, for any given 7
> 0 we have
l(gIQ-,f-f->l
for all E > 0 which are such that (3.11) is true whenever t 3 a / € . Hence, we have established that (3.12)
f-
= w-lim .Ref. €++O
439
3. General Time-Independent Two-Body Scattering Theory
Since I( f - I/ = limt++m/I Q ( t ) f l \ = 11 f 11 2 I/ Q-,f 11, we have shown that the weak limit (3.12) is also a strong limit (see Chapter IV, Lemma 6.1), f-
=
s-limQ-,f, E'tO
i.e., (3.9) is true. T h e proof of (3.10) proceeds along identical lines.
Q.E.D.
As an immediate consequence of the above theorem we obtain the important result (3.13)
which can be considered as a time-independent alternative definition of the wave operators. It should be noted, however, that Theorem 3.1 establishes only that M, C N, , and not that M, 3 N, . Since the conconverse of Theorem 3.1 [which would state that the existence of s-lim6++oQ ,, f implies the existence of s-lim,+,m Q(t)f ] is not known to be true, it has been suggested (see Jordan [1962a, Section 61) that Q* could be defined in a time-independent manner, by taking limits of !2,,f, even in cases when the time-dependent definition (2.1) is not adequate because M, = (0). I n potential scattering, for classes of potentials which have been considered in the last section, the above problem does not arise, since in those cases M, = X , and consequently Q& = s-lim t'irn Q(t) = s-lim c++O Q*,.
From (3.13) we get for any f , g
E
Z
Since a,* are partially isometric with initial domains R, verify that (Q+*ER,f
I g>
= !i$,(EMtQzER+f
, we easily
1 g>
by noting that for f E R, the above relation is equivalent to the relation preceding it, while €or f 1R,, the expressions on both sides of the above relation vanish. I n view of the fact that
440
V. Quantum Mechanical Scattering Theory
we infer from this result (by using Lemma 6.1 of Chapter IV) that Q**
(3.14)
=
s-lim E ~ ~ Q : ~. E ~ + E++O
3.2. INTEGRAL REPRESENTATIONS FOR Go* One of the advantages of the time-independent limits (3.13) for determining a, is that they lead to integral equations which can be used to compute without having to compute O ( t ) . Using the spectral decomposition H
=
j,
X dEF
and recalling that eiHt is a function of H , by employing Theorem 2.5 of Chapter I V we obtain (3.15)
Now, we would like to interchange in the above relation the order of the integrations in h and t , and write (3.16) ( g I Q - , f )
=E
1 dA1 +m
m
--oo
0
exp[-(e
-
ih)t](g I E f e x p ( - i H o t ) f ) dt.
However, before we ask whether such an interchange of the order of integration is correct, we must make clear the meaning of the integration in A occurring in (3.16). We will show that the interchange of the order of integration can be easily justified if the integral in h is taken to be a Riemann-Stieltjes integral.
Definition 3.1.
T h e Riemann-Stieltjes integral
j:w d, 44,
a
< b,
of the complex function F(h) with respect to the complex function u(h), a h b, is said to exist if for any sequence of subdivisions a h(O) < < ... < A(") = b of [ a , b] which is such that the norm 6 = maxk=l,...,m(h(k)- h'k-l)) of the subdivisions in the sequence
<
< <
441
3. General Time-Independent Two-Body Scattering Theory
tends to zero, and for any choice of A'(k) E [X(k-l), A ( k ) ] , K = 1,..., n, the limit of Riemann-Stieltjes sums (3.17) exists, and is independent of the selected sequence of subdivisions; if the integral (3.17) exists, then its value is taken to be (3.18)
1F(h) d,, ~ ( h ) lim f'F ( h ' ( k ) ) [ ~ ( h ( k ) ) b
a
=
6-0
,
-
2=1
>I-
~(h(~-l)
If F(A) and u(A) are defined on R1, then the improper Riemann-Stieltjes integral on R1 is defined as (3.19) provided that the two limits in a and b exist. For functions F(A, A), and o(A, A), of two variables A, A, E W, we define the cross-iterated RiemannStieltjes integral on [a,b] x I , where I is a nondegenerate interval in UP, by requiring that for any of the above sequences of subdivisions of [a, b ] ,
where the above limit should be finite and of a value which is independent of the chosen sequence of subdivisions. It is easy to relate Riemann-Stieltjes integrals to integrals with respect are Bore1 measurable functions (see to measures when F(A) and .(A) Exercise 3.3). It is equally easy to prove in a direct manner that most of the usual properties of Riemann integrals (which are obviously special cases of Riemann-Stieltjes integrals in which a(A) = A) are also properties of Riemann-Stieltjes integrals (see Exercises 3.4-3.8). We find the concept of Riemann-Stieltjes integration of importance because it enables us to give a meaning to the integral in (3.16), namely, the integral in (3.16) can be written as a cross-iterated Riemann-Stieltjes integral with respect to the function (3.21)
~ ( h4) , =
S
A0
0
(g I E f ePiH0tf)dt.
442
V. Quantum Mechanical Scattering Theory
Let us introduce the following convenient notation: (3.22) u((A, A'] x (A,, A,'])
= ~ ( h 'A,', )
- ~ ( h 'A,,)
-
u(A, A);
+ u(A,
4).
We can establish that (3.16) is true by means of Lemma 3.1.
*Lemma 3.1. Suppose the real function o(A, A,) is of bounded variation on [a, b] x [c, d ] , i.e., that for any A E [a, b] and A, E [c, d ] we have
where the supremum is taken over all subdivisions (3.24) a
=
A(')
< A(') < ... < A'"'
- A,
c =
A t ) < A t ) < ... < A t ) = A,
of [a, A] x [c, A,]. If the complex function F(A, A,) is continuous on [a, b] x [c, d ] , then both its cross-iterated Riemann-Stieltjes integrals with respect to o(A, A,) exist and (3.25)
J; 4 idv, A,)
i,4, m, d
A,)
=
4ocl.(~,
Jb
A,)
44,A,).
Proof. I t is easy to see that the functions 4 A , A,)
=
+[.(A,
A,)
+ 4 4 c) * .(A,
A,)]
are nondecreasing in [a, b] x [c, d ] in the sense that a,(h, A,) whenever A A' and A, A,'. Since
<
<
4 4 A,)
=
o+(k A,)
-
< of(A', A,')
4, A,),
we can exploit the linearity properties (see Exercises 6.4 and 6.7) of cross-iterated integrals to reduce the integrals in (3.25) to a sum of integrals of real functions with respect to a+(A, A,) and o-(A, A,). Hence, it is sufficient to prove the lemma for the case of a nondecreasing .(A, A,) of bounded variation and for real F(A, A,). By using the mean-value theorem for Riemann-Stieltjes integrals (see Exercise 3.8), we obtain
443
3. General Time-Independent Two-Body Scattering Theory
<
A;(’”) for some relation (3.20), we get
=
lim
m
< Ah‘“).
Combining this result with the defining
n
CC
o ( ( ~ ( i - lY) ~ ( i ) ]
$’(~(i),
i=l k = l
I)
( ~ t - 1 3) ~ 0( k )
in the limit of finer and finer subdivisions (3.24) of [a, b] x [c, 4. T h e existence of this limit and its independence of the chosen sequence of subdivisions {(A(i), Ah’”))} is obviously a necessary and sufficient condition for the existence of the integral. T o establish the existence of this limit and its independence on the sequence, it is sufficient to note that if {(ACi*j), Ahk,”’>} is a subdivision of [a, b] x [c, 4, which is finer than the subdivision {(Aci), A:’”))}, and if ( A ’ ( i , j ) , A ’ h k s Z ) ’ ) are points within the meshes determined by {(A(i,j), A:’”,’))}, then due to the uniform continuity of F(A, A,) we can achieve for any given E > 0
I q x ( i , d ,A $ k , Z ) )
-
qp, @’)I <
for all sufficiently fine subdivisions {(A(i), Ahk))}. Consequently, if and Ahk*’) denote the points preceding A(iJ) and Ah’”.”, respectively, in the subdivision, then
This establishes the existence of the considered limit, as well as its independence of the sequence of subdivisions. By the same token
444
V. Quantum Mechanical Scattering Theory
also exists. By exploiting again the uniform continuity of F(A, A,) in [a, b] x [c, 4 , we conclude that for any E > 0
1 F(h"'i', A;;@))
-
F(j+, A p ) ]
for all sufficiently fine subdivisions (3.24) of [a, b] x [c, 4 . Hence, the absolute value of the difference of the two integrals in (3.25) is smaller than
i = l k=l
= EU((U,
b] x
(E,
d]).
Since this is true for arbitrarily small values of that (3.25) holds. Q.E.D.
E
> 0, we conclude
Let us compute the total variation function v(A, A,), defined by (3.23), for the function a(A, A,) given in (3.21). By using in the process a decomposition of the type (5.13) in Chapter 111, we easily derive
Since the integrand in the above integral is continuous and therefore integrable, it follows that a(X, A,) is of bounded variation. Consequently, Lemma 3.1 can be applied to infer that
445
3. General Time-Independent Two-Body Scattering Theory
T o establish the validity of (3.16), we have to prove that the above relation remains true in the limit when a + - 00 and b, T -+ 00. I n view of the relations (see Exercise 6.8)
+
so s-, s: s: 1:" :s 1," sT, so s, :s s," sE s: s-, s, sP, 16" ss" 1," +m
(3.27)
+m
t m
-
=
-
=
+m
+
+
+
+
+m
'
+m
and (3.26), we see that this will be the case if and only if the integrals on the right-hand side of the equalities (3.27) exist and vanish when a --f -a and b, T -+ +a. Employing the procedure used in deriving (3.26) we arrive at the following estimate:
I 1;:
dtPt
s"
eiAtd,(g
I Efe-iHoy)l
a0
< 4 1: e-'t{(e-iHOtf + g I E H ( ( ~ ,, b , ] ) ( e - i ~ O y+ g))
+ (e?Ot + ig I EH((a,, b,l)(e-"otf + ig))
+ d ( e - i H o t f I @((a, ,b,])
epiH0tf)
+ 1/2(g I ~ ~ ( ,( b,l)g)) a , dt.
It is obvious that the integral on the right-hand side of the above inequality can be made arbitrarily small for sufficiently large values of either co (for arbitrary values of do), or of a, (for arbitrary values of bo), or of -b, (for arbitrary values of a,). Since, according to Lemma 3.1, the order of integration in X and t can be reversed without changing the value of this integral, we conclude that all integrals on the right-hand sides of the relations (3.27) vanish in the limit when a + --OO or b, T --+ +a;for example, for the representative case of the first of these integrals, we can choose for given 77 > 0 a ~ ( 7 )and a sequence ~(7= ) T, < T~ < T~ < .*. diverging to infinity which is such that
and consequently
V, Quantum Mechanical Scattering Theory
446
Thus, we conclude that (3.26) stays valid in the limit a ---t --GO and b, T -+ $00, and therefore (3.16) is ’ true. Using the fact that e - i H o l is a function of H, , and applying Lemma 3.1 in the same manner as before to interchange orders of integration, we arrive at the result W
0
exp[-(E =
W
0
=
-
ih’)t](g I E y exp[-iH0t] f) dt
dt exp[ -(G
Strn dAo -m
Jm 0
-
ih’)t]
exp[-(E
-
+m
-m
ih‘
exp[ -ihOt] dAo(EFgI E? f)
+ iho)t](EfgI E T f ) dt
Inserting this result in (3.16), we obtain
I n a completely analogous manner we can derive a corresponding relation for !2+G:
T h e two relations (3.28) and (3.29) are equivalent to the single relation
if the parameter E > 0 is replaced by the new parameter 71 # 0, which can assume also negative values. Using the defining formula (3.20) for cross-iterated integrals, we obtain from (3.30) (Q,*g
If) =
where the last integral is dejined by the limit preceding it.”
* See also Definition 3.4 for integrals of this type in Appendix 3.7 to this section.
3. General Time-Independent Two-Body Scattering Theory
447
According to Theorem 2.4 of Chapter IV,
and consequently we have (3.31)
+
+
Since iq(X - Ho i7)-1 and (A - Ho)(X- H,, iq)-l are bounded functions of H , , they are defined everywhere in 2. According to Theorem 2.4 of Chapter IV, we can add and multiply these operator functions as we would ordinary functions, and therefore we have the identity X - H, ;.! =I(3.32) X - H, i7 X - Ho i7
+
+
Substituting the right-hand side of the above equation in (3.31), and using symbolic notation (see also Appendix 3.7 to this section), we arrive at the following expression for Q,*: (3.33)
.n,*
=
1-1
h - H,
+m --na
X
-
H,
+ i7 d A E f .
Consider a free state eciHot!P,Y EM, , which is the incoming asymptotic state of e - i H t Y + or the outgoing asymptotic state of eciH1YI-. Since obviously
we deduce from (3.14) and (2.10)
This result, combined with (3.33) leads, after a rearrangement of terms, to the equations (3.34)
T h e above two relations can be considered to be equations for determining Y- and Y+, respectively. As such, they represent a Hilbert space form of the type I Lippmann-Schwinger equations. I n the next sections we shall see that in the special case of potential scattering these equations
V. Quantum Mechanical Scattering Theory
448
lead to a more commonly encountered form of integral equations for distorted plane waves.
3.3. INTEGRAL REPRESENTATIONS FOR 52, T h e main drawback of (3.34), viewed as equations for finding Y* when Y is given, is that they require a knowledge of the spectral function EF of the total Hamiltonian H . Now, in practice, EF is an unknown quantity which is very difficult to compute. I n fact, if E,H were known, then one could easily compute the time evolution operator U ( t ,to) = exp[-iH(t - to)], and thus solve the dynamics of the problem generally, without even having to resort to scattering theory techniques. A practically more suitable set of equations relating Y and Y*,which yield themselves to perturbation techniques, can be derived from (3.1) by applying the spectral theorem first to H,, and then to H , and otherwise proceeding in the same manner as in the derivation of (3.16):
Hence, we end u p with the relation
An analogous relation can be derived for SZ,, . After reintroducing the parameter 7 # 0, we can write (3.36) and the corresponding relation for fin+€jointly in the symbolic form (3.37)
Naturally, if we exploit the relation (3.38)
-iv H - A,
-
iv
=I-
H
H - A, -
A, - iv'
449
3. General Time-Independent Two-Body Scattering Theory
(3.37) can be recast as (3.39) Hence, if we again write (3.40)
Y*
= Q*Y =
s-lim Q,,Y, a+*,
P ! E M* ,
then in view of (3.39), (3.41)
Y*
=
Y
+ s-lim V+*O
--m
A. - H
+ iq d"E 2 Y . A
T h e above two equations are the type 11 Lippmann-Schwinger equations in Hilbert space. They are usually considered to be solutions of (3.34). However, they actually are not explicit solutions of the type I equations iV)-l, (3.34) since they contain the operator function ( H - h,)(X, - H which is unknown in practice. O n the other hand, we shall see later in this section [see (3.67)] that (3.41) can be solved by iterative procedures. Let us write (3.39) in the form
+
Taking adjoints on both sides, we obtain
I n symbolic notation the above relation assumes the form (3.43)
which will be very useful further on. 3.4.
THETRANSITION AMPLITUDE
It was mentioned in $1 of Chapter IV that if Y + ( t ) and YJt) are states in the Schroedinger picture, the 1 ( Y + ( t ) 1 Y - ( t ) ) j 2 is called the transition probability from YJt) to Y+(t);furthermore, the complex number ( Y + ( t ) I Y-(t)) is called the transition (probability) amplitude from Y J t ) to Y + ( t ) .
450
V. Quantum Mechanical Scattering Theory
Let us assume that Y+(t)and YJt) are scattering states, with asymptotic states Yy(t) and YEx(t),respectively. Since Y*(O) = Q+Y?(O)
= Q_YYt(O),
we easily obtain (Y+(t)I Y-(t)> = (Y+(O) I Y-(O)> = (YYt(0) I SYF(0)).
Let us simplify the notation by writing
YF(o)= YJ~,
yYt(o) = Ye,
where the indices i and f stand for “initial” and “final,” respectively. Using the properties of strong limits, we obtain
=
s-lirn(Q-€YfI s-lim Q+,Yi> €++O
€+fO
Substituting the expression (3.39) for Q-€ in (Q-fYt I Q+Yi), we get
O n the other hand, after using (3.43) to express the identity operator in terms of the other two operators occuring in that equation and then applying it to the vector Q+Yi,we arrive at the relation
After substituting the above expression for Q+Yiin the first term on the right-hand side of (3.45) and carrying out an obvious summation, we obtain
I n view of (3.44) and the fact that if
M+ = M- ,
3. General Time-Independent Two-Body Scattering Theory
45 I
we deduce from (3.47) that
I n terms of the T operator (3.50)
and provided that
M,
=
M-
T = -1 - s 2rri ’ , we can write (3.49) in the form
By taking advantage of the intertwining property (2.4), we immediately obtain from (3.51)
It is interesting to note that on account of the relations (3.53)
Yi
= Y F ( 0 )=
s-lim t-l-m
eiHot
e
-iHt
WO),
the vectors Yf and Yi can be considered to be the asymptotic states of the interaction-picture states [see (3.43) of Chapter IV] (3.55)
Y*(t)
=P o t
e-iHtY*(0),
respectively. Thus, it can be said that (3.52) provides an expression for the transition amplitude (Yf I T Y i ) from the incoming interactionpicture asymptotic state Yi to the outgoing interaction-picture asymptotic state Yf .
3.5. THERESOLVENT OF
AN
OPERATOR
I n the Lippmann-Schwinger equations (3.34) and (3.41) we encountered operators of the form ( A - h - iq)-l, where in the first case A stands for H and in the second case A stands for H , . Operators of the form ( A - {)-l, where { is in general a complex number and A is
452
V. Quantum Mechanical Scattering Theory
a linear operator in #, play a very important role in functional analysis. Since they have been systematically studied, it is advisable to acquaint oneself with the results of such studies, and then apply these results to the special problems at hand.
Definition 3.2. If A is any linear operator in #, the resolvent RA(5)of A is the operator-valued function (3.56)
RA(5)
=
( A - C)-l,
5 C1,
defined at all complex values of 5 at which ( A - <)-l exists. We note that, according to Theorem 4.2 of Chapter 111, ( A - 0-l exists if and only if the equation ( A - 5)f = 0 has the unique solution f = 0, i.e., if and only if 5 is not an eigenvalue of A. We can classify the points 5 in the complex plane C1 in relation to a given linear operator A as in Definition 3.3.
Definition 3.3. T h e set of complex numbers 5 consisting of all points 5 E C1at which the resolvent RA(5)is a bounded operator defined densely in # is called the resolvent set of A. T h e complement of the resolvent set is called the spectrum S A of A. The spectrum S A is the disjoint union of the point spectrum Sg , continuous spectrum S: , and residual spectrum St, which consist of all points which have the following respective properties: Sg of the points 5 E S A at which I?,(<) does not exist; St of all E SA for which RA(c)exists, is unbounded and defined on a dense subset of X ; :S of all 5 E S A at which R,(c) exists but it is not defined in a dense subset of 2. I n the case where A is a self-adjoint operator, the above definition of the spectrum coincides with Definition 6.2 of Chapter 111. T o establish this equivalence, we need to show that
<
(3.57)
Iff belongs to the domain of ( A - 0 - 1 , there is a vector g E g Asuch that f = ( A - S)g. Consequently, we have
Conversely, if
3. General Time-Independent Two-Body Scattering Theory
453
then, according to Theorem 2.5 of Chapter IV (3.58)
is defined. Moreover, g
=
[1/(A - [)I f belongs to g Abecause
Hence, by (2.25) of Chapter IV,
i.e., f is in the domain of (A - [)-l. Thus, (3.57) is established. By comparing (3.58) here and (2.20) of Chapter IV, we see that the above argument also establishes that when A is self-adjoint,
1, h--5 1
(3.59)
RA(-5)
=
dEA A =1 A--5
Hence, for self-adjoint operators, Definition 6.3 of Chapter I11 and Definition 3.3 of the point spectrum Sg of A coincide, since EA({[})# 0, 5 E W,if and only if [ is an eigenvalue of A , in which case (A - [)-l does not exist. We shall prove now the equivalence of the two definitions (6.3 of Chapter I11 and 3.3) for the continuous spectrum :S of a self-adjoint operator. When, for a fixed real 5, E A ( [ [ - E , [ €1) # 0 for all E > 0 and E4({[}) = 0, we must have either E A ( [ [ - E , 5 ) ) # 0 or E A ( ( [ ,5 €1) # 0 for all E > 0. Let us say that the first alternative is the case. Then the equations
+
+
ER("
can be satisfied for all E
- E,
O)f€
= f €2
llfE
/I = 1,
> 0. Inasmuch as
we conclude that R,([) is an unbounded operator. On the other hand, R A ( [ ) is defined on the dense set of all vectors g satisfying EA([t;- E, 5 ~ ] ) g= 0 for some E > 0. Thus, if [ is an element of the continuous spectrum in the sense of Definition 6.2 in Chapter 111, then
+
V. Quantum Mechanical Scattering Theory
454
it belongs to the continuous spectrum in the sense of Definition 3.3. T h e complete equivalence of these two definitions of St for a self-adjoint operator A follows from Lemma 3.2.
Lemma 3.2. T h e resolvent RA(5)of a self-adjoint operator A is a bounded linear operator if and only if either I m 5 f: 0 or 1; is real but - E, 5 + €1) = 0 for some E > 0; in either case 9BA(1) =X.
+
Proof. We have seen earlier that if 5 is real and EA([S - E, 5 E)] # 0 for all E > 0, then RA(5)either does not exist or, if it exists, it is an unbounded operator. Assume now that 5 is real and EA((5 - E,, 5 c0)) = 0 for some e0 > 0. Since the function 1/(5 - A) is bounded on
+
(-a, 5 - €01 u [5
we conclude that
+
€0,
+a),
Hence, it follows from (3.57) that RA(C)is bounded and defined for all f E X . I n the case where I m 5 # 0, we have
s,
1
IA
-5
lP
<
d /I ERf/12
1
jRl d II
T h e above inequality establishes that when Im
11 RA(C)l/
(3.60)
and I?,(()
is defined for all f
E
<
5 # 0,
1 9
2. Q.E.D.
It follows from the above lemma that a self-adjoint operator has no residual spectrum. 3.6.
THERESOLVENT METHODIN SCATTERING THEORY
T h e two resolvents we encounter in time-independent scattering theory are the resolvent R,,(() of the free Hamiltonian H , and the resolvent R(5) of the total Hamiltonian H . However, while Ro(5) is easy to compute in practice, R(5) is an unknown quantity. Thus, Theorem 3.2, which relates R(5) and Ro((),proves to be very useful.
455
3. General Time-Independent Two-Body Scattering Theory
Theorem 3.2. Let H and Ha be any two (not necessarily selfadjoint) operators in the Hilbert space 2, having identical domains of . T h e following second resolvent equations are definition g H= gH0 satisfied by the resolvents R(5) and Ro(S)of H and H, , respectively: (3.61) R(1) - Ro(5) = -Ro(O(H - Ho)R(O = --R(I)(H - Ho) Ro(O for any 5 which belongs to the resolvent sets of both H and H a , i.e., for 5 $ SH u S H o .
Proof. According to Lemma 3.3, R(5) and Ro(5) are defined on the entire Hilbert space 2 whenever 5 does not belong to the spectra of H and Ha . Hence, for every f E 2 there are vectors g ~9~ and h E gH0 such that f = ( H - 5)g = (Ha - 5)h. Since g H= BHo, g also belongs , therefore to g H 0and ROMH - H0)g
= Ro(5)(H - Ho) R(5)f
is defined. Setting above ( H - H,)g the relation Ro(5)(H - H0)g
=
= RO(O(ff - o g
[H - 5 -
- (Ha - 5)]g, we
get
g = RO(0f - R(Of,
which is equivalent to the first of the identities (3.61). T h e second identity in (3.61) can be established by a similar procedure. Q.E.D. Before proceeding with the applications of the second resolvent equations, we shall derive, for the sake of completeness, the first resolvent equations.
Theorem 3.3. If C1 and 5, belong to the resolvent set of the linear operator A, then the first resolvent equations (3.62) are satisfied.
Proof. T h e range of A - 5 is identical to the domain of definition of RA(C)= ( A - <)-l, which coincides with J? when 5 is in the resolvent set of A. Hence, for any f E A?,a vectorg can be found so that f = ( A - 5,)(A - 5,)g, and therefore (3.63)
(51 - 5 2 ) RA(4.1) R A ( S 2 ) f
= (51
-
52)g.
On the other hand, we have (3.64)
[R~(5i)- R~(52)I.f = ( A - 52)g - ( A - I J g
= (51 -
Since (3.63) and (3.64) hold for any f E 2, (3.62) follows.
Q.E.D.
456
V. Quantum Mechanical Scattering Theory
Let us return now to our main task of giving a perturbational method for computing transition probabilities. Using Theorem 3.2 we infer that, if BH = aHo , (3.65)
R(5) = Ro(5) - RO(5)fflR(5),
whenever I m we obtain
5 # 0. After
ffl
=H
-
Ho ,
n successive iterations of the above relation,
where the remainder R, is (3.67)
R,
=
(-1)” (RO(5)Hl)”+l R(1).
+
If the above remainder converges to zero, in some sense, * when n -+ CO, then (3.66) provides us with a means of computing R(5) when R,(<) is known: (3.68) R(5) = Ro(1)- Ro(5)HiRo(5)
+ Ro(O HiRo(5) HiRo(O f- .*.
3
This result combined with (3.67) leads eventually to a “perturbation” solution of the Lippmann-Schwinger equations. This solution is obtained by substituting for R(ho iv) = ( H - A, - iq)-l in (3.41) the series on the right-hand side of (3.68). If the series so obtained for the integral in (3.41) converges, and the order of summation and integration can be interchanged, then Y*can be computed to any order.
+
3.7.
OF VECTORAPPENDIX: INTEGRATION AND OPERATOR-VALUED FUNCTIONS
I n the last two sections we encountered integrals, such as (3.5), which are special cases of Bochner integrals. T h e theory of Bochner integration is based on the following theorem.
Theorem 3.4. Suppose p is a measure in the measurable space % and that f(.$), 4 E R, is a vector-valued function on the measurable set R, assuming values in a Hilbert space 2.If (g I f(4)) is measurable on R for anyg E 2 and if (3.69)
j, I l f ( O l l ~ P ( ~<) +a,
* See also Lemma 5.2 and Theorem 5.9 of this chapter.
3. General Time-Independent Two-Body Scattering Theory
457
then: (a) there is a unique vector h E 2, called the Bochner integral of f ( E ) on R , denoted by
which is such that for all g E 2 (3.70) (b) if A is a bounded operator on 2 and IIAf(f)11is measurable, then (3.71)
A
J f ( 5 ) 440 = J R
R
Af(5)4-43
Proof. (a) T h e linear functional
is bounded on account of (3.69):
Hence, by Riesz' theorem there is a unique vector h which is such that $(g) = ( h 1 g) for all g E Z . (b) T h e Bochner integral of Af(0 exists since (g I Af(<))= (A*g j f(5)) is measurable for any g E 2,and
I n order to prove (3.71) it is necessary and sufficient to show that (3.72)
=
J
R
(g I Af(ED 44'3
for all g E X . Using the definition of the Bochner integral, we get ( g I A h ) = (A*g I h ) =
thus establishing that (3.72) is true.
JR (A*# lf(5)) 440, Q.E.D.
458
V. Quantum Mechanical Scattering Theory
T h e Bochner integral for vector-valued functions can be used to define Bochner integrals for operator-valued functions A([). As a matter of fact, if the Bochner integral AR[ffl
=
1 [A(Oflddf) R
exists for all f E 2,then the mapping f -+A R [ f ]is linear since
can be defined to be the Bochner integral of A ( [ )on R . It is useful to note that the integral in (3.73) has the property (see Exercise 3.9) (3.74)
A,*
=
j
R
A*(.!) d P ( 0
Naturally, all of the above results can be immediately generalized to complex measures p by decomposing such measures into a sum of real measures. A certain kind of generalization of Bochner integrals has been encountered in (3.33), (3.34), (3.40), and (3.41). We define such integrals in a more general context by means of the following concept of integration.
Definition 3.4. Let f,)and A(h) be functions in h which assign to every X E [a, b] a vector f,)in 2 and a bounded operator A(h) on 2, respectively. Suppose that for any sequence of subdivisions a = A(0) < h'1) < ... < = b of [a, b] with norm 6 = maxi h(k) - h ( k - 1 )
converging to zero, and that for any n
(3.75)
E
l
[X(k-l),
459
3. General Time-Independent Two-Body Scattering Theory
exists for some fixed g E 8. If this limit is the same for any choice of sequences of subdivisions with shrinking norm 6 +. 0 and for any appropriate choice of X ' ( k ) , then it is denoted by the integral symbol
A corresponding improper integral is defined as
I+d& I m f , )
(3.77)
=
ijy,
J
b
dA
with a similar definition holding when a + - co. We encountered integrals of the type (3.76) in (3.31) and (3.39). I n those cases, fA is of the form E,f, where EA is a spectral function and f is a fixed vector in Z . For a given interval I C [wl,the function +I(d =
J
I
d,
is obviously linear. If &(g) is defined for all g E .%? and if it is bounded, then by Riesz' theorem there is a unique vector hI which satisfies the equality (hI I g) = $ I ( g ) for all g E 2. We denote this vector by the integration symbol
j, 4 4 4 f A .
(3.78)
Suppose KAis a operator-valued function. It is easy to see that
J 4 4 d,[K,(a,f, + a z f d J 4 4 dA[alK,f, + azKf2l a1 J 4 4 d A W 1 + 144d A W 2 =
I
I
=
I
a2
I
and that the existence of the integrals on the right-hand side of the above relation implies the existence of the integral on the left-hand side. Hence, the mapping which takes f into Adf)=
j
I
4 4 dAKAf
is linear. We denote the linear operator A, by the integration symbol (3.79)
A,
=
1 I
A ( h ) d A K A.
460
V. Quantum Mechanical Scattering Theory
Integrals of this kind have been encountered in formulas for Q, as (3.33) and (3.37).
, such
EXERCISES 0
3.1.
Show that there are unique operators sZ-,(s)
< E \< s, which satisfy the relations (g I Q-,(s)f> = E (g I Q+,(s)f>
=
for allf, g E Z , and that (( sZ,,(s)~~
3.2.
E
and sZ.+E(s),
j’ e-Yg I Q(W>at,
s”
--s
eft(g I Q(W>dt
< 1.
Prove that Qh., = s-lim Q,,(s). W+W
3.3. Suppose that the Riemann-Stieltjes integral of F(X) with respect to a(h) exists in [a, b], that F(h) is Bore1 measurable, and that .(A) is nondecreasing. Show that F(h) is integrable with respect to a measure p(B) which satisfies the relation p(( - co,A ] ) = .(A), and that
p(4444 = L 3.4.
b ,
F(4 444.
Derive from the basic Definition 3.1 that the integrals (3.18)-
(3.20) are linear; for example,
3.5. Prove that if the Riemann-Stieltjes and [c, b] exists ( a < c < b), then
jcF(h) dU(X) = a
a
F(h) &(A)
integral on [a, b], [a, c]
+ J F(h) du(X), b
a
< c < b.
C
Extend the proof to show that for cross-iterated integrals
Remark. If .(A) as well as F(X) are discontinuous at c, the integral on b] might not exist, although the integrals on [u, c] and [c, d ] exist.
[u,
461
3. General Time-Independent Two-Body Scattering Theory
3.6.
Show that if
a(/\)is
nondecreasing and F,(A)
< F2(A),then
<
Using this result, show that if o(A, A,) is nondecreasing [i.e., o(A, A,) A‘ and A. A,’] and Fl(A,A,) Fz(AAO), then
@’, Ao’) whenever A
<
<
<
3.7. Assuming that the respective Riemann-Stieltjes integrals for
u1 and u2 exist, prove that the corresponding integrals for the linear
combinations alul
+ azu2, a, , a2 E C1, exist and
where I and I,,are arbitrary nondegenerate intervals in R1.
3.8. Using the result of Exercise 3.6, prove the mean-value theorem for Riemann-Stieltjes integrals: IfF(A) is real and continuous, and if u(A) is nondecreasing and bounded on [a, b], then there is a point A’ E [a,b] for which
j’F(A) d,o(X) = F ( X ) j a
b
d*u(A) = F(X’)u((a, b ] ) .
a
Show that the above integral exists if F(h) and assumptions made earlier.
o(A)
satisfy the
3.9. Prove that if (3.73) exists, then the relation (3.74) holds. 3.10. Prove that the resolvent RA(() of a self-adjoint operator A can be represented for I m ( < 0 by a Bochner integral: R(5) = i
1’” 0
ei(A-C)t
dt.
462 4.
V. Quantum Mechanical Scattering Theory
Basic Concepts of Time-Independent Scattering Theory for Hamiltonians with Eigenfunction Expansions
4.1. FREEPLANEWAVESIN THREE DIMENSIONS I n $3 we presented a very general framework for time-independent scattering theory. However, in practice one encounters much more specialized versions of that framework, which are, on the other hand, more convenient to deal with from a computational point of view. T o derive these formulations rigorously from the general framework of $3, we need the existence of eigenfunction expansions for the Hamiltonians H , and H . Before formulating the concept of eigenfunction expansions on a general level, we shall elucidate this concept by studying it in the special case of the kinetic energy (or “free”) Schroedinger operator H, on L2([w3), defined by means of the differential operator form
If we consider H i to be an operator acting on the space of all everywhere twice-differentiable functions f (r), then Hi has eigenfunctions @k(r)
1
k2
Gk(r) = -eikr 2m ak(r), (24312
associated with the eigennumbers (4.3)
A(k)
=
k2/2m.
I n physical literature the functions ak(r) are called free plane waves. However, these eigenfunctions are not square integrable on R3 in the Lebesgue measure, and therefore they do not represent eigenvectors of H,. T h e family (cDk(r):k E R3} of eigenfunctions of H i has some remarkable properties, which follow from Theorems 4.5 and 4.6 of Chapter 111. According to Theorem 4.5 of Chapter 111, if 4 EL~([W~), and if B, , B, ,... are any bounded measurable sets, the functions (xB,$)(r) have Fourier transforms UF(xB,+),since for every n = 1,2, ... the function XB,(r)$(r) is integrable as well as square integrable with respect to the Lebesgue measure. On the other hand, if B , = W,then
(J2=l
4. Time-Independent Scattering Theory for Hamiltonians
// xB,+
-
+ 11
-+0 when
in Chapter 111,
463
n -+ +a,and therefore, in view of Theorem 4.6 UF+ = s-lim U&B,+).
(4.4)
n++m
Let us introduce some convenient notation in Definition 4.1, Suppose F ( x , x’) is the kernel of an integral operator
Definition 4.1. on L2(S , v) (4.5)
and that 9 is the family of all functions g(x) for which the integral in , (4.5) exists. Then we shall write for some h € L 2 ( S v) (Fh)(x) = 1.i.m.
Js
F(x, x’) h(x’) dv(x’)
if the linear operator defined by (4.5) has an unique extension f = Fh to h, i.e., if for any sequence g, ,g, ,... E 9 converging strongly to h there is an unique f E L2(S,v) such that
We note that if F happens to be a bounded operator on 9, then by the extension principle of bounded operators (Chapter 111, Theorem 2.6), , and represents the 1.i.m. J”F(x)h(x) dv(x) is defined for all h € L 2 ( S v) extension of the operator F to h. I n particular, if F is the Fourier transform U,, then (4.8)
$(k)
=
(U,+)(k)
=
cikr+(r)dr
( 2 ~ ) -1.i.m. ~/~ JP3
is well defined. Moreover, by virtue of (4.11) in Chapter I11 we also have . .
and consequently (4.10)
I,,
+(r) = ( 2 ~ ) -1.i.m. ~/~
eirk$(k)
.
dk.
464
V. Quantum Mechanical Scattering Theory
T h e formal analogy of (4.10) and (4.8) with the expansion formula of a vector 9 E 2,
*
=
c
(en
n
I 4)en
9
in an orthonormal basis ek becomes obvious if we rewrite these two formulas in the form (4.11) +(r)
=
1.i.m.
3$(k) Gk(r) dk,
$(k)
=
R
1.i.m.
I
Dk*(r) #(r) dr
and introduce the convenient notation (4.12)
(@k I $> = 1.i.m.
I
Qk*(r) +(r) dr
= $(k).
T h e fact that (Gk I f ) is not related to the inner product in L2(R3) is emphasized in this notation by the round bracket in (. [ -). Due to such analogies, the relations (4.11) are referred to as eigenfunction expansions of $(r) by means of free plane waves Qk(r). I t is important to realize that we can easily express the spectral measure EHo(B)of H,, with the help of Qk(r).Indeed, we recall that the momentum operators are essentially multiplication operators when they act on $(k), (4.13) (P'")#)" (k)
= k,$(k),
(PW/J)"(k)
=
k,$(k),
(P'"#)" (k)
= k,$(k),
so that the k variables are in this case identical to the momentum variables p. Since H,, is a function of P,
(4.14)
H
O
-
P2
-=fl(P),
- 2m
we have, in the momentum space,
or equivalently, in the configuration space,
(4.15) 4.2.
DISTORTED PLANEWAVES
Let us now consider a total Schroedinger operator H defined by means of the differential operator form (4.16)
Hr
=
-(1/2m)A
+ V(r).
4. Time-Independent Scattering Theory for Hamiltonians
465
I n $96-7 we shall extend to H’ the above results on H: by proving that, under certain assumptions on the potential V(r), the following statements are true. For each vector k E R3, there is a unique solution”
@k)(r)= (2+3/2
(eikr
+ vk(r))
of the differential equation (4.17)
-
for which, in spherical coordinates r, 0, and tically as follows:
+, vk(r) behaves
asympto-
wheref,(O,+) is a function uniquely determined by vk(r). T h e family {@i+)(r): k E R3} provides an eigenfunction expansion for any element $+ €L;,(R2) in the sense that (4.19)
where Lic(R3) = EH(S z ) L2(R3) denotes the closed linear subspace of L2(R3)corresponding to the continuous spectrum S z of H . The function $(k) appearing in the eigenfunction expansion (4.19) is not the Fourier-Plancherel transform of $+(r),but rather the FourierPlancherel transform of another function related to $+ by the equation $+ = Q+$. In fact, one of the most important results (which will be derived in 97) of time-independent potential scattering is that Q+ is in the present case a partial isometry with initial domain M, = L2(R3)and final domain R+ = Lic(R3). Hence, to every $+ EL~JIW~) corresponds a unique $ E L2(R3)such that $I+ = Q+$. In physical literature the functions @L+)(r)are referred to as the outgoing (or retarded) distorted plane waves, while fk(O,+) is called the scattering amplitude since, as we shall see in 97, it is intimately related to the function F 1 ) ( p ;w , w ’ ) introduced in (1.31): (4.20)
fk(w)
=
(&/I
k I) T(’)(Ik 1;
-w,
w’),
-w
=
(T
- 8, 2n - 4), w’ = (8’, 4’).
* Strictly speaking, in general this function does not satisfy the equation in the set of singularitiesof V(r),which is required to be of Lebesgue measure zero.
V. Quantum Mechanical Scattering Theory
466
T h e justification of the term “distorted plane wave” is that in the special case when V(r) = 0 we have vk(r) = 0, and @:+)(r)becomes the free plane wave Ok(r). I n addition, the function
@k)(r;t)
(4.21)
= @p’(r) exp[--i(k2/2m)t]
provides a solution of the time-dependent Schroedinger equation (4.22)
a
i--@p)(r;t) at
=
[-(1/2m)A
+ V(r)] @p)(r;t ) .
T h e “distorted part,” vk(r) exp[ -i(k2/2m)t] of this wave describes, heuristically speaking, a process which recedes away from the scattering center r = 0. T h e functions (4.23)
@-)(r)
=
@!t*(r)
are called incoming (or advanced) distorted plane waves for analogous we have the reasons. They also satisfy (4.16), and for any $- EL;~([W~), expansion (4.24)
where $- = SZ-$, with $ uniquely determined by I,!- due to the fact that M, = L2([w3)and R- = Lic([w3). A further significant result of $6, which in fact represents a generalization of (4.19) and (4.24), is that the spectral measure EX(B) can be computed on Li,(R3)by means of O;&)(r): (4.25)
(E*(B) # f ) ( r )
=
1.i.m.
@p’(r) $(k) dk.
I n particular, for B = R1, the relations (4.25) assume the form (4.19) or (4.24), respectively. It is important to realize that (4.15) is also a special case of (4.25), since when V(r) = 0, we have SZ, = SZ- = 1 and Li0(R3) = L2(R3). 4.3.
FREEAND DISTORTED SPHERICAL WAVES
T h e free plane waves Ok(r) are not the only eigenfunctions of the differential operator H i . It is easy to verify that the functions (4.26)
akZm(r) = 4 4 -i)yl(kr) Yzm( -6,
-4)
467
4. Time-Independent Scattering Theory for Hamiltonians
are also eigenfunctions of HA :
-L2
(4.28)
=
[-
i a a sine-] sin 0 a0 a0
1 +--sin20 a@
a2
'
for any values (4.29)
0
< k < +a,
1 = 0, 1 , 2,...,
m = --I,
-Z
+ 1 ,..., +Z.
T h e function QkIm(r,8, (6) is called an outgoing free spherical wave. It will be shown in 97 that the family of all functions (4.26) provides us with eigenfunction expansions for any I,!J E L2(R3):
where, using the notation 0,introduced in (7.10) of Chapter 11, we have @zlm(r) +(r, 8,d) r2 sin B dr d0 d+.
(4.31)
I n the case that V(r) is spherically symmetric, i.e., V(r) = V,,(r),the differential operator H' has the eigenfunctions
These eigenfunctions are called, respectively, outgoing and incoming distorted spherical waves and are generalizations of Qklm ; i.e., (4.33)
H'@EA(r,094) = (- 2m A + vow)@ g k0,4) = 2y &A@, 1
and for any
where I,!J+
i,h*
E Lie(R3)
= Q,$,
and J l m ( k )is given by (4.31).
k2
468
V. Quantum Mechanical Scattering Theory
It is well known (see Butkov [1968, Section 9.101) that the asymptotic behavior for large r of the spherical Bessel functionj,(kr) is (4.35)
j,(kr)
-
( I l k ) sin(kr
- h/2),
I --+
+a.
We shall see in 97 that the functions Rk$)(r)have a similar asymptotic behavior
I n physical literature 6,(k) are called phase shifts, since they represent the change of phase from the free spherical waves corresponding to the noninteraction case V ( r ) = 0, to the distorted spherical waves. We shall see in 97 that they play a crucial role in the computation of the S operator, which can be written in the form (4.37)
=
(w;Vb@)
exp[2&(k)l $Zm(k).
T h e function S,(k) = exp[2iSl(k)] is called the S matrix in spherical coordinates. T h e reason for this terminology is that if we use Dirac’s bra and ket notation to write 1 klm) = D k l m nthen , taking in (4.31) and (4.36) # = Dklm and working formally, without paying attention to the real mathematical meaning of these expressions, we get (k’l’m’ I S I klm)
=
6(k’ - k) 6,,, 6,,
exp[2i6,(k)].
I n 97 we shall establish the following generalization of (4.34):
where (4.39)
Ao(k)= k2/2m.
These formulas can be recast in a more compact form if we introduce where the measure po = p ( k ) x p(u)on g3,
469
4. Time-Independent Scattering Theory for Hamiltonians
It is easy to verify (see Exercise 4.1) that, written in terms of the measure
po , (4.38) becomes
(4.41)
(EH(B)&)(r)
= 1.i.m.
'Atl'B)
ah)$(kt 4 & O ( k
km),
where $(k, 1, m) and @$A(r) are arbitrary extensions of the functions q l m ( k )and @j$(r), respectively, outside the support of the measure p,, . In particular, for H = Ho , we have
as a special case of (4.41). 4.4.
EIGENFUNCTION EXPANSIONS AND GREENFUNCTIONS
We are now ready to introduce a general concept of eigenfunction expansions which stems from the special cases occurring in (4.1 I), (4.15), (4.31), and (4.41).
Definition 4.2. Let H; be a differential or integral operator which serves to define a self-adjoint operator H,, in L2(Rm,p). Suppose that there is a measure v on A?" and functions Qa(m), p E R" [not necessarily p ) ] , such that for almost all p E Rn, we have belonging to L2([Wm,
Then we say that the family {Qa:p E [W"} provides an eigenfunction expansion for Ho if for any E L2([Wm, p) we have a unique &,t E L2(R", v) such that
+
(4.43)
Let H be another self-adjoint operator in L2(Rm,p), given by means of a ,differential or integral operator form Ha. Extrapolating from (4.17), (4.25), (4.33), and (4.41), we expect that under certain circumstances
470
V. Quantum Mechanical Scattering Theory
there might be functions @;*)(a) which for almost any /3 are eigenfunctions of H. (4.45)
(HY@)(a) = A(P)
and that for any
EL~JIW~ p),
t,b*
@pya),
= EH(Sf)L2(Rm,p) we have
(4.46) where [tacitly assuming that R+ (4.47)
+(p)
1.i.m.
1
3
R-
= Lgc(Rm,p)]
J ap)*(a)+*(a) dp(a), [Wn
+* = Q*+.
I n keeping with the terminology used in physics, for any fixed p we shall call Q4(a)a free wave, @I;”( an .)outgoing (or retarded) distorted wave, and @;-)(a)an incoming (or advanced) distorted wave. We encountered in $3 another important quantity associated with a self-adjoint operator A , namely its resolvent RA(5)= ( A - [)-l. It can happen that for 5 belonging to the resolvent set of A , i.e., 5 4 S A , the resolvent R,(iJ is an integral operator of the form (4.48)
( h ( C ) + ) ( a )= 1.i.m.
s,
GA(a, a’;
C) +(a’)4401’).
Definition 4.3. If 5 belongs to the resolvent set of the linear operator A inL2(Rm, p), and if GA(a,a’; 5 ) is integrable in the measure p on bounded Borel sets in R2m of finite measure p x p, and is such that
for all t,h eL2(Rrn, p), then GA(a,a’;
5 ) is called a Green function of A
-
5.
It is an easy exercise (see Exercise 4.2) to show that if p is u finite, then the Green function of A - 5 (if it exists) is determined on R2rnby A uniquely up to its values on a set of p x p measure zero. We shall denote the Green functions of H and H,, by G(a, a‘; 5) and G,(a, a’; C), respectively. I t is computationally very helpful when for self-adjoint A the Green function G,(a, a‘; C) has some limiting values when 5 approaches the real axis in the complex plane. However, we can expect that, in general, G(a, a’; 5) will have some singularities in a and a’. Let us denote by D , the Borel set in RZmat which G(a, a‘; 1) is not defined or becomes
4. Time-Independent Scattering Theory for Hamiltonians
471
infinite for some values of 5 E C1, with I m 5 # 0. When 5 approaches the real axis, we must avoid letting a and a’ assume values for which ( a , a‘) E D, . T h e reader will note that this condition appears explicitly in Definition 4.4. We have to exphasize that it will be tacitly assumed throughout this section that D, is of p x p measure zero.
Definition 4.4. T h e function Gj4+)(a,a‘; A) and G$-I(a, a‘; A) are called, respectively, the retarded and advanced Green functions of the self-adjoint operator A if GA(a,a’; h & ie) converges to G$*)(a,a’; A) uniformly in h E SE and in ( a , a’) from any compact set S C R2m for which S n D, = o. It is easy to see that if , $2 E L : ~ ( R p~), = EA(S<)L2(Rm,p ) are functions with compact supports* Si= supp $i, i = 1, 2, of finite measure p for which (8, x S,) n D, = IZI, then
T h e reason for the terminology used in Definition 4.4 is that, as we shall see later, the advanced (retarded) Green functions are closely related to the incoming (outgoing) waves @:*)(a) exp[-iA(p)t].
4.5. THETRANSITION MATRIX When H has an eigenfunction expansion, we can use this expansion to reduce the iterated integral in (3.52) to an ordinary integral on a measure space. We show this in Theorem 4.1.
Theorem 4.1. Suppose that Q0(a) is continuous and bounded almost everywhere with respect to p x v in both variables a E Rm and b E Rn, that the family {Q0 : E Rn} provides an eigenfunction expansion of the free Hamiltonian H, , which acts in L2(Rm,p), and that its eigennumber function A@), defined in (4.42), is continuous in PER”. Moreover, assume that whenever E > 0, H - h - i~ has a Green function G(a, a’; h i ~ )which is p integrable on Rm in CY. for almost all a’, and in a’ for almost all a , and continuous in a, a‘ E Rm, h E R1 - SpH,
+
* Recall that the support supp +h of a function #(a) is the closure of the set of all points
01
E Rm at
which #(a) # 0. A set S in R“ is compact if and only if it is bounded.
472
V. Quantum Mechanical Scattering Theory
with the possible exception of a set So x R1 C RZm+l, where So is a closed subset of R2m having p x p measure zero. If there is a function T(P1 P Z ) , P1 P 2 6 (wn, such thatX 9
9
x
I,,
M a ‘ ) G(a, a’;
&(a) @,*,(a) 08m
+i ~ )
@D~(U’)
for any y!Ji , y!Jf E M, n Lic(Rm,p), M, = M, , which are such that the supports of $,(p) and &(p) are compact and of finite measure v, and if all the integrals in (4.50) exist, then we have
for all such y!Jf and y!Ji. T h e function T(Pl,P2) is called by physicists the transition matrix (or T’matrix) in the variables /3 E W. T h e reason for this terminology lies in the conventionally adopted Dirac notation, in which one writes T(P,, p2) = (pl I T I p2), where p are variables related to some observables (such as in the case when p = k are momentum variables). It has to be realized, however, that T ( p , , p2) is in general a function and not an actual matrix element of the T operator. T h e reader should note that if G(+)(a, a’; A) exists and the limit E + +O can be taken in (4.50) under the integral sign, then (4.52)
T(P, , P Z ) =
1
08”
444 @;(a) j
R”
M a ’ ) G‘+’(a,a’;
4 @&).
In order to prove the above theorem, we have to relate {t+hf j E~sZ+E~t+hi) in (3.52) to the eigenfunctions (Do(.). For this we need Lemma 4.1.
Lemma 4.1. Let En , h E R1, be a spectral function for which there is a function F ( u , /3; A) such that
(4.53)
(.EA+)(a)= 1.i.m.
F(a9
P ; 4 &PI M P ) ,
* From now on dp dv under an integral sign denotes integration with respect to the measure p X v. The reader should check when he reaches 555-7 that the conditions of this theorem are satisfied in potential scattering.
473
4. Time-Independent Scattering Theory for Hamiltonians
where y(P) is a continuous function of /3. Suppose + 1 ( ~ )and $& are I) two functions which vanish outside the respective closed bounded sets D,C Rm, and D, C Rn, and that in addition $&Ihas ) compact support of finite measure Y. If, in addition, F(a, 6 ; A) is continuous on D, x D, x [a, b] and h(A) is continuous on [a, b ] , we have
According to Definition 3.1,
Proof. (4.55)
1:h(A)
dA<#l
I EA#2)
X ['Y(,)
F(% 8; A,)
$UP) 4 8 ) - j Y ( B ) < A ~ - F(% 8; AE-1) $2?1(8) w3)] ~ 9
<
<
where the only limitation in the choice of A,' is Al-, A,' A, . Since F(a, 6; A) is continuous in all three variables 01, and A, it can be shown (see Exercise 4.3) that the limit in (4.55) is identical to
Exploiting the continuity of F(a, /3; y(/3)) in 01 and /3, and the continuity of h(y(/3))in /3, we infer (see Exercise 4.4) that the limit (4.56) is identical to the limit
thus establishing that (4.54) is true.
Q.E.D.
From (3.38) and (3.39) we get
'?#f
+'<
Hcl
ET#i> =
J+*
-03
'A
(Ho
Ez$fI H
-2E A'
- iE E W ~ I L , )
474
V. Quantum Mechanical Scattering Theory
I n view of (4.49), we have, for almost all a E Rm,
where we have used Fubini’s theorem to interchange the order of integration.” We can apply Lemma 4.1 to the last integral in the above k) and relation, since according to the assumptions on G(a, a‘;A cDOZ(a’),the function
+
(4.59)
F6(%P z ; A)
=
1
Rm
G(a, a’; A
+ i4
@BZ(4
44’)
satisfies the conditions required in that lemma. Thus, combining (4.58) with Definition 3.4 we easily get (4.60)
On the other hand, the following equality holds almost everywhere in a,
* Before applying Fubini’s theorem, we have to make sure that the integrand is integrable on the specified set with respect to all its variables simultaneously, i.e., in the present case, with respect to the measure Y x p in the variables (PZ , a’). It is left to the reader to verify, now as well as in later steps of this proof, that whenever Fubini’s theorem is applied, this integrability condition is fulfilled due to the assumptions stated in Theorem 4.1.
4. Time-Independent Scattering Theory for Hamiltonians
475
so that, after repeatedly applying Fubini's theorem, we derive from (4.60) (4.61)
( E z h I Q+, E A ~ W -
dv(/31)$e*(/31 )
where (4.62)
T(P1 ' 8 2 ; E)
=
1,
(4%) - 4%))@,*,(a)Fd%/3z
; 4 / 3 2 ) ) @(a).
Since (4.50) is supposed to hold even when &(flZ) = 0 for Bz > h and $,(/?,) = 0 for /?, > A,, we get by taking the limit E + +O in (4.61) (4.63)
(E?*f
I Q+ ET*i)
where T(& , &) is the function satisfying (4.50). From (3.52) we obtain
Using (4.63) we easily derive (4.65)
by using Lemma 4.1 and Definition 3.4. By the same procedure
476
V. Quantum Mechanical Scattering Theory
from which (4.51) follows immediately. This completes the proof of Theorem 4.1.
THEENERGYSHELLAND
4.6.
THE
S MATRIX
Formula (4.51) provides a practical way of computing the transition amplitude 1 T+,) from the initial state + i to the final state fil, provided that the T matrix can be computed by using, for instance, the relation (4.50). I n practice, the limit E -+ +O can always be taken easily, and has the effect of restricting in (4.51) the integration in p1 and & to the surface A(pl) = A(&). I n physical literature this surface is known as the energy shell. One reason for this terminology could be that Ef = A(&) and Ei = represent the energy eigennumbers of Wo for the respective eigenfunctions @)B,(a)and @8,(ct).
Theorem 4.2. y
E
Suppose that the variables E = A@) E R1 and B E Rn in (4.51), that
Rn-l can be substituted for the variables
(4.66)
dvi(E) dvn(y),
dv(B)
&(E)
= p ( E ) dE,
where p ( E ) is a continuous function, and that the function T(E, , yt ; E , ,y,) [obtained by substituting in T(& , p2) for p1 and p2 the new variables E, , yf and E , , yi , respectively] is continuous in Ef and Ei . If +i(E,y ) and +,(E, y ) have compact supports of finite measure V, and are continuous in E, then the following equality holds: (4-67)
(#n
1
I T#I) = dEp2(E)
dvn(yr) dv,(yi) T(E, yf ; E , YI)
x $f* (E, Yf)&(E,
rz),
where the integration in E , yf , and y , extends over all possible values which these variables can assume. Theorem 4.2 can be easily derived from (4.50) if the following lemma is used.
Lemma 4.2. If the function j ( x ) is continuous at the origin and bounded on any finite interval of R1 and if e(x2 ~ ~ ) - ~ fis( integrable x) eo , then on R1 for any 0 < E
+
<
(4.68) Prooj.
For given 7
> 0, let us
choose N ( 7 ) > 0 so large that
4. Time-Independent Scattering Theory for Hamiltonians
and that for all 0
477
< E ,< E,,
For any w such that 0
+”
(4.71)
E
< N(q),we have
w
dx = 2 arctan - , E
we obtain by using the generalized mean-value theorem of integral calculus and (4.70)
1
2
G ; Y(% w ) arctan
-
f(0)
I + 2N ( d Wrl), 2E
where y ( ~w, ) is some number in between the infimum and the supremum off(x) on [ - w , + w ] . Due to the continuity off(x) at the origin, the first term on the right-hand side of the above inequality does not exceed 7 if w > 0 is chosen sufficiently small and W / E sufficiently large. Then, for such fixed w , the second term can be made smaller than 7 by letting E -+ 0. Hence, taking into account (4.69), we conclude that (4.68) is true. Q.E.D. When (4.66) holds, (4.51) assumes the form
478
V. Quantum Mechanical Scattering Theory
If the functions T , p, #i , and I+$fulfill the requirements of Theorem 4.2, then the above function f ( x ) satisfies the conditions of Lemma 4.2. Consequently, we get the relation f(0) = (& 1 T&), which is identical to (4.67). Loosely speaking, the relation (4.68) can be written in the form (4.75)
It must be realized, however, that (4.75) assumes a precise meaning only when the Dirac function 6(x) is defined as a functional on a given space of functions, i.e., as a distribution. On the other hand, in the theory of distributions, the spaces of functions usually considered are not such as to include all the functions satisfying the conditions of Lemma 4.2. Hence, one has to realize that when one encounters in physical literature the relation (4.67) written in the form
the meaning of S(E, - Ei) in this relation is more general than that of a Schwartz distribution. Naturally, this meaning is actually contained in Theorem 4.2. We recall that the S operator is related to the T operator by the identity S = 1 - 2riT, so that we have
Let us introduce the functional (in the sense of distribution theory) S,(E, , yi ; Ei, yi) defined by the relation
and the functional
Then (4.76) assumes the form
4. Time-Independent Scattering Theory for Hamiltonians
479
I n physical literature, the functional S ( E , , yf ; Ei , yi) is called the scattering matrix or the S matrix.
4.7. THEOPTICALTHEOREM T h e relation S = 1 - 2iriT between the S operator and the T operator leads to the identities
+ 4rr2T*T, 2ni(T - T * ) + 4n2T*T.
(4.80)
S*S
=
1 - 2ni(T
(4.81)
SS*
=
I
-
-
T*)
Hence, S is a unitary operator on M, if and only if (4.82)
T* - T
=
2niT*T
= 2niTT*.
T h e above relations require that the T matrix have certain properties, as stated in Theorem 4.3.
Theorem 4.3 (Generalized optical theorem). Suppose that the function T ( E , , y, ; Ei , yi) is continuous in Ei and E, , that v is cr finite, and that dv(E, y ) = p(E) dE dv,(y), where p ( E ) is a continuous function. Then the S operator is unitary on M, = M, if and only if (4.83)
T*(E, Yl ; E, ye) =
=
-
T(E, yf ; E, Y i )
jT*(E, E, Yf) T ( E , E, y M E ) dVO(Y) 2ni jT ( E ,y ; E, re) T*(E, y ; E, yl)P(E) dV,(Y) y;
y;
for almost all values of E E R1 and yf , yi E Rn--l, provided that for any compact set D C Rn-l of finite measure v, the function
j I T(E,Y ;E , r’)lP(E) dVO(Y’) D
is integrable and square integrable in E and y. T h e above theorem is an easy consequence of (4.82). First of all, we have to observe that due to the conditions imposed on the T matrix and on the measure V , Theorem 4.2, is valid in the present case. Moreover, for t/f , i,hi E M, , satisfying the conditions of that theorem, we have (4.84)
<$f
I (T* - T )$1)
=
(T$f I 41)
- (#f
I T*I)
V, Quantum Mechanical Scattering Theory
480
O n the other hand, under the same conditions we get under the restrictions of the theorem by employing Lemma 4.2 (as in the proof of Theorem 4.2) twice in succession
If T(E,y ; E, y ’ ) satisfies the conditions stated in the theorem, we can apply Tonelli’s theorem (Chapter 11, Theorem 3.14) twice in succession to deduce that the products of integrands in the above relation is integrable with respect to v1 x vo x v,, x v,, . Hence, we can use Fubini’s theorem to interchange the orders of integration in (4.85) and derive
Thus, we see that the expressions on the left-hand sides of (4.84) and (4.86) are equal if and only if (see also Exercise 4.5) the first of the equations (4.83) holds. T h e proof for the second set of equations (4.83) proceeds along similar lines. T h e relation (holding almost everywhere) (4.87)
Im T ( E , y ‘ ; E, y ’ )
= -n
1 T ( E ,y ’ ; E, y)I2p(E)dvo(y)
is known as the optical theorem; it owes its name to its similarity to a corresponding relation derived in classical electromagnetic scattering, which includes scattering of visible light as a special case. We note that the optical theorem is a corollary of the generalized optical theorem, which is obtained when we set yi = yf = y‘ in (4.83). However, for the validity of (4.87) it is sufficient that only the first equation in (4.83) holds. Since this first equation is equivalent to condition that S*S = 1 on M, , the optical theorem is valid under weaker conditions than the
4. Time-Independent Scattering Theory for Hamiltonians
48 1
unitarity (on M,) of the S operator. I n fact, we see from (2.11) that S*S = Q,*s)-Q-*Q+ = Q+*ER-Q+, and therefore we have S*S = EMo if ER 2 ER+, i.e., if R+ C R- . It-is important to observe that the optical theorem, as well as its generalized version (4.83), refers to properties of the T matrix T ( E , , yf ; E , , 7,)on the energy shell, when E - Ei . Under certain .additional assumptions on the T matrix, the equations of the generalized optical theorem can be extended off the energy shell. These off-energyshell equations are known as the Low equations (see Newton [1966, Section 7.2.21). 4.8.
THELIPPMANN-SCHWINGER INTEGRAL EQUATIONS
When H , and H possess eigenfunction expansions, Green functions, as well as advanced and retarded Green functions, then the LippmannSchwinger equations can be recast in the form of integral equations, which are computationally easier to handle than the operator equations of 93. We shall now derive the type I integral equations, pointing out in the course of the derivatives which assumptions are required to obtain the final result. These assumptions are frequently satisfied in practice. Suppose that H , is given by a differential or integral operator Po. Let us further assume that if +*(a) = (Q,+)(a), and if &(p) E %?l(Rn) is a function with support which is of finite measure v, and disjoint from a certain closed set 9,[of possible singularities of @;*)(a)] for which p(9,) = 0, then (4.88)
(H01Cl*)(4=
j,. $((P)(Ho"@f')(4W P )
almost everywhere (with respect to p ) in Rm (see Exercise 4.7). By virtue of the above assumption and (4.46) we have
V. Quantum Mechanical Scattering Theory
482
+
If Go(a,a'; h iq) $@)((A - E')@;*))(a') is integrable (see also Exercise 4.8) in ( a ' , p), then the order of integration in 01' and j3 can be reversed, and afterwards Lemma 4.1 can be applied, thus arriving at the result
x
444 Go(% a'; 4 3 )
JRrn
+ i 7 ) ( 4 P ) - Hf)
.)'%I@
T h e type I Lippmann-Schwinger equation (3.34) leads to the relation
Using (4.43), (4.46), and (4.90), and assuming that the limit 7 --t &O can be carried under the integral sign (see also Exercise 4.9), we can rewrite the above equation in the form (4.9 1)
J
R"
44a) $0*(4
j
s,444 +J
IWm
R"
4 P ) $(PI @f%)
$0*(4
J W P ) $(PI @A4 J W P ) $(P) J
444 $0*(4
R"
R"
444 G%u,
a'; 4 P ) >
x (A@)- H,"')@f'(a'). T h e equation (4.91) holds for any 1cl0(a)E %:(UPwith ) support of finite measure p, and any I,@) E %:(Rn) with support which is of finite measure Y and disjoint from the Yoof p measure zero. Hence, using standard arguments (such as in solving Exercise 4.2), we infer that (4.92)
@r)(a) + J = QB(a)
Rrn
GF)(a, a'; A(P))([A(P) - H,"']
@F))(a') dp(a')
for almost all (with respect to p ) values of 01 E Rm and almost all (with respect to u ) values of p E Rn. T h e two equations (4.92) represent the integral versions of the type I Lippmann-Schwinger equations. I n potential scattering one can easily justify (see Exercises 4.7-4.9) for very large classes of potentials, the manipulations leading from (3.34)
483
4. Time-Independent Scattering Theory for Hamiltonians
to (4.92). We shall adopt, however, a different approach in 96 to derive (4.92) in potential scattering. In this approach functions @;*)(a) are first defined in terms of the full Green function. After that, it is shown that these functions satisfy (4.92). I n the end, it is proved that the functions @;*)(a) are indeed the distorted plane waves by showing that they fulfill the requirements of 54.2. I n the type I1 Lippmann-Schwinger equations (3.41), the roles of H and H,, , as well as Y* and Y, are reversed. Hence, it will be possible to derive the type 11Lippmann-Schwinger integral equations
(4.93)
@’(a)
=
+ J”
0jB(a)
IWm
G(*’(a, a’; A(/3))(A(/3) - Ha’)aB(a’) +(a‘)
by the same methods. All we have to do is reverse the roles of free and full Green functions, as well as of the free and distorted waves. We leave this straightforward task to the reader.
4.9. INTEGRAL FORMS OF
THE
RESOLVENT EQUATIONS
When the appropriate Green functions exist, we can recast the resolvent equations (3.61) and (3.62) as integral equations.
Theorem 4.4. If GA(n,a’; 5 ) is the Green function of the linear operator A , then the first resolvent equation (3.62) is equivalent to the equation (4.94)
GA(% a’; 51)
- GA(%
a’;
52)
holding for almost all 01 and a’, provided that the above integral exists.
Proof. T h e equation (3.62) holds if and only if (4.95) (31 I R4(51) 3 2 )
- (31
I RAG21 32)
= (51 - 5&31
I R4(51) W
5 2 )32)
for any i+bl and i+b2 belonging to a dense domain. Such a dense domain is the set of all functions with compact supports of finite measure, for which (supp i+bl x supp i+b2) n D , = a . For such functions (4.95) is equivalent to
484
V. Quantum Mechanical Scattering Theory
provided that GA(a,a”; 5,) G,,(a’’, a’; 5,) is integrable in a”. It is now easy to establish, by using Fubini’s theorem and already standard procedures (see Exercise 4.2), that (4.96) is equivalent to (4.94) holding almost everywhere. Q.E.D.
A similar procedure establishes Theorem 4.5.
Theorem 4.5. Suppose the Green functions G(a, a’; 5) and GO(m,a’; t;) of H - t; and H , - 5, respectively, exist. Then the second resolvent equations (3.61) are satisfied if and only if (4.97)
G(a,a’; 4 ) - GO(a, a’; 5) =
=
-IRrn Go(ol, c)(H“” H,””)G(ol”, a”;
-1
Rn
-
a’; 5) dp(a”)
G(a,a”; ()(Ha”- H,“”)Go(a”,a‘; 4) dp(01”)
almost everywhere in a and a’, provided that the above two integrals exist. T h e proof of the above theorem is almost identical to the proof of the preceding theorem, and it is left to the reader.
EXERCISES 4.1. Show that the expressions on the right-hand sides of (4.38) and (4.41) are identical. 4.2. Show that if GL1)(a,a’; 5 ) and GLz)(a,a’; 5) are two Green functions of A - 5, and p is u finite, then GI1’(a,a‘; 5 ) = GLz’(a,a‘; 5) almost everywhere in a and 01’ (with respect to the measure p x p). 4.3.
Show that the limit in (4.55) is identical to (4.56).
4.4. Prove that, under the conditions of Lemma 4.1, the limits (4.56) and (4.57) are identical. 4.5. Show that the right-hand side of (4.84) is equal to the right-hand side of (4.86) for any choice of #i(E, y ) and &(E, y ) having compact supports of finite measure if and only if the first equation in (4.83) is satisfied.
4.6. Show that in potential scattering [A(k) - H i ] @F)(r)= V(r) @F’(r),
except at the singularities of V(r).
5. Green Functions in Potential Scattering
485
4.7. Verify that in potential scattering (4.88) holds for distorted is continuous on [w3 - Y o ,where Yo plane waves @:+)(r)if (A@L%))(r) is closed and of Lebesgue measure zero.
Show that Go(r,r’;X -1- iq) V(k’)@:*)(r’)$(k) is integrable on in (r, k) if I @L*)(r)l const for all r, k E [w3, if $(k) E W:(R3), V(r’) is square integrable, and the free Green function is given by (5.6). 4.8.
<
[w6
k) and G(*)(r,r’;A) are 4.9. I n potential scattering G(r, r’; X given by (5.6) and (5.16), respectively. Prove that if V(r) is square o ) some e0 > 0, and if integrable and I V(r)i = O ( l / ~ ~ + ~for I @:*)(r)l const, then
<
lim
W*O
j+O*(r)Go(r,r‘;A + i ~ V(r’) ) @f)(r’)$(k) dr dr‘ dk =
for J0(r)E
1
+,,*(r) Gr)(r, r‘; A) V(r’) @F)(r) $(k) dr dr‘ dk
%t([W3) and $(k) E Vi(R3).
5. Green Functions in Potential Scattering 5.1.
THEFREEGREENFUNCTION
Potential quantum scattering is the special case of quantum scattering theory in which the underlying formalism is that of wave mechanics, and the interaction is determined by a potential. In applying the general methods of scattering theory to potential scattering of two particles, we shall limit ourselves to systems of two spinless, nonidentical particles interacting via a potential V(r), where r = r2 - rl , and rl , r2 are the position vectors of the two particles. This case exhibits all the essential features of all other cases. For example, the case of nonzero-spin particles can be treated essentially by the same methods as the zero-spin case when group-theoretical methods are applied. I n the case of systems of two identical particles one must also take into account the presence of the Fermi-Dirac or Bose-Einstein statistics. From the practical point of view this means that in these last two cases the calculations have to be carried out in the respective subspaces of antisymmetric or symmetric functions of L2([w3), rather than in the space L2(R3)itself.*
* See
particles.
Rys [1965] for a general treatment of the scattering theory of two identical
486
V. Quantum Mechanical Scattering Theory
I n the absence of external fields and mutual interaction, the relative motion Hamiltonian H , of a system of two spinless particles is the kinetic energy operator in the center of mass system. We recall from Theorem 7.3 of Chapter IV that H , = T , is a self-adjoint operator for which (5.1)
(Ho$)(r) =
1 2m A W ,
--
$69
0
9
and that Ho is the only self-adjoint operator having this property. On wave functions $(k) €L2([W3)in the momentum space, H , acts in the following manner:
T h e above operator is obviously self-adjoint and, since it is a function of the momentum operators PCX),P(g),P @ ),it immediately follows that it has a pure continuous spectrum sHo =
(5.3)
s?
=
[O, +m).
Thus, its resolvent (5.4)
is defined for all values of 5 E C1.Moreover, R,([)is a bounded linear operator defined on the entire Hilbert space & = L2(R3)whenever Re 5 < 0 or Im 5 # 0, i.e., when the argument arg 5 of 5 is within the open interval (0, 277).
Theorem 5.1. T h e resolvent R,(()of the free Hamiltonian (5.2) is an integral operator (5.5)
(Ro(5)$)(r) =
R3
Go@, r'; 5) +(T') dr'
for values of 5 restricted to the range 0 < arg 5 G,,(r, r'; l), called the Green function of Ho , is
where
fit is the square root
for which Im
< 27r. T h e kernel
1/2mZ > 0.
487
5. Green Functions in Potential Scattering
Proof.
It is very easy to verify by using (5.2) that
(5.7)
Assume that the wave function #(r) in the configuration space is continuous and of compact support, i.e., E %?:((w3). T h e n #(r) has a Fourier transform (5.8)
$(k)
= (2.rr)-3/2
j,,
e-ikr’
$(r‘) dr‘.
Furthermore, by Theorem 4.5 of Chapter 111, $(k) eikr is integrable* on R3. Since (k2/2m- 0-l is bounded in k over R3 when 0 < arg 5 < 27r, the inverse Fourier transform of (k2/2m- 4)-l $(k)
exists. By virtue of (5.7) and (5.8) (5.9)
1
(&(l)$)(r) = 03
I,,
dk
k2 (x -
eirk
I
R3
dr‘ rikr‘ *(r’)*
Due to the fact that #(r’)is of compact support, the function
is Lebesgue integrable on R6 with respect to the variables r and k. Hence, we can apply Fubini’s theorem to interchange the order of integration in (5.9):
T o carry out the above integration in k, let us write (for fixed r - r’) the variable k in spherical coordinates introduced in such a manner that k(r - r’) = kp = kp cos 8. p =
* Throughout this section, whenever we deal with integration, we have in mind integration with respect to the Lebesgue measure, except if otherwise explicitly stated.
488
V. Quantum Mechanical Scattering Theory
It is very easy to carry out immediately the integrations in 0 and 0 C$ 2n:
< < k2 / (= 5)
(5.11)
-1
-
R
-
so a,
<8
eikpdk
k sin kp dk p(k"2m - 5)
=
-4ma/p
s
+m -m
k sin kp dk , k2 - 2m5
In the last of the above integrals we have to integrate with respect
to k the function
k &kD k sin k p k2 - 2m5 2i(k - f i m l ) ( k
(5.12)
+65)-
k e-akD 2i(k - .\/2m5)(k
+
over the entire real line R1. If we work with the complex variable k E C1, the above functions are analytic everywhere in the complex plane, ,' d<, Im > 0. Let us close except for two simple poles at +% the contour of integration by semicircles in the upper-half and lower-half complex plane, respectively, when integrating on the real line the first and second function on the right-hand side of (5.12). Since the contributions from the respective integrations on the semicircles vanish in the limit of their radii becoming infinite, we easily determine, by using Cauchy's integral formula and computing the respective residuals
,'c
s
+* --m
k sin kp ka - 2 m 5
=-/1 2i
-
+, e-ikn
1 dk - k2 - 2m5 2i k
eikD
k2 - 2m5
dk
k e--ikn -7-r
-
2/2mr
lk=-
-
- -,ei~d\/2mg -
Inserting this result in (5.11), we get from (5.10)
for all I+!IE%~(R~). Since V:((w3) is dense in L2([w3)(see Chapter 11, Theorem 4.9,while both R,,(C) and the integral operator with the kernel (m/27r)I Y - r' 1-l e x p ( i . \ / m 1 r - r' I) are bounded linear operators defined on the entire Hilbert space L2(R3),we infer with the help of the extension principle (Chapter 111, Theorem 2.6) that (5.13) is true Q.E.D. for all I+!I EL~(IW~).
489
5. Green Functions in Potential Scattering
It follows from (5.6) that the free Green function has the following symmetry properties: (5.14)
Go(r, r'; 5)
=
Go*(r, r'; 5*),
(5.15)
GO(^, r'; 5 )
=
Go(r', r; [)
for any 5.2.
5 E C1 not on the positive
real axis.
RETARDED AND ADVANCED FREEGREENFUNCTIONS
It is easy to infer from (5.6) that when 5 approaches the positive real axis from above or from below, the Green function G,,(r, r'; 5) also approaches well-defined respective limits for all r # r', r, r' E R3. These limits turn out to be the advanced and retarded free Green functions Gh*)(r, r'; A).
Theorem 5.2. T h e retarded free Green function G(+)(r,r'; A) and the advanced free Green function exist for all A E R1 when Ho is given by (5.1): (5.16)
G r ) ( r , r'; A )
m
1 e x p ( i i dm 1 r
=
274 r
=
Gi-)(r, r'; A )
-
r'
-
r'
1)
for any h >, 0, and (5.17)
GF)(r, r'; A)
=
Go(r, r'; A )
for h < 0. T h e proof of the above theorem is very straightforward. It consists in checking that the conditions of Definition 4.4 are satisfied when we adopt the expressions (5.16) and (5.17) for G&*)(r,r'; A). I n the present case, the set of singularities of Go(r, r'; 5) is (5.18)
DGo= {r, r': 1 r - r' 1
= 0).
Note that DG0is indeed of Lebesgue measure zero in R6(see Exercise 5.1)! If we introduce the variable K = 4/2m(, we easily see that (5.19)
(*)
I Go(r, r'; 2/2m)- Go (r, r'; A)1
490
V. Quantum Mechanical Scattering Theory
and that the exponential converges uniformly to one on compact sets in R6 when K -+ fd2mh. Hence, the convergence of Go(r, r'; ~ ~ / 2 r n ) to GA*)(r,r'; A) is uniform on any compact domain which does not intersect D G 0 .Thus, Theorem 5.2 is established. Since in the present case we have (5.20)
Gr)(r, r'; A)
= !l%
Go(r, r'; A
r # r',
k),
we deduce from (5.14) and (5.15) (5.21)
Gi*)(r, r'; A)
=
Gr)(r', r; A),
(5.22)
Gp)(r, r'; A)
=
[Gi-)(r, r'; A)]*
for any X E R1. These properties can be also derived directly from (5.16) and (5.17).
EQUATIONS WITH HILBERT-SCHMIDT KERNELS 5.3. INTEGRAL One of the main significant features of the free Green function Go(r, r'; 5) is that it provides an integral equation for the full Green function G(r, r'; c), corresponding to the Schroedinger operator H , for which (5.23)
(fwr)
=
-(l/W M r )
+ W ) $w,
4 E 9"'
This integral equation is the second resolvent integral equation (4.97), which in the present case becomes (5.24)
G(r, r';
5 ) = Go(r, r'; 1) - J Go(r, r"; 5 ) V(r") G(r", r'; 5) dr". R3
This equation can be solved by the Fredholm method if its kernel (5.25)
K(r, r";1) = Go(r, r";t) V(r")
is of Hilbert-Schmidt type." Before attacking the problem of solving (5.24), let us first review some basic facts about integral operators with Hilbert-Schmidt kernels.
Definition 5.1. T h e kernel K(a, a'), a, a' operator K on L2(Rm), (5.26)
(K$)(a)=
I
IWm
K(a, a') $(a') d"a',
E
Rm, of the integral
4 EL2(R"),
* A condensed and very lucid presentation of the Fredholm theory for Hilbert-Schmidt kernels (L2 kernels) is given by Smithies [1958, Chapter VI].
49 1
5. Green Functions in Potential Scattering
is a Hilbert-Schmidt kernel (or L2 kernel) if 1 K(a, a')j2 is integrable on Wm:
(5.27)
+a.
A Hilbert-Schmidt kernel which is continuous and such that K(a, a') is integrable on IWm is called a trace-class kernel. The terminology for kernels of integral operators is related to the terminology for the corresponding operators in a straightforward fashion: integral operators with trace-class kernels are of trace class" with T r K = JK(ar,a') dma, and integral operators with HilbertSchmidt kernels are Hilbert-Schmidt operators (see Exercise 5.3). The Fredholm integral equation is (5.28)
where
is, in general, some complex number. If we search for solutions K(a, a') is the kernel of an integral operator on L2(Rm), and if ~ E L ~ ( [ W then ~ )the , integral equation (5.28) can be written in operator form: w
$(a) which are Lebesgue square integrable, so that
(5.29)
C = f +wKC.
Hence, (5.28) has a unique solution in L2(Rm)if and only if (1 - wK)-l exists and iff is in the domain of definition of (1 - wK)-l. For w # 0, (1 - wK)-l is related in a straightforward manner to the resolvent RK(5)of the operator K : -1
w
If K(a, a') is a Hilbert-Schmidt kernel, then K is a Hilbert-Schmidt operator, and as such, K has a pure point spectrum. Hence, (1 - wK)-l will exist if and only if 1 / w E SK. When this is the case, the domain of definition of (1 - wK)-l is the entire space L2(Rm). The Fredholm theory essentially provides a means for the computation of (1 - wK)-' from the kernel K(a, a'). The main results of this theory for L2 kernels are contained in Theorems 5.3 and 5.4. The reader interested in the proofs of these theorems is advised to consult Smithies [1958, Chapter VI].
* A proof of this statement can be based on Mercer's theorem (see Riesz and Sz. Nagy [1955, Section 981).
492
V. Quantum Mechanical Scattering Theory
Theorem 5.3. Let K(a, a') be a Hilbert-Schmidt kernel and let us introduce the modijed Fredholm determinant of K(a, a')
1
0
K(a, 3.2)
... K(a1
9
an)
dmal
K(a, a') K(a, a,) ..' K(a, a,)
(5.31)
(-1P
I,.
n!
=
1, 2,...; for n
case the series
0
q a , , a')
Dn(a,a') = -
for n
dma, ,
=
*.'
K(a,, a,)
dmal . * * d"an ,
0, do = 1 and D,(a, a ' ) = K(a, a'). In that m
(5.32)
d(w) =
1 d,w"
n=O
is convergent for all complex values of C1,the series
w.
Likewise, for any fixed
w E
c m
(5.33)
D(a, a'; w) =
&(a, a')w"
n=O
is convergent in the mean [in L2((WZm)] for all w . Moreover, the function D(a, a'; w ) is a Hilbert-Schmidt kernel for any fixed complex w .
Theorem 5.4. If d ( w ) # 0 and ~ E L ~ ( R the ~ ) ,integral equation (5.28) has the unique Lebesgue square-integrable solution (5.34)
Theorem 5.5. The function d ( w ) vanishes if and only if eigenvalue of the integral operator K.
is an
5.4. THEFULLGREENFUNCTION We can apply the Fredholm theory to the integral equation (5.24) when its kernel (5.25) is of the Hilbert-Schmidt type. Under these circumstances, (5.24) is a special case of the integral equation (5.28), in
493
5. Green Functions in Potential Scattering
which the variables a, a' E R" become r, r' E R3, and w = - 1 . over, we note that when Z; is not on the positive real axis,
More-
is square integrable in r or r', since, according to Theorem 5.1, we have to choose the square root of 6 for which I m dz > 0. Hence, for any fixed r", (5.24) has a unique solution, provided that d(-I) # 0 for the given kernel (5.25). T o establish that d(-1) is not zero for the kernel K(r, r'; <) in (5.25), note that K(r, r'; 5) is the kernel of the operator
According to Theorem 5.5, d(-1) eigenvalue of this operator, i.e.,
=
0 if and only if
w =
-1
is an
for some nonzero y5 E GBH = BH0. However, the above relation implies that Ht,h = &. Thus, as long as Z; does not belong to the spectrum of H , the Green function G(r, r'; Z;) of H exists, and can be computed by means of (5.34) with w = -1. Let us find out now the restrictions imposed on the potential V(r) by the dehand that K(r, r'; Z;) be square integrable. Applying Fubini's Theorem, we find that R6
I K(r, r'; ( ) I z dr dr'
=
1,.
dr'( V(r')I2
[R3 drl Go(r,r'; <)Iz*
*
Using the expression (5.6) for Go(r,r'; Z;) we easily compute that (5.36)
11 R3
Go(r,r'; <)I2 dr
=
m
1
Im
44 > 0.
Thus, we get the relation
which shows that Go(r,r'; 5) V(r') is for I m Z; # 0 a kernel of HilbertSchmidt type if and only if V(r) is square integrable on R3.
494
V. Quantum Mechanical Scattering Theory
T h e main conclusions of the above discussion are contained in Theorem 5.6.
Theorem 5.6.
If the potential V(r) is square integrable on
R3, then the full Green function G(r, r’; 5 ) of H - 5 [where
+ V(r)] exists and can be written in the form 1 G(r, r‘; 5) = Go(r, r’; 5)- qTq D(r, r”;-l) GO(r”,r‘; 5) dr”,
H 1 - (1/2m)A (5.38)
s.,
where d( - 1; t;) and D(r, r ” ; - 1) are obtained by taking K(r, r’; 1) = Go(r, r’; 1)V(r’)
as a kernel in (5.30)-(5.33).
5.5.
SERIES EXPANSION OF
T h e kernel D(r, r”;- 1 ) by the infinite series (5.39)
THE
FULLGREENFUNCTION
of the integral operator in (5.38) is given
Wr, r’; 5) = K(r, r‘; 5)
+ C Dn(r,r‘; 0, m
n=l
in which
This would imply that the full Green function can be expanded in a series (5.40)
G(r, r’;5) =
m
n=o
GJr, r‘; 0,
where Go(r, r’; 5) is the free Green function, and (5.41)
for n = 1, 2, ..., if the order of integration and summation can be interchanged. T o prove that this is indeed the case, we need Lemma 5. I.
495
5. Green Functions in Potential Scattering
Lemma 5.1. If K(a, a') is a kernel of Hilbert-Schmidt type, then the Hilbert-Schmidt norm
of the modified first Fredholm minor D,(a, a') satisfies the inequality,
and the modified Fredholm determinant d, satisfies the inequality (5.43)
I d, I d
n
{(e/n)1/211K1l2}",
=
1, 2,...,
where 11 KII, is the Hilbert-Schmidt norm of the integral operator with the kernel K(a, a ' ) :
K,
The proof of the above result can be found in the work of Smithies [1965, Section 6.51. Expanding the determinant in (5.31) along the first column, we easily obtain (5.44)
a') = dn+1K(%a')
&(.I,
+J
Rrn
Dn-d", al)K(al
9
4 drn%.
Iterating the above relation, we find that (5.45)
Dn+l(a,4 = dn+.lK(a,4
+1
Rm
+ d, I
Rrn
dma1K(%4
K(a, al)K(al , a ' ) d"a1
1,
d"a,D,-l(%
9
a2M.2
9
4-
Hence, applying the Schwarz-Cauchy inequality to estimate the third integral in (5.45), and using afterwards (5.42), we arrive at the inequality
496
V. Quantum Mechanical Scattering Theory
Using the above inequality, we derive from (5.41)
(5.47) L
3
I Dn+l(rrr”,1) Go(r”,r’; 0 dr“
< I dn+l(-l; I)/ j
+ I dn(-l,
S)l
R
1
I K(r, r“,5)
R3
dr”JGo(r”,r‘; 1)l
r‘; 01 dr“
R
drll K(r, r,; I) K(rl r“;01 112
Naturally, in deriving the above inequality one has to make sure that all the integrals present in it really exist. I n fact, K(r, r”;5 ) is square integrable, while Go(r”,r’; () is integrable as well as square integrable . this it immediately follows that the in r” when 0 < arg 5 < 2 ~ From first integral on the right-hand side of (5.47)exists. Since the integrand of the ,second integral is equal to
we infer from (5.6)and the square integrability of Go(r, rl ; 5) V(rl) in r and rl that this integral also exists for all r, r’ E R3, with the possible exception of a set of measure zero. Finally, the third integral also exists since according to (5.36)
and both V(r”) and G(r”, r’; 5 ) are square integrable in r”. Combining the inequalities (5.43)and (5.47), we obtain the estimate (5.48)
5. Green Functions in Potential Scattering
< n=N+1 f
(c)""I1 1 I KILn
R
K(r, r";
from which we infer that (5.49)
gz I j, [
c m
n=N+1
497
5) Go(r", r'; 5)1 dr"
1
Dn(r, r"; 5) Go(r",r';
This last result leads to the relation
5) dr"
I
= 0.
j,, D(r, r"; 5 ) Go(r",r'; 5) dr" =
JR3
n=O
Dn(r, r"; W
+
L
3
5) GO(r",r'; 5 ) dr"
[,2+, 5) Dn(r, r";
Go(r", r';
1
5) dr"
11 , Dn(r, r"; 5) Go(r",r'; 1)dr", W
=
n=O
which shows that the expansion (5.40) is valid. We summarize these results in the following theorem.
Theorem 5.7. If K ( r , r'; 1;) is a kernel of Hilbert-Schmidt type, then the full Green function G(r, r'; 1;) can be expanded for any 1; I$ SH in the series (5.40), with terms given in (5.41), which is uniformly absolutely convergent relative to the function
498
V. Quantum Mechanical Scattering Theory
We should mention in this context that a sequence sl((), s2(6), ... of functions is said to converge to the function S ( f ) uniforml) relative to the functionF(6) if for any E > 0 there can be found a number N ( E ) such that IS,([) - s(4)I < E F ( ( )for all values of [. Naturally, a series C u,([) is said to converge uniformly with respect to some function R(6) if the sequence of its partial sums s,([) = Z b l u,(f), n = 1, 2,..., converges uniformly relative to F(5). Finally, this series is absolutely uniformly convergent relative to F( 6) if the series C 1 u,( <)I is uniformly convergent relative to F(5). Obviously, if a series is absolutely uniformly convergent relative to F ( f ) ,then it has to be also uniformly convergent relative to F(6).
5.6. SYMMETRY PROPERTIES OF
THE
GREENFUNCTION
T h e two symmetry properties (5.14) and (5.15) of the free Green function are inherited by the full Green function, i.e., for 5 $ SH (5.51)
G(r, r’; 5)
=
G*(r, r‘; <*),
(5.52)
G(r, r’; 5)
=
G(r’, r; 5)
holds almost everywhere in R6. T h e first of these two relations is a straightforward consequence of the formulas (5.39) and (5.40) for G(r, r’; c), resulting from the fact that V(r’) is real, and consequently K(r, r‘; 6)
=
Go(r7r‘; 5) V(r’) = Go*(r,r‘; [*) V(r’) = K*(r, r‘; 5*).
T o prove (5.52) we have to resort to a lengthier argument. First of all, we have to note that the function rdr, r’; 5)
=
1
R3
D(r, r”;-1) Go(r”,r‘; 5) dr“
is the kernel of a Hilbert-Schmidt operator. This follows from the fact that Go(r”,r’; 5 ) is the kernel of the bounded operator ( H , - C)-l, while D(r, r”; -1) is a kernel of the Hilbert-Schmidt type, since (see Exercise 5.4) it is the limit in the mean of the Hilbert-Schmidt kernels (see Exercise 5.5) N
C &(t, r”; --I), n=O
N
=
1,2, ...
Consequently, D(r, r”; - 1) is the kernel of a Hilbert-Schmidt operator
D(- I). Thus, rl(r, r’; l )is the kernel of the operator D( - l)(Ho - ()-I,
499
5. Green Functions in Potential Scattering
which is the product of a Hilbert-Schmidt operator and a bounded operator, and therefore is of Hilbert-Schmidt type (see Exercise 5.6). The function f*(r)g(r’) is square integrable in R6 whenever f,g €L2(R3).Since rl(r,r’; 5) is a kernel of Hilbert-Schmidt type, it is also square integrable in W. Hence, the functionf*(r) Tl(r, r‘; [)g(r’) is square integrable in R6. Consequently, Fubini’s theorem can be applied to infer that
Since G,,(r, r’; 5) is also square integrable in R6, a corresponding relation holds for it. Consequently, in view of (5.38), we have (5.53) =
j,,dr’g(r’) 1w3 dr G(r, r’; I)f*(r).
On the other hand, the relation
in conjunction with (5.51) implies that
=
=
J
R3
dr‘g(r’) J
1, dr’g(r’)
R3
w3
dr G*(r’, r; {*)f*(r) dr G(r‘, r; {)f*(r).
Comparing (5.53) and (5.54), we see that
j,, dr’g(r’) j w3 dr G(r, r‘; 5)f*(r’) =
I
dr’g(r’)
[w3
Iw3
dr G(r’, r; [)f*(r)
for all square-integrable functions g(r’). This is possible if and only if
1 G(r, r’; [)f*(r) dr R3
=
w3
G(r’, r ; [)f*(r) dr
500
V. Quantum Mechanical Scattering Theory
for almost all r' E [w3. Since the above equality holds for all squareintegrable functions f(r), we infer that (5.52) holds almost everywhere in R3 x R3 = R6. This completes the proof of (5.52). OF 5.7. THESPECTRUM
THE
SCHROEDINGER OPERATOR
We have seen already that if K(r, r'; 5 ) is a Hilbert-Schmidt kernel the full Green function exists as long as 5 $ SX. This is certainly the case if I m 5 # 0. T h e following theorem shows that this will be also the case if 5 is real but negative, with the possible exception of a finite or at most countably infinite number of values.
Theorem 5.8. Suppose the potential V(r) is locally square integrable and bounded at infinity. Then the continuous spectrum SE of the Schroedinger operator H 3 -(1/2m)A V(r) contains no negative values; moreover, there are only at most countably many negative eigenvalues, all of finite multiplicity, * and having no negative number as an accumulation point. I n order to prove the above theorem, we need a number of auxiliary results. Since these results are of interest in themselves, we shall present them as lemmas and theorems. We recall from 93 that if H and H , are any two operators (not necessarily self-adjoint) with identical domains of definition and 5 belongs to their resolvent sets, then the resolvent R ( ( ) of H can be expressed in terms of H , = H - Ho and of the resolvent I?,(<) of Ho by making use of the infinite series in (3.68), provided that this series converges in some sense. Lemma 5.2 represents, to some extent, a converse to this result, valid under the additional assumption that HlR0(5) has a bound less than one; in this context, it should be noted that when g H= BHo, the operator H,R0(5) is defined everywhere on iP since R,({) is defined everywhere on Z , and that it maps any f E Z into R , ( 5 ) f ~ 5 3= ~ ,g H 1so, that H,R,(<)f is defined.
+
Lemma 5.2. Suppose that H and H , are two operators (not necessarily self-adjoint) with identical domains of definition g X = gX,, , and that the complex number 5 belongs to the resolvent set of H, . If 11 HlRo(5)/1< I , H , = H - H , , then 5 belongs also to the resolvent set of H , and (5.55)
R(5) = RO(5) u-lim n++m
c (-1)k(KRo(5))k. n
k=O
Recall that the multiplicity of an eigenvalue is the number of linearly independent eigenvectors corresponding to that eigenvalue. #
501
5. Green Functions in Potential Scattering
Proof. Since I/ H,Ro([)/l < 1, the uniform limit in (5.55) exists (see Exercise 5.9). For any f E g Hwe have [RO(O
f
k=O
=
(-1)"(~1Ro(0)k]
(H - 5)f
RO(O(H0 - 5 ) f - Ro(5) fflR,(I)(HO - Of (-1)"Ro(5) ffl ... %(5)(Ho - I;)f
+ + Ro(5)Hlf + (-1)"Ro(5)
=f
Ro(5) HlRO(4-1fflf ffl *.. Ro(5) Hlf
-
+ (-1)"RoK)
+
.*.
+ ..*
Hl ... Ro(5) Hlf z f .
This shows that if A ( [ )denotes the operator on the right-hand side of the equality (5.55), then we have A({)(H - c) f = f for all f E g H . Since g HE gH0and Ro([) has the range g H 0 the , operator ( H - [) R,({) is defined everywhere on X . Consequently, we can write
Thus, we also have ( H - [) A ( [ ) = 1, i.e., A ( [ ) is identical to = I?({). Q.E.D.
( H - [)-'
If A is an unbounded self-adjoint operator, then its spectrum S A is an unbounded set. However, the set S A C [wl could be still bounded from above or from below, in which case we shall say that A is bounded from above or from below, respectively. I n the case where A is bounded from below, we denote by mA the greatest lower bound of the set SA: mA = inf A. A&
We note in passing that A is bounded from below if and only if the set
is bounded from below, with the same greatest lower bound (see Exercise 5.10).
502
V. Quantum Mechanical Scattering Theory
Theorem 5.9. Suppose H and Ho are two self-adjoint operators with identical domains of definition, that Ho is bounded from below, a < I , b 3 0, and that for some 0
<
/I Hlf /I < all H O f I l
(5.56)
for all f
E
+ bllfll,
Hl
=
H - Ho
3
gH0 = g H Then . H is also bounded from below, and
(5.57)
m, 3
mHo
b
- max 11 9 --a
+ a/
mHo
11.
Proof. We shall prove the main statement of the theorem by showing that for any real (5.58)
b
5 < m H o - max 11-a
9
f
mHo
1 1 9
we have 11 H1R,,(5)~~ < 1, and consequently, according to Lemma 5.2, such a 5 E R1 belongs to the resolvent set of H . This means that the spectrum of H contains only such points which do not satisfy (5.58); i.e., it is bounded from below and its greatest lower bound m, satisfies (5.57). T o prove that 11 H,R,(5)11 < 1, we use (5.56) to derive the inequality
Since S H o C [mHo,
+ co),and therefore for any f
we conclude that 11Ro(5)11 <
1 mHo -
5'
E
X and real
5 < mHo,
5. Green Functions in Potential Scattering
503
Combining the above two inequalities with (5.59), we obtain (5.60)
Since it is easily seen that the expression on the right-hand side of the inequality (5.60) is smaller than one when 5 satisfies (5.58), it follows from Lemma 5.2 that such 5 is in the resolvent set of H . Consequently, H is bounded from below by the expression on the right-hand side of (5.60). Q.E.D.
+
We recall from 97 in Chapter IV that when H = T, V, is the Schroedinger operator with a potential satisfying the conditions of Theorem 7.4 in Chapter IV, then BH = gH0 (Chapter IV, Theorem 7.5) if H , = T , is defined by (7.15) in Chapter IV, and also (5.56) is satisfied by any 0 < a < 1. Hence, we can state Theorem 5.10. Theorem 5.10.
T h e n-body Schroedinger operator H
=
H,
+ V,
corresponding to a potential V(rl ,..., r,) satisfying the conditions of Theorem 7.4 in Chapter IV, is bounded from below. As a by-product of the above theorem and (5.57), we can obtain estimates on the lower bound of an n-body Schroedinger operator H by computing the constants a and b, as was done in the proof of Theorem 7.4 in Chapter IV. A single-particle Schroedinger operator H with a potential satisfying the requirements of Theorem 5.8 also satisfies the conditions of Theorem 7.4 in Chapter IV. Hence, such a Hamiltonian H is bounded from below. T h e second resolvent equation (5.62)
( H - A)-1
=
( H o - A)-1
-
( H o - A)-lV(H - A)-1
holds for any value of h in the resolvent set, and therefore, in particular, for any real h < mH . T h e desired result about the spectrum of H , stated in Theorem 5.8, can be then obtained from Lemma 5.3, which will be derived with the help of the following important theorem. *Theorem 5.11 (Weyl's criterion for limit points). A real number A, is a limit point" of the spectrum of a self-adjoint operator A if and
* By definition, a limit point of the spectrum is either an accumulation point of the spectrum, or an eigenvalue of infinite multiplicity.
504
V. Quantum Mechanical Scattering Theory
only if there is a sequence fl ,f2 ,... E aAwhich converges weakly to the zero vector, and is such that Ilf,li = 1 and (5.63)
lim lI(A
n++m
-
h ) f n I1
0.
Proof. Suppose that 11 f, 11 = 1 , that w-limm++mf , = 0, and that (5.63) is true. For an arbitrary open interval (a, , b,) containing A, we can write II(A - An)fn
1/2 =
1
(A
R
~ IEiYn I 11~
-
2 (bn - An)2
jm ~ IEiYn I + (an 112
b0
== (bn
- An)2
IKI
- ER,>fnII
+
(an
- ~2
jao~ --a,
I Effn I
112
- ho12 II EtJn 1 1 ~ .
Now, the expression on the left-hand side of the above inequality approaches zero in the limit n + +a.Hence, we have
and consequently
=
lim 11 Eb”,fn fl+
W
1‘
+
lirn
n++m
11 Efofn /I2
=
1.
This implies that the subspace EA((ao,b n ] ) Z onto which EA((a,, b,]) projects cannot be finite dimensional; in fact, if this subspace were of finite dimension N , we could select an orthonormal basis {el ,..., eN> in it, and write
11 EA((an bol)fn /I2
= I(el
9
lfn>12 + . * *
+I(~N
lfn>l2
+
0,
since lim,,,, (el j f,) = = lirn,,,, (eN If,) = 0 due to the weak convergence to zero of fl ,f2 ,.... Thus, we conclude that An is either an accumulation point of S A or an eigenvalue with an infinite-dimensional characteristic subspace, i.e., that A, is a limit point of S A . Conversely, if A, is a limit point of the spectrum of A and I , 3 1, 3 are open intervals containing A,, then each EA(I,) is the projector on an infinite-dimensional subspace of X . Hence, we can choose an infinite orthonormal system {el , e2 ,...} such that EA(Im)en = e, , n = 1, 2,... . Since we have II(A - An)en
/12
=
1
In
-
An)2
~ IE,Aen I 11< ~ I I n I /I en 1 ’
+
0
5. Green Functions in Potential Scattering
505
when these intervals were so chosen that their lengths shrink to zero, and since limn++, (en I f ) = 0 for any f E 3 because of Bessel’s inequality, we see that the three conditions of the theorem are satisfied . Q.E.D. by this sequence el , e, ,... E 9A
*Lemma 5.3. If A and K are self-adjoint bounded operators on LP and K is also completely continuous, then the sets of the limit points K coincide. of the spectra of A and A
+
Proof. If A, is a limit point of the spectrum of A, according to Theorem 5.1 1 we can find a sequencef, ,f, ,... E LP such that 11 f, (1 = 1, w-lim,,+, f, = 0 and lirnnjfm ll(A - A,)f, 11 = 0. Since f , ,f,,... is a bounded sequence and K is completely continuous, it must contain a subsequence g , ,g , ,... such that Kg, , Kg, ,... converges strongly to some vector h E 2. However, since w-limnj+m g , = 0, we infer (K*h I g,) = 0. Hence, that 11 h 1, = limn,+, ( h 1 Kg,) = lirn,,,, lirn,,,, ll(A K - A,) g , 11 = 0, which in conjunction with 11 g , 11 = 1 and w-lim,++,g, = 0 implies, by virtue of Theorem 5.9, that A, is a K. limit point of the spectrum of A , = A Conversely, if A, is a limit point of the spectrum of A, , then, according to the same argument applied to ( - K ) , we deduce that A, is also a limit point of the spectrum of A = A , - K . Q.E.D.
+
+
We can return now to the proof of Theorem 5.8, and observe first the spectrum of ( H , - A)-,, that since the spectrum of H, is [0, +a), A < 0, is [0, -I/A], as seen from the relation EROyB)
= EHyF-yB)),
F(A’) = (A’
-
Ay.
T h e operator ( H , - A)-, V is an integral operator with the kernel K(r, r’; A) = G,(r, r‘;A) V(r’). Since this kernel is of Hilbert-Schmidt type, ( H , - A)-l V is a Hilbert-Schmidt operator, and therefore it is completely continuous. For A < m H ,the operator ( H - A)-l is bounded, and consequently, by Theorem 8.l(b) of Chapter IV,(H, - A)--, V ( H - A) is also a completely continuous operator. Hence, according to (5.53) and Lemma 5.3, the limit points of the spectrum of ( H - A)-, are the same as the limit points of the spectrum of ( H , - A)-,, i.e., they constitute the interval [0, - l/A]. This implies that the limit points of the spectrum of H constitute the interval [0, +a), i.e., H has no negative limit points. This means that the continuous spectrum of H is contained in [0, a), that any negative eigenvalues of H must be of finite multiplicity, and that these eigenvalues have no accumulation point on the negative real axis. Thus, Theorem 5.8 is established. .Theorem 5.8 deals with the point spectrum of the Schroedinger
+
506
V. Quantum Mechanical Scattering Theory
operator on the negative real axis. For the more restricted (but still very large) class of potentials V(r) for which there is an Ro > 0 such const, that for all r 3 R, the potential V(r) is continuous and I Y V(r)l it can be shown (see Kato [1959, Theorem 11) that the Schroedinger operator has no positive point spectrum, i.e., no positive eigenvalues. T h e reader should now recall that these results on the point spectrum of the Schroedinger operator are well illustrated by the case of the hydrogen atom treated in 97 of Chapter 11. I n that case we were dealing with a point spectrum of the form St = {-a/nz : n = 1, 2,...}, where a is a constant characteristic of the hydrogen atom.
<
5.8.
THERELATIONBETWEEN THE RESOLVENT AND THE SPECTRAL FUNCTION OF A SELF-ADJOINT OPERATOR
We know already that if the spectral function EA of any self-adjoint operator H is given, then its resolvent R(5) = ( H - c)-l can be computed by means of the formula (5.64)
It is not so obvious, however, that the converse of this statement is also true-namely, that the spectral function EAof H can be computed from the resolvent R(5) (i.e., from the Green function of H , if it exists). T o show that EA can always be computed from R(<),we need the following lemma. Lemma 5.4. If the function F(A) has a continuous derivative F’(A) on [a, 61, and is Riemann-Stieltjes integrable on [a, 61 with respect to cr(A), then (5.65)
lbF(h) a
&(A)
= F(b) u(b) - F(a) .(a)
1
b
-
a
F’(h) .(A)
dh,
where the integral on the right-hand side of the above relation is a Riemann integral. Proof. Taking A,‘ = A, in the defining formula (3.18) and recalling that A, = a and A, = b, we get
According to the mean value theorem of differential calculus, F(h,)
=q L 1 )
+
qJk-l)(&
-h l ) ,
4-1
<
Jk-1
< A,
*
507
5. Green Functions in Potential Scattering
Consequently, we have (5.66)
c
k=l
c n
n
F ( A k ) u( Ak)
-
F ( A k ) a( hk- l )
k=l
Since F’(A) is uniformly continuous on [a, b],
and the expression in (5.66) converges to the right-hand side of (5.65).
Q.E.D.
Let us apply the result of the above lemma to the integral in (5.64). For Im 5 # 0 this integral can be considered to be a Stieltjes integral, and since (A - 0 - l is continuously differentiable, we get
Letting a + - 00 and b -+
+
00,
we arrive at the result
from which we can derive Theorem 5.12.
Theorem 5.12. For any fixed f,g E 2, the function ( f I R ( 5 ) g ) is analytic at any complex 5 with I m 5 # 0, and for A, A,
<
where the above integrations should be carried out along any piecewise smooth curves which do not intersect the real axis.
Proof. According to (5.67) the complex derivative
508
V. Quantum Mechanical Scattering Theory
exists at any point 5 E C1 with I m 5 # 0 since the above improper Riemann integral converges absolutely and uniformly in any closed neighborhood of 5 which is disjoint from the real axis. Hence, ( f I R(5)g) is analytic in the upper and in the lower complex plain. Due to the analyticity of ( f I R ( 5 ) g ) the integrals in (5.68) are independent of path. If we carry out the integration along straight lines and use (5.64) and (5.67), we get
T h e integrand in the last of the above integrals is Lebesgue integrable in s and A on the set [A,, A], x [wl, since it is majorized by a function integrable on that set:
I
(s
(f I EAg) -
A
-+ i € ) 2 I
llfllllgll
(s - A)2
+
€2
.
Hence, the order of integration in s and h can be reversed and, after carrying out the integration in s, we obtain
By taking into consideration
from (5.69) we easily derive
509
5. Green Functions in Potential Scattering
It is quite obvious that the integrands of the above two integrals satisfy the conditions under which Lemma 4.2 is valid. Hence, by using that lemma, we arrive at the relation W+0 lim
-
(f I EAz-Og>
f
which is identical to (5.68).
(f I EAl-Og)
=
(f 1 E A z g )
-
(f I EAlg),
Q.E.D.
We shall see in the next section that the above theorem provides an important means of computing the spectral measure of a self-adjoint operator from its Green function.
EXERCISES 5.1. Prove that the set DG0defined in (5.18) is a Bore1 set of Lebesgue measure zero in R6. 5.2. Assuming that JRa, I K(a, a ’ ) / 2&(a) dp(a’) < +a,show that = JOBS K(a, a ’ ) f ( a ’ ) dp(a’) on the adjoint K* of the operator (Kf)(a) L 2 ( W ,p ) is also an integral operator with the kernel (K*)(a,a‘) = K*(a‘, a ) , and that the kernel (K*K)(a,a’) of K*K is equal to JR3 K*(CY“,a ) K(a”,a’) +(a’’).
5.3, Prove that any integral operator K on L2(Rn,p), which has an L2 kernel K(a, a’), is a Hilbert-Schmidt operator, and that
5.4. Give the reason why the limit in the mean K(a, a’) of a sequence of Hilbert-Schmidt kernels Kn(a,a’),
is a kernel of Hilbert-Schmidt type.
5.5, Prove that the kernel D,(a, a’) in (5.31) is of Hilbert-Schmidt type when K(a, a‘) is of Hilbert-Schmidt type. 5.6. Show that if A is Hilbert-Schmidt operator in a separable Hilbert space and B is a bounded operator then B A is a Hilbert-Schmidt operator.
.5.7. Show that for I m 5 # 0 the function F(r, r‘; 5) defined in (5.50) is integrable in r and r’ on a domain of the form D x R3, where D is compact if V(r) is bounded at infinity.
510
5.8.
V. Quantum Mechanical Scattering Theory
Let A , , A,
,... be
a sequence of bounded operators for which
x,",11 A, 11 < + co. Prove that the uniform limit A
=
u-lim n++m
n
1 A,
k=l
exists and is a bounded operator with
k=l
5.9.
Use the results of Exercise 5.8 to show that n
u-lim n++m
exists if 11 A
/I < 1 and I ck 1
c,Ak k=l
< M for all k = 1, 2, ... .
5.10. Prove that the self-adjoint operator A is bounded from below i f a n d o n l y i f ( f 1 Af) > m , , I j f 1 1 2 f o r a l l f ~ 9 ~ .
6. Distorted Plane Waves in Potential Scattering 6.1.
THERELATIONOF DISTORTED PLANEWAVES TO THE GREEN FUNCTION
T h e distorted plane waves @++)(r) can be constructed from the full Green function G(r, r'; t;) by taking its Fourier-Plancherel transform (6.1)
G(r, k; 6) = ( 2 ~ ) -1.i.m. ~/~
G(r, r'; 6) ecikr'dr'
L
3
in the variable r', and then setting
T h e verification of (4.25) is then achieved by resorting to the fundamental formula
-
1
- lim Ti r++O
I,, ha
($ I [R(h
which was derived in Theorem 5.12.
+ ie)
-
R(h - ie)]#) dh,
511
6. Distorted Plane Waves in Potential Scattering
T h e main purpose of the present section is to establish that, under suitable conditions in the potential, the limit in (6.2) really exists, and that the functions defined by (6.2) really qualify as the advanced distorted plane waves introduced in 94. Theorem 6.1 represents the first step in this direction.
Theorem 6.1. If the potential V(r) is integrable and square integrable on R3, then for any complex number K with I m K > 0, the function (6.4) h(r, k; K )
k2 - K'
-2m eirk +
:
(24312
1.i.m.
(24312
s,
G (r, r';
&-)
eikr'
dr'
is well defined by the above expression for almost all" values of r, r' E R3; in addition, for any such K with K~ $ SH and almost all r E R3 and k E R3, this function satisfies the equation (6.5) h(r, k; K )
= g(r,
k; K )
-
m
-
2n
S 3,
exp(iK1 r - r' I r - rI I
in which (6.6) g(r, k; K )
=
m2
- ~-
4/8nb
SR3
exp(iK1 r
-
r'
I r - rt I
I)
I)
V(r') h(r', k; K ) dr',
V(r') exp(ikr') dr'.
Proof. By comparing (6.1) and (6.4) we see that (4.7)
Hence, the existence of h(r, k; K ) hinges on the existence of e(r, k; K2/2m), whose existence, in its turn, can be inferred from the square integrability of G(r, r'; K2/2m) in r' E R3. But we know already that for any fixed r' and K~ 4 SH, G(r, r'; K2/2m) is the square integrable in r E [w3 solution of (5.24). Since for any fixed r and K we have by (5.52) that G(r, r'; K2/2m) = G(r', r; K2/2m),we conclude that G(r, r'; ~'/2m) is square integrable in r' E R3. Therefore, h(r, k; K ) is well defined by (6.4) for almost all k E R3. Due to the existence of G(r, k; K2/2m), we have
* In this section all the measure theoretical concepts refer to the Lebesgue measure, except if otherwise stated.
512
V. Quantum Mechanical Scattering Theory
on account of the unitarity of the Fourier-Plancherel transform. Hence, the second resolvent equation (see Theorem 3.2)
can be written in the form
I t can be easily computed that for I m K > 0 the Fourier transform (with respect to the variable r’) of 1 r - r ’ 1-l exp(iK I r - r’ I) is equal to d G (k2 - K ~ ) -exp(-ikr), ~ so that
Now, it can be shown that the function
K (r, r‘;
c (r’; k; &)$(k)
$)-
is integrable in r’ and k on R6 (see Exercise 6.1). Hence, the order of integration in r’ and k can be inverted in (6.9) (by virtue of Fubini’s theorem), and we obtain R
c (r, k; $-) -
1
P3
$(k) dk
dk +(k)
1
R3
a)c
dr‘ K (r, r’;
(r‘, k; T). K2
6. Distorted Plane Waves in Potential Scattering
513
Since the above equation holds for arbitrary square-integrable functions &k), we infer that the following equation is satisfied for almost all k E R3, (6.11)
K2 G (r, k;-)2m
=
~
2m exp(-ikr) (277)”2 k2 - ~2 exp(iK1 r - r‘ I) _ _m V(r’) 2rS,3 I r - r’ I
e (r’, k; 2) dr’. 2m
After solving (6.7) for e ( r , k; ~ ~ / 2 r and n ) substituting the expression obtained for G(r, k; ~ ~ / 2 rinto n ) (6.11), we arrive at the integral equation (6.5). I n this equation g(r, k; K)is defined by the integral in (6.6), which is essentially the Fourier transform of the function
Ir
-
r‘ 1-l exp(iK1 r - r’ I) V(r’).
Since this function is the product of the two square-integrable functions V(r’) and I r - r ’ 1-l exp(iK I r - r’ I), it is integrable, and therefore its Fourier transform indeed exists. Q.E.D. Though the equation (6.11) has a kernel of Hilbert-Schmidt type, we cannot solve it by the Fredholm method because the inhomogeneous term 4277 I r - r ’ 1) - 1 / 2 exp(ikr) is not square integrable in r. The transition to the new function h(r, k; K) serves the purpose of providing the equation (6.5), which has an inhomogeneous term g(r, k ; K) square integrable in r. Hence, this last equation can be handled by the Fredholm method. 6.2.
EXTENSION OF G(r, k ;
5)
TO THE
REALAXIS
In the following lemma, we investigate the equation (6.5) in the case of I m k = 0.
Lemma 6,1, Suppose the potential V(r) is measurable and locally” square integrable on R3, that the integral (6.12)
* The function V(r) is locally square integrable if and only if it is square integrable on any compact set. 1 V(r)( = O(r-2-Co) if there are constantsR, > 0 and Co > 0 such that r 2 + Q V(r)l < C, for all r > Ro’. These two conditions when combined obviously imply that V(r) is square integrable on Rs.
V. Quantum Mechanical Scattering Theory
514
exists, and that (6.13)
+
for some E , > 0. Then, if H , 1 V 1 has no positive eigenvalues,* the integral equation (6.5) has a unique solution h(r, k; K ) for all complex values of K with Im K 3 0, except for those pure imaginary values of K for which ~ ~ / 2 rE nSf. We know already from Theorem 6.1 that the function K~ $ SH. We observe, however, that this integral equation does not have a kernel of Hilbert-Schmidt type when Im K = 0, since in this case, by Fubini's theorem, Proof.
h(r, k; K) satisfies the integral equation (6.5) for any K for which
exp(iK1 r - r'
I) V(r')
1
2
dr dr'
T o remedy this situation we introduce the auxiliary function
(6.14)
-dl V(r)l vdr)
>o
if
V(r)
if
V(r)
if
V(r)
= 0,
where vo(r) vanishes outside the set {r: V(r) = 0} and it is positive and continuous on that set; in addition, vo(r) is chosen as to make I V(r')l integrable on cW6 in the variables r and r'. vO2(r)lr - r' We rewrite the integral equation (6.5) in the form (6.15)
hl(r, k;K ) = gl(r, k; K )
+
w
&(r, r';
K)
Mr', k;K ) dr',
* The absence of positive eigenvalues of H can be established by studying the dependence of (1 - Kl(~))-'on K , considering in the process (1 - K ' ( K ) ) -as~an operator on a is suitable Banach space (see Ikebe [1960]).In fact, once the existence of (1 - K1(~))-l established by this method (even for the case of real K ) , then the absence of any positive eigenvalues of the Schroedinger operator can be derived as a consequence of the considerations in this section. Furthermore, we already mentioned at the end of $5 that due to a theorem of Kato, the Schroedinger operator has no positive eigenvalues when I V(r) I = O(r-l). Hence, later on, we shall not rentain this redundant condition on the spectrum of H.
6. Distorted Plane Waves in Potential Scattering
515
where
The newly introduced kernel Kl(r, r'; K ) is of Hilbert-Schmidt type whenever Im K 2 0. This is due to the fact that
and that the integral on the right-hand side of the above inequality exists by virtue of the requirement that (6.12) exists. The function g(r, k; K ) is well defined by (6.6) for almost all values of r even when Im K = 0. As a matter of fact, we have (6.17)
the existence of the integral (6.17) is a consequence of the restrictions imposed on V(r) (see Exercise 6.3). We note that the constant C can be chosen t o be the same for all the values of r, k, and K . Since the kernel Kl(r, r; K) is of Hilbert-Schmidt type, it can be solved by applying Theorem 5.4, provided that gl(r, k; K ) is square integrable in r E R3. But the square integrability of gl(r, k; K ) in r E R3 for almost all values of k E 083 and I m K 2 0 can be inferred easily from the existence of the integral
by using Fubini's theorem. Let us now write m
P(K =)C dn(l)(K), n=O
m
D("(r, r'; K )
=
1 Dn(l)(r,r';
K),
n=O
where d , ( l ) ( ~and ) Dn(l)(r,r'; K ) are obtained by taking Kl(r, r'; K ) to be the kernel in (5.30) and (5.311, respectively. According to Theorem 5.4, the equation (6.15) has the unique square-integrable (in r E R3) solution (6.19)
516
V. Quantum Mechanical Scattering Theory
provided that d ( l ) ( ~#) 0. Hence, if we knew that d ( l ) ( ~#) 0 also in the case when I m K = 0, the lemma would be established. ) if and only if the integral operator By Theorem 5.5, d ( l ) ( ~vanishes K,(K)with the kernel Kl(r, r’; K ) has the eigenvalue one. We will show, however, that even when K is real, Kl cannot have the eigenvalue one. The relations (5.6) and (6.16) indicate that for Im K > 0 the operator K l ( ~ )is identical to the operator -v(HO - ~~/2rn)-l I V 1 / 2 , where ZJ and I V 11/ 2 are the operators of multiplication by v(r) and I V(r)11/2, respectively. If K , were a real number for which K,(K,)#, = #,, 11 $, (1 > 0, then we would have (see Exercise 6.4)
#,
= w-lim K+KO
-)2m K2
K,(K)go = -w-lim v K+Ko
where in taking the limit we must keep Im K
$ E gIv, n 9 1 v , 1 / z , we could write (6.20)
(# I I V‘ I’”$Jo)
=
(I I‘P2#I # J
=
-(I
v I#
141) =
-(*
-1
I V‘
> 0.
3
Hence, for any
I I v l#l)>
where $1 appears as the weak limit of ( H , - ~~/2rn)-l I V /1/2#o ; this weak limit exists as a consequence of the existence of the advanced Green z,hl = function Gh+)(r,r’; ~2/2rn). Since, obviously, ( H , - ~,~/2rn) - I V 1 /2#o , we infer from (6.20) that ( H , - ~,~/2rn) = - I V I , i.e.,
P o
+ I V‘
IWl
= (Ko2/2m)h*
+
But, we have already stated that the operator H , 1 V 1 does not have positive eigenvalues. Hence we conclude that d ( l ) ( ~ ,.f ) 0. Q.E.D. The preceding lemma will be essential in deriving the rest of the results in the present section. Hence, it is desirable to understand very well the nature of the condition imposed in this lemma on the potential V(r). The lemma allows potentials which are singular on a set YvC R3, as long as the measure of this set is zero and (6.12) holds. For instance, in the case of the Coulomb potential V(r) = const I r I-l, 9’would consist of the single point r = 0. We note, however, that the Coulomb potential does not satisfy the condition (6.12) of the lemma. On the other hand, the Yukawa potential V(r) = const e-alrl 1 r 1-l satisfies these conditions, and it also has only one singularity which is at the origin.
517
6. Distorted Plane Waves in Potential Scattering
In general, an almost everywhere continuous potential V(r),which is at infinity, and has locally square integrable, vanishes faster than I r only a finite number of singularities r’ in whose neighborhood it diverges more slowly than 1 r - r’ fulfills all the conditions of the lemma. The last of these conditions guarantees, in conjunction with (6.13), the existence of the integral (6.12). It must be realized, however, that these are sufficient but not necessary conditions for Lemma 6.1 to hold true. On the other hand, they are satisfied, for instance, by any bounded and almost everywhere continuous potential, such as the square-well potential. Moreover, all the potentials commonly occurring in practice, with the exception of the Coulomb potential, satisfy such conditions. We shall refer to the set YVmany times in the future. Hence, it is desirable to be more specific and define YVas the set (of Lebesgue measure zero) which is such that V(r) is bounded on any closed subset of R3 which is disjoint from 9’:. This means that Yvcontains all the singularities of V(r), but not also all the points at which V(r) is discontinuous. Thus, for instance, SP, is empty when V(r) is the square-well potential, which has a two-dimensional set of points of discontinuity, but no singularities. 6.3.
c(r,
CONTINUITY PROPERTIES OF k ; 5) I N THE UPPER HALFPLANEOF 5
Our next task is to relate the functions G(r, k; 5) and h(r, k; K ) to the spectral measure EH(B) of the Schroedinger operator H. Such a relation can be established by means of the basic formula (6.3). First of all, let us relate the integrand in (6.3) to G(r, k; f ) . Using the first resolvent equation (3.62), we deduce that
(6.21)
(4I P ( 5 )
- R(5*)1$> = (5 - 5*K$ =
I R(5)R(5*)$)
(5 - 5*)(W*WI R(S*)$).
Since according to (5.52) the Green function G(r, r’; 5) is symmetric in r and r‘, we can write
If we choose $(r) to be continuous and of compact support, then we take the Fourier transform on both sides of the above equation, and thus obtain, after interchanging the order of integration in r and r’,
V. Quantum Mechanical Scattering Theory
518
This interchange of the order of integration can be easily satisfied when the domain of integration D is compact. As a matter of fact, according to Theorem 5.7,
I G*(r, r’; 5) +(r)l = I G(r’, r; 5) +(r)l< constF(r’, r; 511 +(r)l, where F(r’,r; 5)1 +(r)l is a function integrable in r’ and r on R3 x D when $(r) is continuous and of compact support (see Exercise 5.7); hence, G*(r, r’; 5 ) $(r) is integrable on D x R3 and Fubini’s theorem can be applied. Let us introduce the function
$(k; 5 )
(6.23)
= (2m5 - k2) 1.i.m.
1, e*(r,-k; 5)
+(r) dr.
Using this function and (6.22)’ we can write
Setting
5
=
h
+ ie, we easily obtain from (6.21) and the above relation 1
=
a
1,3
2k (k2/2m-
+ e2
I $(k; X
+ ;€)I2
dk.
Hence, in the present case, (6.3) assumes the following form: (6.25)
(+I (EE
+ EE-oM)
-- 1 -
4nim2
-
(4 I (EE + G-o)#)
!$IJ,, dXJ,s Aa
(A
-
+ c2 I $(k; X + ie)12 dk
2ie k2/2m)2
We observe that the reversal of the order of integration in k and h in the above integrals, taken at fixed e > 0, is a direct consequence of Tonelli’s lemma (which guarantees that the integrand is integrable in X and k on [A,, A,] x R3) and Fubini’s theorem. We are now faced with the problem of finding the limit E --f +O in (6.25). At first glance, this would seem to be a straightforward task since the factor .[(A - k2/2rn)2 €21-1 is strongly reminiscent of a similar
+
6. Distorted Plane Waves i n Potential Scattering
519
factor in (4.68). A more careful inspection reveals, however, that the k)12 situation is not quite that simple, since the other factor I $(k; X in the integrand also depends on E. Hence, the following lemma will be needed in computing the limit (6.25).
+
*Lemma 6.2. Suppose the potential V(r) is measurable and locally square integrable on R3, that the integral
exists, and that" (6.26)
Then for any compact domain 9 in the upper complex plane I m K 2 0}, containing no pure imaginary values of K for which ~ ~ / 2Er SF n , the following are true.
{K:
(a) For any (6.27)
E
> 0, there is a a(€, 9)such that I h(r, k;K
+ A K ) - h(r, k;.)I
< eFo(r),
+
for any I AK I < a(€, 9),K A K E 9, k E R3 and r E R3 - Y ; , where 9'"is the set of singularities of V(r), i.e., V(r) is bounded on any compact set disjoint from 9". (b) There is a constant C ( 9 ) such that (6.29)
I h(r, k;41 < C W F 0 ( r )
f o r a n y K E 9 , k ~ R ~ a n d r ~YV. R ~ -
+
* The behavior of V(r) for Y --+ a, is determined by the requirement that the integrals appearing in the definition (6.28) of F,(r) exist. In fact, since the integration in r' E Ra O(l/l r' Icltalla) involves products of three functions which behave as O(l/l r' and O(l/l r' la) (see Exercises 6.5 and 6.6) the integrand behaves for Y' --* +a, as O(l/l r' Icaa+al/z). For the convergence of the integral, we need g(5a 5 ) > 3, i.e., OL >
+
8.
520
V. Quantum Mechanical Scattering Theory
We base the proof of Lemma 6.2 on a detailed analysis of the relation (6.30)
h(r, k; K )
obtained by multiplying the right and left-hand side of (6.19) by (v(r))-l. . . .. First of all, we shall study g(r, k; K ) . If R, > 0 is sucl that r2+,oI V(r)l C, for all r > R, , then we have for any R, R,
I g(r, k;K
+
<
>
- g(r, k; .>I dr'
If R, is chosen large enough, the first integral on the right-hand side of the above inequalities can be made smaller then any given E, > 0 for all r E R3 (see Exercise 6.2). For any such fixed R, , the second
integral can be also made smaller then I AK I so small that
1 exp(i1 r
E,
for all r E [w3 by choosing
1I < exp(-2Ro Im A K ) { cos ~ 2R, 1 AK( -1 - r' 10,)
-
1 + 1 sin 2R0/A K 1 I}
where the constant a is such that
for all r (6.31)
E
R3 (see Exercise 6.3). Hence, we have
I g(r, k;K
+ A K ) - g(r, k;.)I
< 2e1
for all r E R3. Let us prove now that d ( l ) ( ~is)a uniformly continuous function of K in the upper half of the complex plane ( K : I m K O}. Specializing
>
521
6 . Distorted Plane Waves in Potential Scattering
(5.30) to the present case and expanding the determinant representing the integrand, we get
x
fq,...,u,(rl
,***,
rn;
K)
dr1 ... drn
-
In the above formula, ~ ( a,..., , a,) is zero for all even permutations ( a , ,..., a,) and one for all odd permutations. T h e summation extends over all permutations (a1,..., a,), except those in which aJ = j for some j = 1,..., n. T h e functionfal ,...,a,(rl,..., r, ; K ) is of the form
where the product has n factors and the sum has n terms, each vector ri occuring exactly twice. Hence, for given el > 0 and R , > 0 we have (6.33)
I f m l ....,Jr1 ,.-,r n ; K + AK) -fal ,...,a,(rl )..., r,; .)I
<
for all rl ,..., Y, R , and all sufficiently small I A K 1. Let us write the integral in (6.32) in the form
<
<
where Sn(R,) = {rl:rl R,} x ... x {r,: r, R,}. T h e second integral can be made smaller than any given E , > 0 by choosing R1 > R, sufficiently large, while the first integral is continuous in K on account of (6.33). This establishes that &')(K) is uniformly continuous in K in the region { K : I m K 3 O}. Since according to (5.43) (6.34)
Id3K)I
< {(e/41'zll
~l(K)ll2Y
< {(e/n)1'21/
~l(O)llZ)",
the series for P(K) converges uniformly in K. This fact and the uniform continuity of d L l ) ( ~ )for every n imply that &)(K) is uniformly continuous O}. in the complex domain { K : I m K
>
V. Quantum Mechanical Scattering Theory
522
Let us investigate now the terms in the sum of the integrand in = 0, we have
(6.30). For n (6.35)
m exp(-Kl r - r' IT-r'I
1 Do(1)(r, r'; K) = v(r) 2n
I) d -. I V(r'>l*
Let us introduce the function (6.36)
It is easily seen that for any E;,)
<
> 0 and A, > 0, we have
+
R, , I m K , Im(K A K ) 3 0 and all sufficiently small for all I, Y' values of I AK I. To derive a similar formula for Dil)(r,r; K ) , we specialize (5.44) to the present case, thus obtaining (6.38)
D!)(r,
r'; K ) = d:$(~)
Kl(r, r'; K )
+
R
D!(lr,
r,; K ) Kl(rl , r'; K ) dr,
Let us introduce the convenient notation K,"(r, r') =
1
R3"
&(r, rl) &(r1, rZ) Kz(r,, , r') dr,
dr,
.
6. Distorted Plane Waves in Potential Scattering
523
We specialize (6.39) to the case n = 1 and using the estimates (6.34),
(6.37), and (6.40)
I D?(r, r'; .)I
< K2(r,r'),
we easily obtain that for all r, r' (6.41)
(Dp)(r,r'; K
+ AK)
< $K2(r,
r')
-
I Kl(r, r'; K)I
< R, and
Dp)(r, r'; K ) )
+ EP)K:(r,
r')
< Kz(r,r'),
I m K , Im(K
+1
r+R1
+ A K ) 3 0,
Kz(r,rl) Kz(rl , r') drl ,
where for any given R, > 0, we can choose E ( ' ) , d 2 )> 0 small, with corresponding sufficiently small values for I AK 1. From (6.38) we also obtain, by using the estimates (6.34) and (6.40), (6.42)
1 Dp'(r, r'; .)I
< {(e/2)1'211K , ( ~ ) l l ~ } ~ Kr')~ (+r ,KZ2(r,r').
Proceeding by mathematical induction, and using the same type of arguments as in going from n = 0 to n = 1, it is straightforward to R, and I m K , Im(K A K ) 3 0 deduce that for all r, r'
+
<
where ck'), ..., ckn) are constants and EL'), ..., €in) > 0 can be chosen arbitrarily small for adequately small values of I AK 1; in this process of induction one must also generalize (6.42), by establishing that Dk')(r, r'; K ) = &K2(r, r') ... akn+')Kg+'(r,r'),where u;'), ..., a:) are constants. From (6.30) we get
+ +
(6.44)
I h(r, k; K + A K ) - h(r, k; K ) I < I g(r, k;K A K ) - g(r, k;.)I
+
I d(l)(K
f dK)
x I gl(r', k; K
- d(l)(K)I
+ A K ) dr' ~
~
D;)(r, r'; K
+
AK)
524
I
I I,.
+ d”1(K)
V. Quantum Mechanical Scattering Theory
fI
D!)(r, r‘; K
+ AK)
-
v(r)
n=O
X
A K ) dr’ ~
+
n=O
X
A K ) - gl(r‘, k; K ) I dr’
D!)(r, r‘; K )
II
I!
We have already established that the first term on the right-hand side of the inequality (6.44) can be made arbitrarily small, uniformly in r, k, and K . Since O ( K is )continuous and never vanishes for Im K 3 0, it must have a positive minimum in any compact region of the upper complex K plane. T o estimate the integral in this second term, we specialize (5.46) to the present case and easily obtain (6.45)
Since (see also Exercises 6.2 and 6.3)
where (6.46)
for rl for rl for rl
<1 > 1, > 1,
0 c0
< e0 < 1 > 1,
we see that the second term in (6.44) is majorized by EF,,(r), where E > 0 can be chosen arbitrarily small for adequately small values of I LIK I. To deal with the third term on the right-hand side of (6.44), we must split the sum into two parts, the first part containing only n = 1,..., N ,
F,(r) is defined in (6.28) and
525
6. Distorted Plane Waves in Potential Scattering
while in the second part N < n (6.45) and (6.46) that for given E we can achieve
x I gl(r’, k; K
+
< co. It is easy to establish by using > 0 and compact 3 C I m K 2 0}, {K:
+ O K ) ]dr‘ < EFo(r)
for all K , K AK E 9, k E R3, r E R3 - Y V by , choosing N sufficiently large. For any such N , it is then possible to make the integral
over the region S = {r’: r’ 3 R,} arbitrarily small for sufficiently large R, 3 R, ; this result easily follows from the estimates (6.45) and (6.46)
(see also Exercises 6.5 and 6.6). Finally, it follows from (6.43) and (6.46) that by choosing R, > 0 adequately large (to make the last term in (6.43) as small as desired) and I AK I sufficiently small, we can make the integral (6.47) over the region S = {r’: r’ I?,} smaller than E,F,(r) for any a priori given E , > 0. Thus, the third term in (6.44) also conforms to the pattern leading to the first statement (6.27) of the lemma. Moreover, it is evident upon an inspection carried out in the same vein that the last term in (6.44) also conforms to this pattern on account of (6.31) and (6.45). Hence, (6.27) is established. T h e proof of (6.29) proceeds along similar lines, and is left to the reader.
<
6.4.
OF THE GREENFUNCTION RELATION TO THE SPECTRAL MEASURE
By combining (6.7) and (6.23), we arrive at the relation
which is valid for any complex value of K with I m K # 0. The function F,(r) in (6.28) is bounded on any compact domain disjoint from YV(see Exercise 6.13). Hence, if we choose #(r)E %:([w3) to have a support disjoint from Yv , we can infer from Lemma 6.2 that, given E,, > 0, we have
V. Quantum Mechanical Scattering Theory
526
for all k E R3 and A E W, and for any sufficiently small value of This result, in conjunction with (6.25), implies that
E
> 0.
Let us take now A,, A, E:S with A, > A,. I n view of (6.48) and C, (where C, Lemma 6.2, $(k; A) is continuous in A and I $(k; A)/ is a constant) for all k E R3 and all A E [A, , A,]. Thus, we can infer that
<
< Clz (arctan
A,
-
k2/2m E
-
arctan
Hence, it follows from the properties of the function arctan x that for < E 6 and given 6 > 0 we can choose a constant C , such that
0
<
where the newly introduced function F,(k) is obviously integrable on R3. On the other hand, by using a slightly altered version of Lemma 4.3 (see Exercise 6.8) we infer that
6. Distorted Plane Waves in Potential Scattering
527
Hence, we can apply Lemma 3.1 of Chapter IV to deduce that the limit f O can be taken under the first integral sign in (6.49). Thus, we obtain E -+
after taking into consideration that A, , A, E Sf and therefore Eh,-o = EAi, i = 1, 2. We shall extend the relation (6.50) to more general functions a,h(r), thus arriving at the following theorem, in which we slightly reiax the restrictions imposed in Lemma 6.2 on the potential V(r) (see Exercise 6.9).
Theorem 6.2. Suppose that the potential V(r) is measurable and locally square integrable, that it is bounded on any compact set disjoint from a certain closed set YVof measure zero [which contains all the V(r') is locally" singularities of V(r)], that the function V(r)l r - rf integrable in r and rf in Iw6 and that I V(r)l = O(r--2--co)for some c0 > i . Then the function
= @Li)*(r),
exists for every k E [w3; in addition, for any a,h E L ~ J R ~=) EH(Sf)L2(Iw3) we have for any Bore1 set B C Sf C [0, 00)
+
where
By comparing (6.51)-(6.53) with (6.23), (6.49)) and (6.50) we see that we have essentially already proved Theorem 6.2 for the case when $(r)
is continuous and has compact support which is disjoint from SP,. Now, the set 9 0, of all functions which are continuous and of compact is dense in L2(Iw3); this easily follows from the support disjoint from Yv
* See Exercise 6.9.
V. Quantum Mechanical Scattering Theory
528
fact that %i(lw3) is dense in L2(R3),while any given function in VE(R3) can obviously be approximated arbitrarily well in the mean by an element of %; (see Exercise 6.14). Hence, if $(r) is any given squareE %'$ which integrable function, we can find a sequence $(l)(r),$(2)(r),... converges to $(r) in the mean. Now let us show that $y)(k), @)(k),... converges in the mean to a limit $+(k) which is the same for any sequence {$(")(r)} converging strongly to $(r). First of all, we note that $(i)(r)- $(j)(r) belongs to VO, when both $(i)(r)and $(j)(r) belong to Vb . Hence, (6.52) holds for yYt)(r)- $(j)(r),
Noting that
and then taking A, = 0 and letting Aa + fco, we obtain
11 #(i) - #(') 1 '
=
/ ,I $$'(k) R
- $y)(k)l2 dk.
This shows that i,&)(k), $F)(k),... is a Cauchy sequence in L2(R3). Consequently
$+(k) = 1.i.m. $f'(k) exists. Moreover, if {$in)(r)} is another sequence from %; converging to $(r) in the mean, then $(n)(r) - $in)(r)also belongs to V; . By the above procedure we easily obtain again
I1 #(n)(r)- #?)(r)Il'=
s ,I R
$p'(k)
- $k)(k)I2 d
k
and consequently {t,@(k)} converges to the same limit as {Jy)(k)}. By already standard methods (see Chapter 111, $5) we infer from (6.52) that for any Bore1 set B C [0, co)
+
Taking in both sides of the above relation the limit n + fa, we arrive at (6.52). Hence, Theorem 6.2 is established.
529
6. Distorted Plane Waves in Potential Scattering
6.5.
THELIPPMANN-SCHWINGER EQUATIONS FOR PLANE WAVES
We introduced the functions Ok+)(r) by means of the formula (6.2). Afterwards, we related these functions to the spectral measure of the Schroedinger operator. We have not yet established, however, whether @i+)(r)defined by (6.51) are indeed the outgoing distorted plane waves defined in 94. T h e first step in establishing the physical nature of Ok+)(r) is to show that these functions are indeed eigenfunctions of the Schroedinger operator. Let us introduce the functions wk(r) = [ g I 3 I 2 h ( r ,k; 1 k I)
(6.55)
for all k E R3. Combining (6.51) with (6.7) and using the above definition, we get
+
(6.56)
@F)(r)= ( 2 ~ ) - ~ / ~ ( e i k ' ok(r)).
According to (6.5) and Lemma 6.1, uk(r) satisfies the integral equation
where K (6.58)
I k I and
=
m exp(ik1 r - r' I) gk(r) = - V(r') exp(ikr')dr'. I r - r'I 277 S R 3
Expressing in (6.57) vk(r) in terms of result.
@i+)(r),we
obtain the following
Theorem 6.3. If the potential V(r) satisfies the conditions imposed on it in Theorem 6.2, then the functions @:*)(r) defined in (6.51) satisfy (almost everywhere in r E W) the integral equations (6.59) @k(*t)
(r) =
~
(243'2
exp(ikr) - m 2rr
I
R3
exp(+ikl r - r'
I r-r'I
I) V(r') @L*)(r')dr'
for all k E R3. T h e inhomogeneous term in the above integral equations is a free plane wave. Hence, after taking a look at (4.92), we easily recognize in (6.59) the type I Lippmann-Schwinger integral equation. Since we have already seen that the solution of (6.57) [and therefore also that of (6.59)]
V. Quantum Mechanical Scattering Theory
530
is unique, we can infer from Theorem 6.3 that if outgoing plane waves with the properties required in 94 exist, then they coincide with the functions @i+)(r)defined by (6.51). AND 6.6. THESCATTERING AMPLITUDE
THE
BORN APPROXIMATIONS
We can derive from the Lippmann-Schwinger equations (6.59) the asymptotic behavior of ok(r) which was predicted in (4.18).
Theorem 6.4. Suppose that the potential V(r) is measurable and locally square integrable in R3, that it is bounded on any compact set disjoint from a given closed set YV[which contains all the singularities V(r’) is locally integrable on R6, and of V(r)], that V(r)l r - r’ that for some e0 > 0 (6.60)
Then ok(r)satisfies* the following estimate for large values of r : (6.61)
where 9 and r/r and
rj5
are the two spherical coordinates of the unit vector
Proof. T h e conditions imposed on V(r) in the present theorem are such that all the conditions of Theorem 6.2 are also satisfied. Consequently, zlk(r)satisfies the integral equation (6.57). Let us introduce the function (6.63)
m 2n
wk(r) = - -(ezkr
+ nk(r)) V(r)
in terms of which (6.64) Strictly speaking, we should explicitely require that k2/2mE Sr. However, relying on the results of Kato [1959] (where Kato shows that, for his class of potentials, H has no positive eigenvalues), we can state that for potentials V(r) satisfying the conditions of Theorem 6.4, we have k2/2m4: Sp”.
531
6. Distorted Plane W a v e s in Potential Scattering
We note that zIk(r) is certainly uniformly bounded at infinity (see Exercise 6.10), so that for fixed k (6.65)
I zuk(r)I< const I V(r)l = O(l/r3+%).
Let us write now (6.64) in the form
According to (6.65), there is a constant M such that for suitably large R
Consequently, for such values of R
By selecting, for fixed r, the x1 coordinate axis in the direction of the vector r, and working in spherical coordinates, we arrive at the following estimate of the above integral:
Furthermore, using Taylor's formula at r,
1
Ir-rlI
1
=y
[l
=
+0( 3 1 . 2
0, we get
V. Quantum Mechanical Scattering Theory
532 Consequently, we can write (6.68)
exp(ik( r - rl Ir-rlI
1
Let us take in (6.66) R = ~ ( l - a ) / ~ with , some fixed 6 < 1, and then choose r so large that (6.67) becomes valid. Inserting (6.68) in IR’, we obtain, after noting that for the selected R we have YJY < ~ - ( l + ~ ) / ~ ,
Now, by virtue of (6.67), we also have
and consequently
On the other hand, for the chosen value of R,(6.67) yields IR” =
O
(
1 yl+[(1-6)/2]c0
) +O
1 (=)2
so that we obtain
+
Setting in the above estimate 6 = ~,,/(2 (6.61). Q.E.D.
< E,, , we
E ~ )
finally obtain
T h e main significance of the above theorem lies in the fact that it provides us with the explicit formula (6.62) for the scattering amplitude fk(e,#). This formula requires, however, a full knowledge of @L+)(r).
6. Distorted Plane Waves in Potential Scattering
533
On the other hand, even in the case where @)ljri)(r) is not known, we can still obtain an approximation of f k ( e , + ) by using (6.56) and neglecting the second term vk(r). Then we obtain the jirst Born approximation of the scattering amplitude: m f k ( 8 , c ) m - 2n
(6.69)
i,,exp
(-z%
7)r . r‘
v(r’)exp(ikr‘) dr’
Higher Born approximations can be obtained by an iteration method in which we start by setting, on the right-hand side of (6.59), @i+)(r’)w ( 2 ~ r ) - ~eikr’, / ~ thus obtaining the second Born approximation for @&+)(r): (6.70)
@k)(r)m
___
(243’2
exp(ikr) exp(ik1 r
-+IR. Ir-r’I
-
r‘ I)
V(r’) exp(ikr’) dr‘.
Inserting the above approximate expression for @$+)(r) in (6.62), we obtain the second Born approximation for the scattering amplitude
f (4 4).
can be inserted on the T h e second Born approximation for @)ll”(r) right-hand side of (6.59) to obtain the third Born approximation for @i+)(r),which in turn can be used in (6.62) to obtain the third Born approximation for fk(B, $), etc. For certain potentials, this iterative procedure converges, and provides a practical method” for the computation of fk(8, +). 6.7.
THEGENERALIZED PARSEVAL’S EQUALITY
We recall from $5.8 that under the restriction imposed in this section on the potential V(r), the Schroedinger operator H has a point spectrum SE only on the negative real axis. Moreover, by Theorem 5.8, all the eigenvalues of H are of finite multiplicity, and SF has no negative accumulation points. These results imply that we can choose an orthonormal basis (6.71)
of eigenvectors @ k , v of H , (6.72)
* See Massey [1956] for criteria of convergence of the above method, as well as for estimates on the degree of accuracy of the first Born approximation.
V. Quantum Mechanical Scattering Theory
534 which is such that for any
+ €L2(jW3)we have
Hence, by combining the above formula with the relation (6.52) in which we take B = ,S: we obtain
II 4 112
(6.74)
=
I1 E H ( S ; ) $ 112
+ I1
EH(S:)+
112
T h e above equality is strongly reminiscent of (4.16) in Chapter I. Consequently, we can expect that other formal features of Theorem 4.6 in Chapter I are retained in the present case. This hope is realized in Theorem 6.5. Theorem 6.5. Suppose the potential V(r) satisfies the conditions in Theorem 6.2. Then for any $ €L2([W3)and any Bore1 set B C S : C [ O , +m)
&(k)
= 1.i.m.
@f)*(r) $(r) dr, L
and for any
3
z/A2) eL2([W3)
(6.76)
Proof. One can easily derive (6.76) from (6.52) by setting first + $(2) and then $ = $(I) + i$@), and using formula (5.13) in Chapter IV with f = +(I) and g = $(".
+
= +(l)
Let us select any +'(r) from the family %?; of all continuous functions with supports which are compact and disjoint from the set .SP, of singularities of V(r). From the last statement of Lemma 6.2 we infer that for such functions
(6.77)
I $+'(k)I =
15
R3
@p)*(r)$'(r) dr
I<
const
s,1
+'(r)ldr.
535
6. Distorted Plane Waves in Potential Scattering
We know from Theorem 6.2 that a function $+(k), related by (6.53) to an arbitrary square-integrable function $(r), is square integrable on R3. Hence, $(k) is integrable on any compact Borel set B' C OX3. This statement in conjunction with (6.77) implies that the function $'*(r) @L+)(r) $+(k) is integrable in the variables (k, r) on any set of the form {k: k2/2m EB,} x R3, if the Borel set B, C Sf is compact. Consequently, the reversal of the order of integration in the following derivation,
-
Sk2/2msB,,
=
dk $+(k)
1 dr $'*(r) R3
j
R3
kz12meBn
@L+)(r) $'*(r)dr
@F)(r) $+(k) dk,
can be justified by using Fubini's theorem. Since (6.78) holds for arbitrary $'E%?:, and since we already know that V: is dense in L2(R3),we conclude that (6.79)
whenever B, C S z , is compact. If the Borel set B C Sf is not bounded, we can introduce the sets B, = B n [0, n] which are bounded, and to which (6.79) can be applied. Since EH(B,)$ converges strongly to EH(B)$ when n ---t 00, the relation (6.75) immediately follows as a consequence of (6.79). Q.E.D.
+
6.8. DISTORTED PLANEWAVESAS EIGENFUNCTIONS OF THE SCHROEDINGER OPERATOR For the complete identification of @:+)(r)as outgoing distorted plane waves, we still have t o establish that @:+)(r)is an eigenfunction of the Schroedinger operator with the eigennumber k2/2m. Theorem 6.6 provides a set of precise criteria under which @:+)(r)is indeed an eigenfunction of the Schroedinger operator.
Theorem 6,6. Suppose that the potential V(r) is continuous in the neighborhood of a point r E R3, and that it satisfies the conditions
536
V. Quantum Mechanical Scattering Theory
imposed on it in Theorem 6.2. Then the distorted plane waves @:*)(r) satisfy the time-independent Schroedinger differential equation (6.80)
at that point r E R3. Before proceeding with the proof of the above theorem, let us make a few remarks. For the proof of this theorem we need, essentially, only to know that V(r) and @;*'(r) are continuous in some neighborhood of r, and that
T h e restrictions on @:*)(r)are fulfilled automatically if V(r) satisfies the conditions of Theorem 6.2 and the restrictions imposed on it in the present theorem. It can happen, however, that @:*)(r)satisfies these restrictions even when V(r) does not obey all the conditions of Theorem 6.2 (see Ikebe [1960]). Hence, the validity of Theorem 6.6 is somewhat independent of the validity of Theorem 6.2. T h e proof of Theorem 6.6 is straightforward, and it essentially consists in applying the Laplacian A to the right-hand side of equation (6.59) and computing the result. This computation is easily carried out with the help of Lemma 6.3.
Lemma 6.3, Suppose f (r) = O(r-2-eo) for some c0 > 0, and that there is a neighborhood A'" of the point r E R3in which f (r) is continuous and the function (6.81)
is well defined by the above integral. Then g(r) is twice differentiable
at that point in each of the variables x, y, and x, and
(6.82) Proof.
(-Ag)(r)
=
+ 47d(r).
k2g(r)
Let S be a neighborhood of r and write
We shall compute separately the two terms on the right-hand side of the above identity.
537
6. Distorted Plane Waves in Potential Scattering
Straightforward differentiation shows that
a exp(ik1 r - r' I) (6.83) ax Ir-r'I a2
(6.84) -
exp(ik1 r
Ir
ax2
-
-
~
(x
-
x')
exp(ik1 r - r' I)
Ir
-
r'
+
(ik
l2
r' I) r' I -
exp(ik1 r - r' I) Ir-r')
x (ik
+
1 - k2(x - x')2 - ik(x - x') Ir-r'I Ir-r'I2
-
Ir-r'I3
On any bounded domain which does not contain some closed neighborhood of r, we have
Now, the function on the right-hand side of the above inequality is integrable on account of the existence of the integral in (6.81) and the fact that if a function is integrable then it is also absolutely integrable. Hence, we conclude that f(r) is locally integrable. I t is easy to infer from (6.83) and (6.84) that for sufficiently large R, > 0 and all rl from some open neighborhood S C N of r const
an exp(ik1 r - r' I) I r' 13+ro ' ax. I rl - r' I If (r')l G constf(r'), for n
=
r' 2 R,
r' E R3 - S and r' R,
<
1, 2. Thus, we deduce that Ss,-
an exp(ik1 r - r' I) I r - r' I f (r') dr',
ax.
n
=
1,2,
exist for r E S. Consequently, a/ax and a2/ax2 can be carried under the integral sign when applied to the right-hand side of (6.81) (see Exercise 6.1 1). Since the same obviously holds for the variables y and x , and since
538
V. Quantum Mechanical Scattering Theory
we obtain (6.86) -A
exp(ik)r
Ir
SR3-,
-
rl
-
r'
I
I)
f(r') dr'
=
k2
exp(ik1 r JR3-s
Ir
-
-
r'
r' I)
I
f(r') dr'.
Let us choose
(6.87)
S = ( ~ 1 I:
x
-
I < a, I y1 - y I < a, I ~1 - x I < a } .
If a > 0 is sufficiently small, then the generalized mean-value theorem of the integral calculus can be applied to obtain
+a
exp(ik1 r - r' I)
-a
-
Ir
f(r')
I
r'
I
x,=x+-a
,
x =x
dy'dz' dy' dz'
I
x'=x+a
x =x--a
,
where / y o- y 1 < a and j x,, - z j < a. Using (6.83) and the meanvalue theorem of integral calculus, we get
La +a
+a
8 exp(ik1 r
a~
1r
-
-
r'
r' I)
1
1
x'=a
dy' dz'
x'=--a
where j 7 I < a and 1 f I < a. This shows that, due to the continuity o f f ( r ) in M , by taking the limit a + 0 we shall get in (6.88) a contribution of the form (6.89) f ( r ) lim
a-t+O
i_,.La +a
fa
-2a exp(ik[a2 (a2
+ ( y - y')2 + (Z
+ (y
-
y')2
-
z')2]1/2)
+ (z - z')2)3/2
dy' dz'.
To compute the above limit, we must compute the following integral explicitly: +-a exp[ili(a2 y12 z12)1/7 dY1 dz, -a (a2 y12 212)3/2
:s
i
+ +
+ +
539
6 , Distorted Plane Waves in Potential Scattering
where rav is obtained by applying again the generalized mean-value theorem to the first of the above two integrals, and it satisfies the inequality 1 rav 1 < a&'. Hence, the limit (6.89) is equal to -(477/3)f(r). Due to the symmetry of the considered expressions under the exchange of x, y , and z, when S is the box in (6.87), we have
=
3 lim a+o
a2
~
ax2
1
exp(ik1 r - r' I r - r' I
I) f(r') dr' = 47~f(r).
Hence, the desired result in (6.82) is obtained by combining (6.90) fO. Q.E.D. with the result (6.86) in which we let a ---f
In order to be able to apply the above lemma to derive (6.80) from (6.59), we have to establish that (6.91)
and that the above function is continuous in some neighborhood of the considered point r E R3. Since I @i*)(r)l= O(1) (see Exercise 6.10), it immediately follows that (6.91) is true. T h e continuity of Oi*)(r) is an easy consequence of (6.59) (see Exercise 6.12). Hence, in computing the effect of applying the Laplacian A to the right-hand side of (6.59), we can use Lemma 6.3; then (6.80) easily follows. EXERCISES 6.1 +
Show that if the potential V(r) is square integrable, the integral
J,. I K(r, r'; 5) e(r', k; 5) &k)l dr' dk
exists for every r E [w3 when I m
drml > 0.
6.2. Prove that the integral in r' of the function 1 r' ) 2 + E o 1 r - r' I over the region {r': I r' 1 3 R,} exists when R, > 0 and c0 > 0, and 1, and as O(l/r) that when r + co, it behaves as O(l/fo) if 0 < c0 if co > 1.
+
<
6.3. Prove that if the potential V(r) obeys the restriction of Lemma 6.1, then
where the constant C is independent of r.
540
V. Quantum Mechanical Scatering Theory
6.4. Show that K ( K )is a weakly continuous vector-valued function ) = s-limAK+,K ( K A K ) +’ of K in the upper half plane, i.e., K ( K +’ whenever Im K , 2 0 and +’ E for all K in some neighborhood
+
Of K O .
6.5. Show that if the potential V(r) satisfies the conditions of Lemma 6.1, then
and that the above integral defines a locally integrable function of r.
6.6, Show that if the potential V(r) satisfies the conditions of Lemma 6.1, then for any fixed r I$ SP,
and the above function is locally square integrable in r l . 6.7. Prove that if the potential V(r) satisfies the conditions of Lemma 6.2, then the integrals appearing on the right-hand side of (6.28) exist.
6.8. lim
Show that iff(x) is continuous on [a, b], then
1
l b
r++O.X
a
E
x2
+
€2
f(x) dx
if a t O < b if a > O or b < O , a < b .
=
6.9. Show that if the potential V(r) is locally square integrable, Co~-2-fo,c0 > 0, for almost everywhere continuous, and I V(r)l Y 2 R o , then the function V(r) I r - r’ V(r) is integrable on R6 if it is integrable on the set {(r, r’):Y R, , Y‘ Ro}.
<
< <
6.10. Prove that when the potential V(r) satisfies the conditions of Lemma 6.2 then the function vk(r) defined by (6.55) is uniformly const) for Y 3 Ro and k assuming values from bounded (i.e., 1 uk(r)l a compact set D, containing no points for which k2/2m E.:S
<
7. Wave and Scattering Operators in Potential Scattering
54 1
6.11. Let h(r, u ) be a function with a continuous (in r and u ) partial derivative hu(r,u ) = ah(r, u)/au, and assume that for r 3 R,
where p(r) is integrable on {r: r 3 Ro}. Show that if hu(r,u)f(r) and f ( r ) are integrable over the sphere {r: r R,},then
<
6.12.
Show that the function
is continuous in r if If(r)i
=
O ( T - ~ - ~for O )some
e0
> 0.
6.13. Prove that the function Fo(r) defined in (6.28) is bounded on any closed set X , disjoint from Yv[provided that the potential V(r) satisfies the conditions of Lemma 6.21. 6.14. Show that the family YF of all functions in %?;([w2) with supports disjoint from the set YV(mentioned in Theorem 6.2) is dense in L2(R3). 7.
Wave and Scattering Operators in Potential Scattering
7.1. THEEXISTENCE OF ASYMPTOTIC STATES IN
POTENTIAL SCATTERING
In the study of a particular scattering model it is crucial to determine from the start the magnitude of the family of asymptotic states. I n other words, we have to establish the size of the initial domain M, of 52, . I n potential scattering, under very mild restrictions imposed on the potential, this question is answered by Theorem 7.1. Theorem 7.1. Suppose the potential V(r) is locally square integrable and that for some E > 0 V(r)
=
0
1 . L+J -
542
V. Quantum Mechanical Scattering Theory
Then the initial domain M, of the wave operators 52, corresponding V(r) coincides with the to H,,> -(1/2m)A and H 3 -(1/2m)A entire Hilbert space L2(R3). T h e proof of this theorem is based on Theorem 2.8, and it will be carried out in several stages. It is important, however, to verify before starting with the proof that in the present case H is indeed self-adjoint. This self-adjointness of N is a consequence of the restrictions imposed on V(r), which has to be locally square integrable and bounded at infinity to satisfy these restrictions. Consequently, -( 1/2m)A V(r) determines a unique self-adjoint operator by virtue of Theorem 7.5 in Chapter IV. The starting point of the proof of the above theorem consists in choosing an adequate domain g1 which satisfies the conditions of Theorem 2.8. We shall prove now that we can choose for g1the linear manifold spanned by all functions
+
+
qp(k) = k,k,k, exp[-(1/2m) k2 - ikp],
(7.2)
pE
R3,
where p varies over all vectors in R3 and k, , k, , and k, are the Cartesian components of the vector k E R3. As a straightforward consequence of (5.2) we obtain
= exp (-it
--k2I
&(k)
2m
= k,k,k,
exp
( -1 2m- i t
k2
- ikp).
Hence, by virtue of Theorems 7.3 and 7.5 in Chapter IV we have" n g H and , consequently exp(-itH,) 9, C g Hn 0gH. exp(-itHo) $, E gH0 The second condition imposed on g1in Theorem 2.8 is that when t , -+ - co and t, + 00, the strong limits of Wl(tl , t,) $, exist. We have already pointed out, at the end of $2, that the existence of the integrals (2.41) and (2.42) is sufficient for the fulfillment of this condition. I n view of this fact, we shall settle the question by showing that
+
(7.4)
for all
pE
R3.
* Note that Ts = Ho and Ts +
Vs = H.
7. Wave and Scattering Operators in Potential Scattering
543
By taking the Fourier transform of both sides of the relation (7.3), we obtain for any real 6 (7.5)
I(exp( -itffo)#P)(r)l
=
const
If we now take 0 < 6 < 712, so that 712 - 6 > 0, we get, by multiplying both sides of the above inequality by V(r),
where C is a constant which majorizes the function
and is obviously independent of r, p, and t. If we choose 6 so that in addition 6 < E , then by virtue of (7.1) the function on the right-hand side of (7.5) is square integrable. Consequently, we can write
where Cp is constant with respect to the variable t E R1. Thus, (7.4) holds and the second condition of Theorem 2.8 is fulfilled. This establishes that the closure g1of the linear manifold 9,spanned by all functions (7.2) is contained in M, .
544
7.2.
V. Quantum Mechanical Scattering Theory
WIENER'STHEOREM ON THE CLOSURE OF TRANSLATIONS
T o conclude the proof of Theorem 7.1, we still have to show that Gf, = L2(R3), and thus establish that M, = L2(R3).T o arrive at this result we need the following lemma, which is a straightforward generalization, to arbitrarily many dimensions, of a theorem by Wiener.
*Lemma 7.1. Let us denote by Jv; the set on which the Lebesgue square-integrable function f ( x ) , x E Rn, vanishes. T h e Lebesgue measure of Nfis zero if and only if the linear manifold (Ff) spanned by the family gf of all functions exp[ip x ] f ( x ) , obtained by varying p over all vectors p E Rn, is dense in L2(R"). Proof. We set out to prove that, given any g E L2(Rn) and E > 0, we can find vectors p , ,...,p , E Rn and complex numbers a, ,..., a, such that
1,. I
g(x) - f ( x )
We note first that for any (7.7)
s
(I dnx < 166.
uyeiz'9v "=l
E
> 0 we can satisfy the inequality
J" B , I g(x)I2 dnx <
6,
B'
=
R"
-
B,
with a compact Bore1 set B , by choosing that set sufficiently large. Let us define the functions
for any positive integer M ; note that h,(x) is almost everywhere well defined, sincef(x) vanishes only on a set of measure zero. I t is easy to see that by virtue of Theorem 3.10 in Chapter I1 g(x)I2dnx = 0.
Consequently, we can find an M for which
7. Wave and Scattering Operators in Potential Scattering
545
Since (7.7) obviously implies that
we easily obtain from this inequality and (7.8), by using in the process the triangle inequality in L2(Rn),
Let L be a box L
=
{x: -1
< xi < I, i = 1, ...,n}
which contains B, and let g l ( x ) be a function equal to xB(x) hM(x)in L and periodic in each of the variables x, ,..., x, , with the period 21. For sufficiently large 1 we have
where L' = Rn - L, and the prime on the summation sign indicates = k, = 0. Comthat the sum does not include the term with k, = bining (7.9) and the above inequality we obtain (7.10)
by making use of the triangle inequality. Let us expand g l ( x ) in a Fourier series,
c.=--J
where k
=
1 (21)"
L
. k-) * x
gz(x)exp (-in
1
dnx,
(k, ,..., k,) and the convergence of the above series is in
546
V. Quantum Mechanical Scattering Theory
the mean. A straightforward computation yields, for any positive integers N and N o < N ,
where the summation C” extends over all K = ( K , ,..., K,) in which 1 k, I > N o for at least one value o f j = I , ..., n. Due to the convergence of G j ck 12, the second sum on the right-hand side of the above inequality can be made arbitrarily small by choosing No large enough, while the first sum can be then made arbitrarily small by taking N sufficiently larger than No . Hence, the sequence of functions
N
=
1 , 2,...,
+
converges in the mean to zero when N + co. Thus, it follows from , Theorem 4.5 of Chapter I1 that there is a subsequence f,,(x), f N 2 ( x )..., which converges to zero for almost all values of x E Rn. Now we can apply Lemma 3.1 of Chapter I V to infer that (7.11)
after noting that the use of this lemma is justified by virtue of the inequality I.fN(x)f(x)I
< {I gL(x)l + M)i.f(x)l’
I n fact, the above inequality follows from the estimate
547
7. Wave and Scattering Operators in Potential Scattering
the estimation of the last integral in the above relation could be carried out due to the formula
which can be proven by mathematical induction in N . Since (7.1 1) is true, we can choose one N, = N' for which
x (1 -
9)
k.x
exp (inT)l
dnx
< 6.
Combining (7.10) and (7.12), we immediately obtain, by means of the triangle inequality,
I
N'
g(x) - f ( x )
k,.
... k,=-N
, (I
-
$!)!
**.
(1 -
!$!)
I
k * x T) dnx < 166.
exp (in
I
Since the above inequality becomes identical to (7.6) after an appropriate change of symbols, we have proven the statement that the linear manifold (gj) spanned by Fj is dense in L2(R3). Conversely, if (Fj) is dense in L2(R3),then the measure of JV; is zero. In fact, if this were not so, then by choosing g(x) = xX,(x) we would have
.s, I
C a,eiPY'" S
g(x) - f ( x )
"=l
so that (7.6) would not be satisfied for all
andp, ,..., p s E R n . Q.E.D.
E
> 0 by
any a, ,..., a, E C1
548
V. Quantum Mechanical Scattering Theory
I f f ( p ) is the Fourier-Plancherel transform off(x), then it is easy to Psee that the Fourier-Plancherel transform of f(x)eiPu.z is T h e function f(p - p”) is usually called the translation of f ( p ) by the amount p , . By taking the Fourier-Plancherel transform of the integrand in (7.6) we obtain
PC
P Y ) ’
Hence, we conclude that the linear manifold spanned by all translations of 3 ( p ) is dense in L2(Rn).This statement constitutes Wiener’s theorem on the closure of translations. Let us return now to the proof of Theorem 7.1. If we set
then (7.2) assumes the form
$,(k)
= f(k) e c i k p .
Sincef(k) vanishes only on a set of Lebesgue measure zero, we infer from Lemma 7.1 that the linear manifold g1spanned by all function (7.2) is dense in L2(R3). Consequently, 6,= L2(R3),and since we already know that M, 3 g1, we finally obtain Ma = L2(R3). This concludes the proof of Theorem 7.1. We note that this theorem guarantees that Ma = L2(R3) for all almost everywhere continuous potentials V(r) which decrease at infinity faster than the Coulomb potential. However, at this stage we cannot say anything about the unitarity of the S operator in potential scattering. Only when additional restrictions on the potential are imposed, as required in the theorems of the preceding section, will we be able to prove that in potential scattering the scattering operator S is unitary. ON THE POTENTIAL FOR PHYSICALLY 7.3. CONDITIONS SATISFACTORY POTENTIAL SCATTERING THEORIES
T h e next theorem will relate wave operators to distorted plane waves, providing at the same time the answers to all the basic questions regarding R,, M a , and the unitarity of the scattering operator S in potential scattering.
Theorem 7.2. Suppose the potential V(r) is measurable and locally square integrable, that it is bounded on any compact set disjoint
I . Wave and Scattering Operators in Potential Scattering
549
from the closed set 9“ of measure zero [containing all the singularities of V(r)], that V(r) I r - r’ V(r’) is locally integrable in R6, and that (7.13) for some > 6. Then the initial domain Ma of 52, is L2(R3),the final ) the domains R, of 52, are both identical to Lic(R3)= E H ( S 3 L 2 ( R 3and S operator is unitary on R = R, = R- ; furthermore, (7.14)
(Q*+)(r)= 1.i.m. J.3
@f)(r) &k) dk,
where (7.15)
$(k)
=
1.i.m.
I.,
+(r) ecikrdr,
and @L+)(r) are the outgoing distorted plane waves, while @L-)(r)= @L+k)*(r) are the incoming distorted plane waves. T h e validity of statements which are essentially equivalent to the assertions made in the above theorem about M a , R, , S, and 52, has been taken for granted by physicists from the very beginning of quantum scattering theory. In proving the above theorem, we confirm that this faith was justified only to the extent that the potential satisfies the conditions imposed on it at the beginning of the theorem. Consequently, it is important to realize that these conditions are essential only to the extent of making applicable all the theorems on distorted plane waves in the preceding section. We discussed these conditions in $6, mentioning at the same time that those theorems are also valid under Ikebe’s condition requiring that V(r) is locally Holder continuous and locally square integrable, and that (7.16) for some P > 0. Thus, the assertions of Theorem 7.2 are also valid under Ikebe’s conditions, which are in some respects more stringent than the conditions of Theorem 7.2, but are slightly more relaxed in their restriction of the behavior of V(r) at infinity (by the 6 power in r). As a matter of fact, the proof of Theorem 7.2, which will follow, uses the same method which Ikebe devised to prove that the above assertions about M a , R,, S , and 52, are true when the potential fulfills his conditions (see Ikebe [1960, $10, $111).
550
V. Quantum Mechanical Scattering Theory
I n the case when the potential is spherically symmetric, the above conditions can be further relaxed by replacing (7.16) with
and making the additional requirement that r V ( r ) is locally integrable on [0, +a)(see Green and Lanford [1960]). Since the assertions of Theorem 7.2 are not true for the Coulomb potential, it is clear that for spherically symmetric potentials one cannot hope to do any better by further relaxing (7.17). However, in the general case, it is an open question whether one could relax (7.16) any further. 7.4.
DISTORTED PLANEWAVESAS KERNELS OF INTEGRAL OPERATORS
We shall prove Theorem 7.2 in a few stages. T h e first step consists in verifying that the integral operator in the right-hand side of the equation (7.14) is a well-defined mathematical entity.
*Lemma 7.2. If @:*)(r) are distorted plane waves belonging to a Schroedinger operator H with a potential satisfying the conditions of Theorem 7.2, then the operators W+ and W- are defined by the relations (7.18)
(W,+)(r)
=
1.i.m.
on all I)€L2([W3), and satisfy the relations
(7.20)
W**E"(B) = EHo(B) W**,
B E B1.
Proof. We shall prove the lemma only for W - , since the corresponding results for W+ easily follow from the relation @:-)(r) = *(r). First of all, we note that under the present restrictions on the potential, Lemma 6.2 holds. I t follows that for any given bounded closed set X,, which is disjoint from the set Yvof singularities of V(r) we can find a constant C such that
@s)
(7.21)
1
I @F'(r)l= ( 2 ~ ) -eikr ~ /+ ~ h(r, k; I k I) 2m 1
~
I<
C
7. Wave and Scattering Operators in Potential Scattering
551
as long as r E X,, and k assumes values from a bounded set (see Exercise 6.13). Let us consider the bilinear form
(7.23)
$(k)
=
( 2 ~ ) - ~ / ~ecikr#(r)dr, JR3
(7.24)
$+’(k) =
1 @F)*(r)+’(r)dr. R3
With the help of Theorem 6.2 we show that this bilinear form is bounded:
Consequently (see Chapter 111, Exercise 2.5), there is a unique bounded linear operator W- satisfying the relation
W+#I *’) = (* I $7
(7.25)
for all t,h, $’ €L2([W3). Let us choose now $(k)E$?i([W3)and $ ’ ( r ) E $ ? t , where VO, is the family of all functions in $?;([w3) with supports disjoint from Y;.For such $(k), “1.i.m.” in (7.18) can be dropped, and (7.22) can be written in the form
Since for such functions $’(r), (7.21) is satisfied, we immediately find $’(r) is integrable in R6. Consequently, by Fubini’s that $*(k) @L+)*(r) theorem, we can interchange the order of integration in (7.26), thus arriving at the result (7.27)
(W+* I $7 =
J
R
dr P
W j JR3 @W) $(k) dkl*.
Since 2‘ 7; is dense in L2(R3)(see Exercise 6.14), equality (7.27) holds for all $’(r) E 2‘ 7; if and only if (7.28)
V. Quantum Mechanical Scattering Theory
552
almost everywhere. Hence, (7.18) is established for $ E %?:((w3). Moreover, due to the fact that qE([w3) is dense in L2((w3),the general validity of (7.18) for W, and arbitrary $ €L2(R3)is an immediate consequence of (7.28). I n order to express W+* as an integral operator, let us take in (7.22) +(r) E %?;((w3) and $+'(k) G V;([w3),and write
(# I W+*#') =
=
( 2 ~ ) - J~ / ~dk$+'(k) R3
)I R3
e-ikr#(r)
&I*.
Since, under the present conditions, the function $*(r) eikr$+'(k) is integrable in r and k on [wB, we can invoke Fubini's theorem to invert the order of integration in k and r:
(# 1 W+*#')= ( 2 ~ ) - ~ / dr ~ #*(r) R3
lj eikr$+'(k)dkl. R3
Now, the type of reasoning earlier applied to (7.27) leads us to the conclusion that (7.29)
(W+*#)(r) = ( 2 ~ ) - 1.i.m. ~ / ~J
R
eikr$+(k) dk
for arbitrary $ E ~ 2 ( ~ 3 ) . Combining (7.18) and (7.29), we obtain (W+W+*#)(r) = [W+(W+*#)](r) = 1.i.m. =
1.i.m.
JR3
1., &)(r) $+(k) dk
@Pk(+'(r)(W+*#)" (k) dk
=
(EH(SF)#)(r),
where the last step follows from Theorem 6.5, and in particular from (6.75). Hence (7.19) is established. Finally, combining (6.75) with (7.29), we easily derive ( W+*EH(B))(r) = ( 2 ~ ) - 1.i.m. ~/~
eikr(EH(B)#);(k) dk L
=
( 2 ~ ) - 1.i.m. ~/~
3
eikr$+(k) dk, 'kZ/2mEB
= ( 2 ~ ) -1.i.m. ~/~
eikr$+(k) dk. ' kZ/ ZmE B
This establishes (7.20).
Q.E.D.
553
7. Wave and Scattering Operators in Potential Scattering
7.5.
THERELATION OF WAVEOPERATORS TO PLANEWAVES
We are now ready to carry through the proof of Theorem 7.2 to its completion. Let us adapt the relation (2.37) to the present case, setting t, = 0, t, = T, and f = $ to obtain (7.30)
((Q(7) -
I)# I g )
=i
j T((eKiHot $ I He-iHtg) - (HoediHot I e - i H t g ) ) dt )I
0
for any g E B H. Since the restrictions imposed at present on the potential meet the demands of Theorem 7.5 in Chapter IV, we infer that if # E g H, then th, E g Hn 9fH0 = 9". Hence, for E gH0( H - Ho)e-iHot # = Ve-a*ot O . *, and the relation (7.30) assumes the form )
(7.31)
(Q(,)*
I g> = (+ I g> + i
1' 0
(eiHtVe--iHot
*I
g> dt.
The above relation can be extended to arbitrary g E L2(R3).Setting g = W-4, €L2([W3),and using the relation W-*eiH1 = eiHol W- *, which is a direct consequence of (7.20) (see also Exercise 2.1)) we obtain
+
Since it can be readily shown (see Exercise 7.1) that (7.32)
we easily arrive at the result
It should be emphasized that, in this context, all the integrals over the infinite interval [0, +m) are improper Riemann integrals, and not Lebesgue integrals.
V. Quantum Mechanical Scattering Theory
554
We can calculate the integrand in the above relation by using (7.29) and (4.8),
We resort now to the relation (6.53), which for +(r) E Wz(R3) yields
We note that the limit in the mean in (6.53) could be immediately taken when deriving the above relation, since the integrand in (7.35) is integrable on R3. T o arrive at this conclusion, we combine (6.29), (6.55), and (6.56) to derive the estimate
SinceF,,(r) 1 V(r)/ is locally integrable (see Exercise 7.2)while ( c t H o t +)(r) is uniformly bounded for all t E R1 (see Exercise 7.3), we conclude that F,(r) V(r)(eriHot$)(r) is locally integrable. Furthermore, since IFo(.)] const when r 2 R and R is sufficiently large (see Exercise 6.13), we obtain for such values of r E R3
<
T h e function on the right-hand side of the above inequality is the product of two square-integrable functions and therefore is integrable. Thus, the integrand in (7.35) is integrable for r 3 R, and consequently it is integrable on R3. By combining (7.34) and (7.35) we obtain (7.38)
(4 I exp(iH,,t) =
.s,
W-*V exp(--iH,,t)+)
dk $*(k)
s
R3
dr &'*(r)V(r)(exp[-i(H,
Let us show now that when +(r) E %;(R3),
we have
- k2/2m)t]$)(r).
555
7. Wave and Scattering Operators in Potential Scattering
where C , is a constant independent of k E R3 and t E R1. T o see that, consider separately the above integral over B, and B,‘, where B, is a closed set disjoint from SP,, and its complement B,’ is bounded. Since the argument leading to (7.37) can be applied when r varies over B, (see Exercise 6.13), we have
LoI
&)*(r) V(r)(exp[-i(Ho
~
k2/2m)t]+)(r)ldr
where 11 V I/ and 11 # 11 denote the L2 norms of the square integrable functions V(r) and $(r), respectively. Furthermore, since (exp[-i(H, - k2/2m)t]#)(r) is uniformly bounded in k E [w3 and t E R1 (see Exercise 7.3), by using (7.36) we get
f
Bo’
I &’*(r)
V(r)(exp[-i(H,
- k2/2m)t]+)(r)ldr
< const
thus establishing the validity of (7.39). Let us insert (7.38) in (7.33) and choose &k)
where (7.41)
&(+, +)
=
is+“ dt eft 0
x
s,
E
1,.
%:(R3),
(1
+ Fo(r))dr,
thus obtaining
dk$*(k)
jR3 dr @i-’*(r) V(r)(exp[-i(H,
- k2/2m)t]#)(r).
Using (7.39), we easily infer that the above integrand is integrable in t E [0, CO) and k E R3. Consequently, the reversal of the order of integration in t and k is justified by Fubini’s theorem. Hence, for closed B disjoint from Y Vwe , have
+
(7.42)
M,4 4) = I?(#, 4)
+ i I,, dk4*(k)
dt (F: I exp[-i(Ho - k2Pm - ic)tl$>,
556
V. Quantum Mechanical Scattering Theory
where (7.43)
I;(#,
4) = i j dk$*(k) Iw
x (7.44)
Is,
j+m dt ecGt 0
dr &)*(r) V(r)(exp[-i(Ho
-
k2/2m)t]#)(r),
Ff(r) = XB(r)@(i)(r) V(r).
We note that the inner product under the integral sign in (7.42) is indeed defined since Ff €L2(R3),due to the fact that I @L-)(r)l const on B and that V(r) is square integrable. Using Theorem 3.1 of Chapter IV, we infer that
<
Consequently, we easily compute
= i lim r++m
=
-i (F'
(Ff 1 (exp [-i
1
R,
(H,
-
k2 (= + i c ) #)
k2
- - ic
2m
=
-i
k2 (Ho(- 2m
-k)Ff
If we combine (5.5) and (5.6) with the above result, we get
On the other hand, using the fact that the function
1 #).
557
7. Wave and Scattering Operators in Potential Scattering
is uniformly bounded with respect to following estimate:
k E R3 and t E R1, we obtain the
This estimate shows that for any sequence B, C B, C we have
*-.,(J, B,
= R3,
Let us combine this last result with (7.45) in order to take the limit B, --t R3 on the right-hand side of (7.42). Since the integrand on the right-hand side of (7.45) is integrable on R9 in r , r’, k E R3, the limit B, --t R3 can be carried under the integral sign. Hence, we have
L(A4) = $J
R
IJ
R3
exp[i(k2 - 2 r n i ~ ) lI /r~ - r‘ Ir-r’)
x V(r’) @i-)(r’)dr‘
1* $*(k)
11
+(r)d r dk.
Inserting this result in (7.40), and noting that exp[i(k2- 2 r n i ~ ) l1 /r~ - r‘ I] Ir-r’I
=
V(r’) @L-)(r’)dr‘
I r - r’ I) V(r’) @L-)(r‘)dr‘, r --‘I
I,exp(-ikI
and that the convergence in the above limits is uniform for r f SUPP and k E supp $ (see Exercise 7.4), we arrive at the following result: exp(-ik I r - r‘ I) x V(r’) @A-)(r’)dr‘
I*
$*(k) +(r)dr dk.
From (7.18) we obtain (7.48)
1 dr +(r)1 dk @i-)*(r)$*(k) R
J,
R
@;-)*(r) $*(k) +(r) dr dk,
*
V. Quantum Mechanical Scattering Theory
558
where the second equality follows from Fubini’s theorem; this theorem can be applied, since” by (7.36)
where C , and C , are two constants and consequently the integrand in (7.48) is integrable on Re by Tonelli’s theorem. Substituting the expression for (W-4 I $) from (7.48) in (7.47), we obtain
x =(
i R 3
exp(-ik I r - r’ I) V(r’) @i-)(r)dr I r - r’I
1
2 ~ ) - ~ /dk$*(k) ~ R3
1
R3
dr cikr $(r)
I*
=
by using in the process the Lippmann-Schwinger equation (6.59) (r). for @i-) I n the above relation $(r) and the Fourier transform $(k) of d(r) can assume any value in YC(R3). Since Y:(R3) is dense in L2(R3),it follows that W-*SZ- = 1. Consequently, by virtue of (7.19) and the fact that E,EH(Sf),we conclude that
<
w-= w_w-*sz-= EJys;)Q-
= Q-
.
Completely analogous considerations lead to the conclusion that
W+ = SZ, . Hence, we have thus established the representation (7.14)
for the wave operators SZ, in potential two-body scattering for arbitrary $ eL2(R3).This shows that in the present case M, = L2(R3).Moreover, in view of (7.19), we have also established that R, = R- = Ltc(R3). Consequently, the proof of Theorem 7.2 has been completed.
7.6. THERELATIONOF TO THE
THE S MATRIX SCATTERING AMPLITUDE
In scattering theory, the object of principal interest to the physicist is the scattering matrix, since it is related in a relatively straightforward
* Note that according to (6.29) the constant in (7.36) can be chosen to be independent of k on the compact support of &k).
559
7. Wave and Scattering Operators in Potential Scattering
manner to the differential scattering cross section, which is directly measurable. T h e computation of the scattering matrix is made comparatively easy by the following theorem which relates it to the distorted these in their turn can be computed by solving plane waves @J+)(r); the integral equation (6.59) or the differential equation (6.80).
Theorem 7.3. Suppose the potential V(r) satisfies all the conditions imposed on it in Theorem 6.4. Then the scattering operator S can be expressed by the following integral formula: (7.49)
( S $ ) - ( k ) = $(k)
-
r* 1"
i d/z;J d+ 0
0
dB sin ef-k(B,
4) $1(k,
8, d),
wherefk(B, #) is the scattering amplitude defined in (6.62), and (7.50)
yT1(k, B, 4)
= $(K
cos 4 sin B, k sin 4 sin B, K cos 8).
As a starting point for the derivation of (7.49) we take formula (7.31) O g Hand g = 52-4) at T = t , and T = t , , (with V = H , , # E g H = and subtract the corresponding sides of the two equations, thus obtaining
By letting t, -+
-
co and t,
--f
+ 00 we arrive at the relation
where the above integral is an improper Riemann integral. On the other hand, we have
since in the present case Q-*Q-= EMo= 1 . Combining the above relation with (7.51), and using the intertwining property (2.3), we arrive at the result
By making use of (7.14), we easily derive (Q- exp( -iHot)$)(r) = 1.i.m.
I,.
@k;)(r)exp[ -i(k12/2m)t]$($) dk, .
560
V. Quantum Mechanical Scattering Theory
Inserting the above formula and the relation (V eXP(-~KJt)W) =
exp[ --i(k,2/2m)t] exp(ik,r) $(k,) dk,
V(r) 1.i.m. JR3
in (7.52), and taking &k), $(k) E %‘g(R3) (so that the integration extends, effectively, over compact domains, and therefore 1.i.m. becomes Lebesgue integration over R3), we obtain
(7.53)
(4 I s+>- (4 I *)
[ (- kZ2 - K)t] exp(ik,r) +*(k,)$(k,).
x exp --i
2m
2m
A potential V(r) obeying the restrictions imposed on it in Theorem 6.4 is obviously integrable on R3. Since I @i-)(r)lis uniformly bounded at k, E supp $, we infer infinity (see Exercise 6.10) for all” k, E supp from the inequality
6,
that the integrand in (7.45) is integrable on R9 in r, k, , and k, . Hence, the order of integration in (7.53) in the variables r, k, , and k, can be interchanged on account of Fubini’s theorem. Therefore, we conclude that
x $*(k,) $(k,)
J
Iw
dr exp(ik,r) V(r) @i;)*(r).
* Here we take it for granted that H does not have any positive eigenvalues. However, even if H had positive eigenvalues, the proof would have to be altered only slightly by choosing &kl) E Vr(R3) and &kz) E V?(R3) with supports disjoint from the sets {kl : k12/2mE S,”} and {kz : kZ2/2mE S,”}. Since these sets are closed sets of measure zero, the totality of such functions is still dense in L2(R3),as can be proven by the method employed in solving Exercise 6.14. T h e reader can then easily make the slight alterations which are necessary t o carry the proof of the present theorem under these conditions.
56 1
7. Wave a n d Scattering Operators i n Potential Scattering
For fixed 7 > 0, the above function is obviously integrable in t , k, , and k, on [-r, +TI x [ws. Hence, we can interchange in the above integral the order of integration in t and (k,, k,), thus arriving at the relation
(4 I S+) - (4 I+) = 2i(2~)-3/2lim
r++m
k2'/2m)7 1 dk, dk, sin(k12/2m k12/2m k,2/2m -
x $*(k,) J(k2) J dr eikzrV(r) @i;)*(r). RS
The above relation can be written in the form
r" 1"
where (k, , Oz , C#Lare J the spherical coordinates of k, and (7.55)
h(k2 , k 1)
= 2 i ( 2 ~ ) - ~ / ~ $ * ( k ~d4, ) 0
x sin 8, &k2) The above limit for r -+
do2
0
1 dr eikzrV(r) @i;)*(r). R3
+ co can be calculated by using Lemma 7.3."
*Lemma 7.3. If the function f ( u ) is integrable and Holder continuous in [wl, i.e., if for some O > 0, (7.56)
I f(u +4
- f(4l < CO(4 I flu
for sufficiently small values of 1 d u (7.57)
1 f ( u ) = - lim T
r++m
+m
IS
I, then sin T(U - V )
f (4dv.
* At first glance, it would seem that this lemma states that lim,,,, in the sense of distribution theory, i.e.,
(1 /?r)(sini&)/.% = 8(x)
However, the usual proofs of the above relation require not only that f ( x ) be Holder continuous, but that it be at least once continuously differentiable.
562
V. Quantum Mechanical Scattering Theory
Proof. We note that 1 dv=-[ 77
fm --m
sin w dw w
___
=
1,
and consequently (7.58)
Let us split up the domain R1 of integration into the three parts 61, and (u 6, +a), and study each part separately. Using the Holder continuity condition (7.56), we obtain for any given E > 0
(-a, u - a), [u - 6, u
+
+
for all T 2 0 and sufficiently small 6 > 0. The integral over the interval (u 6, CO) is equal to
+ +
+m
When
T
+ f co ,
T(U
- v)
dv
+-
f ( v ) dv.
the first of the above integrals converges to zero, i.e.,
for sufficiently large T o see that, write (7.61)
sin
T
> 0. The second integral f ( v ) dv =
+"
[
rL4
also converges to zero.
sin TW
f
(u - w> dw. W
The right-hand side of (7.61) converges to zero according to the Riemann-Lebesgue lemma (see Chapter 111, Lemma 4.1). Thus, the 3 sufficiently large T. expression in (7.60) can be made smaller than ~ / for
563
7. Wave and Scattering Operators in Potential Scattering
A completely analogous argument leads to the conclusion that the u - 8) to the contribution of the integration over the interval (-a, integral on the right-hand side of (7.58) is smaller than r / 3 if 7 is sufficiently large. Consequently, (7.57) is true. Q.E.D. Let us show now that the above lemma can be applied to the computation of the limit in (754). By setting u =
kZ2/2m,
we obtain
(7.62)
(4 I S 4 ) - (d 14) = lim
dk,
-
-03
du
sin(k12/2m- U ) T k12/2m- u
f (u, kl),
where we have extended the integration in u to the entire real line by introducing the function m
(7.63)
u A(&,
k)
for u
>0
for u
< 0.
It can be proved (see Exercise 7.5) that h(k, , k,) is Holder continuous min(1, E,,}: in k, for any 0 < 6
<
I t follows that f ( u , k,) is also Holder continuous for those values of B which are also smaller than one. This can be seen from the following inequality:
If(u + Au, k d - f ( ~ ,ki)l < n- 4 2 \(I u + AU 1 /’ ~ 3 1 2
-
I h ( f i K , k1)l
1 1’2)
+
in which u1 is some value between u and u Au obtained by applying the mean-value theorem of differential calciilus to
564
V. Quantum Mechanical Scattering Theory
I u + du 1 / 2
- Iu
11/ 2 .
Consequently, Lemma 7.3 can be applied to
derive the fact that
1 lim -
(7.64)
7++m
T
j
+m
sin(k12/2m- U ) T kI2/2m- u f (%kl) du
However, we cannot insert this result immediately in (7.62), since we do not know yet whether it is permitted to invert the order of integration in k, and of taking the limit T -+ co. T o justify the taking of the limit T -+ f c o under the integral sign, we employ (7.58) to derive
+
<
Then, using the fact that I f ( u , kl)l C, , where C, does not depend 1, by on k E supp and u E [wl (see Exercise 7.6), we obtain for T means of (7.59)-(7.61),
6
Since the function on the right-hand side of the above inequality is integrable in k, over supp6, and is independent of T, we can apply Lemma 3.1 in Chapter IV and infer that the expression in (7.65) converges to zero when T -+ +a.Hence, the limit T -+ +co can be taken under the integral sign in (7.62). Thus, in view of (7.64), we arrive at the result (7.66)
(4 I S*) =
-
(9 I *)
6j iWz dk, k16*(k,) j2nd+2
i 25
0
0
do, sin O2 &(kl , w2)
565
7. Wave and Scattering Operators in Potential Scattering
where SZ, = [0, 771 x [0, 27~1,dw, = sin 8, do, db2 and w, is the unit vector in R3 with the spherical coordinates O2 and b 2 , while f k ( w z ) is the scattering amplitude defined in (6.62). and %?;(R3) is dense in L2(R3),the relation Since &k), $(k) E g;([w3), (7.66) can be immediately extended to the entire Hilbert space L2(R3). Thus, we arrive at (7.49), and at the conclusion that Theorem 7.3 is valid. 7.7.
RELATIONOF THE SCATTERING AMPLITUDE TO THE SCATTERING CROSSSECTION
T h e formula (7.49) assumes a simpler form when the S operator is replaced by the T operator, (7.67)
(T$)(k)
=
1-s ( T ) ( k )
1
= -=
d277
4
kf-k(w’) y$(k, w‘) dw‘.
Comparing this result with (1.31), we see that the T matrix T ( l ) ( kw, ; is related to the scattering amplitude f k ( w ’ ) by the formula (7.68)
T‘l’(k;W,
w’) =
w‘)
1 kf-k(O’), 2/27
where w = k / k . Formula (1.47) for the differential scattering cross section is valid in a frame of reference with its x axis pointing in the direction w’ of the relative momentum of the incident particles. I n such a frame of reference the relative momentum k after the collision points in the direction w. Thus, (1.47) can be written in the form (7.69)
u(k2/2m,w)
=
8a3 l f k ( e J 2 ,
where e, is the unit vector pointing in the direction of the z axis. T h e above relation (7.69) is not the familiar formula for the differential cross section. T o recast it in the familiar form we have to use the following symmetry property of f k ( 0 ’ ) : (7.70)
f ( k w , ( w ’ ) = focw‘dw).
This property follows from the obvious formal symmetry in of the right-hand side of the relation
w’
(7.71)
m fk(0‘)= - 27
I,,
exp[--ik(w’
- r) V(r) @g(r)dr
w
and
566
V. Quantum Mechanical Scattering Theory
This relation is obtained by using (6.59) to eliminate exp[--ik(w’ * r)] from the integral in the first line of (7.71). T h e usual procedures establish that
is integrable in r and r’ on [w6 (see Exercise 7.7). Hence, Fubini’s theorem can be applied to infer that in (7.71) the iterated integration in r, r’ E [w3 can be reduced to an integration on R6 in (r,r’). Thus, the formal symmetry in w and w’ of the right-hand side of (7.71) is an actual symmetry, in the sense that the expression on the right-hand side of (7.71) remains unchanged under the intercharge of the vectors w and w‘. From (7.69) and (7.70) we immediately obtain
where obviously (7.73)
Formula (7.72) is the very familiar relation expressing the differential cross section in terms of the scattering amplitude. T h e only unusual feature in (7.72) is the presence of the factor 87r3, which usually does not appear in physical literature. Its presence is due to a different “normalization” of the distorted plane waves (6.56), which in physical literature are chosen to be
I n fact, the scattering amplitude fk’(r) obtained by taking in (6.62) Ok(+)(r)instead of @$+)(r) is related to fk(r) by
In terms of fk’(r)(7.72) assumes the form
567
7. Wave and Scattering Operators in Potential Scattering
7.8.
THEPHASE-SHIFT FORMULA FOR
THE
SCATTERING OPERATOR
vs) defined in $7 of Chapter I1 Let us introduce in the space L2(Qn,, [see (7.16) of Chapter 111 the operators
(7.74)
S ( k ) = 1 - 2m' T(k),
(7.75) (T(k)u)(w)=
k
dzJ,
~
f(k",)(-w)
u(w') dw',
u EL2(-%
, PSJ.
The unitarity of the S operator imposes the condition that S ( k ) should be unitary in L2(Qn, , ps) for all k 3 0: (7.76)
S*(k) S ( k ) = S ( k ) S*(k) = 1.
As a matter of fact, for arbitrary u E L ~ (, Q ps)~ and normalized
p eL2(Qn,, p r ) [see (7.16) of Chapter 111 it easily follows from (7.67),
(7.70), and (7.75) that
(S*Sp * u)(k, W )
= p(k)(S*(k)S(k)u)(w).
Consequently, we have for arbitrary u., u €L2(QS, p s ) v"(w)(S*(k)S(k)u)(w)dw =
JrnI p(k)I2k2 dk J",
v*(w)(S*(k)S(k)u)(w)dw
0
= =
J, J"
(p
. ~ ) * ( kw)(S*Sp , . u ) ( w ) k2 dk dw
(p
. v)*(k,w ) ( p . u)(k,w ) dw
=
QP
This means that S*(k) S ( k ) = 1 . A similar argument establishes that S ( k ) S*(k) = 1. The unitarity of the operators S ( k ) , 0 k < +a,enable us to derive Theorem 7.4.
<
Theorem 7.4. Suppose the potential V(r) satisfies all the conditions imposed on it in Theorem 6.1. Then for each k E [0, co), there is an orthonormal basis {w,(k),w2(k),...} in L2(Qs, p s ) such that
+
568
V. Quantum Mechanical Scattering Theory
where the numbers 6,(k), 6,(k),... are real and such that (7.78)
Tr[T*(k) T ( k ) ] =
m
1 I sin S,(k)I2 < +a.
”=l
Proof. T h e kernel T(l)(R; w, w’) of the operator T ( k ) is square integrable in (w, w’) on fin x Q S with respect to the measure ps x pa (see Exercise 7.8). Hence, the operator T ( k ) on L2(fis,pa)is of the Hilbert-Schmidt type” and therefore completely continuous. Since S(k) is unitary, we can apply Theorem 6.1 of Chapter I11 and write (7.79)
S(k) =
2a
eiAdE,(k),
0
where the above integral converges in the uniform sense of (6.1) in Chapter 111. We shall prove now that the complete continuity of the operator T ( k ) implies that the spectral function E,(k) is constant in A except for an at most countable number of discontinuities Aio’(k), Aio)(k),..., where EAF)(k)# EAy)-,,(k). Suppose to the contrary that there is a point A, E (0, 27r) such that EAo(k)= EAO-,,(k)but Ea(k) # E,(k) for any a < A, < b such that a, b E (0, 27r). Then we can choose a sequence A, , A, ,... + A, which is # EA,+,(K). monotonically increasing or decreasing, and such that EAm(k) T o be specific, let us say that A1 < A, < < A,, and let us choose - EAn(k))f,= f , vectors fl ,f2 ,... E A? which are such that (EAn+l(k) and 11 f , 11 = 1. We shall prove that no subsequence of T ( k )fl , T ( k )f , ,... is convergent, thus contradicting the complete continuity of T ( k ) . I n fact, it is easily seen that T ( k ) f , 1 T ( k )f , for m # n (see Exercise 7.9), and consequently
I1 W
) f m- W f n
On the other hand,
II W
f
Il2
=
I1 W
) f W ,
/I2
+ /I W ) f n l 1 2.
11 T ( k )f , 11 3 C > 0 for all n n
/I2
=
=
1 , 2,...,
1
p ((W- 1)fn I ( S ( 4 - l)fn>
1 2inf I sin A, l2 > 0. 4772 n=1,2....
* This can be easily inferred by setting p s ( B ) = 0 for B C [w2 - Q, , thus extending p s to RZ,and then applying the result of Exercise 5.3 and Theorem 8.6 of Chapter IV.
7. Wave and Scattering Operators in Potential Scattering
569
Thus, no subsequence of T ( k ) f , , T ( k ) f , ,... is a Cauchy sequence, which is impossible if T ( k ) is of Hilbert-Schmidt type. Hence, E,(k) can increase only in a discontinuous manner. , p s ) is separable, there can be at most countably many Since L2(Qn, such points A:O)(k), Aio)(k),... of discontinuity. Thus, (7.79) becomes
where it is easily seen that the convergence of the series is in the strong sense, but not necessarily in the uniform sense (think of the case S ( k ) = 1, when the convergence is not uniform). If we introduce an orthonormal basis in each one of the mutually orthogonal subspaces on which E,,;)(k)(k)- E,;)(k)-o(k), n = 1,2,..., project, we obtain an orthonormal basis in L2(Qs,,us) by taking the union of all these bases. After adequately labeling the elements of these bases, and replacing Ac)(k)with the adequately labeled 6,(k) = iAc)(k), we obtain (7.77). T h e corresponding expression for T ( k ) is (7.80)
1 T ( k ) = -( S ( k )- 1) 2ni
1 "
=-
1 I w,(k)) ei'u(k)
u=l
sin S,(k) (w,(k)l.
Thus, we have
and since T ( k ) is a Hilbert-Schmidt operator, (7.78) follows.
Q.E.D.
When the potential V(r) is spherically symmetric, physicists call the numbers 6,(k) phase shifts. We will apply this terminology (see also Ikebe [1965]) to the general case, when V(r) is not necessarily spherically symmetric. Such a name for a,,(,+) seems appropriate, since the absence of interaction is characterized by vanishing phase shifts 6,(k) = 0; in fact, when there is no interaction we have S = 1, and therefore
cI m
S(4 = 1=
W"(k))(WU(k)l.
u=l
If we expand f ( k e , ) ( w )in the orthonormal basis {wl, w p,...} and compare the Fourier coefficients of this expansion with (7.80), we get (7.81)
570
V. Quantum Mechanical Scattering Theory
Hence, the total scattering cross section a(k2/2m) at a given energy k2/2m can be expressed conveniently in terms of phase shifts:
Naturally, both formulas (7.81) and (7.82) are unambiguous only if T ( k ;w', w) and w y ( k ;w') are continuous functions of w', so that (1.47) is valid.
7.9. PARTIAL-WAVE ANALYSIS FOR SPHERICALLY SYMMETRIC POTENTIALS If the potential V(r) is spherically symmetric,
W) = Vok),
(7.83)
r
=
I r I,
the functions w,(k;w) can be easily computed. Using the well-known Bauer's formula (see Exercise 7.10), m
exp(ikx) = exp(ikr cos 19) - C i1(2Z l)j,(kr) P,(cos e),
(7.84)
z=o
+
and combining it with the addition theorem for spherical harmonics, (7.85)
we obtain (7.86)
exp[--ikr(w . w')]
=
1 1 iz(2Z+ l)jz(kr) yzm(w)(ytm(wl))*,
1=0 m=-1
where w = (0,d) and w1 = (el,dl) are spherical coordinates which correspond to the unit vectors w E R3 and w1 E R3, respectively. Let us insert the above series in the integral in (6.62) defining fk(w). Due to the uniform convergence of the series for w, w' E Q8 (see Exercise 7.10) we can integrate term by term, thus arriving at the following relation: (7.87)
fk(w) =
-2nm
m
1
1
1=0 ni=-Z
x
J0s
dw,(
m
izYzm(w) dr rVo(y) 0
Ylm(wl))*@k+)(rl).
7. Wave and Scattering Operators in Potential Scattering
571
From the Lippmann-Schwinger equations (6.59) we immediately see that when V(r) is spherically symmetric @L+)(r) is a function of only Y, k, and cos 0 = r k/rk: (7.88)
=
@(I.)
@ P ) ( r ,cos 0).
Hence, by carrying out a change of variables in which w1 is replaced by spherical coordinates of r with respect to k , we obtain
+ 1) s'
for m # 0
-1
@(+) k ( r , 24) P l ( U )
for m = 0.
du
Consequently, (7.87) can be written in the form (7.90)
T"'(k; w',
W)
=
k
6
-fk0+--w)
=
1 47+7 1 a,@) P,(-W
z=o 21 ~
where the convergence is in the mean on SZ, x (7.91)
az(k)= -kmiz(l
*
w')
a,,and
+ 4)3/21 dr rzVo(r)/" du @ P ) ( r ,u ) Pz(u). co
0
-1
Thus, we have arrived at the following expansion for S(k): (7.92)
m
z
S ( k ) = 1 - 2 ~ -T(k) i = 1C
Z=O m=-2
1 Y,")(l
-
2 n i ~ l ( k ) ) ( Y1. z ~
Comparison with (7.77) yields the following explicit expression for the phase shifts: (7.93)
eZiS,(lC) = I - 2niac(k).
T h e reader can compare the above results with the considerations in 94.3, and easily arrive at the conclusion that (7.94)
S,(k)
= e2iS,(k)
is the S matrix in spherical coordinates.
572
V. Quantum Mechanical Scattering Theory
The great importance of the above formula for the S matrix lies in the fact that in conjunction with (7.91) and (7.93) it enables us to analytically continue S,(k) in the complex planes of the variables I and k. T h e properties of the resulting complex S matrix S ( k , I) provide the foundations of a very fruitful theory of the S matrix, which has become the fundamental stepping-stone of much of the theoretical physics in the past two decades. T h e reader is urged to consult the References for literature dealing with this very important subject."
EXERCISES 7.1. Show that if f ( t ) is continuous and the improper Riemann integral I - lim r f ( t ) dt - oi-tm
=
/
+m
f ( t ) dt
0
exists, then the improper Riemann integral Ic =
+m
ecftf(t) dt
0
exists for every e
> 0, and lirnE++,Ie= 1, .
7.2, Prove that the function F,(r) j V(r)j is locally integrable on R3, where F,(r) is the function given in (6.28).
7.3. Show that for any given $(r) E V;([w3) there is a constant C , such that I(e-iHot $)(r)l C, for all r E R3 and all t E R1.
<
7.4. Prove that the convergence of the limit in (7.46) is uniform with respect to k E Do and r E D if Do and D are any two compact sets in [w3.
7.5. Show that when V(r) satisfies the conditions of Theorem 6.4, the function h(k, , k,) defined in (7.55) is Holder continuous in the sense that
I h(kz for any given 0
+ 4 , ki)
<8
-
h ( 4 , ki)l
< min{ 1, eo>.
< Cdkz
ki) I 4 Is
<
C, 7.6. Establish the existence of a constant C, such that I f ( u , k,)l for all u E R1 and k, E R3, where f ( u , k,) is the function introduced in (7.63) in the course of proving Theorem 7.2.
* A mathematically rigorous treatment of analyticity properties of the S matrix in potential scattering is given in the work by de Alfaro and Regge [1965].
573
I . Wave and Scattering Operators in Potential Scattering
7.7.
Show that the function h(r, r’) = @2)(r)V(r) I r - r‘ 1-1 V(r’)@k’(rr)
is integrable in (r, r’) on R6. 7.8.
Prove that for fixed K
exists, where T ( l ) ( k ;w, 7.9,
w’)
E
[0, + co) the integral
is defined in (7.68).
Suppose [EAir(k) - EAj(K)]fi = fi, i
(4 > A,’] n (A2 ,A,]’
=
1 , 2 for
= 0,
where EA(k)is the spectral function of S(K),and A,, Prove that T(K)f, T(K)f,.
Al’, A,‘
A,,
Prove that
7.10.
and that the convergence is uniform in u E [- 1, Show that for any
7.11.
(exp(-iHOt) =
+ I].
> 0,
+ko)(r)
(2p2
+ -)mit
-312
[
t exp i rko - -b2)] exp -
[(
2m
if y5ko(r)is a wave packet with a Fourier transform of the form $ko(k)
= exp[--P2(k
-
b)21*
Use this result to show that for any bounded Bore1 set B
= B
I(e-iHot$ko)(r)12 dr < #(B)
[4 (isz
+
-&)]-3’2
E
(0, 2 ~ ) .
574
V. Quantum Mechanical Scattering Theory
7.12.
Any free state $:, $$(.I
= (e-i'fot
d)(r)9
corresponding to E L2([w3),displays a tendency of "spreading out" in time. This conclusion can be reached by establishing the evanescence of the wave packet $i(r) from any bounded Borel set B :
Combine the result of Exercise 7.1 1 with Lemma 1.2 and Lemma 7.1 to prove that the above limit is indeed equal to zero. 7.13. packet,
Prove that any interacting state &(r) = (ciH1 1Crn)(r),
$t
represented by a wave
$n E L ~ ( R ~ ) ~
which has an incoming (outgoing) asymptotic state $f($SUt) evanesces from any bounded Borel set B in the configuration space when t + - 00 (t +a). +
8.
Fundamental Concepts in Multichannel Scattering Theory
8.1. THECONCEPTOF CHANNEL T h e outcome of the mutual interaction of three or more particles can lead to results which are qualitively different from the possibilities encountered in two-particle interactions. Indeed, a system of two particles can either be found after interaction in a bound state, or the two particles become eventually free. However, in the case of three particles P , , P , , and P , , there already are other possibilities in addition to the two alternatives of P I , P , , P , becoming eventually free or constituting a bound system (Pl-P2-P3). For example, it is possible that PI and P, will stay bound forever, thus building a new system P l - P 2 , while P, becomes free, etc. Each one of these possibilities determines a particular outgoing arrangement channel of the scattering process between P I , P , , and P , . I n general, an arrangement channel in the scattering process of n particles P, , P , ,..., P , is a particular partitioning of the set { P I ,P , ,..., P,} into a number of subsets, called fragments, where all
8. F u n d a m e n t a l Concepts i n Multichannel S c a t t e r i n g T h e o r y
575
the particles in a fragment are bound together, and where no interaction takes place between particles in different fragments. Clearly, we can talk about incoming channels and outgoing channels, depending on whether the particular channel we are considering refers to the system before or after the scattering had taken place. I n the case where some of the particles in the system { P , , P2 ,..., P,} are identical, not all partitions of the set { P , , P2,..., P,} will represent distinct arrangement channels. This is due to the presence of BoseEinstein or Fermi-Dirac statistics, respectively (see Chapter IV, 94) which rules that identical particles are indistinguishable. For example, if Pl , P , , and P , are identical, then the channel ( P l - P z , P 3 ) in which Pl and P2 are bound, while P , is free, is identical to the channels ( P , , P,-P,) in which P , is free and P, is bound to P , . As a matter of fact, in this example both above arrangement channels are also identical to ( P 2, P,-P,) so that we have only three distinct arrangement channels: (P1-p2-p3), ( P I , p2-p3), and (P1 , P2 , P3). If each fragment 9is in an eigenstate of a set 9, of commuting observables which are constants of motion in that particular arrangement channel, then the system is said to be in the channel determined by eigenvalues of the observables in the family Lo,. I n practice, it is very convenient to take an 0, containing the internal (binding) energy of the fragment, its internal angular momentum (called the spin of the fragment), its angular momentum with respect to the reference system, etc. If 9 , contains only the internal energy of then we call a channel determined by the given internal the fragment 9, energy eigenvalues an energy channel.*
u,
8.2. ELASTIC AND INELASTIC SCATTERING I n the case of many-body (three or more particles) scattering we can distinguish between elastic and inelastic scattering+ by looking at the distribution of energy among fragments before and after the scattering
* We depart somewhat from conventional terminology, in which a channel with respect to some of the above-specified observables is simply called a channel. However, since there does not seem to be complete agreement among different authors o n the precise contents of the set 03 when a channel is defined, the above general approach seems quite desirable. t T h e distinction can be made also in the case of two-body scattering in an external field. In fact, the external field describes the interaction of the two particles with one or more additional particles which are not explicitly included in the system, but are effectively represented in an approximative manner by the “external” field. I n that case, each one of the particles in the system can be bound to the field. However, it is clear that two-body scattering in an external field is, in fact, many-body scattering.
576
V. Quantum Mechanical Scattering Theory
has taken place. Elastic scattering is, by definition, the scattering process in which all the fragments and their internal energies are preserved; for example, if the initial energy channel consists of a particle Pl being scattered from a bound system P2-P, of two particles, then the scattering is elastic if after the scattering has taken place the particles P, and P, are still in a bound state of the same energy as the initial bound state, i.e., if no energy has been transferred from Pl to P,-P,. Naturally, any scattering which is not elastic is called inelastic. Suppose all the fragments in an energy channel are in their internal energy ground states and that A E is the smallest energy gap between the internal energies of these fragments and the energetically nearest eigenstate of their respective internal energies. Then, obviously, no inelastic scattering can take place as long as there is not sufficient kinetic energy to be transfered to the k,th fragment to raise it to an A E . This amount of energy “excited” state of internal energy E T is then called the threshold of the inelastic process in that particular energy channel. If in an inelastic scattering the fragments themselves are decomposed and new fragments result as an outcome of the scattering process, then we talk of a rearrangement collision; for instance, in the above example this is the case when a bound state of PI and P, is formed, while P, goes free after the interaction has taken place. I t should be immediately realized that a rearrangement collision cannot take place at any distribution of energy between the different fragments. For example, in an energy channel of (P,, P2-P,) a minimal energy equal to the internal binding energy of P2-P, is necessary for the decomposition of the fragment P,-P, occurring in the initial state. This energy is then the threshold energy for a rearrangement collision.
+
8.3. CHANNEL HAMILTONIANS I N THREE-BODY SCATTERING We shall illustrate the mathematical counterpart of some of the physical concepts introduced above with the case of three-particle potential scattering. For this purpose we consider three distinct particles which interact with one another via two-body forces. Here, the statement that we are dealing with two-body force means the total potential can be written in the form
and therefore each one of the particles interacts with each other separately. I n addition, we simplify our considerations by assuming that these forces ptj . are of finite ranges pij , i.e., Vij(r), i < j = 1, 2,3, vanishes for r
<
577
8. Fundamental Concepts in Multichannel Scattering Theory
T h e total Hamiltonian of the above system is given by the Schroedinger operator (8.2)
+ +
= HO
‘12
f
‘23
v13
7
where Ho is the free Schroedinger operator
and Vij , i < j , is the potential energy operator between the ith and the j t h particle, (8.4)
, rz rs)
(vij#)(rl
9
=
Vik-i - rj) w . 1 rz 9
9
r3).
Naturally these operators act in the Hilbert space L2(R9). T h e arrangement channel (Pl , P, , P3) in which all three particles are free, is characterized by the fact that all these particles are so distant from one another as to be outside the ranges of the forces with which they interact on one another. Hence, for such distances V(rl , r, , r3)vanishes and the Hamiltonian H(1,2,3)of the system is given effectively by the operator H o , which is said to be the Hamiltonian of that particular arrangement channel. In the arrangement channel (Pl, P2-P,), the particles P, and P3 are within the range p23 of their mutual interaction forces, while PI is outside the range plz from P, , and outside the range p13 from P, . Consequently, in this case V(rl , r 2 ,r3) becomes equal to V , 3 ( r 2 - r3), and the Hamiltonian of the arrangement channel is H(1,2-3)= H, Vz3. We see that in the present situation we are dealing with a much more intricate framework then in two-body scattering: instead of a single “free” Hamiltonian, we have a different Hamiltonian for each one of the five arrangement channels. We can write the Hamiltonian of each one of the arrangement channels as the sum of the Hamiltonians of each one of the fragments in that channel. Thus, for example,
+
(8.5)
H(1.2-3)
= H(l)
+
H(2-3)
7
where, in general, H ( i j denotes the kinetic energy operator of the ith particle, (8.6)
W(i)W(P1
3
Pz > P 3 )
= Pi2/2mi$(Pl
Y
P 2 > P3),
i = 1,2,3,
and H ( 2 - 3 ) describes the system (P2-P3);we can write H(2-3) in the form (8.7)
H(2-3)
= N(2.3)
$-
‘23
3
578
V. Quantum Mechanical Scattering Theory
where H(2,3)is the kinetic energy of the fragment (P2-P,),
and V,, describes the interaction between the two particles in the fragment. When the system, in some arrangement a: and there is no interaction or negligible interaction between the fragments, then the state Y ( t ) = e-iHt Y(0) of the system is described quite well by the asymptotic state YF(t) = p
(8.9)
a
t
YF(0)
in the sense that (8.10)
lim Ij Y(t)- YF(t))I
t+Fm
= 0,
+
where “ex” stands for “in” ( t ---f - CO) or “out” ( t -+ CO), depending on whether the channel is an incoming or outgoing channel, respectively. I n two-body scattering we could formulate a general consistent timedependent scattering theory for any two self-adjoint operators H a n d H,, . However, in multichannel scattering one has to be careful and establish from the very beginning whether the channel description is indeed consistent, since it is not a priori clear that some interacting state might not have two or more distinct asymptotic states satisfying (8.10). I n order to be able to find the conditions under which such ambiguities in the asymptotic description do not occur, we have to understand more fully the structure the arrangement channel Hamiltonian Ha . 8.4. THEINTERNAL ENERGY OF FRAGMENTS IN THREE-BODY SCATTERING
Let us consider the arrangement channel ( P I ,P,-P,) in the scattering of the three particles PI , P2 , and P, interacting via the two-body potential (8.1). T h e fragment P2-P, is asymptotically in a bound state (8.1 I )
Y(2-3)(t) = exp( -iN(2-.3)t) Y(2-3)(0) E L2((wB)
of P, and P, . Let us assume that Y(2-3)(0) is represented by the wave function (8.12)
where according to 95 of Chapter 11, $’ describes the state of the center of mass of the system P2-P, , while $“(r3- r,) describes the state of the
8. Fundamental Concepts in Multichannel Scattering Theory
579
two particles in relation to one another. The two particles are in a bound state when #”(r, - r2)is an eigenvector (8.13)
Ht2n43)#”
E(2-3)
E(2-3)#“2
< O,
of the internal energy operator
where A is the Laplacian in the variable r = r3 - r 2 .Naturally, the most general Y(2-,)(0)has to be an element of the closed subspace of L2(Rs) spanned by all such vectors (8.12), corresponding to all eigenvectors $’’ of H&,”_t,) . When P2-P, is asymptotically in the state Y(2-3)(t) E?(W) and PI is asymptotically in the state Y(l)(t), then the entire system is in the state (8.15)
! F ( t ) = Y(,)(t) @
Y(2-&)
ELZ(R9).
If Y(2-,)(0) is of the form (8.12) and #” is represented by the eigenvector in (8.13), then it is easily seen that
(8.16)
H(1,2-3)
yex(0) = ( H ( l ) = (ff(1)
+ f
HEf3)
f
Hf,n43))
Htjf3)
f
E(2-3))
yex(o)
yex(0),
where H(tk;) is the kinetic energy operator of the fragment P2-Pa,
The quantity
P,-P,.
1 E(2-3)1 is called the internal energy of the fragment
Naturally, it is often desirable to consider energy channels with respect to such internal energy operators. This is possible since in general, as well as in the above particular case, these internal energy operators commute with the arrangement channel Hamiltonian, and ther’efore they are constants of motion. Let us assume that (Pl, P2-P,) is an incoming energy channel, and that the kinetic energy part $’(pZ3)of #(r2,r3) has a sharp distribution of momentum, i.e., that the incoming momentum of P2-P3 is prepared very accurately. We can choose the system of reference in such a manner that p,, m 0, and consequently
580
V. Quantum Mechanical Scattering Theory
Hence, we see that in this case the incoming particle P, must have an amount of kinetic energy at least equal to I E(,-,) I in order to be able to impart it to P,-P, in the collision process and break up that fragment, thus giving rise to a rearrangement collision. Consequently, in this case -E(2-3)represents the threshold energy for a rearrangement collision of the system {P, , P, , P3}in the given incoming energy channel. T h e reader can easily verify that the Hamiltonian H , of any of the above arrangement channels of the system {Pl , P, , P3} is the sum of a kinetic energy part HFinof the center of mass of the fragment, and of an internal energy part Hpt. For example, in the arrangement channel (P, , P,-P,),
in the arrangement channel (P, , P, , P,),
in the arrangement channel (Pl-P2-P3), H&!:-,) is the kinetic energy operator of the center of mass of the entire system, and (8.21)
int
H(1-2-3)
‘12
f- ‘23
+
‘13
It is obvious that the kinetic energy operators of all channels commute with one another since they are all functions of the compatible momentum observables of the three particles. We also know from $5 of Chapter I1 that each center-of-mass kinetic energy operator commutes with the corresponding internal energy operator. However, in general, the Hamiltonians for different arrangement channels do not commute among themselves. This is easily seen to be so in the case of (P, , P, , P3) and (Pl-P2-P,) when the channel Hamiltonians are H,, and H = Ho V,, V,, V13, respectively.
+
+
+
8.5. WAVEOPERATORS IN MULTICHANNEL SCATTERING Following the guidelines set up by the above example, we postulate that in any multichannel scattering process a unique channel Hamiltonian H, is attached to every arrangement channel 01. Let us denote by MZ) the sets of all vectorsf which are not eigenvectors of H , and for which the respective strong limits
8. Fundamental Concepts in Multichannel Scattering Theory
581
exist. By applying Lemma 1.2, we immediately conclude that M$) are closed subspaces of Z which might contain only the zero vector, as often happens in practice (see Exercises 8.1 and 8.2). Hence, we can introduce the projectors EM:) onto MF) and define the wave operators ):?L for the channel a by the formula (8.22)
The above formulas are obviously straightforward generalizations of (2.1). Hence, as mathematical objects,):?L will have for each channel 01 the same properties as Q,. I n particular are partially isometric operators with initial domains M): and final domains R$), which coincide with the ranges of these operators. T h e intertwining properties (2.3)-(2.5)
Qz)
~k)
(8.23)
eitHa - e i t H
(8.24)
Q).!
(8.25)
Q!.',
tER',
E ~ ~ (= B E) ~ ( BQ).! , Q'.!
H, f
=
HQk)f,
f
B E @, E
BH,,
and the relations (2.10) and (2.11)
Qy*Qy= EM ( d
(8.26)
Qn'd
(8.27)
f
Qy*= ER(d
I
Y
will hold in the present case. Theorem 3.1 is also still valid, so that
Moreover, the above two formulas can be taken to be the starting point of the time-independent approach to multichannel scattering. 8.6.
THEUNIQUENESS OF ASYMPTOTIC STATES AND THE
GENERAL STRUCTURE OF CHANNEL HAMILTONIANS
T a k e f E Mg) = MY) n ME) and writef,
=
Qzy.Since
582
V. Quantum Mechanical Scattering Theory
we see that e r i * m 1 f is the incoming asymptotic state of the interacting state e c i H t f ):, and the outgoing asymptotic state of the interacting state e - i H 1 f 2). T he questions now arises, however, whether is the only incoming(outgoing) asymptotic state of e c i H 1 f $ ()e r i H t f ? ) ) ; namely, there certainly cannot be another incoming asymptotic state e+ff=t g of e c i H 1 f $ ) in the same arrangement channel, but, in general, there could be some other incoming asymptotic state e - - i H @ g of e c i H t f : ) in some other arrangement channel /3 # 01. I t was pointed out earlier (in the three-body case) that we cannot expect that such a uniqueness of asymptotic states would hold for any arrangement channel Hamiltonians H , picked u p at random from the family of self-adjoint operators in A?.T o prove such a uniqueness of asymptotic states we have to restrict the families of candidates for arrangement channel Hamiltonians, by requiring that such operators obey certain conditions, which are dictated by the physical situation at hand. Extrapolating from the example considered earlier of three-body scattering, it seems sensible to require that every channel Hamiltonian H , on MY) and on Me) be the sum eriHWt
(8.30)
H,x = Htin + H P t
of a center-of-mass kinetic energy part Hfin and an internal energy part H F t ; here Hakin is the operator which represents the sum of the kinetic energies of the centers of mass of all fragments, while the operator HFt represents the sum of the internal energies of all fragments in the arrangement channel a. Judging from the wave mechanical three-body problem, it is also reasonable to postulate that HFt has a pure point spectrum and commutes with Hakin,while Hnkin has only a continuous spectrum, and that all kinetic energy operators Hfin commute among themselves. When the channel Hamiltonians have the above indicated structure we can prove that incoming and outgoing asymptotic states of any interacting state are uniquely determined by that interacting state, and consequently the description of the scattering experiment in terms of the asymptotic behavior of the system is completely unambiguous. Let us first understand better the problem of uniqueness in mathematical terms. We have seen in (8.19) that if Y(t) = e c i H t Yois an interacting state, and if Yo = L?y)Yp,then Yin(t) = e - i H o l Yin is an incoming asymptotic state of Y(t). If Y y ( t ) = exp(-iH,,t) !Pin were another incoming asymptotic state of Y(t),then we would have Yo= l2:’)Y.P. Thus, Yin(t) could be a state different from Yp(t)only if 01 # 01’ and lj7 - Q(a)Yin= f p+ ’ ) y i n , i.e., . if Yobelonged to both ranges Rf) and 0 + 0
8. Fundamental Concepts in Multichannel Scattering Theory
583
R(a’) + of ace) + and a$’), respectively. Consequently, the necessary and sufficient condition which has to be fulfilled in order to have a unique asymptotic state corresponding to an interacting state is that the different ranges RY) should have no state vectors in common. Naturally, a similar conclusion holds for the ranges RE), which are also required to have no state vectors in common. 8.7.
A THEOREM ON UNIQUENESS OF ASYMPTOTIC STATES
T h e two ranges R’: and RY‘), 01 # a‘, will have no state vectors in common if, in particular, they are orthogonal to one another; in fact, in that case R!,? and RY’) have in common only the zero vector which is not a state vector. T he following theorem shows that if the channel Hamiltonians have the earlier mentioned structure, which is reflected in (8.30), then R’: 1 RY’) for all a # a’ and RY) 1- RF‘)for all p # p’. Consequently, under those conditions every interacting state has at most one incoming and one outgoing asymptotic state.
Theorem 8.1. Suppose the arrangement-channel Hamiltonians H , , acting in the separable Hilbert space H,are of the form H,
=
Hahin-1- H,nt,
where all Htin commute among themselves, while each HFt has a pure point spectrum when restricted to MF) and it commutes with the corresponding H:in. If 01 and p are two distinct arrangement channels and Hakin- Hfinhas no point spectrum, then RY) 1 RY) and R(a) 1 R?.
Proof. It follows from the basic definitions of wave operators that (8.31)
Since H p t has a pure point spectrum on M:), the integral in the spectral decomposition of Hint can be written as a sum: (8.32)
584
V. Quantum Mechanical Scattering Theory
here the summation extends over all eigenvalues of HLnt corresponding to eigenvectors in Mg), and Ea({A}) is the projector in the eigenspace corresponding to the eigenvalue A. It should be observed that in this context it is indeed possible to sum over all eigenvalues X on account of the fact that there can be at most countably many eigenvalues of a selfadjoint operator in a separable Hilbert space. We immediately get from (8.32), by using in the process the commutativity of H:in and HEt,
where the above sum, if infinite, converges in the strong sense. Since a similar relation can be derived for EM?)eiHgt, and since HFin and HFin commute, we can write for any f E Mf) and g E MF)
wheref,, = E(@)({A})fandg,= E(=)({X}) g. Let us take any two vectorsf E MY) and g E M Y which have only a finite number of nonzero components%, andg, , respectively. Then, in view of the fact that (f 1 eiHPte-i H a ' g ) is continuous and therefore integrable in t on any finite interval, we have (8.34) dt
=
c 1 J" A'
A
T
( f A , I exp[i(H;ln
0
+ A' - ~t~~ ~ ) tgA') ] dt.
+
-
Since Hiin A' - H:in - X is a self-adjoint operator, we can apply the mean ergodic theorem (see Theorem 8.3 in Appendix 8.10 to this section) to each term in the sum of (8.34). According to this theorem
=
(fA,
kin
I EH8
kin
({A - x } ) g A )= 0,
8. Fundamental Concepts in Multichannel Scattering Theory
585
where the above inner product is zero because the projector
on the characteristic subspace of Htin- H:in corresponding to the eigenvalue h - A' has to be zero on account of the assumption that Hfin- HFin has no eigenvalues. By using (8.31), we obtain (see also Exercise 8.3)
(f
If f E M Y and g E ME) have only a finite number of nonvanishing components fA* and gA, respectively, then in view of (8.34) and (8.35), we conclude that
for all suchf and g. Since the set of all such vectorsf€ MLB) and 6 E M Y is dense in Mf) and MY), respectively, it follows that
for all f,g E A?.This establishes the assertion that RP) 1 Rf) for 01 # /!I. Q.E.D. A similar argument yields that RY) 1 Ri!) for 01 # p. T h e above theorem is valid under the restriction that Hakin- HFin should have no point spectrum when 01 # p. It should be noted that this condition is certainly satisfied if HFinand Htinare two distinct functions of the momentum operators of the n particles partaking in the scattering process. T h e reader can easily convince himself that such is the case, for example, in potential scattering.
8.8. INTERCHANNEL SCATTERING OPERATORS In multichannel scattering we are faced with the possibility that a system prepared in some given arrangement channel does not have to end up in the same arrangement channel. Hence, obviously, no single scattering operator S could describe all the transitions which the system can undergo. Instead, to any pair (p, a ) consisting of an incoming
586
V. Quantum Mechanical Scattering Theory
arrangement channel 01 and an outgoing arrangement channel to attach a different scattering operator, (8.36)
we have
= R -(8)*9,(a),
I s ,
describing the transitions between these two channels. In fact, if Y(t)has Yfi(O), then the transition amplitude the incoming state YE(t) = to the free state Yyt(t) = e-iHd Y:ut(0) is eWiHUt
(8.37)
lim (Yyt(t) I Y-(t))== t++m lim (e-iHat YY'(0)I eiHt Y-(O))
t++m
-
(Qyy o u t (0) I Y-(O))= (98' YFt(0) I 9:) Y!!yO))
=
< Y y ( o I) s,, Y'"(0)).
+
From (8.23) we easily obtain for all t E R1
s,, eiHet
(8.38)
* Q?) p
=~i_") - eiH8t -
a t =Q )!
* eiHt 92)
= eiHst
Q!)*
s,,
*
We can also derive in similar manner (8.39)
S,, EHa(B)= EHB(B)S,,
,
B
E
a',
One of the most important properties of the single-channel scattering operator is its unitarity. T o establish the presence of some analogous property for multichannel scattering we have to consider simultaneously the entire array of all interchannel scattering operators. Then, using (8.27) we obtain (8.40)
c Y
Sy*S,,, =
c (Q~'*Q!?'')("?''"Q~') Y
I n the case of single channel scattering the identity of R, and R- is a necessary and sufficient condition for the unitarity of S on M, (see Theorem 2.5). A reasonable generalization of that condition to the multichannel case would be to require that (8.41)
@ Rt) Y
= @ R?" = Y
R,
8. Fundamental Concepts in Multichannel Scattering Theory
587
where we tacitly assume that the conditions of Theorem 8.1 are satisfied, and therefore RY) 1 RY’ and Re) 1 R1_8)for a # p. T h e assumption (8.41) implies that
since QY) is a partial isometry with final domain RY). Hence, (8.40) yields
2 K3,,
(8.42)
Y
=
Qi Q, a)*
(8)
-8 E - a5 R$ ? l)
where a,, = 1 for 01 = /3 and a,, = 0 for a # /I. A similar procedure leads to the conclusion that
when (8.41) is satisfied. 8.9.
THEEXISTENCE OF ASYMPTOTIC STATES IN
PARTICLE WAVEMECHANICS
Let us consider the wave mechanical case of n particles without spin interacting via two-body forces, so that
where
Let us choose an arrangement channel a containing the fragments
FL1),...,F:n,). T h e free motion of each one of these fragments FLk’is described by the Hamiltonian
(8.45)
588
V. Quantum Mechanical Scattering Theory
where the summation is over the indices of the particles in that fragment. T h e channel Hamiltonian is (8.46)
H,
==
H:)
+ ... + H?).
Take R, to be the center-of-mass position vector of the lzth fragment, and let pk stand for the rest of the internal motion coordinates, so that (Rk, p k ) completely describes the positions of all the particles in the fragment. Then proceeding in the manner indicated in $5 of Chapter 11, we can write (8.47)
Hik) = -1/2Mk ARk
+ Ha(lc)int,
where M k is the mass of the fragment 9Lk and ) HLk)int is the Hamiltonian of the internal motion of the particles in that fragment. If +:(pk), k = I , ..., n, , represent eigenvectors of Hik)int,then e--iHat Y(O), with Y(0) given by (8.48)
k=l
yt(Rk) #:(fk),
y : )...,Vl,". EL2(R3),
is a "free state" in the channel a. Obviously, the closed linear manifold N(,) spanned by all the functions of the form (8.48) coincides with the set of all free state vectors in the channel 01.
Theorem 8.2. Suppose the functions Vij(r), i < j , i, j = 1,..., n, are Lebesgue square integrable on R3. Then the initial sets M;) of the channel wave operators Sz?) coincide with the set N'") of all free state vectors in the arrangement channel a. I n order to avoid very involved notation, we shall prove Theorem 8.2 for the special case of the arrangement channel {PI , P2-P,} in three-body scattering. I n this case
where m2, is the reduced mass of the system P2-P, and (8.50)
T he space N ( n )of free state vectors in this channel is the closed linear manifold spanned by all the functions (8.51)
d(r1 r2 r) = Ydr1) FAR) 1Cro(r23> 9
3
8. Fundamental Concepts in Multichannel Scattering Theory
589
corresponding to all Yl , Y2€L2([W3)and all eigenvectors # of
On account of Wiener's theorem (see Lemma 7.1) the family of all functions (8.51) with Yk, R = 1, 2, represented by functions which are Fourier transforms of Gaussian wave packets exp[-(p - P ~ ) ~k] ,= 1 , 2, is dense in N(a)when p1 and pz are allowed to vary over OX3. For such functions we can easily compute
Hence, according to Theorem 2.8, the strong limits
E ( ~if) the improper Riemann integral for all exist for all Y Y N
is convergent for all 4 of the above indicated form. Now, using (8.52) we get the following kind of estimate:
11 vlZepiHwt # 112
x exp[ --a,(R
-
bJ2#02(r23)dr,, dr2, dR t
< const E4 (1 + (2m,)l
2 -312 *
590
V. Quantum Mechanical Scattering Theory
Hence, we conclude that (8.53) converges, and consequently N ( a )= M(*)= M(n) Mk).
+
~
8.10. APPENDIX: VON NEUMANN’S MEANERCODIC THEOREM T h e following theorem was first derived by von Neumann [I9321 in order to prove the quasi ergodic hypothesis of classical Hamiltonian mechanics, and it was used by us to obtain (8.35).
Theorem 8.3. space Z and
Suppose A is a self-adjoint operator in the Hilbert
(8.55)
irrespective of the mode in which t , - t , tends to infinity. We note that there is a unique bounded linear operator B(t,, t,) (a Bochner integral) for which the relation (8.56)
holds for all f,g E A? (see Exercise 8.7). Hence, (8.55) states that E({O)) = tw;lim B(t, , tz). 2-
p + m
I n order to prove Theorem 8.3, we write (8.57)
( f I B(t, > t 2 ) E ) =
< f I B(t,
From (8.54) we get
3
tz)
E({O))g)
+ ( f I B(t,
3
tz)
E(R1- {OHg).
59 1
8. Fundamental Concepts in Multichannel Scattering Theory
Thus, we see from (8.57) and (8.58) that (8.55) is true if and only if (8.59)
lim
t,-t,++m
(fl
B(t, , t2)h) = 0,
h
=
E(R1 - (0))g.
We shall prove that (8.59) holds by showing that
converges to zero when t2 - t, , we get Since U,I* =
-+
+
00.
After introducing in the above integral the new variables s and t = t’ t”, we arrive at the following relations:
+
=
t” - t’
Since the integral
obviously exists, we can apply Fubini’s theorem to interchange the order of integration in t and A. Thus, we obtain, after carrying out the integration in 1,
=
s,
(
sin 3(t2 - tl)A 3(tz - t l ) A
)
11 EAh 112*
592
V. Quantum Mechanical Scattering Theory
Let us split the domain R1 of integration in three parts (-00, -71, m), and majorize the integrand by 1 on (-7, +7] and by ( * ( t z - t,)h)-, on the other two intervals. Then we arrive at the estimate (-7, +7], and (7,
+
By choosing 7 small enough, we can make the first term on the right-hand 2 any a priori given E > 0; side of the above inequality smaller than ~ / for this is due to the fact that
Then, for such a value of 7, the second term in (8.60) can be made smaller than ~ / by 2 choosing I t , - t , 1 sufficiently large. Hence lim
t2--t,++m
11 B(t, , t,)h /I
= 0.
Thus, (8.59) holds, and therefore (8.55) is ’ true.
EXERCISES 8.1. (a) How many arrangement channels can there be in the scattering process of three distinct particles ? (b) How many arrangement channels have actually been observed in the scattering of a protonp, a neutron n, and an electron e, i.e., for which channels 01 do we have Mk’ = {O}? (c) Give the conventional names of the fragments in all experimentally realizable arrangement channels in the scattering of n, p , and e interacting by means of the nuclear and the Coulomb force.
8.2. (a) Count the arrangement channels in four-particle scattering in which there are only two pairs of distinct particles. (b) How many of these channels are not empty in the scattering of two protrons and two neutrons (as far as our present experimental knowledge extends) ? 8.3, Suppose that f ( t ) is a continuous function for t and that limt++mf(t)= a exists. Show that
E
[0, +a),
593
References for Further Study
8.4. Consider multichannel scattering with a finite number of arrangement channels a. Show that the operator
commutes with e i H t and with EH(B). Remark. This operator was introduced by Jauch [1958b] as a candidate for a scattering operator in multichannel scattering. However, its inadequacy for this role is reflected in the fact that its knowledge is not sufficient to compute transition probabilities for scattering processes in which we have transitions between distinct arrangement channels (see also Exercise 8.6). 8.5. Show that if RY' 1 RY' and R!? addition (8.41) holds, then S'*S'
=
S'S'*
=
1 RE'
for a #
6, and
if in
ER.
8.6. Show that if Y J t ) and Y+(t)have the respective asymptotic states Y?(t) = e-iHal YE(0)and Yyt(t) = e--iHf16YOut + (0),then the transi= (YYt(0) I S,,Yp(O)) can be written tion amplitude (YJO)I Y+(O)) in the form
(Y+(O) I Y-(O))= (YYt(0) I QJB'*S~Qnj.'Y!n(o)) =
(YYt(0) I Ql"'*SfQ'.'!P
(O)),
provided that the conditions stipulated in the preceding exercise are satisfied. 8.7, Prove that for any family U , of unitary operators defined by (8.54) there is a unique bounded operator B(t, , t z ) which satisfies (8.57)
for all f,g E X .
References for Further Study The basic ideas of two-body time-dependent scattering theory in Hilbert space are contained in the work of Jauch [1958a]. These basic definitions have to be somewhat modified [PrugoveEki, 1971al in order to be applicable to potential scattering with longrange potentials or other more general cases. This has been pointed out by Dollard [1964, 19691, Jauch et al. [1968], and others. The time-independent approach to scattering theory in Hilbert space has been discussed on a general level by Jordan [1962a]. Examples of nonunitary S operators are given by
594
V. Q u a n t u m Mechanical Scattering T h e o r y
Kato and Kuroda [1959]. T h e Lippmann-Schwinger equations in Hilbert space are derived by Prugoverki [1969b, 1971bl. Some further papers on mathematical questions related to two-body scattering theory are by Jauch and Zinnes [1959], Birman and Krein [1962], Lirnid [1963], and Belinfante [ 19641. A subject closely related to scattering theory is the perturbation theory of linear operators. A general survey can be found in the work of Kato [1967]. A reference book with an extensive bibliography is the one by Kato [1966]. Some other papers on the subject are by Kuroda [1959b, c, 19671, Wilcox [1966], and Howland [1967]. On the subject of Green functions and eigenfunction expansions the reader is referred to the works of Titchmarsh [1962] and Berezanskii [I9681 as general reference books. T h e last one also contains an extensive list of references. Representative papers dealing with the quantum mechanical aspects of this subject are by Kodaira [1949], Povzner [1953, 19553, Green and Lanford [1960], and van Winter [1964]. Much of the treatment of potential scattering given in $56-7 has been inspired by Ikebe [1960, 19651. This reference adopts, however, a Banach space approach in solving the basic integral equation for the full Green function rather than the more restrictive Hilbert space approach. For a general physicist’s treatment of the subject the reader is referred to Newton [1966], which also contains a very extensive bibliography. A mathematically rigorous exposition of complex analysis methods in potential scattering can be found in the book by de Alfaro and Regge [1965]. T h e reader desiring a better insight in the mathematical problems of multichannel scattering theory is referred to the following sample of articles: Jauch [1958b], Hack [1959], Zhislin [1960], Jordan [1962b], Hunziker [1964, 19661, Fadeev [1965], van Winter [1965], and van Winter and Brascamp [1968].
v
Quantum Mechanical Scattering Theory
I n the preceding four chapters we have given the basic mathematical tools needed for a mathematically rigorous study of conventional nonrelativistic quantum mechanics. T h e actual applications have been limited either to obtaining very general, albeit fundamental, results of quantum mechanics or to computing the bound states of specific systems (e.g., the hydrogen atom). Within the context of nonrelativistic quantum mechanics, scattering theory deals with collision phenomena of particles moving at velocities small by comparison with the speed of light. Since scattering theory is concerned with completely or partially free particles, it is of utmost interest from the experimental point of view. From the theoretical point of view, scattering theory is still the object of intensive research for the physicist as well as the mathematician. We shall give in this chapter the most basic ideas of nonrelativistic quantum scattering theory. I n illustrating these ideas, we shall limit ourselves to two-particle scattering, which is the only case which has received a thorough treatment until now. T h e scattering of n particles, usually referred to as the n-body problem, is a field of active current research, which deserves at least one entire volume for a minimally adequate treatment. A very short introduction to this general case will be given in the last section of the present chapter. 1.
Basic Concepts in Scattering Theory of Two Particles
THEORY AND 1 .l. SCATTERING
THE
INITIAL-VALUE PROBLEM
I n studying in Chapter I1 n-particle systems in quantum mechanics, we were concerned exclusively with problems involving bound states. 391
392
V. Quantum Mechanical Scattering Theory
Such states are related to the point spectrum of the internal energy operator of the system, as described in $5, Chapter 11. We shall now turn our attention to two-particle states, which are not bound states. T h e solution of the general initial-value problem for an n-particle system is in essence contained in Axiom S3 of Chapter IV, §3-namely, if we could prepare by some means a state Yo of the system at the instant t o , and if the Hamiltonian H were given, then, in principle, we could compute the state* Y(t)= e x p [ - i H ( t
-
to)]Yo
at all later instants t . However, such a computational job, besides being in most practical cases unmanageable due to its extremely great computational complexity, is not warranted by the needs of the experimentalist who carries out experiments at the present level of technical sophistication. T h e experiments carried out at present in molecular, atomic, nuclear, and elementary particle physics which involve, from the theoretical point of view, an initial value problem are exclusively scattering experiments. Such experiments are characterized by the fact that the system undergoes a preparatory measurement at some instant t o , a determinative measurement at a later instant t , , and that for most of the time span t , - to of the experiment there is no interaction (or practically negligeable interaction) between the different constituent parts of the system. As a matter of fact, these parts interact with one another only for a very small part of the duration t , - to of the experiment. For the rest of the time they are spatially too far apart to have the very shortrange forces make themselves felt. This and some other features of scattering experiments, which will be mentioned later, make it unnecessary to solve completely the initial-value problem in order to relate the theoretical predictions to the available experimental data. It is clear from the above remarks that for the greater part of a scattering experiment, the different constituent parts of the physical system under observation move independently from one another, i.e., they are “almost free.” I n order to elucidate this intuitive concept of “almost free” motion in a mathematically clear and concise form, it is desirable to consider this concept first in the framework of classical mechanics, which readily appeals to intuition.
* In this chapter we adopt a system of units in which Fz
=
1.
1 . Basic Concepts in Scattering Theory of Two Particles
393
1.2. ASYMPTOTIC STATESI N CLASSICAL MECHANICS Let us consider a two-particle classical system. I n the Newtonian formalism of classical mechanics, the state of such a system is given by a vector-valued function
T h e trajectory of the kth particle, k = 1, 2, can be described by a threedimensional vector-valued function rk(t),which geometrically is a curve in three dimensions. If the two particles of the system undergo the kind of motion which was depicted earlier as typical in a scattering experiment, then the trajectories rk(t),k = 1, 2, should have as characteristic the main qualitative features of the curves in Fig. 4: at the beginning t w to
FIG. 4. Asymptotic states in classical mechanics.
of the experiment and for a relatively long time afterwards the motion of each particle should be “almost free,” i.e., “almost” along a straight line rF(t) at a uniform speed vf, k = 1, 2; then, during a relatively short period, while the particles are interacting with one another, the motion is, in general, very complex; finally, at the end, t w t, , of the experiment and for some time before that, the motion of the particles should be again along straight lines rBut(t)and at “uniform speed” viUt. Now, in classical mechanics a particle is said to be in free motion if its state is of the form r(t) = ro
+ vt,
so that r ( t ) = 0. Hence, at times t w t o , we expect that
V. Quantum Mechanical Scattering Theory
394
and similarly for t (1.3)
M
t,
rk(t)m
ryt + vFtt,
k
=
1,2.
The vectors r? and v p characterize a free state of the kth particle, k = 1, 2, which is called the incoming asymptotic state. Similarly, the free
state characterized by riutand vPt is called the outgoing asymptotic state of the kth particle. It is not a priori clear in what precise sense the approximation signs in (1.2) and (1.3) should be understood. One obvious choice is to take them to mean that (1.4)
I rk(t)- (rp + vpt)~w 0,
I i.k(t)
- vf
I
w 0,
t w to, t
M to,
where I * I above denotes the length of three-dimensional vectors. Since one can reasonably expect that the situation does not intrinsically change if we increase t , - t o , and, moreover, the above approximations might hold with increasing accuracy when to --+ - 00, it seems natural to give to (1.4) the following precise meaning:
lim
t-1-m
1 r k ( t )- Vkin I = 0.
Similar considerations lead to the following corresponding requirements:
The conditions (1.5) and (1.6) define in a precise way the asymptotic states of a single particle. Naturally, the asymptotic states xin(t) and xout(t) of the entire system is the aggregate of the asymptotic states of its two constituent particles. Incoming asymptotic states defined by (1.5) can be thought of as being states in which the particles of the system would be at all times if there were no interaction between the two particles. Similarly, the outgoing states would correspond to the fictitious case of particles which have interacted in the “infinite past,” and afterwards move independently.
395
1. Basic Concepts in Scattering Theory of Two Particles
We note that according to the definitions in (1.5) and (1.6), if asymptotic states exist at all for a given classical physical process, then they are unique for each state, since vf and viUtare uniquely determined by the second of the respective relations in (1.5) and in (1.6), and then rin and riutare uniquely determined by the first relations in (1.5) and (1.6), respectively. It is, however, natural to ask whether asymptotic states in the above sense exist for every classical physical process of importance. The answer to the above question is negative. I n the case of longrange forces, i.e., forces which are of significant intensity even at arbitrarily large separations between particles, asymptotic states in the above sense do not exist. A notable example of such a force is the Coulomb force (see Exercises 1.1 and 1.2). The intuitive physical explanation of this phenomenon is that long-range forces never sufficiently loosen their grip on the particles to allow them to travel in a free manner when they are at large distances from one another. A generalization of the above concept of asymptotic state is obviously necessary to deal with this situation. However, the already introduced concept is of sufficient generality to cope with most of the practically important cases. Even the Coulomb potential is in practice most often “screened,” i.e., instead of having to deal with the Coulomb potential V ( r ) l / r , we are faced with a potential which at large distances decreases much faster than l/r, and for which asymptotic states, in the above sense, exist.
-
STATESAND SCATTERING STATES 1.3. ASYMPTOTIC IN THE SCHROEDINGER PICTURE OF QUANTUM MECHANICS It is straightforward to adapt the concept of asymptotic states defined in the classical two-particle case by (1.5) and (1.6) to the case of quantum mechanics. We can achieve this by stating that in the Schroedinger picture, the “free states” Yin(t) and Yout(t) are, respectively, the asymptotic incoming and outgoing states of a two-particle system in state Y(t)if and only if lim 11 Y(t)- Y’”(t)lI= 0,
t+-m
In order to define precisely the meaning of the concept of a “free state” we must assume that in addition to the “total” Hamiltonian W 2 ) of the two-particle quantum mechanical system we have at our disposal a “free” Hamiltonian Hi2),which would describe a system consisting of two particles with the same physical characteristics (mass, intrinsic
396
V. Quantum Mechanical Scattering Theory
spin, etc.), but which do not interact among themselves. For instance, in the case where Hc2)is given essentially” by
then Hi2) is taken to be essentially 1 2m1
A -1
1 2%
- -A2
3
where the domains of definition of the above differential operators are adequately defined,t as was done in 97 of Chapter IV. However, if both particles move in an external force field, given by a potential Vext(rl , r2 , t), then instead of (1.8) we have
and instead of (1.9) we have
This will be the case when, for example, the system under consideration consists of two spinless charged particles moving in the force field generated by a much heavier particle (such as the heavy nucleus of some atom). Since the heavier particle is practically unaffected by the motion of the lighter particles, such a problem can be treated fairly accurately as a two-body problem in an external field rather than a three-body problem. Consider now the general case when the “free” Hamiltonian Hh2) and the total Hamiltonian W 2 )are given by the self-adjoint operators associated with the system. We say acting in the Hilbert space 3‘(2) that a nonzero vector-valued function Y(f)(t),t E R1, is a free state in the Schroedinger picture if and only if it satisfies the relation Y q t ) = exp( -iH(y’t)
Y‘f’(O), t E R1,
and if Y(r)(O)is orthogonal to the closed linear subspace 3‘;;)of bound of the system with Hamiltonian Hi’). In the absence of states in i?F2)
* See the discussion in $7 of Chapter IV on the essential self-adjointness of the Schroedinger operator. t We note that in the preceding two operators, as well as throughout this chapter, we have adopted a system of units in which fi = 1.
397
1. Basic Concepts in Scattering Theory of Two Particles
external forces both Hi" and its internal energy part Hit), obtained after subtraction from Hi2) of the center of mass motion, are operators = (0). For example, with a pure continuous spectrum so that 2;;) this is the case when Hi2)is given by (1.9), and therefore Hit) is essentially -(1/2m0)A acting on wave functions +(R, r), where A is the Laplacian containing derivatives with respect to the relative position coordinates of r and m, = m,m,(m, mz)-l. I n reality, the particles interact, and any Schroedinger picture state Y(t)in which they can be found, which will be called occasionally an interacting state, has to satisfy the relation
+
Y(t)= exp(--iHW) Y(O),
t E R1.
Due to the unitarity of the operator exp(-2P2)t) we have (1.10) 11 ~ ( t-) ~ " " ( t ) l l= 11 exp(--iH(2)t) Y(O) - exp(--i@)t) Ul""(0)ll
11 Y(O)- exp(iH(%) exp(--i@)t) Y ~ " ( O ) ~ I = 11 exp(i@t) exp(--iH(2)t) Y(O) - Y""(O)\~, =
where "ex" is a symbol which will be frequently used in the future, and which stands for "in" and i'out." In view of the preceding notation, we see that (1.7) implies that
or, equivalently, (1.12)
lim
t+Tm
11 ~ ~ " (0 exp(iH?)t) ) exp(--iH(2)t) Y ( O )= I~ 0,
where the interpretation of the above notation is that one should let t --f -co when "ex" stands for "in," and t -+ +co when "ex" stands for "out." Since, conversely, (1.11) or (1.12) implies that (1.7) is true, we can state the following theorem.
Theorem 1.1. T h e state Y(t)= exp(-iH(%) coming and outgoing asymptotic states Yex(t):
Y(0) possesses in-
(a) if and only if the strong limits (1.13)
Yex= s-lim exp(iHt)t) exp(--iH%) Y(o), t+Tm
exist and belong to
0;)&:;
398
V. Quantum Mechanical Scattering Theory
E 2P @ (b) if and only if there are vectors Yex(0) such that
%At),ex = in, out,
Y(O) = s-lim exp(iH(2)t)exp( - i ~ p ’ t )~ “ ( 0 ) .
(1.14)
t+Tm
An interacting state Y(t)which has an incoming as well as an outgoing asymptotic state is called a scattering state. We recall from relation (3.43) of Chapter IV that
represents in the interaction picture the same state which is represented in the Schroedinger picture by Y(t)= exp(--iH(2)t) Y(0).Thus, pex, ex = in, out, can be thought of as being the asymptotic states of p(t) in the interaction picture.
1.4.
WAVE
MBLLER
OPERATORS
Denote by RY) and R!? the set of all vectors which are the limits (1.14) for some Yex(0) E J f ( 2@ ) Jfi2)when t -+ - 00 and t .--t 00, respectively, i.e.,
+
Rf’ (1.15)
= {f+ : f+ =
~ ‘ 2= ) {f-
~<~i~Q‘”(t)f,f~S OXP’},
:j- = s-lim d 2 ) ( t ) f f , E t++m
s 0~ f ) } ,
~ ( ~ ’ (=t e) x p ( i ~ ( ~ exp(-iHp)t). )t)
We see from (1.14) that if a nonzero vector g belongs to either RY) or R?), then exp(--iH%)g does not represent a bound state, since it is intuitively obvious that a bound state cannot be asymptotically free. Hence, it seems natural to assume that in any realistic quantum mechanical two-particle theory
Rt) = RL2) = %?)A, where 2i2’ is the space of bound states. We shall see later that the conclusion of this heuristic argument is confirmed when dealing with potentials which usually occur in practice.
Lemma 1.1.
If A(t) is a uniformly bounded family of operators,
It A(t)tl < c,
--co
< t < +a,
1. Basic Concepts in Scattering Theory of Two Particles
399
then the set M of all vectors f E & for which s-limA(t)f, t+tO
-a < to < +a,
exists is a closed linear subspace of
&.
Proof. Since the existence of the strong limits of A(t)f and A(t)g for t -+ to implies the existence of the strong limits of A(t)(af bg) = aA(t)f + b A ( t ) g ,it follows that M is a linear subspace of x. Let f be a vector in &? which is the strong limit of a sequence f, ,f , ,... of vectors from M. If
+
gn = s-lim A ( t ) f n , t+to
then we can write
for all t sufficiently close” to to ,we see by glancing at the right-hand side of (1.16) that
II A(t1)f
- A(t2)fII
<
f
for all such t , , t , which are sufficiently close to t o . Hence, the strong limit of A ( t ) f for t -+ to exists, and thereforefE M. Q.E.D. The operator function L P ( t ) defined in (1.15) assumes as value a unitary operator for every value o f t E W, and consequently it is uniformly bounded: -a < t < +a. 11 rn(2’(t)lI = 1,
* If to =
&a,sufficiently “close” to to means sufficiently large.
400
V. Quantum Mechanical Scattering Theory
Thus, we can apply Lemma 1.1 to conclude that the families MT’ of ) Afb? for which the strong limits Q c z ) ( t ) f for all vectors f~ X c z0 t -+ -a and t -+ +a,respectively, exist are closed linear subspaces 02;;). Hence, the family of Mf)
=
MF) n MF)
of all vectors f for which both limits, s-lim W ) ( t ) f = j+ , t+Tm
exist is a closed linear subspace of X ( 2 ) . Let EM?’be the projectors onto MY’, (1.17)
E M y % = MF’.
Then, according to the above definition of MY), the limits
exist for all vectors f E X , because E $ ’ ~ EMF). Then the operators (1.18)
exist and are defined on the entire Hilbert space X ( z T ) .h e operator L??) is called the incoming M0ller wave operator and QF) is called the outgoing M0ller wave operator. We shall refer to these two operators as the ‘Lin’’ and “out” wave operators. 1.5. THESCATTERING OPERATOR T h e scattering operator (or Soperator) S for the problem at hand is defined in terms of the wave operators
Qz),
Qz)
Since the wave operators are strong limits of uniformly bounded operators Qcz’(t), /I Q(z’(t)\\ 1, they are bounded. Furthermore, we have seen that Q, are defined everywhere in 2. Hence, the adjoints QF)*exist and are bounded (see Chapter 111, Theorem 2.7). Thus, S2) is a bounded linear operator, defined on the entire Hilbert space iFZ).
<
1. Basic Concepts in Scattering Theory of Two Particles
401
The importance of the scattering operator lies in the fact that many quantities which are readily measurable by scattering experiments are easy to compute if we know S ( z ) . T o provide an illustration, assume that a scattering state Y-(t) of the system has been prepared at the beginning of the scattering experiment, and that at the instant to of the preparatory measurement Y(t)was essentially a free state YF(t),
Furthermore, assume that a determinative measurement at an instant t , , long after the two particles of the system have interacted, can determine the value of the observable Etl of the form
which is represented by the projector on the scattering state Y-(t) at t = t , . Then, if Y+(t) is normalized, themean value of Etl is given by the expression (1.20)
PY++Y-
=
W+WI Etl Y+(t1)>= IV+(t1) I y--(h))IZ*
We recall from the discussion at the end of $1 in Chapter IV that the above expression is usually referred to as the transition probability from Y+(t)to Y-(t),due to the usual assumption that if the outcome of the determinative measurement of Etl is X = 1, the system is left afterwards in the state Y-(t). We note that the expression in (1.20) is actually independent of t , , (1.21)
due to the fact that the operator exp(-zW2)t) is unitary. Let Yout(t) be the outgoing asymptotic state of Y-(t): lim
t++m
11 ~ - ( t) ~ ? ~ ~ (= t >0,l l
Since according to (1.14)
Y-out( t ) = e x p ( - - i ~ t ’ t )YP(o>.
402
V. Quantum Mechanical Scattering Theory
we arrive at the following expression for the transition probability:
This expression is very convenient and helpful in practice, since it involves free states which in the typical scattering experiment are computationally easy to derive from the available experimental data.
SCATTERING CROSSSECTION 1.6. THEDIFFERENTIAL In practice, the typical two-particle scattering experiment is of the kind depicted in Figure 5 : a beam of particles 6, impinges on a target ector
INCIDENT B E A M COLLIMATING
'* Yo
FIG. 5. Scattering of a beam of particles on a target.
consisting of particles 6,. Assuming that the particles in the beam do not interact with one another, and that each particle 6, in the beam interacts with only one particle 6, in the target, the experiment can be viewed as consisting of a large ensemble of independent scattering experiments of two-particle systems" 6 = @I 9 6 2 > .
* The two mentioned conditions can be satisfied by taking a "weak" beam of particles (i.e., a beam with few particles per unit volume), and taking for the target a slab of material which is sufficiently thin to eliminate the possibility of a particle 6,scattering from more than one particle in the target.
403
I . Basic Concepts in Scattering Theory of Two Particles
The above experimental setup is ideally suited to determine the percentage of the particles in the beam which can be found in a given volume of space, occupied by a detector, after they have been scattered by the target. I n any scattering experiment the detector is placed as depicted in Fig. 5, i.e., sufficiently far from the target so that it never interferes with the scattering process. Hence, the experiment obviously provides information about the probability that a particle of the beam, coming towards the target from the direction 0 <40
wo = (+0,4)),
< 277,
0
< 80 < -,
will be scattered within the solid angle dw around the direction = (4, el,
(1.23)
o < + G 2-,
o G e < T.
If N denotes the number of two-particle scatterings per unit time, and dE] of incident if the solid angle dw and the energy range [ E , E particles are sufficiently small, then this probability can be written in the form N n ( E , w,, , w ) dE dw, since it is obviously proportional to N as long as the two earlier mentioned conditions on beam and target are satisfied. It can be expected that T(E,wo , w ) is dependent on the energies E and E’ of the particles of the system 6 = (6,, S,} before and after collision. However, since energy is conserved, we must have E = El, so that the dependence on E‘ does not have to be displayed. I t has become customary to orient the frame of reference so that y50 = Oo = 0, and to call the quantities
+
(1.24)
o(E, w )
=
( N / J oT(E, ) 0,w ) ,
a(E) =
1
a(E, w ) dw
QS
(where Jo is the incident JEux, i.e., the number of incident particles per unit time and unit beam cross section) the dzjferential cross section and the total cross section, respectively. The notation du/dw instead of a(E, w ) is often encountered in literature. 1.7. THETRANSITION OPERATOR In potential scattering W 2 and ) Hi2’ are given by (I .8) and (1.9), and one can easily relate o(E, w ) to the S operator. T o establish this relation, it is computationally advantageous to express S(2)in terms of a new operator, (1.25)
S‘2’
=
1 - 2Ti77‘2’
where the operator F 2 )is referred to in physical literature as the transition operator or T operator. I t owes its name to the fact that in the case when
404
V. Quantum Mechanical Scattering Theory
there is no interaction present, the system stays in the same free state (i.e., there are no “transitions”), and the operator F 2 )is zero. We easily establish that T ( 2 )= 0 when there is no interaction by noting that the “absence of interaction” means that = Hi2) and therefore LF2)(t)= 1 , i.e., Qf)
~
fp= s(2)= 1.
We have seen already in 97 of Chapter I1 that in the absence of external forces the Schroedinger operator Hi2’ of a two-particle system can be written in the form H p
=
H p
+z p
where HA2)is the center-of-mass energy operator and Hi2)is the internal energy operator. T h e Hilbert space A?(2)= L2([w6)of wave functions #(R, r) is obviously the tensor product ( 1.26)
x(2) =
#t’@
#(I)
of the Hilbert spaces A?‘,“ = L2([w3)and &(I) = L2([w3)of wavefunctions $,(R) and #(r), respectively, where R and r are the center of mass variables given in (7.2) of Chapter 11. It becomes evident from looking at the differential operators (1.27)
-
and (7.3) in Chapter I1 that HA’) = HA’) @ 1 and H12) = 1 @Hi1), where HL1) and Hi1) act in XL1)and &?(I), respectively. It is natural to require that, in general and when no external forces are present, the free and total Hamiltonians can be written in the form
Since the operators HL2’ and HAY’, as well as HA2) and Hj2’, obviously commute, we can use Theorem 2.4 of Chapter IV to infer that exp( -i~,’2’t) = exp(--iH,(2)t) exp( - i ~ $ ) t ) , exp(iH‘”t) = e x p ( i ~ f ) t )exp(iHi2)t).
1. Basic Concepts in Scattering Theory of Two Particles
405
The above two relations are derivable directly from (2.13) in Chapter I V by noting that in the above case all operators are bounded and defined on the entire Hilbert space A?(2). From (1.28) we get (1.29)
d ” ( t ) = exp(iH(2)t)exp( -iHf’t)
Inserting the above expression for L P ( t ) in (1.18) and (1.19), we easily arrive at the following result.
Theorem 1.2. If the free and total Hamiltonians Hi2) and H ( 2 )of a two-body system can be decomposed into sums (1.28) of commuting Hamiltonians for center of mass and internal motion, then
where QF) and S1) are operators in the internal-motion Hilbert space of the tensor product A?(2)= 26)8 A?(1). The above theorem shows that in case of two-body potential scattering in which no external forces are involved, we can work in the Hilbert space 2 ( l ) which is related to the internal motion of the system. If we work in momentum space, we can introduce the relative momentum variables p, related to the relative position vector r. We might hope that operator could be written as an integral operator in which, the S1) in case of spinless particles, would be of the form (1.30)
However, we shall see in the $2 that S1) commutes with the free Hamiltonian Hi;). Lemma 1.2 shows that, on account of this fact, the representation (1.30) is not feasible.
Lemma 1.2,
Let K be a bounded integral operator on L2(R3)
406
V. Quantum Mechanical Scattering Theory
with a kernel K(p, p’) which is Borel measurable in Rs. If K commutes with the self-adjoint operator Hi:),
then K is the zero operator.
Proof. If f(p) is a function of compact support A , then E p ( A ) f =
f E gH;;) , where Ep(d) is the spectral measure obtained as the product
(1.13) in Chapter IV of the spectralmeasuresof the three momentum component observables: (EP(4g)- (P)
= XA(P)g’(P)
T h e operator HA:)Ep(d)is obviously bounded, and commutes with K ,
=0
(almost everywhere).
Since the above expression has to vanish almost everywhere for all f(p) of compact support, we have =0
(almost everywhere in R3 x R3 = Re).
Now, the set of points in R6 for which I p I = I p’ I (called the energy shell) is of Lebesgue measure zero. Hence, we deduce that K(p,p’) vanishes almost everywhere in R6. Iff (p) and g(p), f,g E L ~ ( R ~are ) , bounded functions of compact support, then g*(p) K(p, p’)f (p’) is certainly integrable on R6 since K(p, p’) is Borel measurable and vanishes almost everywhere. From the relation
we get by applying Fubini’s theorem
(g I K f )
=
j
R
dPL?*(P)
j
R
dP’K(P9 P’MP’) = 03
for all f,g E WE(R3), Since W 3 W ) is dense in L2((a3),we get Kf = 0 for all f E VE(R3). Thus, K is a bounded linear operator which is zero on a dense subset V@3) of L2((a3),and therefore K = 0. Q.E.D.
407
1. Basic Concepts in Scattering Theory of Two Particles
0
Let
us
introduce
spherical
< +GO,0 < 8 < T , 0 < 4
coordinates
(p, w)
=
( p , 8, C$),
< 27r, in the space R3 of the variables
p. We shall see later in this chapter that the T ( l )operator can be written as an integral operator with respect to the variable w, (1.31)
over the set Q,, defined in (7.10) of Chapter 11, which is essentially isomorphic to the unit sphere in three dimensions. T h e function T ( l ) ( p ;w, w’) is uniquely determined by the operator T ( l ) (see Exercise 1.6) if we restrict ourselves to continuous kernels for the integral operator in (1.3 1). 1.8. THET-MATRIXFORMULA FOR THE DIFFERENTIAL CROSSSECTION T h e differential cross section o(E, w) can be computed from the p ; w, w’) which occurs in the integral representation (1.31) function F)( of the T operator. I n order to arrive at an expression relating o(E, w ) and T ( l ) ( p w, ; w ’ ) , we have to express in theoretical terms those essential features of a scattering experiment which enter in the definition of the differential cross section. I n any such scattering experiment, a beam of particles scatters from the particles of a target. T h e experimental setup is such that the relative momentum p of the incoming particles is well defined. Let us introduce spherical coordinates ( p , 8,+) in the p space, and suppose that the incoming particles are prepared in a scattering state Y(t),which is such that p lies within the set (1.32)
Denote by Yin(t) the incoming asymptotic state of Y(t).Suppose that Yin(0)is represented by the function (see Exercise 1.5) (1.33)
$in(p,P) = 4@) $dPL
of the center of mass momentum P and relative momentum p. Let us denote by E‘P’(B)the spectral measure (1.34)
(~(pYw)(P, P)
= XB(P)
$w,PI-
408
V. Quantum Mechanical Scattering Theory
It can be shown (see Exercise 1.8) that for any Bore1 set B
E
g3
Since the support of the function +(P, p; t ) representing Y(t)is within R3 x C(O),we deduce from the above relation that
Hence, $o(p) vanishes almost everywhere outside C(O). Denote by Yout(t)the outgoing asymptotic state of Y(t).It is easy to show (see Exercise 1.4) that ( I .36)
Yout(o)= S'2'Yi"(0) = ( 1 x S'")Yin(o).
Hence, by applying the same type of argument which led to (1.34) to Y o u t ( t ) , we obtain
Consequently, the probability P(lmo+ Iu0,)of finding that the relative momentum p of the system, a long time t after the scattering had taken place, is within the cone
c(+) = {P: (0, c) EL,^+,
L ~=,[e,', eo' + 4~x
[+ot,
+or
+4
1
is given by (see also Exercise 1.12) (1.37)
l(S(1'#o)(P)12dP.
We shall see in the next section that S1) commutes with EHA?(B) (see Theorem 2.3), where (E
(B )#)-
(PI
=
X{pa,2m,EB)(P) $(PI.
409
1 . Basic Concepts in Scattering Theory of Two Particles
It follows that for the complement B, E 39, of the set {p2/2m:p we have EHdt’(Bl) s(l’#o = S‘l’EHd?(B,) #o
=
E
I,,},
0
since E%’(B1) t,!Jo = 0 because $,(p) vanishes outside e(O).This means vanishes when p2/2m$ B, , i.e., when 1 p I 4 Ipo. Therethat (,S1)t,!J0)(p) fore, if the integral in (1.37) is expressed in spherical coordinates, the integration in p extends in effect only over the set Ipo. Thus, we can write
Let us consider the case when the direction of relative motion of the two particles in the system has changed as a result of intersection, i.e., n IUo, = 0 . If we use the definition (1.25) of when wo # wo’ and IUo the T operator to write S“’#, = #o - 27TiT‘1’#,
and recall that result
go(p,w ) vanishes when w is outside Iwo, we arrive at the
Hence, we get for the probability density P(wo‘)(per unit solid angle) of scattering in the direction wo’
provided that ( T ( l ) # o ) ( pw, ’ ) is continuous in w’ in some neighborhood its integral of wet. Substituting in the above relation for the operator F1) representation (1.31), we obtain
410
V. Quantum Mechanical Scattering Theory
where the integrations in o1and w, extend over the unit sphere in three dimensions. The request that ( T(ljh0)(9, 0’) be continuous can be met by demanding that T ‘ l ) ( p ;w’, w) be continuous in w‘. At this stage one must remember that the expression (1.41) is the probability density of scattering of two particles in the relative direction wo‘ with respect to one another. We have already indicated, however, that in a scattering experiment a beam of particles is being scattered on a target consisting of a large number of particles. Thus, in a scattering experiment, a great many two-body scatterings are observed simultaneously. These scatterings between particles in the beam and particles in the target are taking place at different locations with respect to the laboratory frame of reference. Hence, in order to arrive at an expression for the differential cross section, an averaging of P(wo’) over all these two-body scatterings at different ‘locations has to be carried out, If a particle 6,” in the target is at a location ro in relation to another particle G,’, while 6,’ is in position r with respect to a particle G1 in is at the location r ro in relation to the incident the beam, then 6,” particle 6,. Since, with the exception of location, all the other variables are the same, we conclude that if of 6,’ and 6,”
+
*m represents Yin(0)for {G1, G,’}, the wave function
represents Yin(0) for { G I , G,”}. By taking Fourier transforms, for {GI, G2”}we easily get &,(PI
J~(P),
= ex~(--~ro)
where J0(p) refers to {G1, G,’}, and JrJp) refers to {GI, G,”}. Hence, the probability Pro(wo’) for {Gl, S,”} is (1.42) Pro(w,,’) = 477’
J”
J”
d p p 2 dwl d o z T*(p;w,,’, 1%
wl)
instead of (1.41). We shall now make the assumption that the scatterers in the target are uniformly distributed, so that No scatterers interact with the incident particles per unit target volume and unit time. Then the average number
1. Basic Concepts in Scattering Theory of Two Particles
411
of particles scattered* per unit time and target volume in the direction is
w,'
(1.43)
4-%7= No J V P,(W,')
dro
1,dr, 1dppa 1
= 4?raN,
dw,
x Tc')(p;WO',
w8)
dw, exp[iro(p, - p,)] F1)* ( p ; w',
$O*(Pl) $O(o(pBI)IP1-P~-o
wl)
9
assuming that the scatterers in the target are sufficiently dense that we may, with sufficient accuracy, replace the sum over the individual scatterers with an integral over the volume V of the target. We shall evaluate the integral (1.43)under the assumption that the target is a slab of thickness 6 , which extends to infinity in the directions x, and yo (see Fig. 5). Setting k = pa - p1 and reversing the orders of integration in (1.43), we obtain (1.44)
Y(W0))
= 47r2No
I dw, I dr, j d p dw8p2eikroF(w,';p,
w , ,ma),
where we have introduced the function (1.45)
WJ;; P , w1 ,w2) =
T"'*(p; wgl,
4T ' Y p ; w;, 4$ O * ( P , 4 JO(P'
WZ).
Since I p1 I = I p2 I = p, only two of the components of k are independent. We can substitute for w 2 the variable w which represents the spherical coordinates 0 and of p2 in a coordinate system with its z axis in the direction of p1 . An easy computation yields
+
By substituting for o
p2 dp dw
= p 2 sin ll
=
k,
=p
sin ll cos+,
k,
=p
sin ll sin 4.
(8,+) the variables k, and
k, , we obtain
dp dll d+ = p2 sin 0 a@, a(e2 ' +') dp dk, dk, k,) 9
=
1 dp dk, dk, . cos e
* In practice, the scatterers in the target are approximately at rest with respect to the laboratory frame, so that the relative scattering direction w i computed in the center-ofma88 frame is very often almost equal to the scattering angle measured in the laboratory reference frame.
412
V. Quantum Mechanical Scattering Theory
Using the properties of Fourier transforms applied to the variables
k, , k, and their conjugates xo and yo , we obtain dxo dY0 = 4n2
J dk, dk, eirokjdP , B1F ( w o ’ ; P ,
j F b o ’ ;P ,
w12
w2)
cos 0
1.0
I
k,=ky=O
w1
, w2)
P2 dP.
Since obviously dp dw, = dp dw, the above relation can be used in = computing the integral in (1.44). When k, = k, = 0, we have and O1 = 8, or = and 8, = 7~ - 0 2 . Since F(wo’;p , wl, w,) vanishes when w, or w2 are outside Iwo, it follows that when Iwois sufficiently small only the first alternative = and = 8, yields a nonzero result. This means that k, = 0, 8 = 0, and the integration in zo in (1.44) can be immediately performed:
+, +,
+, +,
+,
&C
We note that No6 = N is the total number of scatterings per unit time and target area, i.e., the incident flux Jo . Let us choose dp, , Ad,, and A+o very small, and the direction of the incident beam along the z axis, so that 8, = +o = 0. Then, since
we get by the mean-value theorem of the integral calculus
We note that the application of the mean-value theorem is warranted ; w ) is continuous on account of the implict assumption that T ( l ) ( p w’, in some neighborhood of w = (0,O). Since v(wo’)is the number of scatterings in the direction wo’ per unit time, we have” according to the definition of a(E, wo) v(-w0‘) a(E, wo’) = ~.
* The minus sign is due to the choice 8,
Jo
0. Since the origin of the coordinate system is at the target, this choice means that the beam travels in the -e, direction. However, in physics the convention is that in the formula for o(E, w ) angles should be measured from the direction of the incident beam, i.e., in a coordinate system with its zoaxis pointing in the -e, direction (see Fig. 5). = #o =
1. Basic Concepts in Scattering Theory of Two Particles
413
Thus, we have obtained the expression (1.47)
-ur, 0)(2
for the differential scattering cross section under the assumption that T ( l ) ( p w; l ' , wl) is continuous in wl' and w1 when wl' is in a neighborhood of w' and w1 in a neighborhood of (0,O).
EXERCISES
+
1.1. Let rEx vIXt,R = 1, 2, be the asymptotic states for the state rk(t),k = 1 , 2, of a classical two-particle system. In the center of mass coordinates (7.2) of Chapter I1 the state of the system is given by R(t) and r(t), where R(t) = R, V,t describes the uniform motion of the 1 - TiX, center of mass of the system. Show that if rex = rex vex = vFx - vgx, then I r(t) - (rex vex t)l ---t 0, (l/t) r(t) -+vex and I r(t)l - t I vex I rex- vex/Ivex I when t -+ ~ C Qrespectively. ,
+
+
--f
1.2. Prove that a system of two particles interacting via a potential V ( r ) = l / r (gravitational or Coulomb potential) does not possess any asymptotic states satisfying (1.5) and (1.6).
1.3, Show that the adjoint K* of a bounded integral operator K ,
defined on L2(Rn), is an integral operator of the form
1.4. Prove that if Yex(t)E M,(ex = in, out) are the asymptotic states of Y(t),and S is the scattering operator, then !Pout(0)= S!Pin(0).
1.5. Show that the formula (1.47) for the scattering cross section stays valid in the general case when Yin(0)is represented by an'arbitrary wave function p ( P , p ) , which is not necessarily in the form of the product occurring in (13 ) . 1.6, Prove that if T ( l ) ( p w ; , w ' ) and 7'A1)(p;w , w ' ) are two kernels satisfying (1.31) for a given operator P ) ,then T f ) ( p ;w , w ' ) = T ( l ) ( pw; , w ' ) almost everywhere with respect to the measure p!') x p p x p p on R1 x sZp x 12,.
414
V, Quantum Mechanical Scattering Theory
1.7. Suppose Y(t) and Y,,(t),t E W,are vector-valued functions assuming values in the Hilbert space %, and that there is a constant C such that I/ Y(t)lI C and 11 Y,,(t)ll C for all t E W.Let A be a bounded 11 Y(t)- Y,,(t)ll= 0 implies that operator on 2. Show that liml+FFm
<
<
1.8.
Use the result of Exercise 1.7 to prove that ‘t
lim (Y(t)I E(p)(B)Y(t)>= (Yex(0) I E(D)(B)Yex(O)), w
+
where (HA2)$)“(P, p) = (P2/2M0 p2/2m,) $(P, p), with M,, and mo denoting, respectively, the total and the reduced mass of the system. 1.9.
Consider the case of one-particle wave mechanics, when
H, 2 (1/2m)A. Using the well-known relation (exp(-iH,t)+)“(k)
=
exp[ -i(k2/2m)t] $(k)
derive that, for functions $(r) which are Lebesgue integrable as well as square integrable, (exp(--iH,,t)+)(r)
=
J
( m / 2 ~ i t ) ~ / ~exp[--i(m/2t)(r
-
r’)2]+(r’)dr‘
R3
almost everywhere in R3. 1.10.
Verify that the operators Uil) and U\2’, defined for real t # 0
by ( ut(l)+)(r) = (m/it)3/2 exp[-im(r2/2t)l
(U,(2)+)(r)= exp[-im(r2/2t)l
$(rnr/t),
+(TI,
are unitary. Use the result of the preceding exercise to show that for H , 2 -(1/2m)h p f O t -
-
u, u, . (1)
(2)
1.11. Use the result of Exercise 1.10 to prove that the operator families eciHol and U f ’ defined there satisfy Iim ll(e-iHOt - U,(1))+ 11
t+Fm
for all,$ EL~(W).
=
o
415
2. General Time-Dependent Two-Body Scattering Theory
1.12.
Consider the cones
c(+) = {(r, e,+): y 2 o , ~ G , e G 0,
+ ~ 84, , G + G + A+) +o
and C(-) = {r: -r E C+}in R3. Use the results of Exercises 1.7 and 1.1 1 to prove that for H, 2 -(1/2m)A
Remark. This result indicates that asymptotically in the future the probability of finding the particle spactially located in the cone C(+) is equal to the probability of finding its momentum within F+).
2. General Time-Dependent Two-Body Scattering Theory 2.1.
THEINTERTWINING PROPERTY OF WAVEOPERATORS
In $1 we defined the Moller wave operators SZ, by means of strong limits of SZ(t), Q* =
(2.11
:diz Q(t)
~ ( t=) eiHt
EM+
e-iHot
,
H
=
H*,
Ho = Ho*,
in the time parameter t . In this definition, the closed linear subspaces M, of X are spanned by all vectorsf which are such that eeiHot f is a free state and the respective strong limits s-limt-,TmSZ(t)f exist." T h e version of scattering theory based on this procedure of computing the wave operators, and eventually the scattering operator (2.2)
s = Q-*Q+,
is called time-dependent scattering theory. In this section we shall derive some of the basic properties of wave and scattering operators, which follow from the general definitions (2.1) and (2.2), and which do not require the specification of Ho and H within
* A more common way of defining Q+ features EH(SL)instead of EMi in (2.1),where S,", is the absolutely continuous spectrum of H (in practice, Sg = S,", though in general S,",C S,"). Since, in general, the strong limit of Q(t)EH(S)E for t co might not --f
exist, this alternative definition gives rise to the question of existence of wave operators. In our version, the wave operators always exist, but one must check in each particular case whether M i is large enough to warrant a physical meaning to Q+ and S.
416
V. Quantum Mechanical Scattering Theory
the confines of some particular model. It can be easily seen by comparing (2. I), (2.2) with (1.1 8), ( I .28), and (1.29) that the derived theorems apply and S 2 as ) well as to their counterparts Qg)and to the operators in the “internal motion” Hilbert space %’(l). Naturally, in the first case one should take H = and H , = Hi2),while in the second case one should take H , = Hi1)and H = H(l).
Qz)
Theorem 2.1. properties
T h e wave operators 62, have the intertwining
(2-3)
Q*pHO
=
eitHQ+ ,
t
E
R1,
Proof. If fl belongs to the respective subspaces M, , then according to the definition of M, , the respective strong limits Q ( t ) f l , t -+ co, exist. Thus, we can write
11 exp(iHt) Q,fl =
lim
t14m
= lim
tl-’Fm
=
lim
tl+FW
- s-lim[exp(iHt,) exp( -iHOtl)] exp(iH,t) fl I/ tlpTm
(1 exp(iHt) Q*fl
- (exp(iHt,) exp(--iHotl)) exp(iH,t)fl
I/ exp(iHt) Q+fl - exp(iHt) exp[iH(t, 11 Q+fl - exp[iH(t,
-
t)] exp[-iH,(tl
Hence, exp(iHot)fl also belongs to M,
- t)] exp[-iHo(tl -
- t)]fl
(1
t)]fl j j = 0.
, and since
s-lim exp(iHtl) exp( -iH0t,) exp(iHot)fl tl+kC
(1
= Q+ exp(iHot)fl
,
we have proved that
for all t E R?. Now, if f2 E M i fact, for allf, E M,
=
S 0 M, , exp(iH,,t)f, also belongs to M i . I n
= 0,
2. General Time-Dependent Two-Body Scattering Theory
417
since exp(iHot)f, E M, according to the preceding discussion. Thus, due to definition (2.1) of Q, , we have Q, fi = 0 and Q, exp(iHot) fi = 0. Consequently (2.6)
Q*
exp(iHot)fz
=
fi E M i .
exp(iHt) 0,fi = 0,
Since every f E 2 can be decomposed in the form
we immediately obtain (2.3) from (2.5) and (2.6). The relation (2.4) can be easily derived from (2.3) by methods frequently employed in Chapter I V (see Exercise 2.1). Finally, by making use of the resdlt (3.6) in Theorem 3.1, Chapter IV, and of the fact that 52, are bounded and therefore continuous, we easily derive that for f E aH0 Q*Hof = Q* s-lim t+O
= s-lim t+O
1 zt
1
zt (exp(iH,t)
1
= s-lim t-0
zt
1
at
1)f
(Q* exp(iHot)- 521) f
= s-lim 7-(exp(iHt) 0, t+O
-
-Q ),
(exp(iHt) - 1) Q*f
f =
HQ*f. Q.E.D.
2.2. THEPARTIALISOMETRYOF WAVEOPERATORS We recall that in the last section M, were defined to be the sets of all for which the respective limits vectors f E 2 02$)
f*
= s-lim Q ( t )f t+im
exist, while R, are the sets of all vectorsf, , According to Lemma 1.1, the sets M, are a closed linear subspace of 2.We shall prove the same statement for R, , by showing that Q, are partially isometric operators.
Definition 2,1, A bounded linear operator U defined on the Hilbert space lf is said to be partially isometric with initial domain M and final domain N if U maps M isometrically onto N and maps M I = lf 0 M into the zero vector, i.e., if 11 Ufll = llfll f o r f E M and 11 Ufll = 0 f o r f e Z 0 M.
418
V. Quantum Mechanical Scattering Theory
Theorem 2.2. The sets R, are closed linear subspaces of H ;the wave operators SZ, are partially isometric with initial domains M, and final domains R, , respectively. Proof. According to the definition of R, , any vector f, E R, is the strong limit f + =A ;T Q(t)f = Q+f
. Moreover, since D(t) is unitary,
for some vector f E M,
II Q+f II
(2.7)
II Q(4f II
= t-1-m lim
= llfll
iff E M, . Hence, the restriction 0, of SZ, to M, has an inverse 0;' , which is bounded on account of (2.7). T o prove that R, is closed, let h E X be the strong limit of a sequence h, , h, ,... E R, . By (2.7), f, = 6$z1 , f , = ,... is a Cauchy sequence of elements from M, . Since M, is closed; fi ,f, ,... has a limit f E M, . Therefore, h = 0,f E R, , which proves that R, is a closed set. The linearity of R, is an obvious consequence of the linearity of 0,. Thus, R, is a closed linear subspace of X,and 0,maps M, isometrically onto R, . Moreover, if g _L M, , then Q+g = s-lim Q(t) En+g = 0. t+-m
Thus, 0, is partially isometric with initial domain M, and final domain R, . The corresponding result for R- is established in a similar manner. Q.E.D.
Lemma 2.1, If U is a partially isometric operator on # with initial domain M and final domain N, then U* is a partial isometric operator with initial domain N and final domain M, and the following relations are true: (2.8)
U*U
= EM,
(2.9)
UU*
=
Proof. If g 1M, then Ug Hence, iff E M, then (g
=
EN.
0 and consequently U*Ug = 0.
I u * w =
for all g E MI, and therefore U* UfE M. Since M is linear, we also have f - U*Uf E M for any f E M. On the other hand, (h
If-
U * U f ) = ( h I f ) - (Uh I U f ) = 0
2. General Time-Dependent Two-Body Scattering Theory
419
for all h, f E M (see Exercise 2.2), which implies that f = U*Uf. Thus, U*Uf = f for all f E M and U*Ug = 0 for all g 1M, i.e., U*U = EM and (2.8) is established. We shall prove now that U* is partially isometric. If h E N, there is a unique f E M such that h = Uf, 11 f I( = 11 h 11. Consequently, U*h = U*Uf = f by (2.8). On the other hand, if h, 1N, then (U*hl I fl) = (h, I Ufl) = 0 for all fi E %, since Ufl E N always. Consequently, U*hl = 0 whenever h, 1N. Thus, U* is partially isometric with initial domain N and final domain M. By reversing the roles of U and U* in (2.8) we show that (2.9) is true. Q.E.D.
If we apply this lemma to the operators Q, relations :
, we arrive at the following
(2.10) (2.1 1)
In particular, if M, = M- = M, = Z , we would have Q+*Q* = 1. We shall see in later sections that in problems of physical interest Mi’) coincides with the subspace #(2) 0Xh;) of all vectors which are = ZL1) 0&?hi), where [see (1.26)] orthogonal to the subspace C Z(l)is the space spanned by the eigenvectors of the “internal” free Hamiltonian Hit) given in (1.28). Ordinarily, Hit) has a pure = (0) and Mi2) = X ( z ) . continuous spectrum, in which case
%A2)
2.3. PROPERTIES OF THE S OPERATOR We turn our attention now to the S operator, giving special notice to the practically very important property of its “unitarity.”
Theorem 2.3. The S operator and the T operator commute with the free Hamiltonian Ho . Proof. Let EHo(B)be the spectral measure of Ho . Using the intertwining property (2.2) and its equivalent (2.12)
Qi*EH(B)= EHo(B)Qi*, B E B’,
derived from (2.2) by taking adjoints, we obtain E ~ O ( B )=S E ~ O ( B SZ-*Q+ ) = Q - * E ~ ( BQ+ ) = Q - * Q + E ~ ~ (= B )S E ~ O ( B ) .
Thus, S commutes with Ho . I t is evident from (1.25) that T also commutes with H , . Q.E.D.
420
V. Quantum Mechanical Scattering Theory
We recall that Theorem 2.3 has been used in proving that (1.30) is not feasible. I n Theorems 2.4 and 2.5 we show that S*S = EM, if M, = M, .Thus, the condition that S # O coincides with the condition that M, # {0}, i.e., that there are asymptotic states and that the wave operators do not vanish.
Theorem 2.4. The S operator maps the subspace M, into M-, and maps its orthogonal complement Z 0 M, into zero; moreover it maps M, isometrically into M- (i.e., I/ Sfll = 11 f 11 for all M,), if and only ifQ R, C R- . Proof. According to Lemma 2.1, the final domain of Q-* is identical to the initial domain M- of Q- . Therefore, Sf = Q-*(Q+f) E M- for a l l f e 2.However, iff 1M- , then Q,j = 0 and consequently Sf = 0, which establishes the first statement in the theorem. According to Theorem 2.2, Q, maps M, isometrically onto R,, i.e., 11 Q+gII = I / g11 for g E M, , By Lemma 2.1, Q-* maps R- isometrically onto M- . Thus, we have 11 Q-*Q+g I/ = /I Q+g/I = I( g 11 for all g E M, if R, C R- . However, if R, 0(R- n R,) # {0}, then the elements of R, 0(R- n R,) are mapped by Q-* into zero, and S = Q-*Q+ is not isometric on M, . Q.E.D.
The above theorem indicates that the S operator might be unitary if M, = M- = M, and S is restricted to M,. Theorem 2.5 gives an important necessary and sufficient condition for this case to occur.
Theorem 2.5. The S operator is partially isometric with initial domain M, and final domain M- and satisfies the relations (2.13)
S*S
=
EM+,
SS*
=
EM_
if and only if R, = R- . Proof. Assume that R,
= R - . Using ER&Qi= Q,
and the identity (2.1 I), we obtain that (2.13) holds: S*S
= Q+*Q-Q-*Q+ = Q+*ER-Q+ = Q+*ER,Q+
SS* = Q-*Q,Q+*Q-= Q-*ER +Q- = Q-*ER-Q-
= Q+*Q+ =
EM+,
= Q-*Q- = E M - .
* In theories which are invariant under time reversal M, = M- = M, and there is a one-to-one antilinear mapping of R- 0 R, onto R, 0 R - . Hence, in such theories either R, E R- or R, is not a subset of R- , and S is not isometric on M, .
42 1
2. General Time-Dependent Two-Body Scattering Theory
Now, by Theorem 2.4, when R- = R, , the S operator is partially isometric, with initial domain M, and final domain M- equal to its range 9Zs . Since, according to the same theorem, 9Zs C M- , we obtain by using (2.13) and Lemma 2.1 that gs= M- . Conversely, assume that the relations (2.13) are true. We get by expressing in (2.13) S*, S, and EM* in terms of the wave operators Q, :
Q-*Q+Q+*Q= Q-*Q-.
Q+*Q-Q-*Q+ = Q,*Q,
Multiplying the above two relations from the left by Q+ and from the right by Q+*, respectively, we obtain, after using (2.10) and (2.11),
Thus, we have for any f E 2
11 E R + f l / 2
=
(f 1 E R + f )
= (ER+fl
=
(f 1 E R + E R - E R + f )
ER-ER+f)
= 11 E R - E R + f 1 / 2 *
The above equality implies (see Exercise 2.3) that ER+f = ER ER+f for a l l f e 2,i.e., ER-ER+ = ER+ = El+ = ER+E,- . Inserting the above relation in (2.15), we get ER-
= ER_ER+ER-= Ei-ER+ = ER-ER+ = ER+ .
Q.E.D.
I n the case of two-body potential scattering, i.e., scattering theory in which H and H , are essentially of the form (1.8) and (1.9), we will be able to prove that indeed M, = M- = M, and R, = R- when the potential V(r) fulfills certain conditions. However, it is still interesting to investigate the physical consequences of a theory in which R, # R- , i.e., when the S operator is not a partial isometry from M, to M- . Let us assume that go is a vector which belongs to R, but does not belong to R- . According to the definition of R, , the fact that go E R, implies that there is a vector gin E M, such that go = Q+gi, , i.e.,
Hence, the interacting state
e-iHt
go has an incoming asymptotic free
422
V. Quantum Mechanical Scattering Theory
state. This means that e-iHlg, is not a bound state. However, since go 4 R- F lim eiHt
t++m
-Mot
e
go
does not exist. Hence, by Theorem 1.1 e--iHtgo has no outgoing asymptotic state, Thus, e--iHtgois a “quasi-bound” state, which is approximately a free state in the distant past, but which does not become a free state again at any time in the future. On the other hand, if there is a vector g, E R- which does not belong to R,, then the theory possesses an interacting state e-{Htg, which becomes a free state in the distant future, but wich has never been a free state in the past. Thus, the condition of “unitarity” of the S operator, which is usually imposed on scattering theories, is equivalent to the requirement of not having any quasi-bound states in the theory.
2.4. PROPERTIES OF R,
AND
M,
Thus far in this section we have never used the basic assumption that a state eciHotg is called a free state only if it is not a bound state of the Hamiltonian Ho . As a matter of fact, all the theorems of this section stay valid if we define M, to be the sets of all vectors f E Y? for which the respective strong limits of Q ( t ) f , t ---f oc), exist. If M, are defined this way and we choose H identical to H , , we would have Q(t) = 1 and M, = M, = A?,always. However, if Ho has bound states, then not all e-i’foif, f E &, are free states, and M, defined in $1 is a nontrivial subspace of X . Let MI = EH(S : ) Z denote the closed subspace of Y? onto which the spectral measure E H ( Sf) belonging to the continuous spectrum of H projects.
Theorem 2.6. T h e closed subspaces R, of Y? are also subspaces E of M, = EH(Sf)A?, and are left invariant by EH(B), i.e., E H ( B ) ~ R, for all B E 98, whenever f~ R, .
Proof. Since R, are the ranges of Q, , we can find for any g , vectors f* , such that g , = Q,f, . Hence, using (2.4), we get
E
R,
which establishes that R, are left invariant by H. We shall prove that R, C M, by showing that Q J , is orthogonal to all the eigenvectors of H .
423
2. General Time-Dependent Two-Body Scattering Theory
Let g be an eigenvector of H with the eigenvalue A,, EH({Ao})g= g. Given any f, E M, , we can write (g
so that
I Qifl) = <-w{AoHg I Qih)= (Q**EH({Ao))gIfl) I fi>
=
= (g I Q*@%AO))fl)
= 0,
where we have made use of (2.12) to derive the third equality; the last f, = 0, since no f, E M, is an inner product vanishes because EHo({Ao}) eigenvector of Ho . Since Q,fZ = 0 whenever f2 1M, , we derive, by setting f =fl f 2 , f l E Mo , f 2 IM* 9
+
( g I Q * f > = ( g I Q*fl>
for any f E X . Hence, R, g o f H . Q.E.D.
+( g I
Qif2>
=0
are orthogonal to any eigenvectors
We will be able to prove in later sections that in potential scattering with certain classes of potentials, we have R, = R- = M, . I n order to arrive at results of this kind, one usually starts by proving first that R, = R- . This establishes immediately that the S operator is unitary. Now we shall prove a theorem which states a sufficient condition for the identity of R, and R- . This theorem applies to operators with a simple continuous spectrum. The concept of one operator A with a simple spectrum was introduced in $5 of Chapter IV: the self-adjoint operator A has a simple spectrum if the one-element set {A}is a complete set of operators. According to the theorems of $5 in Chapter IV it is equivalent to say that A has a simple spectrum if there is a vector fo which is cyclic with respect to A. If SA, and S;t denote the continuous and point spectrum of A, then the closed subspaces #A, = EA(SA,)X and 8:= EA(S;t)X onto which E A () :S and E A ( Sf:)project, respectively, are left invariant by A. T o see that, note that X = X t 0Xf:and that
(f I A g )
=
I+" d ( f I E$g> -m
A
=0
iff E $?A, a n d g E S;t n g A or , iff E X;t a n d g E #A, n BA. The restrictions A, and A, of A to Scand X,., respectively, are obviously self-adjoint operators with spectral resolutions (2.16) (2.17)
A
.-I AdE?,
A,
-
=
j h dEp,
EAc(B) = EA(B n Sf), EAp(B) = EA(Bn S,"),
424
V. Quantum Mechanical Scattering Theory
respectively. When A , or A, are operators with a simple spectrum, we say that A has a simple continuous spectrum or a simple point spectrum, respectively. I t is easy to establish that A has a simple spectrum if and only if both its continuous spectrum and its point spectrum are simple (see Exercise 2.4).
*Lemma 2.2. Let A be a self-adjoint operator with a simple spectrum, and let M, , k = 1, 2, be invariant subspaces of A , i.e., for any f E M, A a,,,we have Af E M, , k = 1 , 2. If the restrictions A , , k = 1, 2, of A to M, have identical spectra, then M, = M, . Proof. Since M, is left invariant by A, the operators A and EM, commute (see Exercise 2.5). Hence, by Theorem 5.6 of Chapter IV, the projector EMk is a function of A, i.e., EMk= Fk(A). From the properties EM,= E* and E;, = EM,, by using (2.26) of Chapter IV Mk we prove that Fk(h) is a real function satisfying
and therefore F,(h) can assume only the values 0 and 1 for X E SA. Thus, if we set F,(h) = 0 for h $ S A , we show F,(h) is the characteristic function of a Bore1 subset B, of SA. We obviously have Al, = AEM,
= AF,(A).
Hence, the spectrum S A k of A, coincides with B , . Consequently, Q.E.D. S A 1 = S A a implies that F l ( A ) = F,(A), i.e., EM, = E M , . The following theorem can be very useful in establishing that S is unitary.
Theorem 2.7. R, = R - .
If the continuous spectrum of H is simple, then
Proof. Let H , be the restriction of H to the closed subspace M, = E H ( S F ) X . T h e operator H , is obviously a self-adjoint operator with a pure continuous spectrum which is identical to the continuous spectrum of H . Hence, according to the assumption, the spectrum S H 1 of Hl is simple. According to Theorem 2.6, R, are closed subspaces of M, which H , leaves invariant. Since the spectrum of H, is simple, we can apply Lemma 2.2 and infer that R, = R- . Q.E.D.
2. General Time-Dependent Two-Body Scattering Theory
425
2.5. DYSON’S PERTURBATION THEORY We saw in 91 that the knowledge of the S operator is sufficient for the calculation of the differential cross section and of transition probabilities. This shows that a complete solution of the dynamical problem, which would consist in computing the evolution operator U(t, to) introduced in 93 of Chapter IV, is not necessary for the theoretical description of a scattering experiment. Since the computation of Y(t)= U(t, to)Yo for given Yo can be a difficult and intricate computational task, it is desirable to develop methods for computing the S operator in a direct manner. Dyson’s perturbation theory provides one such method. I t follows from (2.1) that for anyf, g E 2 (2.18)
(f 1 Q-*g>
=
( Q - f I g>
=
!jE
Since Q*(t) is unitary, we have 11 Q*(t)g 11 = 11 g 1 . On the other hand, if g E R- , it follows from the fact that Q-* is a partially isometric operator with initial domain R- (see Theorem 2.2 and Lemma 2.1) that 11 Q-*g (1 = 11 g (I, so that I/ Q*(t)g /I = 11 Q-*g 11. According to Lemma 6.2 of Chapter IV, this result in conjunction with (2.18) implies that Q-*g = s-lim E,-Q*(t)g,
g
t++m
S
R-
.
= R- , we have
Hence, if S is partially isometric, so that R, (2.19)
E
=
EM_ s-lim Q*(t)(s-lim Q(t,) EM+)
=
EM- s-lim(s-lim Q*(t)Q(to)) EM,,
t++m t++m
to+m
to-l-m
where the operator (2.20)
V(t, 2,)
=
Q*(qQ(to)= eiHot
e--iH(t--to)
e-iHoto
is obviously unitary for any t, to E R1. Comparison of
*
- eiHot
QO-
e
-iHt
with (3.43) of Chapter IV shows that Q*(t) is the operator which determines the time dependence of the state p(t)in the interaction picture. Hence, we have (see Chapter IV, Exercise 3.4)
(2.22)
H,(t) = eiHOt(H- H,)
for any f E g Hn B H o .
e-iHot
426
V. Quantum Mechanical Scattering Theory
We have seen in 97 of Chapter IV that in potential scattering, under , very reasonable assumptions on the potential, g Hcoincides with gH0 When this is the case, and the vector f belongs to g H= g H 0we , have exp(--iHoto) f E gH0and exp(iHt,) exp( -iHoto) f E g H (see Exercise 2.6), so that Q(to)f E g H= BHn gH0 and (2.23)
av, at
~
to+
=
s-lim V(t At-0
+4
At
v(t,to) f = - -iHI(t)v,t 0 ) f .
--i@
I Hdt)
to) -
Thus, for any vector h E 2? (2.24)
(a/at)(h
I
w,t o m
=
v,
t0)f).
If the function ( h I H I ( t ) V ( t ,to)f) is integrable in t, we obtain from (2.24), after having observed that V(to, to) = 1, the following relation: (2.25)
(h I
v(t,t o ) f ) = ( h If)
-
-i
f( h I HI(~I)
If the integral --i
t
Jto
v(t1 t01.f)
t0
( h I HI(t1) V
l
3
t o m dtl
is defined for any h E X , it determines for each fixed f E gHo a continuous linear functional in h, and therefore by Riesz's Theorem (Theorem 2.3 of Chapter 111), it defines a mapping f -+ Wl(t,t o )f. I t is easy to see that Wl(t,t o ) is a linear operator defined on BH0. We write, symbolically,
I n this notation (2.25) is equivalent to the statement that (2.27)
V ( t , to)f = (1 - -i
St Hdtd $0
v(t1 , 10) 4)f,
f E
If we treat the above integral equation recursively in V(t,to),we obtain after n steps
v(t,to)f
=
+ vi(t,t o ) + + vn-i(t, + W d t , to)
"'
tO)lf,
f E gHo
Y
where Vk(t,t o ) and W J t , to) are defined by the relations
(2.29) W,(t, to) = (-iy
1 dt, St'dt, t
to
to
r-' to
dt, HI(tl)... HI(t,-l) V(t, , to),
427
2. General Time-Dependent Two-Body Scattering Theory
whose precise meanings are established in the same manner as in the case of W,(t, to). It has become customary to introduce a chronological ordering operation T by setting (2.30)
T[HI(~,)HI(tz)
H~(t,)l= Hdt,,) HdtiJ
* a *
til 2 tia >, * * * 2 tt, .
HdtiJ,
This operation orders the time-dependent operators in the product H,(t,) ... HI(tk)in a chronological sequence of increasing values of the time variable from right to left, so that the integrand in (2.28) becomes symmetric under permutations of t, ,..., t, when T is set in front of it. Since there are n! such permutations, it is easy to check that
Suppose that
exists for every k
=
(2.33)
1, 2, ..., and that W,f
=
w-lim(w-lim W,(t, t o ) f ) t-fm
to+-m
also exists and converges (at least in a weak sense) to the zero vector Under these assumptions we obtain from (2.19), when n -+ +GO. (2.20), (2.30), and (2.31) the following expansion: S
+ + vs + * . . ) E ~ l + f *
EFI_(~ vi
The above expansion is known as a perturbation expansion of the S operator. In the zero order for the case M, = M,, it becomes S = E;, = EM,,i.e., it describes the case when there is no scattering. The computation to higher orders is facilitated in practice by an explicit knowledge of the commutator [HI(tl), H,(t,)] = D(tl , t z ) . This makes possible the computation of the integrand of V , by successive use of the relation T[HI(tl) HI(tZ)l = HI(t1) W,)-
wz
-
4) Wl > tz).
V. Quantum Mechanical Scattering Theory
428
Computations of this nature occur very frequently in quantum field theory, and techniques have been developed for coping with them in a systematic manner. OF ASYMPTOTIC STATES 2.6. THEEXISTENCE
There are cases even in potential scattering in which M, = {0}, i.e., when there are no asymptotic states in the sense of (1.7). One notable example is the case of Coulomb scattering, when we are dealing with the Schroedinger operator H with a Coulomb potential. Namely, it can be shown that quantum Coulomb scattering shares with the classical Coulomb scattering the feature of not having asymptotic states in the conventional sense (1.7), though asymptotic states can still be introduced in a somewhat different sense of the word." This particular example indicates that it is desirable to produce some criteria for the existence of asymptotic states. T h e rest of this section deals with the derivation of Theorem2.8,first proved by Jauch andZinnes [1959], which will be found very useful in establishing the existence of asymptotic states in potential scattering (see Exercise 2.8 and $7).
*Theorem 2.8. Suppose 9,, 9,C g Hn g H ois, a linear submanifold of 2 which is such that e-'iHot f E g Hn gH0 for all f E 3, . Then the closure 6, of 9, belongs to the initial domains M, of the wave operators 52, if and only if when t, ---t - 00 and t, ---t co the respective strong limits of
+
Il(tl , t 2 ) f =
(2.34)
t2
eiHtH,epiH0tfdt,
Hl
=H
-
H, ,
tl
exist for every f E 9, . A sufficient condition for the existence of these limits is that for some t , , t, E [wl
sf", II
ffl
exp(--iHot)fII &
< fa,
/:p
II Hl exp(--iHot)fIl dt < f a ,
respectively. T o understand the above theorem, we must comprehend the meaning of the integral appearing in (2.34). This integral is defined by the requirement that
* For details, see Dollard [1964].
429
2. General Time-Dependent Two-Body Scattering Theory
determines a continuous linear functional in g. I n fact, if the integral in (2.35) exists, then, by Riesz' theorem (Chapter 111, Theorem 2.3), there is a unique vector f, = Il(tl, t z )f such that (fi I g ) = +r(g;t , , tz). Thus, in order to know that Il(tl,te) is well defined we only have to establish that the Riemann integral in (2.35) really exists. For this task we need Lemma 2.3. For any two vectors f E gH0 and g E g Hthe function
Lemma 2.3.
(2.36) F ( t ) = (exp( -iHot)f I H exp( -iHt)g) - ( H , exp( -iHot)f I exp( -iHt)g) is continuous in t E R1, and (2.37)
1(
i
(
-
I exp(--iHt)g)) dt
<[-Q(t,)- -Q(W I g).
Proof. T o establish the continuity of the first term in (2.36), we note that for any T E R1 (2.38)
+
+
I(exp[--iH0(t .)If I H exp[-iH(t T ) ] g) - (=p( -iHot)f I H exp(--iHt)g)l d I(exp[--iHo(t .)If1 H(exp[--iH(t T ) ] - exp(-iHt))g)I I((exp[--iHo(t T ) ] - exp(--iH,t))f I H exp(--iHt)g)l d llfll I1 H(exp[--iH(t .)I - .xp(--iHt))g I1 ll(exp[-iHo(t .)I - exp(--iH,t))fI/ I/ Hexp(--iHt)gl/.
+
+ +
+ + +
+
We recall now that by Theorem 2.7 of Chapter IV, the operator H(exp[-iH(t T ) ] - exp(-iHt) is a function of the self-adjoint operator H , and consequently
+
H(exp[-iH(t
+
T)]
- exp(-iHt)) g = (exp[-iH(t
+
T)] -
exp(-iHt)) Hg.
Since exp(-iHt) and exp(--iHot) are strongly continuous operatorvalued in t (see Chapter IV, §6.3), the expression on the right-hand side of (2.38) converges to zero when T -+ 0. T h e continuity in t of the second term in (2.36) is even easier to prove by following the same line of reasoning, since we have
(Ho exp[-iHo(t f .)If[ exp[--iH(t f T ) ] g) - (Ho exp(-iHot)fI exp(-iHt)k')l d I/ Ho(exp[--iHo(t 41 - exp(--iHot))fI/ II g /I /I Ho exp(--iHot)fII Il(exp[--iH(t dl - exp(--iW)g I/.
+
+
+
430
V. Quantum Mechanical Scattering Theory
The cqntinuity of F ( t ) in the variable t implies the existence of the integral in (2.37) for any given t , , t, E R1. Thus, the first part of the lemma is established. T o establish the second part, we recall from Theorem 3.1 of Chapter IV that
s-l.l-n ;(exp[-iA(t + 2
T)]
- exp( -iAt))g = A exp(-iAt)
g
for any A = A* in general, and consequently for A = H and A = H , in particular. I n view of this fact, a glimpse at (2.36) suffices to show that iF(t) = (d/dt)(e-iHOtf 1 f?-"""g>.
Q.E.D.
Hence, (2.37) immediately follows.
Let us return now to the proof of Theorem 2.8. When
f Egl then exp(--iH,t)fE
gH
gHo >
g1C g H so , that
1 H exp( -iHt)g)
=
(exp(iHt) H exp( -iHot)f
1 g),
and F ( t ) can be written in the form F ( t ) = (exp(iHt) Hl exp( -iHot)f ] g).
Thus, according to (2.37), we have (2.39) ([Q(t,) - C!(tl)]f I g)
=i
fa
(exp(iHt) Hl exp( -iHot)f
I g) dt
tl
for any f E g1and g E g H Since . g His dense in 2,the integral on the right-hand side of the above relation can be said to determine a unique linear functional in g E Z . Thus, we can write symbolically (2.40)
[L?(t2)- Q(t,)]f
=i
1 exp(iHt) Hl exp( -iHot) f dt. t2
tl
It follows from the above relation that in the case s-lim Q ( t ) f exists for t -+ -co or t -+ +a,the respective strong limits of the integral 00 also exist. in (2.34) for t , -+ - co or t , -+ Conversely, let us assume that
+
s-lim t,-+-m
~11
e x p ( i ~ t H~ ) exp(--iH,,t)fdt
43 1
2. General Time-Dependent Two-Body Scattering Theory
exist for all f E 9, . Then, it follows from (2.40) that s-lim Q(t,)f also exists when t , -+- co and f E 9,. However, by Lemma 1.1, the set of all vectors f for which this last limit exists is closed, and consequently the closure G, of 9,must be contained in it. Thus, the theorem is established. For all practical purposes, the investigation of the existence of the strong limits in t, and t, of the integral in (2.34) reduces to the investigation of the existence of integrals (2.41)
for some t, E W, and of
for some t , E R1. I n fact, by using the Schwarz-Cauchy inequality we get (2.43)
Hence, the existence of (2.41) implies, in conjunction with (2.39), that l l f l l = \I S+fII, w-lim Q(t,)f exists when t , -+ -a.Since / / Q ( t l ) f i l we infer by using Lemma 6.2 of Chapter IV that s-lim Q(t,)f also exists for t1 -+ -a. T h e argument for t, -+ + co is completely analogous.
<
2.7. THEPHYSICAL ASYMPTOTIC CONDITION T h e condition which the Schroedinger-picture asymptotic states Y e x ( t ) (ex = in, out) of the interacting state Y(t)have to satisfy, (2.44)
lim /I Y(t)- Y e x ( t ) ( I = 0,
t+*m
has no direct physical meaning, since the above vector norm is not a measurable quantity. We recall from $41 and 2 of Chapter IV that the directly measurable quantities are the probabilities
432
V. Quantum Mechanical Scattering Theory
for any set of compatible fundamental" observables. Hence, we expect asymptotic states to satisfy some kind of physical asymptotic condition which would specify that the probabilities (2.45) for the interacting state Y(t)approximate arbitrarily closely the corresponding probabilities for the respective asymptotic states Yinpout(t),when t -+ fCO. That this is actually the case is stated in Theorem 2.9. Let the Hamiltonian H and a set 0,of additional fundamental observables be such that any other observables of the interacting system or the free system is a function of compatible observables from 0, = { H } u 0, or OFx = {H,} u 0,,respectively. Furthermore, assume that whenever { H , A,', ..., An'}, A,', ..., A,' E 8,, is a set of compatible observables, then H , , A,', ..., A,' are also compatible. Under these assumptions, the asymptotic states Y e x ( t )(ex = in, out) of the scattering state Y ( t ) satisfy the physical asymptotic conditions
*Theorem 2.9.
(2.46) (2.47)
for any sets {A, ,..., A,) C 0,, {H, A,', ..., An'}C 0, of compatible observables. Proof. I t is straightforward (see Exercise 1.7) to prove that (2.46) holds, i.e., lim /(Y(t)I EA1s...BAm(B) Y ( t ) )- ( Y e x ( t ) I
t+*m
( B )Y e x ( t ) ) I
EA1s...*Am
=0
when (2.44) is satisfied. Since A,', ..., A,' commute with H as well as with H, , we infer that for arbitrary B E 9Yn eiHt
e-iHotEAl'
,...,A , , ' ( B , )
= EA''
....,an'(^,) e i H t e --iHot .
By applying the operators on both sides of the above equation to vectors f E M, and letting t -+ f03, we obtain immediately EAl',...,An'(B,) f E M, and (2.48)
Q&Ai'..
.. an'(^,)f = *A''..
...An' (B,) Q*f
*
* See the discussion in the beginning of 92 in Chapter IV concerning fundamental observables.
433
2. General Time-Dependent Two-Body Scattering Theory
IM, , then (EA”*-.’A”’(B2)g If)
Furthermore, if g
( g I EA”....*A”’(B2)f) = 0
=
for all f E M, , and therefore EA1’,...,A,‘g M, . Consequently, (2.48) holds for arbitraryf E 2.Combining this relation with (2.4))we conclude that (2.49)
QiEHO(B,) EA1’....*An’(B2) = EH(B,)EA1’,,,,*A*‘(fj2) Qi
for any B , E B1 and B , E gn. As a consequence of the basic properties of spectral measures (see Chapter 111, the proof of Theorem 5 4 , we can write for any B E gn+l EHo.AI‘s...*An‘(B) = inf
1 1EHO(BP))EAL’-....An‘(BF)): Bc k
u BP) x
B$)/,
k
where the projector infimum is taken only over finite unions of sets Bik) x BLk’) with Bik)E 491) BLk)E gn. Hence, EHo3*1*...,*-(B)is the strong limit of some sequence of finite sums
1EHo(BP))
EAi’
.....An‘(Bp))
k
corresponding to a sequence of sets Uk(Bik)x BLk)),with B as an intersection. This result in conjunction with (2.49) implies that Q*EHO’Al‘
(2.50)
I....
A”@)
= EH.A”
.....A”’@)
Qk
for any B E gn+l. We recall that (2.51)
where ex
Y(0) = Q*YeX(O), =
in for 52, and ex
=
out for 52-
. Hence,
after noting that
434
V. Quantum Mechanical Scattering Theory
where the last step follows from the fact that Q,*Q,?Pex(0) = Yex(0), since Yex(0)E M, . Hence, (2.47) is established. Q.E.D. All nonrelativistic quantum mechanical theories satisfy the conditions of the above theorem. For example, in wave mechanics for a single particle interacting with a potential V(r), one can take
W C -(1/2m)A
+ V(r)
and 0,consisting of L2,L,(n E R3, I n I = l), X , and P,(K = 1,2,3); here, L2 denotes the total angular momentum operator (encountered in 97 of Chapter 11), L, = n * L is the angular momentum projection in the direction n, and X , , P , are the position and momentum components
with respect to the Kth axis of a Cartesian inertial reference frame. From the physical point of view, it is sufficient if asymptotic states satisfy the physical asymptotic conditions (2.46) and (2.47), rather than (2.44). As a matter of fact, albeit that for long-range potentials (such as the Coulomb potential or V ( r ) r - ~ 0, < cy. < 1) there are no asymptotic states satisfying (2.44), there are free states satisfying (2.46) and (2.47). One could call such states physical asymptotic states, and develop" a scattering theory based on the existence of these states. As a matter of fact, in this case one could define the wave operator Q, to be the operator which maps an incoming asymptotic state into the corresponding interacting state, i.e., as the operator satisfying (2.51); Q- could be defined in a similar manner. As soon as the wave operators are defined, one can proceed in the usual manner and introduce the scattering operator by means of (2.2) and the transition operator by the old formula
-
(2.53)
T
z=
(1
-
S)/2ni.
For the rest of the quantities of importance in scattering theory (transition probabilities, differential scattering cross section, etc.), one can also retain the old definitions.
EXERCISES 2.1. Assume that C is a bounded operator in A?, and A, , A, are self-adjoint operators in H with spectral measures EA1(B)and EA2(B), respectively. Prove that the relation eiAltC= CeiAat is satisfied for all t E R1 if and only if EA1(B)C = CEA2(B)for all B E 9P.
* For details see the article by PrugoveEki [1971a], which also contains further references.
435
3. General Time-Independent Two-Body Scattering Theory
2.2, Show that if U is a partially isometric operator with initial domain M, then ( U f I Ug) = ( f I g ) for all f,g E M. 2,3. Show that if E is a projector and )I Ef then f = Ef.
11
=
llfll
for some f
E
2,
2.4. Show that f o is a cyclic vector with respect to A if and only if EA():S f o is a cyclic vector of A , and EA( S;) f o is a cyclic vector of A, . 2.5, Prove that if a self-adjoint operator A leaves the closed linear , M ]= 0 for all B E 9 I l . subspace M invariant, then [ E A ( B ) E 2.6. Show that if A is self-adjoint and f all t E R1.
E
g A ,then
eiAi
f
E
g Afor
2.7. Suppose that the potential V(r) is Lebesgue square integrable Apply the result of Exercise 1.9 to prove on R3 so that J) V ) )< +CQ. that if $(r) is Lebesgue integrable as well as square integrable on R3, then
almost everywhere in R3, and consequently
where ( Ve+Hof $)(r)
=
V(r)(e-iHOt$)(r).
If the potential V(r) is Lebesgue square integrable on R3, then = L2(R3)in potential scattering for V(r). Explain how this statement follows from Theorem 2.8 and Exercise 2.7. 2.8.
M,
=
M-
3. General Time-Independent
Two-Body Scattering Theory
3.1.
THERELATIONOF TO THE
THE
TIME-INDEPENDENT
TIME-DEPENDENT APPROACH
I t is possible to construct the Mdler wave operators Q, by taking limits with respect to parameters other than the time parameter t . Scattering theory based on such time-independent procedures for computing the basic operators Q, , S, and T is called time independent. The relation of the time-dependent to the time-independent approach is best exhibited by showing that Q, can be obtained as strong ,, for E -+ +0, where the parameter E , limits of operator functions Q
436
0
V. Quantum Mechanical Scattering Theory
+
CO, is not related to the time parameter t and has no physical meaning, and the operators SZ,, satisfy the equations
Let us show that the relations (3.1) can play the role of definitions of ,, . linear operators Q Since e i H t and are strongly continuous functions of t (see $7 of Chapter IV), Q(t) = e i H t is also strongly continuous (see Chapter 111, Exercise 5.6). Hence, for anyf, g E A?,the function ( g I Q ( t ) f > = ( g I eiHt i?iH~"f>, t
E
UP,
is continuous, as well as bounded in t :
l ( g I e i H t e-iHotf >I
< It g II /I eiHte-iHOtfII = l l f
I/ It g II.
Consequently, the first improper Riemann integral in (3.I), (3.4
converges absolutely for any f,g E A?:
Thus, the functional ( g If) is defined on A? x 2. It is straightforward to verify that ( g If ), f,g E Z, is a bilinear form. Furthermore, we see from ( 3 . 3 ) that (. I *) is bounded. Hence, there is a unique bounded linear operator (see Chapter IV, Exercise 2.5), which we denote by Q-, , such that (3.4)
(g I Q J >
m
=
( g ~ f =) E
0
e-Et(g I eiHte- i H ~ft > d t
for allf, g E 2,We write the above relation in the symbolic form
(3.5)
Q-, = E
1 m
0
eCt eiHtePiHQtd t ,
E
> 0,
3. General Time-Independent Two-Body Scattering Theory
437
under the agreement that its real meaning is given by (3.4). I n fact, (3.5) is an example of a Bochner integral (see Theorem 3.4 in Appendix 3.7 to this section). It can be shown in a completely analogous manner that for any E > 0 there is a unique bounded linear operator Q,, which satisfies
T h e above relation is written in the symbolic form for all f, g E 8. 0 Q ,
=
J"
ert e i H t
--m
e-iH t dt,
E
> 0.
We are interested in the existence of the strong limits of Q,,f E -+
$0.
as
From (3.3) we immediately infer that
By applying Lemma 1.1 we deduce that the sets N, and N- of all vectors f E %' for which the respective limits of Q,,f exist, (3.8)
N*
= {f:s-lim Q*, E++O
f E if]
are closed linear subspaces of 3. We shall prove now that M, C N, and M- C N- .
Theorem 3,l. If for some f E 2, s-limt-,+m52(t)f exists, then dim6++,, 52-, f also exists, and (3.9)
s-lim Q(t)f t++m
= s-lim Q-, f. f+fO
Similarly, if for some g E S, s-limt-,.-m Q(t) g exists, then s-limE++,,52+Fg also exists, and (3.10)
s-lim Q(t)g t+-m
=
s-lim .R+,g. f+,O
Proof. Let us write f-
=
s-lim Q(t)f t++m
438
V. Quantum Mechanical Scattering Theory
when the above limit exists. Then, for any g E % Kg
I
Q-€f
-f-h
I < €1
m
=
E
0
e-ct(g I Q ( t ) f ) dt
-E
/
m
0
e-€$(g If-)
dtl
m
0
= /I g
II
e-€tl(gISZ(t)f-f-)ldt<€IIgl/ J m e - € t ~ ~ ~ ( t ) f - f - ~ ~ d t
J
0
m
e-u I/ Q(u/.)f
-f- /I du.
We set out to prove that the above integral is smaller than any given q > 0 for all sufficiently small c > 0. Choose some a > 0 and split the interval of integration:
s," 1," j: =
+
*
For the first integral we obtain the following upper bound:
Now, take some
(Y.
satisfying o
and then choose
E
1 rl 2 ltfll
+ Ilf-I1 '
> 0 so small that
(3.11)
for all t 2 a / € . For any such
j
m
e-" II Q(u/.>f
E,
we compute that
-f- I/ du
Consequently, for any given 7
> 0 we have
l(gIQ-,f-f->l
for all E > 0 which are such that (3.11) is true whenever t 3 a / € . Hence, we have established that (3.12)
f-
= w-lim .Ref. €++O
439
3. General Time-Independent Two-Body Scattering Theory
Since I( f - I/ = limt++m/I Q ( t ) f l \ = 11 f 11 2 I/ Q-,f 11, we have shown that the weak limit (3.12) is also a strong limit (see Chapter IV, Lemma 6.1), f-
=
s-limQ-,f, E'tO
i.e., (3.9) is true. T h e proof of (3.10) proceeds along identical lines.
Q.E.D.
As an immediate consequence of the above theorem we obtain the important result (3.13)
which can be considered as a time-independent alternative definition of the wave operators. It should be noted, however, that Theorem 3.1 establishes only that M, C N, , and not that M, 3 N, . Since the conconverse of Theorem 3.1 [which would state that the existence of s-lim6++oQ ,, f implies the existence of s-lim,+,m Q(t)f ] is not known to be true, it has been suggested (see Jordan [1962a, Section 61) that Q* could be defined in a time-independent manner, by taking limits of !2,,f, even in cases when the time-dependent definition (2.1) is not adequate because M, = (0). I n potential scattering, for classes of potentials which have been considered in the last section, the above problem does not arise, since in those cases M, = X , and consequently Q& = s-lim t'irn Q(t) = s-lim c++O Q*,.
From (3.13) we get for any f , g
E
Z
Since a,* are partially isometric with initial domains R, verify that (Q+*ER,f
I g>
= !i$,(EMtQzER+f
, we easily
1 g>
by noting that for f E R, the above relation is equivalent to the relation preceding it, while €or f 1R,, the expressions on both sides of the above relation vanish. I n view of the fact that
440
V. Quantum Mechanical Scattering Theory
we infer from this result (by using Lemma 6.1 of Chapter IV) that Q**
(3.14)
=
s-lim E ~ ~ Q : ~. E ~ + E++O
3.2. INTEGRAL REPRESENTATIONS FOR Go* One of the advantages of the time-independent limits (3.13) for determining a, is that they lead to integral equations which can be used to compute without having to compute O ( t ) . Using the spectral decomposition H
=
j,
X dEF
and recalling that eiHt is a function of H , by employing Theorem 2.5 of Chapter I V we obtain (3.15)
Now, we would like to interchange in the above relation the order of the integrations in h and t , and write (3.16) ( g I Q - , f )
=E
1 dA1 +m
m
--oo
0
exp[-(e
-
ih)t](g I E f e x p ( - i H o t ) f ) dt.
However, before we ask whether such an interchange of the order of integration is correct, we must make clear the meaning of the integration in A occurring in (3.16). We will show that the interchange of the order of integration can be easily justified if the integral in h is taken to be a Riemann-Stieltjes integral.
Definition 3.1.
T h e Riemann-Stieltjes integral
j:w d, 44,
a
< b,
of the complex function F(h) with respect to the complex function u(h), a h b, is said to exist if for any sequence of subdivisions a h(O) < < ... < A(") = b of [ a , b] which is such that the norm 6 = maxk=l,...,m(h(k)- h'k-l)) of the subdivisions in the sequence
<
< <
441
3. General Time-Independent Two-Body Scattering Theory
tends to zero, and for any choice of A'(k) E [X(k-l), A ( k ) ] , K = 1,..., n, the limit of Riemann-Stieltjes sums (3.17) exists, and is independent of the selected sequence of subdivisions; if the integral (3.17) exists, then its value is taken to be (3.18)
1F(h) d,, ~ ( h ) lim f'F ( h ' ( k ) ) [ ~ ( h ( k ) ) b
a
=
6-0
,
-
2=1
>I-
~(h(~-l)
If F(A) and u(A) are defined on R1, then the improper Riemann-Stieltjes integral on R1 is defined as (3.19) provided that the two limits in a and b exist. For functions F(A, A), and o(A, A), of two variables A, A, E W, we define the cross-iterated RiemannStieltjes integral on [a,b] x I , where I is a nondegenerate interval in UP, by requiring that for any of the above sequences of subdivisions of [a, b ] ,
where the above limit should be finite and of a value which is independent of the chosen sequence of subdivisions. It is easy to relate Riemann-Stieltjes integrals to integrals with respect are Bore1 measurable functions (see to measures when F(A) and .(A) Exercise 3.3). It is equally easy to prove in a direct manner that most of the usual properties of Riemann integrals (which are obviously special cases of Riemann-Stieltjes integrals in which a(A) = A) are also properties of Riemann-Stieltjes integrals (see Exercises 3.4-3.8). We find the concept of Riemann-Stieltjes integration of importance because it enables us to give a meaning to the integral in (3.16), namely, the integral in (3.16) can be written as a cross-iterated Riemann-Stieltjes integral with respect to the function (3.21)
~ ( h4) , =
S
A0
0
(g I E f ePiH0tf)dt.
442
V. Quantum Mechanical Scattering Theory
Let us introduce the following convenient notation: (3.22) u((A, A'] x (A,, A,'])
= ~ ( h 'A,', )
- ~ ( h 'A,,)
-
u(A, A);
+ u(A,
4).
We can establish that (3.16) is true by means of Lemma 3.1.
*Lemma 3.1. Suppose the real function o(A, A,) is of bounded variation on [a, b] x [c, d ] , i.e., that for any A E [a, b] and A, E [c, d ] we have
where the supremum is taken over all subdivisions (3.24) a
=
A(')
< A(') < ... < A'"'
- A,
c =
A t ) < A t ) < ... < A t ) = A,
of [a, A] x [c, A,]. If the complex function F(A, A,) is continuous on [a, b] x [c, d ] , then both its cross-iterated Riemann-Stieltjes integrals with respect to o(A, A,) exist and (3.25)
J; 4 idv, A,)
i,4, m, d
A,)
=
4ocl.(~,
Jb
A,)
44,A,).
Proof. I t is easy to see that the functions 4 A , A,)
=
+[.(A,
A,)
+ 4 4 c) * .(A,
A,)]
are nondecreasing in [a, b] x [c, d ] in the sense that a,(h, A,) whenever A A' and A, A,'. Since
<
<
4 4 A,)
=
o+(k A,)
-
< of(A', A,')
4, A,),
we can exploit the linearity properties (see Exercises 6.4 and 6.7) of cross-iterated integrals to reduce the integrals in (3.25) to a sum of integrals of real functions with respect to a+(A, A,) and o-(A, A,). Hence, it is sufficient to prove the lemma for the case of a nondecreasing .(A, A,) of bounded variation and for real F(A, A,). By using the mean-value theorem for Riemann-Stieltjes integrals (see Exercise 3.8), we obtain
443
3. General Time-Independent Two-Body Scattering Theory
<
A;(’”) for some relation (3.20), we get
=
lim
m
< Ah‘“).
Combining this result with the defining
n
CC
o ( ( ~ ( i - lY) ~ ( i ) ]
$’(~(i),
i=l k = l
I)
( ~ t - 1 3) ~ 0( k )
in the limit of finer and finer subdivisions (3.24) of [a, b] x [c, 4. T h e existence of this limit and its independence of the chosen sequence of subdivisions {(A(i), Ah’”))} is obviously a necessary and sufficient condition for the existence of the integral. T o establish the existence of this limit and its independence on the sequence, it is sufficient to note that if {(ACi*j), Ahk,”’>} is a subdivision of [a, b] x [c, 4, which is finer than the subdivision {(Aci), A:’”))}, and if ( A ’ ( i , j ) , A ’ h k s Z ) ’ ) are points within the meshes determined by {(A(i,j), A:’”,’))}, then due to the uniform continuity of F(A, A,) we can achieve for any given E > 0
I q x ( i , d ,A $ k , Z ) )
-
qp, @’)I <
for all sufficiently fine subdivisions {(A(i), Ahk))}. Consequently, if and Ahk*’) denote the points preceding A(iJ) and Ah’”.”, respectively, in the subdivision, then
This establishes the existence of the considered limit, as well as its independence of the sequence of subdivisions. By the same token
444
V. Quantum Mechanical Scattering Theory
also exists. By exploiting again the uniform continuity of F(A, A,) in [a, b] x [c, 4 , we conclude that for any E > 0
1 F(h"'i', A;;@))
-
F(j+, A p ) ]
for all sufficiently fine subdivisions (3.24) of [a, b] x [c, 4 . Hence, the absolute value of the difference of the two integrals in (3.25) is smaller than
i = l k=l
= EU((U,
b] x
(E,
d]).
Since this is true for arbitrarily small values of that (3.25) holds. Q.E.D.
E
> 0, we conclude
Let us compute the total variation function v(A, A,), defined by (3.23), for the function a(A, A,) given in (3.21). By using in the process a decomposition of the type (5.13) in Chapter 111, we easily derive
Since the integrand in the above integral is continuous and therefore integrable, it follows that a(X, A,) is of bounded variation. Consequently, Lemma 3.1 can be applied to infer that
445
3. General Time-Independent Two-Body Scattering Theory
T o establish the validity of (3.16), we have to prove that the above relation remains true in the limit when a + - 00 and b, T -+ 00. I n view of the relations (see Exercise 6.8)
+
so s-, s: s: 1:" :s 1," sT, so s, :s s," sE s: s-, s, sP, 16" ss" 1," +m
(3.27)
+m
t m
-
=
-
=
+m
+
+
+
+
+m
'
+m
and (3.26), we see that this will be the case if and only if the integrals on the right-hand side of the equalities (3.27) exist and vanish when a --f -a and b, T -+ +a. Employing the procedure used in deriving (3.26) we arrive at the following estimate:
I 1;:
dtPt
s"
eiAtd,(g
I Efe-iHoy)l
a0
< 4 1: e-'t{(e-iHOtf + g I E H ( ( ~ ,, b , ] ) ( e - i ~ O y+ g))
+ (e?Ot + ig I EH((a,, b,l)(e-"otf + ig))
+ d ( e - i H o t f I @((a, ,b,])
epiH0tf)
+ 1/2(g I ~ ~ ( ,( b,l)g)) a , dt.
It is obvious that the integral on the right-hand side of the above inequality can be made arbitrarily small for sufficiently large values of either co (for arbitrary values of do), or of a, (for arbitrary values of bo), or of -b, (for arbitrary values of a,). Since, according to Lemma 3.1, the order of integration in X and t can be reversed without changing the value of this integral, we conclude that all integrals on the right-hand sides of the relations (3.27) vanish in the limit when a + --OO or b, T --+ +a;for example, for the representative case of the first of these integrals, we can choose for given 77 > 0 a ~ ( 7 )and a sequence ~(7= ) T, < T~ < T~ < .*. diverging to infinity which is such that
and consequently
V, Quantum Mechanical Scattering Theory
446
Thus, we conclude that (3.26) stays valid in the limit a ---t --GO and b, T -+ $00, and therefore (3.16) is ’ true. Using the fact that e - i H o l is a function of H, , and applying Lemma 3.1 in the same manner as before to interchange orders of integration, we arrive at the result W
0
exp[-(E =
W
0
=
-
ih’)t](g I E y exp[-iH0t] f) dt
dt exp[ -(G
Strn dAo -m
Jm 0
-
ih’)t]
exp[-(E
-
+m
-m
ih‘
exp[ -ihOt] dAo(EFgI E? f)
+ iho)t](EfgI E T f ) dt
Inserting this result in (3.16), we obtain
I n a completely analogous manner we can derive a corresponding relation for !2+G:
T h e two relations (3.28) and (3.29) are equivalent to the single relation
if the parameter E > 0 is replaced by the new parameter 71 # 0, which can assume also negative values. Using the defining formula (3.20) for cross-iterated integrals, we obtain from (3.30) (Q,*g
If) =
where the last integral is dejined by the limit preceding it.”
* See also Definition 3.4 for integrals of this type in Appendix 3.7 to this section.
3. General Time-Independent Two-Body Scattering Theory
447
According to Theorem 2.4 of Chapter IV,
and consequently we have (3.31)
+
+
Since iq(X - Ho i7)-1 and (A - Ho)(X- H,, iq)-l are bounded functions of H , , they are defined everywhere in 2. According to Theorem 2.4 of Chapter IV, we can add and multiply these operator functions as we would ordinary functions, and therefore we have the identity X - H, ;.! =I(3.32) X - H, i7 X - Ho i7
+
+
Substituting the right-hand side of the above equation in (3.31), and using symbolic notation (see also Appendix 3.7 to this section), we arrive at the following expression for Q,*: (3.33)
.n,*
=
1-1
h - H,
+m --na
X
-
H,
+ i7 d A E f .
Consider a free state eciHot!P,Y EM, , which is the incoming asymptotic state of e - i H t Y + or the outgoing asymptotic state of eciH1YI-. Since obviously
we deduce from (3.14) and (2.10)
This result, combined with (3.33) leads, after a rearrangement of terms, to the equations (3.34)
T h e above two relations can be considered to be equations for determining Y- and Y+, respectively. As such, they represent a Hilbert space form of the type I Lippmann-Schwinger equations. I n the next sections we shall see that in the special case of potential scattering these equations
V. Quantum Mechanical Scattering Theory
448
lead to a more commonly encountered form of integral equations for distorted plane waves.
3.3. INTEGRAL REPRESENTATIONS FOR 52, T h e main drawback of (3.34), viewed as equations for finding Y* when Y is given, is that they require a knowledge of the spectral function EF of the total Hamiltonian H . Now, in practice, EF is an unknown quantity which is very difficult to compute. I n fact, if E,H were known, then one could easily compute the time evolution operator U ( t ,to) = exp[-iH(t - to)], and thus solve the dynamics of the problem generally, without even having to resort to scattering theory techniques. A practically more suitable set of equations relating Y and Y*,which yield themselves to perturbation techniques, can be derived from (3.1) by applying the spectral theorem first to H,, and then to H , and otherwise proceeding in the same manner as in the derivation of (3.16):
Hence, we end u p with the relation
An analogous relation can be derived for SZ,, . After reintroducing the parameter 7 # 0, we can write (3.36) and the corresponding relation for fin+€jointly in the symbolic form (3.37)
Naturally, if we exploit the relation (3.38)
-iv H - A,
-
iv
=I-
H
H - A, -
A, - iv'
449
3. General Time-Independent Two-Body Scattering Theory
(3.37) can be recast as (3.39) Hence, if we again write (3.40)
Y*
= Q*Y =
s-lim Q,,Y, a+*,
P ! E M* ,
then in view of (3.39), (3.41)
Y*
=
Y
+ s-lim V+*O
--m
A. - H
+ iq d"E 2 Y . A
T h e above two equations are the type 11 Lippmann-Schwinger equations in Hilbert space. They are usually considered to be solutions of (3.34). However, they actually are not explicit solutions of the type I equations iV)-l, (3.34) since they contain the operator function ( H - h,)(X, - H which is unknown in practice. O n the other hand, we shall see later in this section [see (3.67)] that (3.41) can be solved by iterative procedures. Let us write (3.39) in the form
+
Taking adjoints on both sides, we obtain
I n symbolic notation the above relation assumes the form (3.43)
which will be very useful further on. 3.4.
THETRANSITION AMPLITUDE
It was mentioned in $1 of Chapter IV that if Y + ( t ) and YJt) are states in the Schroedinger picture, the 1 ( Y + ( t ) 1 Y - ( t ) ) j 2 is called the transition probability from YJt) to Y+(t);furthermore, the complex number ( Y + ( t ) I Y-(t)) is called the transition (probability) amplitude from Y J t ) to Y + ( t ) .
450
V. Quantum Mechanical Scattering Theory
Let us assume that Y+(t)and YJt) are scattering states, with asymptotic states Yy(t) and YEx(t),respectively. Since Y*(O) = Q+Y?(O)
= Q_YYt(O),
we easily obtain (Y+(t)I Y-(t)> = (Y+(O) I Y-(O)> = (YYt(0) I SYF(0)).
Let us simplify the notation by writing
YF(o)= YJ~,
yYt(o) = Ye,
where the indices i and f stand for “initial” and “final,” respectively. Using the properties of strong limits, we obtain
=
s-lirn(Q-€YfI s-lim Q+,Yi> €++O
€+fO
Substituting the expression (3.39) for Q-€ in (Q-fYt I Q+Yi), we get
O n the other hand, after using (3.43) to express the identity operator in terms of the other two operators occuring in that equation and then applying it to the vector Q+Yi,we arrive at the relation
After substituting the above expression for Q+Yiin the first term on the right-hand side of (3.45) and carrying out an obvious summation, we obtain
I n view of (3.44) and the fact that if
M+ = M- ,
3. General Time-Independent Two-Body Scattering Theory
45 I
we deduce from (3.47) that
I n terms of the T operator (3.50)
and provided that
M,
=
M-
T = -1 - s 2rri ’ , we can write (3.49) in the form
By taking advantage of the intertwining property (2.4), we immediately obtain from (3.51)
It is interesting to note that on account of the relations (3.53)
Yi
= Y F ( 0 )=
s-lim t-l-m
eiHot
e
-iHt
WO),
the vectors Yf and Yi can be considered to be the asymptotic states of the interaction-picture states [see (3.43) of Chapter IV] (3.55)
Y*(t)
=P o t
e-iHtY*(0),
respectively. Thus, it can be said that (3.52) provides an expression for the transition amplitude (Yf I T Y i ) from the incoming interactionpicture asymptotic state Yi to the outgoing interaction-picture asymptotic state Yf .
3.5. THERESOLVENT OF
AN
OPERATOR
I n the Lippmann-Schwinger equations (3.34) and (3.41) we encountered operators of the form ( A - h - iq)-l, where in the first case A stands for H and in the second case A stands for H , . Operators of the form ( A - {)-l, where { is in general a complex number and A is
452
V. Quantum Mechanical Scattering Theory
a linear operator in #, play a very important role in functional analysis. Since they have been systematically studied, it is advisable to acquaint oneself with the results of such studies, and then apply these results to the special problems at hand.
Definition 3.2. If A is any linear operator in #, the resolvent RA(5)of A is the operator-valued function (3.56)
RA(5)
=
( A - C)-l,
5 C1,
defined at all complex values of 5 at which ( A - <)-l exists. We note that, according to Theorem 4.2 of Chapter 111, ( A - 0-l exists if and only if the equation ( A - 5)f = 0 has the unique solution f = 0, i.e., if and only if 5 is not an eigenvalue of A. We can classify the points 5 in the complex plane C1 in relation to a given linear operator A as in Definition 3.3.
Definition 3.3. T h e set of complex numbers 5 consisting of all points 5 E C1at which the resolvent RA(5)is a bounded operator defined densely in # is called the resolvent set of A. T h e complement of the resolvent set is called the spectrum S A of A. The spectrum S A is the disjoint union of the point spectrum Sg , continuous spectrum S: , and residual spectrum St, which consist of all points which have the following respective properties: Sg of the points 5 E S A at which I?,(<) does not exist; St of all E SA for which RA(c)exists, is unbounded and defined on a dense subset of X ; :S of all 5 E S A at which R,(c) exists but it is not defined in a dense subset of 2. I n the case where A is a self-adjoint operator, the above definition of the spectrum coincides with Definition 6.2 of Chapter 111. T o establish this equivalence, we need to show that
<
(3.57)
Iff belongs to the domain of ( A - 0 - 1 , there is a vector g E g Asuch that f = ( A - S)g. Consequently, we have
Conversely, if
3. General Time-Independent Two-Body Scattering Theory
453
then, according to Theorem 2.5 of Chapter IV (3.58)
is defined. Moreover, g
=
[1/(A - [)I f belongs to g Abecause
Hence, by (2.25) of Chapter IV,
i.e., f is in the domain of (A - [)-l. Thus, (3.57) is established. By comparing (3.58) here and (2.20) of Chapter IV, we see that the above argument also establishes that when A is self-adjoint,
1, h--5 1
(3.59)
RA(-5)
=
dEA A =1 A--5
Hence, for self-adjoint operators, Definition 6.3 of Chapter I11 and Definition 3.3 of the point spectrum Sg of A coincide, since EA({[})# 0, 5 E W,if and only if [ is an eigenvalue of A , in which case (A - [)-l does not exist. We shall prove now the equivalence of the two definitions (6.3 of Chapter I11 and 3.3) for the continuous spectrum :S of a self-adjoint operator. When, for a fixed real 5, E A ( [ [ - E , [ €1) # 0 for all E > 0 and E4({[}) = 0, we must have either E A ( [ [ - E , 5 ) ) # 0 or E A ( ( [ ,5 €1) # 0 for all E > 0. Let us say that the first alternative is the case. Then the equations
+
+
ER("
can be satisfied for all E
- E,
O)f€
= f €2
llfE
/I = 1,
> 0. Inasmuch as
we conclude that R,([) is an unbounded operator. On the other hand, R A ( [ ) is defined on the dense set of all vectors g satisfying EA([t;- E, 5 ~ ] ) g= 0 for some E > 0. Thus, if [ is an element of the continuous spectrum in the sense of Definition 6.2 in Chapter 111, then
+
V. Quantum Mechanical Scattering Theory
454
it belongs to the continuous spectrum in the sense of Definition 3.3. T h e complete equivalence of these two definitions of St for a self-adjoint operator A follows from Lemma 3.2.
Lemma 3.2. T h e resolvent RA(5)of a self-adjoint operator A is a bounded linear operator if and only if either I m 5 f: 0 or 1; is real but - E, 5 + €1) = 0 for some E > 0; in either case 9BA(1) =X.
+
Proof. We have seen earlier that if 5 is real and EA([S - E, 5 E)] # 0 for all E > 0, then RA(5)either does not exist or, if it exists, it is an unbounded operator. Assume now that 5 is real and EA((5 - E,, 5 c0)) = 0 for some e0 > 0. Since the function 1/(5 - A) is bounded on
+
(-a, 5 - €01 u [5
we conclude that
+
€0,
+a),
Hence, it follows from (3.57) that RA(C)is bounded and defined for all f E X . I n the case where I m 5 # 0, we have
s,
1
IA
-5
lP
<
d /I ERf/12
1
jRl d II
T h e above inequality establishes that when Im
11 RA(C)l/
(3.60)
and I?,(()
is defined for all f
E
<
5 # 0,
1 9
2. Q.E.D.
It follows from the above lemma that a self-adjoint operator has no residual spectrum. 3.6.
THERESOLVENT METHODIN SCATTERING THEORY
T h e two resolvents we encounter in time-independent scattering theory are the resolvent R,,(() of the free Hamiltonian H , and the resolvent R(5) of the total Hamiltonian H . However, while Ro(5) is easy to compute in practice, R(5) is an unknown quantity. Thus, Theorem 3.2, which relates R(5) and Ro((),proves to be very useful.
455
3. General Time-Independent Two-Body Scattering Theory
Theorem 3.2. Let H and Ha be any two (not necessarily selfadjoint) operators in the Hilbert space 2, having identical domains of . T h e following second resolvent equations are definition g H= gH0 satisfied by the resolvents R(5) and Ro(S)of H and H, , respectively: (3.61) R(1) - Ro(5) = -Ro(O(H - Ho)R(O = --R(I)(H - Ho) Ro(O for any 5 which belongs to the resolvent sets of both H and H a , i.e., for 5 $ SH u S H o .
Proof. According to Lemma 3.3, R(5) and Ro(5) are defined on the entire Hilbert space 2 whenever 5 does not belong to the spectra of H and Ha . Hence, for every f E 2 there are vectors g ~9~ and h E gH0 such that f = ( H - 5)g = (Ha - 5)h. Since g H= BHo, g also belongs , therefore to g H 0and ROMH - H0)g
= Ro(5)(H - Ho) R(5)f
is defined. Setting above ( H - H,)g the relation Ro(5)(H - H0)g
=
= RO(O(ff - o g
[H - 5 -
- (Ha - 5)]g, we
get
g = RO(0f - R(Of,
which is equivalent to the first of the identities (3.61). T h e second identity in (3.61) can be established by a similar procedure. Q.E.D. Before proceeding with the applications of the second resolvent equations, we shall derive, for the sake of completeness, the first resolvent equations.
Theorem 3.3. If C1 and 5, belong to the resolvent set of the linear operator A, then the first resolvent equations (3.62) are satisfied.
Proof. T h e range of A - 5 is identical to the domain of definition of RA(C)= ( A - <)-l, which coincides with J? when 5 is in the resolvent set of A. Hence, for any f E A?,a vectorg can be found so that f = ( A - 5,)(A - 5,)g, and therefore (3.63)
(51 - 5 2 ) RA(4.1) R A ( S 2 ) f
= (51
-
52)g.
On the other hand, we have (3.64)
[R~(5i)- R~(52)I.f = ( A - 52)g - ( A - I J g
= (51 -
Since (3.63) and (3.64) hold for any f E 2, (3.62) follows.
Q.E.D.
456
V. Quantum Mechanical Scattering Theory
Let us return now to our main task of giving a perturbational method for computing transition probabilities. Using Theorem 3.2 we infer that, if BH = aHo , (3.65)
R(5) = Ro(5) - RO(5)fflR(5),
whenever I m we obtain
5 # 0. After
ffl
=H
-
Ho ,
n successive iterations of the above relation,
where the remainder R, is (3.67)
R,
=
(-1)” (RO(5)Hl)”+l R(1).
+
If the above remainder converges to zero, in some sense, * when n -+ CO, then (3.66) provides us with a means of computing R(5) when R,(<) is known: (3.68) R(5) = Ro(1)- Ro(5)HiRo(5)
+ Ro(O HiRo(5) HiRo(O f- .*.
3
This result combined with (3.67) leads eventually to a “perturbation” solution of the Lippmann-Schwinger equations. This solution is obtained by substituting for R(ho iv) = ( H - A, - iq)-l in (3.41) the series on the right-hand side of (3.68). If the series so obtained for the integral in (3.41) converges, and the order of summation and integration can be interchanged, then Y*can be computed to any order.
+
3.7.
OF VECTORAPPENDIX: INTEGRATION AND OPERATOR-VALUED FUNCTIONS
I n the last two sections we encountered integrals, such as (3.5), which are special cases of Bochner integrals. T h e theory of Bochner integration is based on the following theorem.
Theorem 3.4. Suppose p is a measure in the measurable space % and that f(.$), 4 E R, is a vector-valued function on the measurable set R, assuming values in a Hilbert space 2.If (g I f(4)) is measurable on R for anyg E 2 and if (3.69)
j, I l f ( O l l ~ P ( ~<) +a,
* See also Lemma 5.2 and Theorem 5.9 of this chapter.
3. General Time-Independent Two-Body Scattering Theory
457
then: (a) there is a unique vector h E 2, called the Bochner integral of f ( E ) on R , denoted by
which is such that for all g E 2 (3.70) (b) if A is a bounded operator on 2 and IIAf(f)11is measurable, then (3.71)
A
J f ( 5 ) 440 = J R
R
Af(5)4-43
Proof. (a) T h e linear functional
is bounded on account of (3.69):
Hence, by Riesz' theorem there is a unique vector h which is such that $(g) = ( h 1 g) for all g E Z . (b) T h e Bochner integral of Af(0 exists since (g I Af(<))= (A*g j f(5)) is measurable for any g E 2,and
I n order to prove (3.71) it is necessary and sufficient to show that (3.72)
=
J
R
(g I Af(ED 44'3
for all g E X . Using the definition of the Bochner integral, we get ( g I A h ) = (A*g I h ) =
thus establishing that (3.72) is true.
JR (A*# lf(5)) 440, Q.E.D.
458
V. Quantum Mechanical Scattering Theory
T h e Bochner integral for vector-valued functions can be used to define Bochner integrals for operator-valued functions A([). As a matter of fact, if the Bochner integral AR[ffl
=
1 [A(Oflddf) R
exists for all f E 2,then the mapping f -+A R [ f ]is linear since
can be defined to be the Bochner integral of A ( [ )on R . It is useful to note that the integral in (3.73) has the property (see Exercise 3.9) (3.74)
A,*
=
j
R
A*(.!) d P ( 0
Naturally, all of the above results can be immediately generalized to complex measures p by decomposing such measures into a sum of real measures. A certain kind of generalization of Bochner integrals has been encountered in (3.33), (3.34), (3.40), and (3.41). We define such integrals in a more general context by means of the following concept of integration.
Definition 3.4. Let f,)and A(h) be functions in h which assign to every X E [a, b] a vector f,)in 2 and a bounded operator A(h) on 2, respectively. Suppose that for any sequence of subdivisions a = A(0) < h'1) < ... < = b of [a, b] with norm 6 = maxi h(k) - h ( k - 1 )
converging to zero, and that for any n
(3.75)
E
l
[X(k-l),
459
3. General Time-Independent Two-Body Scattering Theory
exists for some fixed g E 8. If this limit is the same for any choice of sequences of subdivisions with shrinking norm 6 +. 0 and for any appropriate choice of X ' ( k ) , then it is denoted by the integral symbol
A corresponding improper integral is defined as
I+d& I m f , )
(3.77)
=
ijy,
J
b
dA
with a similar definition holding when a + - co. We encountered integrals of the type (3.76) in (3.31) and (3.39). I n those cases, fA is of the form E,f, where EA is a spectral function and f is a fixed vector in Z . For a given interval I C [wl,the function +I(d =
J
I
d,
is obviously linear. If &(g) is defined for all g E .%? and if it is bounded, then by Riesz' theorem there is a unique vector hI which satisfies the equality (hI I g) = $ I ( g ) for all g E 2. We denote this vector by the integration symbol
j, 4 4 4 f A .
(3.78)
Suppose KAis a operator-valued function. It is easy to see that
J 4 4 d,[K,(a,f, + a z f d J 4 4 dA[alK,f, + azKf2l a1 J 4 4 d A W 1 + 144d A W 2 =
I
I
=
I
a2
I
and that the existence of the integrals on the right-hand side of the above relation implies the existence of the integral on the left-hand side. Hence, the mapping which takes f into Adf)=
j
I
4 4 dAKAf
is linear. We denote the linear operator A, by the integration symbol (3.79)
A,
=
1 I
A ( h ) d A K A.
460
V. Quantum Mechanical Scattering Theory
Integrals of this kind have been encountered in formulas for Q, as (3.33) and (3.37).
, such
EXERCISES 0
3.1.
Show that there are unique operators sZ-,(s)
< E \< s, which satisfy the relations (g I Q-,(s)f> = E (g I Q+,(s)f>
=
for allf, g E Z , and that (( sZ,,(s)~~
3.2.
E
and sZ.+E(s),
j’ e-Yg I Q(W>at,
s”
--s
eft(g I Q(W>dt
< 1.
Prove that Qh., = s-lim Q,,(s). W+W
3.3. Suppose that the Riemann-Stieltjes integral of F(X) with respect to a(h) exists in [a, b], that F(h) is Bore1 measurable, and that .(A) is nondecreasing. Show that F(h) is integrable with respect to a measure p(B) which satisfies the relation p(( - co,A ] ) = .(A), and that
p(4444 = L 3.4.
b ,
F(4 444.
Derive from the basic Definition 3.1 that the integrals (3.18)-
(3.20) are linear; for example,
3.5. Prove that if the Riemann-Stieltjes and [c, b] exists ( a < c < b), then
jcF(h) dU(X) = a
a
F(h) &(A)
integral on [a, b], [a, c]
+ J F(h) du(X), b
a
< c < b.
C
Extend the proof to show that for cross-iterated integrals
Remark. If .(A) as well as F(X) are discontinuous at c, the integral on b] might not exist, although the integrals on [u, c] and [c, d ] exist.
[u,
461
3. General Time-Independent Two-Body Scattering Theory
3.6.
Show that if
a(/\)is
nondecreasing and F,(A)
< F2(A),then
<
Using this result, show that if o(A, A,) is nondecreasing [i.e., o(A, A,) A‘ and A. A,’] and Fl(A,A,) Fz(AAO), then
@’, Ao’) whenever A
<
<
<
3.7. Assuming that the respective Riemann-Stieltjes integrals for
u1 and u2 exist, prove that the corresponding integrals for the linear
combinations alul
+ azu2, a, , a2 E C1, exist and
where I and I,,are arbitrary nondegenerate intervals in R1.
3.8. Using the result of Exercise 3.6, prove the mean-value theorem for Riemann-Stieltjes integrals: IfF(A) is real and continuous, and if u(A) is nondecreasing and bounded on [a, b], then there is a point A’ E [a,b] for which
j’F(A) d,o(X) = F ( X ) j a
b
d*u(A) = F(X’)u((a, b ] ) .
a
Show that the above integral exists if F(h) and assumptions made earlier.
o(A)
satisfy the
3.9. Prove that if (3.73) exists, then the relation (3.74) holds. 3.10. Prove that the resolvent RA(() of a self-adjoint operator A can be represented for I m ( < 0 by a Bochner integral: R(5) = i
1’” 0
ei(A-C)t
dt.
462 4.
V. Quantum Mechanical Scattering Theory
Basic Concepts of Time-Independent Scattering Theory for Hamiltonians with Eigenfunction Expansions
4.1. FREEPLANEWAVESIN THREE DIMENSIONS I n $3 we presented a very general framework for time-independent scattering theory. However, in practice one encounters much more specialized versions of that framework, which are, on the other hand, more convenient to deal with from a computational point of view. T o derive these formulations rigorously from the general framework of $3, we need the existence of eigenfunction expansions for the Hamiltonians H , and H . Before formulating the concept of eigenfunction expansions on a general level, we shall elucidate this concept by studying it in the special case of the kinetic energy (or “free”) Schroedinger operator H, on L2([w3), defined by means of the differential operator form
If we consider H i to be an operator acting on the space of all everywhere twice-differentiable functions f (r), then Hi has eigenfunctions @k(r)
1
k2
Gk(r) = -eikr 2m ak(r), (24312
associated with the eigennumbers (4.3)
A(k)
=
k2/2m.
I n physical literature the functions ak(r) are called free plane waves. However, these eigenfunctions are not square integrable on R3 in the Lebesgue measure, and therefore they do not represent eigenvectors of H,. T h e family (cDk(r):k E R3} of eigenfunctions of H i has some remarkable properties, which follow from Theorems 4.5 and 4.6 of Chapter 111. According to Theorem 4.5 of Chapter 111, if 4 EL~([W~), and if B, , B, ,... are any bounded measurable sets, the functions (xB,$)(r) have Fourier transforms UF(xB,+),since for every n = 1,2, ... the function XB,(r)$(r) is integrable as well as square integrable with respect to the Lebesgue measure. On the other hand, if B , = W,then
(J2=l
4. Time-Independent Scattering Theory for Hamiltonians
// xB,+
-
+ 11
-+0 when
in Chapter 111,
463
n -+ +a,and therefore, in view of Theorem 4.6 UF+ = s-lim U&B,+).
(4.4)
n++m
Let us introduce some convenient notation in Definition 4.1, Suppose F ( x , x’) is the kernel of an integral operator
Definition 4.1. on L2(S , v) (4.5)
and that 9 is the family of all functions g(x) for which the integral in , (4.5) exists. Then we shall write for some h € L 2 ( S v) (Fh)(x) = 1.i.m.
Js
F(x, x’) h(x’) dv(x’)
if the linear operator defined by (4.5) has an unique extension f = Fh to h, i.e., if for any sequence g, ,g, ,... E 9 converging strongly to h there is an unique f E L2(S,v) such that
We note that if F happens to be a bounded operator on 9, then by the extension principle of bounded operators (Chapter 111, Theorem 2.6), , and represents the 1.i.m. J”F(x)h(x) dv(x) is defined for all h € L 2 ( S v) extension of the operator F to h. I n particular, if F is the Fourier transform U,, then (4.8)
$(k)
=
(U,+)(k)
=
cikr+(r)dr
( 2 ~ ) -1.i.m. ~/~ JP3
is well defined. Moreover, by virtue of (4.11) in Chapter I11 we also have . .
and consequently (4.10)
I,,
+(r) = ( 2 ~ ) -1.i.m. ~/~
eirk$(k)
.
dk.
464
V. Quantum Mechanical Scattering Theory
T h e formal analogy of (4.10) and (4.8) with the expansion formula of a vector 9 E 2,
*
=
c
(en
n
I 4)en
9
in an orthonormal basis ek becomes obvious if we rewrite these two formulas in the form (4.11) +(r)
=
1.i.m.
3$(k) Gk(r) dk,
$(k)
=
R
1.i.m.
I
Dk*(r) #(r) dr
and introduce the convenient notation (4.12)
(@k I $> = 1.i.m.
I
Qk*(r) +(r) dr
= $(k).
T h e fact that (Gk I f ) is not related to the inner product in L2(R3) is emphasized in this notation by the round bracket in (. [ -). Due to such analogies, the relations (4.11) are referred to as eigenfunction expansions of $(r) by means of free plane waves Qk(r). I t is important to realize that we can easily express the spectral measure EHo(B)of H,, with the help of Qk(r).Indeed, we recall that the momentum operators are essentially multiplication operators when they act on $(k), (4.13) (P'")#)" (k)
= k,$(k),
(PW/J)"(k)
=
k,$(k),
(P'"#)" (k)
= k,$(k),
so that the k variables are in this case identical to the momentum variables p. Since H,, is a function of P,
(4.14)
H
O
-
P2
-=fl(P),
- 2m
we have, in the momentum space,
or equivalently, in the configuration space,
(4.15) 4.2.
DISTORTED PLANEWAVES
Let us now consider a total Schroedinger operator H defined by means of the differential operator form (4.16)
Hr
=
-(1/2m)A
+ V(r).
4. Time-Independent Scattering Theory for Hamiltonians
465
I n $96-7 we shall extend to H’ the above results on H: by proving that, under certain assumptions on the potential V(r), the following statements are true. For each vector k E R3, there is a unique solution”
@k)(r)= (2+3/2
(eikr
+ vk(r))
of the differential equation (4.17)
-
for which, in spherical coordinates r, 0, and tically as follows:
+, vk(r) behaves
asympto-
wheref,(O,+) is a function uniquely determined by vk(r). T h e family {@i+)(r): k E R3} provides an eigenfunction expansion for any element $+ €L;,(R2) in the sense that (4.19)
where Lic(R3) = EH(S z ) L2(R3) denotes the closed linear subspace of L2(R3)corresponding to the continuous spectrum S z of H . The function $(k) appearing in the eigenfunction expansion (4.19) is not the Fourier-Plancherel transform of $+(r),but rather the FourierPlancherel transform of another function related to $+ by the equation $+ = Q+$. In fact, one of the most important results (which will be derived in 97) of time-independent potential scattering is that Q+ is in the present case a partial isometry with initial domain M, = L2(R3)and final domain R+ = Lic(R3). Hence, to every $+ EL~JIW~) corresponds a unique $ E L2(R3)such that $I+ = Q+$. In physical literature the functions @L+)(r)are referred to as the outgoing (or retarded) distorted plane waves, while fk(O,+) is called the scattering amplitude since, as we shall see in 97, it is intimately related to the function F 1 ) ( p ;w , w ’ ) introduced in (1.31): (4.20)
fk(w)
=
(&/I
k I) T(’)(Ik 1;
-w,
w’),
-w
=
(T
- 8, 2n - 4), w’ = (8’, 4’).
* Strictly speaking, in general this function does not satisfy the equation in the set of singularitiesof V(r),which is required to be of Lebesgue measure zero.
V. Quantum Mechanical Scattering Theory
466
T h e justification of the term “distorted plane wave” is that in the special case when V(r) = 0 we have vk(r) = 0, and @:+)(r)becomes the free plane wave Ok(r). I n addition, the function
@k)(r;t)
(4.21)
= @p’(r) exp[--i(k2/2m)t]
provides a solution of the time-dependent Schroedinger equation (4.22)
a
i--@p)(r;t) at
=
[-(1/2m)A
+ V(r)] @p)(r;t ) .
T h e “distorted part,” vk(r) exp[ -i(k2/2m)t] of this wave describes, heuristically speaking, a process which recedes away from the scattering center r = 0. T h e functions (4.23)
@-)(r)
=
@!t*(r)
are called incoming (or advanced) distorted plane waves for analogous we have the reasons. They also satisfy (4.16), and for any $- EL;~([W~), expansion (4.24)
where $- = SZ-$, with $ uniquely determined by I,!- due to the fact that M, = L2([w3)and R- = Lic([w3). A further significant result of $6, which in fact represents a generalization of (4.19) and (4.24), is that the spectral measure EX(B) can be computed on Li,(R3)by means of O;&)(r): (4.25)
(E*(B) # f ) ( r )
=
1.i.m.
@p’(r) $(k) dk.
I n particular, for B = R1, the relations (4.25) assume the form (4.19) or (4.24), respectively. It is important to realize that (4.15) is also a special case of (4.25), since when V(r) = 0, we have SZ, = SZ- = 1 and Li0(R3) = L2(R3). 4.3.
FREEAND DISTORTED SPHERICAL WAVES
T h e free plane waves Ok(r) are not the only eigenfunctions of the differential operator H i . It is easy to verify that the functions (4.26)
akZm(r) = 4 4 -i)yl(kr) Yzm( -6,
-4)
467
4. Time-Independent Scattering Theory for Hamiltonians
are also eigenfunctions of HA :
-L2
(4.28)
=
[-
i a a sine-] sin 0 a0 a0
1 +--sin20 a@
a2
'
for any values (4.29)
0
< k < +a,
1 = 0, 1 , 2,...,
m = --I,
-Z
+ 1 ,..., +Z.
T h e function QkIm(r,8, (6) is called an outgoing free spherical wave. It will be shown in 97 that the family of all functions (4.26) provides us with eigenfunction expansions for any I,!J E L2(R3):
where, using the notation 0,introduced in (7.10) of Chapter 11, we have @zlm(r) +(r, 8,d) r2 sin B dr d0 d+.
(4.31)
I n the case that V(r) is spherically symmetric, i.e., V(r) = V,,(r),the differential operator H' has the eigenfunctions
These eigenfunctions are called, respectively, outgoing and incoming distorted spherical waves and are generalizations of Qklm ; i.e., (4.33)
H'@EA(r,094) = (- 2m A + vow)@ g k0,4) = 2y &A@, 1
and for any
where I,!J+
i,h*
E Lie(R3)
= Q,$,
and J l m ( k )is given by (4.31).
k2
468
V. Quantum Mechanical Scattering Theory
It is well known (see Butkov [1968, Section 9.101) that the asymptotic behavior for large r of the spherical Bessel functionj,(kr) is (4.35)
j,(kr)
-
( I l k ) sin(kr
- h/2),
I --+
+a.
We shall see in 97 that the functions Rk$)(r)have a similar asymptotic behavior
I n physical literature 6,(k) are called phase shifts, since they represent the change of phase from the free spherical waves corresponding to the noninteraction case V ( r ) = 0, to the distorted spherical waves. We shall see in 97 that they play a crucial role in the computation of the S operator, which can be written in the form (4.37)
=
(w;Vb@)
exp[2&(k)l $Zm(k).
T h e function S,(k) = exp[2iSl(k)] is called the S matrix in spherical coordinates. T h e reason for this terminology is that if we use Dirac’s bra and ket notation to write 1 klm) = D k l m nthen , taking in (4.31) and (4.36) # = Dklm and working formally, without paying attention to the real mathematical meaning of these expressions, we get (k’l’m’ I S I klm)
=
6(k’ - k) 6,,, 6,,
exp[2i6,(k)].
I n 97 we shall establish the following generalization of (4.34):
where (4.39)
Ao(k)= k2/2m.
These formulas can be recast in a more compact form if we introduce where the measure po = p ( k ) x p(u)on g3,
469
4. Time-Independent Scattering Theory for Hamiltonians
It is easy to verify (see Exercise 4.1) that, written in terms of the measure
po , (4.38) becomes
(4.41)
(EH(B)&)(r)
= 1.i.m.
'Atl'B)
ah)$(kt 4 & O ( k
km),
where $(k, 1, m) and @$A(r) are arbitrary extensions of the functions q l m ( k )and @j$(r), respectively, outside the support of the measure p,, . In particular, for H = Ho , we have
as a special case of (4.41). 4.4.
EIGENFUNCTION EXPANSIONS AND GREENFUNCTIONS
We are now ready to introduce a general concept of eigenfunction expansions which stems from the special cases occurring in (4.1 I), (4.15), (4.31), and (4.41).
Definition 4.2. Let H; be a differential or integral operator which serves to define a self-adjoint operator H,, in L2(Rm,p). Suppose that there is a measure v on A?" and functions Qa(m), p E R" [not necessarily p ) ] , such that for almost all p E Rn, we have belonging to L2([Wm,
Then we say that the family {Qa:p E [W"} provides an eigenfunction expansion for Ho if for any E L2([Wm, p) we have a unique &,t E L2(R", v) such that
+
(4.43)
Let H be another self-adjoint operator in L2(Rm,p), given by means of a ,differential or integral operator form Ha. Extrapolating from (4.17), (4.25), (4.33), and (4.41), we expect that under certain circumstances
470
V. Quantum Mechanical Scattering Theory
there might be functions @;*)(a) which for almost any /3 are eigenfunctions of H. (4.45)
(HY@)(a) = A(P)
and that for any
EL~JIW~ p),
t,b*
@pya),
= EH(Sf)L2(Rm,p) we have
(4.46) where [tacitly assuming that R+ (4.47)
+(p)
1.i.m.
1
3
R-
= Lgc(Rm,p)]
J ap)*(a)+*(a) dp(a), [Wn
+* = Q*+.
I n keeping with the terminology used in physics, for any fixed p we shall call Q4(a)a free wave, @I;”( an .)outgoing (or retarded) distorted wave, and @;-)(a)an incoming (or advanced) distorted wave. We encountered in $3 another important quantity associated with a self-adjoint operator A , namely its resolvent RA(5)= ( A - [)-l. It can happen that for 5 belonging to the resolvent set of A , i.e., 5 4 S A , the resolvent R,(iJ is an integral operator of the form (4.48)
( h ( C ) + ) ( a )= 1.i.m.
s,
GA(a, a’;
C) +(a’)4401’).
Definition 4.3. If 5 belongs to the resolvent set of the linear operator A inL2(Rm, p), and if GA(a,a’; 5 ) is integrable in the measure p on bounded Borel sets in R2m of finite measure p x p, and is such that
for all t,h eL2(Rrn, p), then GA(a,a’;
5 ) is called a Green function of A
-
5.
It is an easy exercise (see Exercise 4.2) to show that if p is u finite, then the Green function of A - 5 (if it exists) is determined on R2rnby A uniquely up to its values on a set of p x p measure zero. We shall denote the Green functions of H and H,, by G(a, a‘; 5) and G,(a, a’; C), respectively. I t is computationally very helpful when for self-adjoint A the Green function G,(a, a‘; C) has some limiting values when 5 approaches the real axis in the complex plane. However, we can expect that, in general, G(a, a’; 5) will have some singularities in a and a’. Let us denote by D , the Borel set in RZmat which G(a, a‘; 1) is not defined or becomes
4. Time-Independent Scattering Theory for Hamiltonians
471
infinite for some values of 5 E C1, with I m 5 # 0. When 5 approaches the real axis, we must avoid letting a and a’ assume values for which ( a , a‘) E D, . T h e reader will note that this condition appears explicitly in Definition 4.4. We have to exphasize that it will be tacitly assumed throughout this section that D, is of p x p measure zero.
Definition 4.4. T h e function Gj4+)(a,a‘; A) and G$-I(a, a‘; A) are called, respectively, the retarded and advanced Green functions of the self-adjoint operator A if GA(a,a’; h & ie) converges to G$*)(a,a’; A) uniformly in h E SE and in ( a , a’) from any compact set S C R2m for which S n D, = o. It is easy to see that if , $2 E L : ~ ( R p~), = EA(S<)L2(Rm,p ) are functions with compact supports* Si= supp $i, i = 1, 2, of finite measure p for which (8, x S,) n D, = IZI, then
T h e reason for the terminology used in Definition 4.4 is that, as we shall see later, the advanced (retarded) Green functions are closely related to the incoming (outgoing) waves @:*)(a) exp[-iA(p)t].
4.5. THETRANSITION MATRIX When H has an eigenfunction expansion, we can use this expansion to reduce the iterated integral in (3.52) to an ordinary integral on a measure space. We show this in Theorem 4.1.
Theorem 4.1. Suppose that Q0(a) is continuous and bounded almost everywhere with respect to p x v in both variables a E Rm and b E Rn, that the family {Q0 : E Rn} provides an eigenfunction expansion of the free Hamiltonian H, , which acts in L2(Rm,p), and that its eigennumber function A@), defined in (4.42), is continuous in PER”. Moreover, assume that whenever E > 0, H - h - i~ has a Green function G(a, a’; h i ~ )which is p integrable on Rm in CY. for almost all a’, and in a’ for almost all a , and continuous in a, a‘ E Rm, h E R1 - SpH,
+
* Recall that the support supp +h of a function #(a) is the closure of the set of all points
01
E Rm at
which #(a) # 0. A set S in R“ is compact if and only if it is bounded.
472
V. Quantum Mechanical Scattering Theory
with the possible exception of a set So x R1 C RZm+l, where So is a closed subset of R2m having p x p measure zero. If there is a function T(P1 P Z ) , P1 P 2 6 (wn, such thatX 9
9
x
I,,
M a ‘ ) G(a, a’;
&(a) @,*,(a) 08m
+i ~ )
@D~(U’)
for any y!Ji , y!Jf E M, n Lic(Rm,p), M, = M, , which are such that the supports of $,(p) and &(p) are compact and of finite measure v, and if all the integrals in (4.50) exist, then we have
for all such y!Jf and y!Ji. T h e function T(Pl,P2) is called by physicists the transition matrix (or T’matrix) in the variables /3 E W. T h e reason for this terminology lies in the conventionally adopted Dirac notation, in which one writes T(P,, p2) = (pl I T I p2), where p are variables related to some observables (such as in the case when p = k are momentum variables). It has to be realized, however, that T ( p , , p2) is in general a function and not an actual matrix element of the T operator. T h e reader should note that if G(+)(a, a’; A) exists and the limit E + +O can be taken in (4.50) under the integral sign, then (4.52)
T(P, , P Z ) =
1
08”
444 @;(a) j
R”
M a ’ ) G‘+’(a,a’;
4 @&).
In order to prove the above theorem, we have to relate {t+hf j E~sZ+E~t+hi) in (3.52) to the eigenfunctions (Do(.). For this we need Lemma 4.1.
Lemma 4.1. Let En , h E R1, be a spectral function for which there is a function F ( u , /3; A) such that
(4.53)
(.EA+)(a)= 1.i.m.
F(a9
P ; 4 &PI M P ) ,
* From now on dp dv under an integral sign denotes integration with respect to the measure p X v. The reader should check when he reaches 555-7 that the conditions of this theorem are satisfied in potential scattering.
473
4. Time-Independent Scattering Theory for Hamiltonians
where y(P) is a continuous function of /3. Suppose + 1 ( ~ )and $& are I) two functions which vanish outside the respective closed bounded sets D,C Rm, and D, C Rn, and that in addition $&Ihas ) compact support of finite measure Y. If, in addition, F(a, 6 ; A) is continuous on D, x D, x [a, b] and h(A) is continuous on [a, b ] , we have
According to Definition 3.1,
Proof. (4.55)
1:h(A)
dA<#l
I EA#2)
X ['Y(,)
F(% 8; A,)
$UP) 4 8 ) - j Y ( B ) < A ~ - F(% 8; AE-1) $2?1(8) w3)] ~ 9
<
<
where the only limitation in the choice of A,' is Al-, A,' A, . Since F(a, 6; A) is continuous in all three variables 01, and A, it can be shown (see Exercise 4.3) that the limit in (4.55) is identical to
Exploiting the continuity of F(a, /3; y(/3)) in 01 and /3, and the continuity of h(y(/3))in /3, we infer (see Exercise 4.4) that the limit (4.56) is identical to the limit
thus establishing that (4.54) is true.
Q.E.D.
From (3.38) and (3.39) we get
'?#f
+'<
Hcl
ET#i> =
J+*
-03
'A
(Ho
Ez$fI H
-2E A'
- iE E W ~ I L , )
474
V. Quantum Mechanical Scattering Theory
I n view of (4.49), we have, for almost all a E Rm,
where we have used Fubini’s theorem to interchange the order of integration.” We can apply Lemma 4.1 to the last integral in the above k) and relation, since according to the assumptions on G(a, a‘;A cDOZ(a’),the function
+
(4.59)
F6(%P z ; A)
=
1
Rm
G(a, a’; A
+ i4
@BZ(4
44’)
satisfies the conditions required in that lemma. Thus, combining (4.58) with Definition 3.4 we easily get (4.60)
On the other hand, the following equality holds almost everywhere in a,
* Before applying Fubini’s theorem, we have to make sure that the integrand is integrable on the specified set with respect to all its variables simultaneously, i.e., in the present case, with respect to the measure Y x p in the variables (PZ , a’). It is left to the reader to verify, now as well as in later steps of this proof, that whenever Fubini’s theorem is applied, this integrability condition is fulfilled due to the assumptions stated in Theorem 4.1.
4. Time-Independent Scattering Theory for Hamiltonians
475
so that, after repeatedly applying Fubini's theorem, we derive from (4.60) (4.61)
( E z h I Q+, E A ~ W -
dv(/31)$e*(/31 )
where (4.62)
T(P1 ' 8 2 ; E)
=
1,
(4%) - 4%))@,*,(a)Fd%/3z
; 4 / 3 2 ) ) @(a).
Since (4.50) is supposed to hold even when &(flZ) = 0 for Bz > h and $,(/?,) = 0 for /?, > A,, we get by taking the limit E + +O in (4.61) (4.63)
(E?*f
I Q+ ET*i)
where T(& , &) is the function satisfying (4.50). From (3.52) we obtain
Using (4.63) we easily derive (4.65)
by using Lemma 4.1 and Definition 3.4. By the same procedure
476
V. Quantum Mechanical Scattering Theory
from which (4.51) follows immediately. This completes the proof of Theorem 4.1.
THEENERGYSHELLAND
4.6.
THE
S MATRIX
Formula (4.51) provides a practical way of computing the transition amplitude 1 T+,) from the initial state + i to the final state fil, provided that the T matrix can be computed by using, for instance, the relation (4.50). I n practice, the limit E -+ +O can always be taken easily, and has the effect of restricting in (4.51) the integration in p1 and & to the surface A(pl) = A(&). I n physical literature this surface is known as the energy shell. One reason for this terminology could be that Ef = A(&) and Ei = represent the energy eigennumbers of Wo for the respective eigenfunctions @)B,(a)and @8,(ct).
Theorem 4.2. y
E
Suppose that the variables E = A@) E R1 and B E Rn in (4.51), that
Rn-l can be substituted for the variables
(4.66)
dvi(E) dvn(y),
dv(B)
&(E)
= p ( E ) dE,
where p ( E ) is a continuous function, and that the function T(E, , yt ; E , ,y,) [obtained by substituting in T(& , p2) for p1 and p2 the new variables E, , yf and E , , yi , respectively] is continuous in Ef and Ei . If +i(E,y ) and +,(E, y ) have compact supports of finite measure V, and are continuous in E, then the following equality holds: (4-67)
(#n
1
I T#I) = dEp2(E)
dvn(yr) dv,(yi) T(E, yf ; E , YI)
x $f* (E, Yf)&(E,
rz),
where the integration in E , yf , and y , extends over all possible values which these variables can assume. Theorem 4.2 can be easily derived from (4.50) if the following lemma is used.
Lemma 4.2. If the function j ( x ) is continuous at the origin and bounded on any finite interval of R1 and if e(x2 ~ ~ ) - ~ fis( integrable x) eo , then on R1 for any 0 < E
+
<
(4.68) Prooj.
For given 7
> 0, let us
choose N ( 7 ) > 0 so large that
4. Time-Independent Scattering Theory for Hamiltonians
and that for all 0
477
< E ,< E,,
For any w such that 0
+”
(4.71)
E
< N(q),we have
w
dx = 2 arctan - , E
we obtain by using the generalized mean-value theorem of integral calculus and (4.70)
1
2
G ; Y(% w ) arctan
-
f(0)
I + 2N ( d Wrl), 2E
where y ( ~w, ) is some number in between the infimum and the supremum off(x) on [ - w , + w ] . Due to the continuity off(x) at the origin, the first term on the right-hand side of the above inequality does not exceed 7 if w > 0 is chosen sufficiently small and W / E sufficiently large. Then, for such fixed w , the second term can be made smaller than 7 by letting E -+ 0. Hence, taking into account (4.69), we conclude that (4.68) is true. Q.E.D. When (4.66) holds, (4.51) assumes the form
478
V. Quantum Mechanical Scattering Theory
If the functions T , p, #i , and I+$fulfill the requirements of Theorem 4.2, then the above function f ( x ) satisfies the conditions of Lemma 4.2. Consequently, we get the relation f(0) = (& 1 T&), which is identical to (4.67). Loosely speaking, the relation (4.68) can be written in the form (4.75)
It must be realized, however, that (4.75) assumes a precise meaning only when the Dirac function 6(x) is defined as a functional on a given space of functions, i.e., as a distribution. On the other hand, in the theory of distributions, the spaces of functions usually considered are not such as to include all the functions satisfying the conditions of Lemma 4.2. Hence, one has to realize that when one encounters in physical literature the relation (4.67) written in the form
the meaning of S(E, - Ei) in this relation is more general than that of a Schwartz distribution. Naturally, this meaning is actually contained in Theorem 4.2. We recall that the S operator is related to the T operator by the identity S = 1 - 2riT, so that we have
Let us introduce the functional (in the sense of distribution theory) S,(E, , yi ; Ei, yi) defined by the relation
and the functional
Then (4.76) assumes the form
4. Time-Independent Scattering Theory for Hamiltonians
479
I n physical literature, the functional S ( E , , yf ; Ei , yi) is called the scattering matrix or the S matrix.
4.7. THEOPTICALTHEOREM T h e relation S = 1 - 2iriT between the S operator and the T operator leads to the identities
+ 4rr2T*T, 2ni(T - T * ) + 4n2T*T.
(4.80)
S*S
=
1 - 2ni(T
(4.81)
SS*
=
I
-
-
T*)
Hence, S is a unitary operator on M, if and only if (4.82)
T* - T
=
2niT*T
= 2niTT*.
T h e above relations require that the T matrix have certain properties, as stated in Theorem 4.3.
Theorem 4.3 (Generalized optical theorem). Suppose that the function T ( E , , y, ; Ei , yi) is continuous in Ei and E, , that v is cr finite, and that dv(E, y ) = p(E) dE dv,(y), where p ( E ) is a continuous function. Then the S operator is unitary on M, = M, if and only if (4.83)
T*(E, Yl ; E, ye) =
=
-
T(E, yf ; E, Y i )
jT*(E, E, Yf) T ( E , E, y M E ) dVO(Y) 2ni jT ( E ,y ; E, re) T*(E, y ; E, yl)P(E) dV,(Y) y;
y;
for almost all values of E E R1 and yf , yi E Rn--l, provided that for any compact set D C Rn-l of finite measure v, the function
j I T(E,Y ;E , r’)lP(E) dVO(Y’) D
is integrable and square integrable in E and y. T h e above theorem is an easy consequence of (4.82). First of all, we have to observe that due to the conditions imposed on the T matrix and on the measure V , Theorem 4.2, is valid in the present case. Moreover, for t/f , i,hi E M, , satisfying the conditions of that theorem, we have (4.84)
<$f
I (T* - T )$1)
=
(T$f I 41)
- (#f
I T*I)
V, Quantum Mechanical Scattering Theory
480
O n the other hand, under the same conditions we get under the restrictions of the theorem by employing Lemma 4.2 (as in the proof of Theorem 4.2) twice in succession
If T(E,y ; E, y ’ ) satisfies the conditions stated in the theorem, we can apply Tonelli’s theorem (Chapter 11, Theorem 3.14) twice in succession to deduce that the products of integrands in the above relation is integrable with respect to v1 x vo x v,, x v,, . Hence, we can use Fubini’s theorem to interchange the orders of integration in (4.85) and derive
Thus, we see that the expressions on the left-hand sides of (4.84) and (4.86) are equal if and only if (see also Exercise 4.5) the first of the equations (4.83) holds. T h e proof for the second set of equations (4.83) proceeds along similar lines. T h e relation (holding almost everywhere) (4.87)
Im T ( E , y ‘ ; E, y ’ )
= -n
1 T ( E ,y ’ ; E, y)I2p(E)dvo(y)
is known as the optical theorem; it owes its name to its similarity to a corresponding relation derived in classical electromagnetic scattering, which includes scattering of visible light as a special case. We note that the optical theorem is a corollary of the generalized optical theorem, which is obtained when we set yi = yf = y‘ in (4.83). However, for the validity of (4.87) it is sufficient that only the first equation in (4.83) holds. Since this first equation is equivalent to condition that S*S = 1 on M, , the optical theorem is valid under weaker conditions than the
4. Time-Independent Scattering Theory for Hamiltonians
48 1
unitarity (on M,) of the S operator. I n fact, we see from (2.11) that S*S = Q,*s)-Q-*Q+ = Q+*ER-Q+, and therefore we have S*S = EMo if ER 2 ER+, i.e., if R+ C R- . It-is important to observe that the optical theorem, as well as its generalized version (4.83), refers to properties of the T matrix T ( E , , yf ; E , , 7,)on the energy shell, when E - Ei . Under certain .additional assumptions on the T matrix, the equations of the generalized optical theorem can be extended off the energy shell. These off-energyshell equations are known as the Low equations (see Newton [1966, Section 7.2.21). 4.8.
THELIPPMANN-SCHWINGER INTEGRAL EQUATIONS
When H , and H possess eigenfunction expansions, Green functions, as well as advanced and retarded Green functions, then the LippmannSchwinger equations can be recast in the form of integral equations, which are computationally easier to handle than the operator equations of 93. We shall now derive the type I integral equations, pointing out in the course of the derivatives which assumptions are required to obtain the final result. These assumptions are frequently satisfied in practice. Suppose that H , is given by a differential or integral operator Po. Let us further assume that if +*(a) = (Q,+)(a), and if &(p) E %?l(Rn) is a function with support which is of finite measure v, and disjoint from a certain closed set 9,[of possible singularities of @;*)(a)] for which p(9,) = 0, then (4.88)
(H01Cl*)(4=
j,. $((P)(Ho"@f')(4W P )
almost everywhere (with respect to p ) in Rm (see Exercise 4.7). By virtue of the above assumption and (4.46) we have
V. Quantum Mechanical Scattering Theory
482
+
If Go(a,a'; h iq) $@)((A - E')@;*))(a') is integrable (see also Exercise 4.8) in ( a ' , p), then the order of integration in 01' and j3 can be reversed, and afterwards Lemma 4.1 can be applied, thus arriving at the result
x
444 Go(% a'; 4 3 )
JRrn
+ i 7 ) ( 4 P ) - Hf)
.)'%I@
T h e type I Lippmann-Schwinger equation (3.34) leads to the relation
Using (4.43), (4.46), and (4.90), and assuming that the limit 7 --t &O can be carried under the integral sign (see also Exercise 4.9), we can rewrite the above equation in the form (4.9 1)
J
R"
44a) $0*(4
j
s,444 +J
IWm
R"
4 P ) $(PI @f%)
$0*(4
J W P ) $(PI @A4 J W P ) $(P) J
444 $0*(4
R"
R"
444 G%u,
a'; 4 P ) >
x (A@)- H,"')@f'(a'). T h e equation (4.91) holds for any 1cl0(a)E %:(UPwith ) support of finite measure p, and any I,@) E %:(Rn) with support which is of finite measure Y and disjoint from the Yoof p measure zero. Hence, using standard arguments (such as in solving Exercise 4.2), we infer that (4.92)
@r)(a) + J = QB(a)
Rrn
GF)(a, a'; A(P))([A(P) - H,"']
@F))(a') dp(a')
for almost all (with respect to p ) values of 01 E Rm and almost all (with respect to u ) values of p E Rn. T h e two equations (4.92) represent the integral versions of the type I Lippmann-Schwinger equations. I n potential scattering one can easily justify (see Exercises 4.7-4.9) for very large classes of potentials, the manipulations leading from (3.34)
483
4. Time-Independent Scattering Theory for Hamiltonians
to (4.92). We shall adopt, however, a different approach in 96 to derive (4.92) in potential scattering. In this approach functions @;*)(a) are first defined in terms of the full Green function. After that, it is shown that these functions satisfy (4.92). I n the end, it is proved that the functions @;*)(a) are indeed the distorted plane waves by showing that they fulfill the requirements of 54.2. I n the type I1 Lippmann-Schwinger equations (3.41), the roles of H and H,, , as well as Y* and Y, are reversed. Hence, it will be possible to derive the type 11Lippmann-Schwinger integral equations
(4.93)
@’(a)
=
+ J”
0jB(a)
IWm
G(*’(a, a’; A(/3))(A(/3) - Ha’)aB(a’) +(a‘)
by the same methods. All we have to do is reverse the roles of free and full Green functions, as well as of the free and distorted waves. We leave this straightforward task to the reader.
4.9. INTEGRAL FORMS OF
THE
RESOLVENT EQUATIONS
When the appropriate Green functions exist, we can recast the resolvent equations (3.61) and (3.62) as integral equations.
Theorem 4.4. If GA(n,a’; 5 ) is the Green function of the linear operator A , then the first resolvent equation (3.62) is equivalent to the equation (4.94)
GA(% a’; 51)
- GA(%
a’;
52)
holding for almost all 01 and a’, provided that the above integral exists.
Proof. T h e equation (3.62) holds if and only if (4.95) (31 I R4(51) 3 2 )
- (31
I RAG21 32)
= (51 - 5&31
I R4(51) W
5 2 )32)
for any i+bl and i+b2 belonging to a dense domain. Such a dense domain is the set of all functions with compact supports of finite measure, for which (supp i+bl x supp i+b2) n D , = a . For such functions (4.95) is equivalent to
484
V. Quantum Mechanical Scattering Theory
provided that GA(a,a”; 5,) G,,(a’’, a’; 5,) is integrable in a”. It is now easy to establish, by using Fubini’s theorem and already standard procedures (see Exercise 4.2), that (4.96) is equivalent to (4.94) holding almost everywhere. Q.E.D.
A similar procedure establishes Theorem 4.5.
Theorem 4.5. Suppose the Green functions G(a, a’; 5) and GO(m,a’; t;) of H - t; and H , - 5, respectively, exist. Then the second resolvent equations (3.61) are satisfied if and only if (4.97)
G(a,a’; 4 ) - GO(a, a’; 5) =
=
-IRrn Go(ol, c)(H“” H,””)G(ol”, a”;
-1
Rn
-
a’; 5) dp(a”)
G(a,a”; ()(Ha”- H,“”)Go(a”,a‘; 4) dp(01”)
almost everywhere in a and a’, provided that the above two integrals exist. T h e proof of the above theorem is almost identical to the proof of the preceding theorem, and it is left to the reader.
EXERCISES 4.1. Show that the expressions on the right-hand sides of (4.38) and (4.41) are identical. 4.2. Show that if GL1)(a,a’; 5 ) and GLz)(a,a’; 5) are two Green functions of A - 5, and p is u finite, then GI1’(a,a‘; 5 ) = GLz’(a,a‘; 5) almost everywhere in a and 01’ (with respect to the measure p x p). 4.3.
Show that the limit in (4.55) is identical to (4.56).
4.4. Prove that, under the conditions of Lemma 4.1, the limits (4.56) and (4.57) are identical. 4.5. Show that the right-hand side of (4.84) is equal to the right-hand side of (4.86) for any choice of #i(E, y ) and &(E, y ) having compact supports of finite measure if and only if the first equation in (4.83) is satisfied.
4.6. Show that in potential scattering [A(k) - H i ] @F)(r)= V(r) @F’(r),
except at the singularities of V(r).
5. Green Functions in Potential Scattering
485
4.7. Verify that in potential scattering (4.88) holds for distorted is continuous on [w3 - Y o ,where Yo plane waves @:+)(r)if (A@L%))(r) is closed and of Lebesgue measure zero.
Show that Go(r,r’;X -1- iq) V(k’)@:*)(r’)$(k) is integrable on in (r, k) if I @L*)(r)l const for all r, k E [w3, if $(k) E W:(R3), V(r’) is square integrable, and the free Green function is given by (5.6). 4.8.
<
[w6
k) and G(*)(r,r’;A) are 4.9. I n potential scattering G(r, r’; X given by (5.6) and (5.16), respectively. Prove that if V(r) is square o ) some e0 > 0, and if integrable and I V(r)i = O ( l / ~ ~ + ~for I @:*)(r)l const, then
<
lim
W*O
j+O*(r)Go(r,r‘;A + i ~ V(r’) ) @f)(r’)$(k) dr dr‘ dk =
for J0(r)E
1
+,,*(r) Gr)(r, r‘; A) V(r’) @F)(r) $(k) dr dr‘ dk
%t([W3) and $(k) E Vi(R3).
5. Green Functions in Potential Scattering 5.1.
THEFREEGREENFUNCTION
Potential quantum scattering is the special case of quantum scattering theory in which the underlying formalism is that of wave mechanics, and the interaction is determined by a potential. In applying the general methods of scattering theory to potential scattering of two particles, we shall limit ourselves to systems of two spinless, nonidentical particles interacting via a potential V(r), where r = r2 - rl , and rl , r2 are the position vectors of the two particles. This case exhibits all the essential features of all other cases. For example, the case of nonzero-spin particles can be treated essentially by the same methods as the zero-spin case when group-theoretical methods are applied. I n the case of systems of two identical particles one must also take into account the presence of the Fermi-Dirac or Bose-Einstein statistics. From the practical point of view this means that in these last two cases the calculations have to be carried out in the respective subspaces of antisymmetric or symmetric functions of L2([w3), rather than in the space L2(R3)itself.*
* See
particles.
Rys [1965] for a general treatment of the scattering theory of two identical
486
V. Quantum Mechanical Scattering Theory
I n the absence of external fields and mutual interaction, the relative motion Hamiltonian H , of a system of two spinless particles is the kinetic energy operator in the center of mass system. We recall from Theorem 7.3 of Chapter IV that H , = T , is a self-adjoint operator for which (5.1)
(Ho$)(r) =
1 2m A W ,
--
$69
0
9
and that Ho is the only self-adjoint operator having this property. On wave functions $(k) €L2([W3)in the momentum space, H , acts in the following manner:
T h e above operator is obviously self-adjoint and, since it is a function of the momentum operators PCX),P(g),P @ ),it immediately follows that it has a pure continuous spectrum sHo =
(5.3)
s?
=
[O, +m).
Thus, its resolvent (5.4)
is defined for all values of 5 E C1.Moreover, R,([)is a bounded linear operator defined on the entire Hilbert space & = L2(R3)whenever Re 5 < 0 or Im 5 # 0, i.e., when the argument arg 5 of 5 is within the open interval (0, 277).
Theorem 5.1. T h e resolvent R,(()of the free Hamiltonian (5.2) is an integral operator (5.5)
(Ro(5)$)(r) =
R3
Go@, r'; 5) +(T') dr'
for values of 5 restricted to the range 0 < arg 5 G,,(r, r'; l), called the Green function of Ho , is
where
fit is the square root
for which Im
< 27r. T h e kernel
1/2mZ > 0.
487
5. Green Functions in Potential Scattering
Proof.
It is very easy to verify by using (5.2) that
(5.7)
Assume that the wave function #(r) in the configuration space is continuous and of compact support, i.e., E %?:((w3). T h e n #(r) has a Fourier transform (5.8)
$(k)
= (2.rr)-3/2
j,,
e-ikr’
$(r‘) dr‘.
Furthermore, by Theorem 4.5 of Chapter 111, $(k) eikr is integrable* on R3. Since (k2/2m- 0-l is bounded in k over R3 when 0 < arg 5 < 27r, the inverse Fourier transform of (k2/2m- 4)-l $(k)
exists. By virtue of (5.7) and (5.8) (5.9)
1
(&(l)$)(r) = 03
I,,
dk
k2 (x -
eirk
I
R3
dr‘ rikr‘ *(r’)*
Due to the fact that #(r’)is of compact support, the function
is Lebesgue integrable on R6 with respect to the variables r and k. Hence, we can apply Fubini’s theorem to interchange the order of integration in (5.9):
T o carry out the above integration in k, let us write (for fixed r - r’) the variable k in spherical coordinates introduced in such a manner that k(r - r’) = kp = kp cos 8. p =
* Throughout this section, whenever we deal with integration, we have in mind integration with respect to the Lebesgue measure, except if otherwise explicitly stated.
488
V. Quantum Mechanical Scattering Theory
It is very easy to carry out immediately the integrations in 0 and 0 C$ 2n:
< < k2 / (= 5)
(5.11)
-1
-
R
-
so a,
<8
eikpdk
k sin kp dk p(k"2m - 5)
=
-4ma/p
s
+m -m
k sin kp dk , k2 - 2m5
In the last of the above integrals we have to integrate with respect
to k the function
k &kD k sin k p k2 - 2m5 2i(k - f i m l ) ( k
(5.12)
+65)-
k e-akD 2i(k - .\/2m5)(k
+
over the entire real line R1. If we work with the complex variable k E C1, the above functions are analytic everywhere in the complex plane, ,' d<, Im > 0. Let us close except for two simple poles at +% the contour of integration by semicircles in the upper-half and lower-half complex plane, respectively, when integrating on the real line the first and second function on the right-hand side of (5.12). Since the contributions from the respective integrations on the semicircles vanish in the limit of their radii becoming infinite, we easily determine, by using Cauchy's integral formula and computing the respective residuals
,'c
s
+* --m
k sin kp ka - 2 m 5
=-/1 2i
-
+, e-ikn
1 dk - k2 - 2m5 2i k
eikD
k2 - 2m5
dk
k e--ikn -7-r
-
2/2mr
lk=-
-
- -,ei~d\/2mg -
Inserting this result in (5.11), we get from (5.10)
for all I+!IE%~(R~). Since V:((w3) is dense in L2([w3)(see Chapter 11, Theorem 4.9,while both R,,(C) and the integral operator with the kernel (m/27r)I Y - r' 1-l e x p ( i . \ / m 1 r - r' I) are bounded linear operators defined on the entire Hilbert space L2(R3),we infer with the help of the extension principle (Chapter 111, Theorem 2.6) that (5.13) is true Q.E.D. for all I+!I EL~(IW~).
489
5. Green Functions in Potential Scattering
It follows from (5.6) that the free Green function has the following symmetry properties: (5.14)
Go(r, r'; 5)
=
Go*(r, r'; 5*),
(5.15)
GO(^, r'; 5 )
=
Go(r', r; [)
for any 5.2.
5 E C1 not on the positive
real axis.
RETARDED AND ADVANCED FREEGREENFUNCTIONS
It is easy to infer from (5.6) that when 5 approaches the positive real axis from above or from below, the Green function G,,(r, r'; 5) also approaches well-defined respective limits for all r # r', r, r' E R3. These limits turn out to be the advanced and retarded free Green functions Gh*)(r, r'; A).
Theorem 5.2. T h e retarded free Green function G(+)(r,r'; A) and the advanced free Green function exist for all A E R1 when Ho is given by (5.1): (5.16)
G r ) ( r , r'; A )
m
1 e x p ( i i dm 1 r
=
274 r
=
Gi-)(r, r'; A )
-
r'
-
r'
1)
for any h >, 0, and (5.17)
GF)(r, r'; A)
=
Go(r, r'; A )
for h < 0. T h e proof of the above theorem is very straightforward. It consists in checking that the conditions of Definition 4.4 are satisfied when we adopt the expressions (5.16) and (5.17) for G&*)(r,r'; A). I n the present case, the set of singularities of Go(r, r'; 5) is (5.18)
DGo= {r, r': 1 r - r' 1
= 0).
Note that DG0is indeed of Lebesgue measure zero in R6(see Exercise 5.1)! If we introduce the variable K = 4/2m(, we easily see that (5.19)
(*)
I Go(r, r'; 2/2m)- Go (r, r'; A)1
490
V. Quantum Mechanical Scattering Theory
and that the exponential converges uniformly to one on compact sets in R6 when K -+ fd2mh. Hence, the convergence of Go(r, r'; ~ ~ / 2 r n ) to GA*)(r,r'; A) is uniform on any compact domain which does not intersect D G 0 .Thus, Theorem 5.2 is established. Since in the present case we have (5.20)
Gr)(r, r'; A)
= !l%
Go(r, r'; A
r # r',
k),
we deduce from (5.14) and (5.15) (5.21)
Gi*)(r, r'; A)
=
Gr)(r', r; A),
(5.22)
Gp)(r, r'; A)
=
[Gi-)(r, r'; A)]*
for any X E R1. These properties can be also derived directly from (5.16) and (5.17).
EQUATIONS WITH HILBERT-SCHMIDT KERNELS 5.3. INTEGRAL One of the main significant features of the free Green function Go(r, r'; 5) is that it provides an integral equation for the full Green function G(r, r'; c), corresponding to the Schroedinger operator H , for which (5.23)
(fwr)
=
-(l/W M r )
+ W ) $w,
4 E 9"'
This integral equation is the second resolvent integral equation (4.97), which in the present case becomes (5.24)
G(r, r';
5 ) = Go(r, r'; 1) - J Go(r, r"; 5 ) V(r") G(r", r'; 5) dr". R3
This equation can be solved by the Fredholm method if its kernel (5.25)
K(r, r";1) = Go(r, r";t) V(r")
is of Hilbert-Schmidt type." Before attacking the problem of solving (5.24), let us first review some basic facts about integral operators with Hilbert-Schmidt kernels.
Definition 5.1. T h e kernel K(a, a'), a, a' operator K on L2(Rm), (5.26)
(K$)(a)=
I
IWm
K(a, a') $(a') d"a',
E
Rm, of the integral
4 EL2(R"),
* A condensed and very lucid presentation of the Fredholm theory for Hilbert-Schmidt kernels (L2 kernels) is given by Smithies [1958, Chapter VI].
49 1
5. Green Functions in Potential Scattering
is a Hilbert-Schmidt kernel (or L2 kernel) if 1 K(a, a')j2 is integrable on Wm:
(5.27)
+a.
A Hilbert-Schmidt kernel which is continuous and such that K(a, a') is integrable on IWm is called a trace-class kernel. The terminology for kernels of integral operators is related to the terminology for the corresponding operators in a straightforward fashion: integral operators with trace-class kernels are of trace class" with T r K = JK(ar,a') dma, and integral operators with HilbertSchmidt kernels are Hilbert-Schmidt operators (see Exercise 5.3). The Fredholm integral equation is (5.28)
where
is, in general, some complex number. If we search for solutions K(a, a') is the kernel of an integral operator on L2(Rm), and if ~ E L ~ ( [ W then ~ )the , integral equation (5.28) can be written in operator form: w
$(a) which are Lebesgue square integrable, so that
(5.29)
C = f +wKC.
Hence, (5.28) has a unique solution in L2(Rm)if and only if (1 - wK)-l exists and iff is in the domain of definition of (1 - wK)-l. For w # 0, (1 - wK)-l is related in a straightforward manner to the resolvent RK(5)of the operator K : -1
w
If K(a, a') is a Hilbert-Schmidt kernel, then K is a Hilbert-Schmidt operator, and as such, K has a pure point spectrum. Hence, (1 - wK)-l will exist if and only if 1 / w E SK. When this is the case, the domain of definition of (1 - wK)-l is the entire space L2(Rm). The Fredholm theory essentially provides a means for the computation of (1 - wK)-' from the kernel K(a, a'). The main results of this theory for L2 kernels are contained in Theorems 5.3 and 5.4. The reader interested in the proofs of these theorems is advised to consult Smithies [1958, Chapter VI].
* A proof of this statement can be based on Mercer's theorem (see Riesz and Sz. Nagy [1955, Section 981).
492
V. Quantum Mechanical Scattering Theory
Theorem 5.3. Let K(a, a') be a Hilbert-Schmidt kernel and let us introduce the modijed Fredholm determinant of K(a, a')
1
0
K(a, 3.2)
... K(a1
9
an)
dmal
K(a, a') K(a, a,) ..' K(a, a,)
(5.31)
(-1P
I,.
n!
=
1, 2,...; for n
case the series
0
q a , , a')
Dn(a,a') = -
for n
dma, ,
=
*.'
K(a,, a,)
dmal . * * d"an ,
0, do = 1 and D,(a, a ' ) = K(a, a'). In that m
(5.32)
d(w) =
1 d,w"
n=O
is convergent for all complex values of C1,the series
w.
Likewise, for any fixed
w E
c m
(5.33)
D(a, a'; w) =
&(a, a')w"
n=O
is convergent in the mean [in L2((WZm)] for all w . Moreover, the function D(a, a'; w ) is a Hilbert-Schmidt kernel for any fixed complex w .
Theorem 5.4. If d ( w ) # 0 and ~ E L ~ ( R the ~ ) ,integral equation (5.28) has the unique Lebesgue square-integrable solution (5.34)
Theorem 5.5. The function d ( w ) vanishes if and only if eigenvalue of the integral operator K.
is an
5.4. THEFULLGREENFUNCTION We can apply the Fredholm theory to the integral equation (5.24) when its kernel (5.25) is of the Hilbert-Schmidt type. Under these circumstances, (5.24) is a special case of the integral equation (5.28), in
493
5. Green Functions in Potential Scattering
which the variables a, a' E R" become r, r' E R3, and w = - 1 . over, we note that when Z; is not on the positive real axis,
More-
is square integrable in r or r', since, according to Theorem 5.1, we have to choose the square root of 6 for which I m dz > 0. Hence, for any fixed r", (5.24) has a unique solution, provided that d(-I) # 0 for the given kernel (5.25). T o establish that d(-1) is not zero for the kernel K(r, r'; <) in (5.25), note that K(r, r'; 5) is the kernel of the operator
According to Theorem 5.5, d(-1) eigenvalue of this operator, i.e.,
=
0 if and only if
w =
-1
is an
for some nonzero y5 E GBH = BH0. However, the above relation implies that Ht,h = &. Thus, as long as Z; does not belong to the spectrum of H , the Green function G(r, r'; Z;) of H exists, and can be computed by means of (5.34) with w = -1. Let us find out now the restrictions imposed on the potential V(r) by the dehand that K(r, r'; Z;) be square integrable. Applying Fubini's Theorem, we find that R6
I K(r, r'; ( ) I z dr dr'
=
1,.
dr'( V(r')I2
[R3 drl Go(r,r'; <)Iz*
*
Using the expression (5.6) for Go(r,r'; Z;) we easily compute that (5.36)
11 R3
Go(r,r'; <)I2 dr
=
m
1
Im
44 > 0.
Thus, we get the relation
which shows that Go(r,r'; 5) V(r') is for I m Z; # 0 a kernel of HilbertSchmidt type if and only if V(r) is square integrable on R3.
494
V. Quantum Mechanical Scattering Theory
T h e main conclusions of the above discussion are contained in Theorem 5.6.
Theorem 5.6.
If the potential V(r) is square integrable on
R3, then the full Green function G(r, r’; 5 ) of H - 5 [where
+ V(r)] exists and can be written in the form 1 G(r, r‘; 5) = Go(r, r’; 5)- qTq D(r, r”;-l) GO(r”,r‘; 5) dr”,
H 1 - (1/2m)A (5.38)
s.,
where d( - 1; t;) and D(r, r ” ; - 1) are obtained by taking K(r, r’; 1) = Go(r, r’; 1)V(r’)
as a kernel in (5.30)-(5.33).
5.5.
SERIES EXPANSION OF
T h e kernel D(r, r”;- 1 ) by the infinite series (5.39)
THE
FULLGREENFUNCTION
of the integral operator in (5.38) is given
Wr, r’; 5) = K(r, r‘; 5)
+ C Dn(r,r‘; 0, m
n=l
in which
This would imply that the full Green function can be expanded in a series (5.40)
G(r, r’;5) =
m
n=o
GJr, r‘; 0,
where Go(r, r’; 5) is the free Green function, and (5.41)
for n = 1, 2, ..., if the order of integration and summation can be interchanged. T o prove that this is indeed the case, we need Lemma 5. I.
495
5. Green Functions in Potential Scattering
Lemma 5.1. If K(a, a') is a kernel of Hilbert-Schmidt type, then the Hilbert-Schmidt norm
of the modified first Fredholm minor D,(a, a') satisfies the inequality,
and the modified Fredholm determinant d, satisfies the inequality (5.43)
I d, I d
n
{(e/n)1/211K1l2}",
=
1, 2,...,
where 11 KII, is the Hilbert-Schmidt norm of the integral operator with the kernel K(a, a ' ) :
K,
The proof of the above result can be found in the work of Smithies [1965, Section 6.51. Expanding the determinant in (5.31) along the first column, we easily obtain (5.44)
a') = dn+1K(%a')
&(.I,
+J
Rrn
Dn-d", al)K(al
9
4 drn%.
Iterating the above relation, we find that (5.45)
Dn+l(a,4 = dn+.lK(a,4
+1
Rm
+ d, I
Rrn
dma1K(%4
K(a, al)K(al , a ' ) d"a1
1,
d"a,D,-l(%
9
a2M.2
9
4-
Hence, applying the Schwarz-Cauchy inequality to estimate the third integral in (5.45), and using afterwards (5.42), we arrive at the inequality
496
V. Quantum Mechanical Scattering Theory
Using the above inequality, we derive from (5.41)
(5.47) L
3
I Dn+l(rrr”,1) Go(r”,r’; 0 dr“
< I dn+l(-l; I)/ j
+ I dn(-l,
S)l
R
1
I K(r, r“,5)
R3
dr”JGo(r”,r‘; 1)l
r‘; 01 dr“
R
drll K(r, r,; I) K(rl r“;01 112
Naturally, in deriving the above inequality one has to make sure that all the integrals present in it really exist. I n fact, K(r, r”;5 ) is square integrable, while Go(r”,r’; () is integrable as well as square integrable . this it immediately follows that the in r” when 0 < arg 5 < 2 ~ From first integral on the right-hand side of (5.47)exists. Since the integrand of the ,second integral is equal to
we infer from (5.6)and the square integrability of Go(r, rl ; 5) V(rl) in r and rl that this integral also exists for all r, r’ E R3, with the possible exception of a set of measure zero. Finally, the third integral also exists since according to (5.36)
and both V(r”) and G(r”, r’; 5 ) are square integrable in r”. Combining the inequalities (5.43)and (5.47), we obtain the estimate (5.48)
5. Green Functions in Potential Scattering
< n=N+1 f
(c)""I1 1 I KILn
R
K(r, r";
from which we infer that (5.49)
gz I j, [
c m
n=N+1
497
5) Go(r", r'; 5)1 dr"
1
Dn(r, r"; 5) Go(r",r';
This last result leads to the relation
5) dr"
I
= 0.
j,, D(r, r"; 5 ) Go(r",r'; 5) dr" =
JR3
n=O
Dn(r, r"; W
+
L
3
5) GO(r",r'; 5 ) dr"
[,2+, 5) Dn(r, r";
Go(r", r';
1
5) dr"
11 , Dn(r, r"; 5) Go(r",r'; 1)dr", W
=
n=O
which shows that the expansion (5.40) is valid. We summarize these results in the following theorem.
Theorem 5.7. If K ( r , r'; 1;) is a kernel of Hilbert-Schmidt type, then the full Green function G(r, r'; 1;) can be expanded for any 1; I$ SH in the series (5.40), with terms given in (5.41), which is uniformly absolutely convergent relative to the function
498
V. Quantum Mechanical Scattering Theory
We should mention in this context that a sequence sl((), s2(6), ... of functions is said to converge to the function S ( f ) uniforml) relative to the functionF(6) if for any E > 0 there can be found a number N ( E ) such that IS,([) - s(4)I < E F ( ( )for all values of [. Naturally, a series C u,([) is said to converge uniformly with respect to some function R(6) if the sequence of its partial sums s,([) = Z b l u,(f), n = 1, 2,..., converges uniformly relative to F(5). Finally, this series is absolutely uniformly convergent relative to F( 6) if the series C 1 u,( <)I is uniformly convergent relative to F(5). Obviously, if a series is absolutely uniformly convergent relative to F ( f ) ,then it has to be also uniformly convergent relative to F(6).
5.6. SYMMETRY PROPERTIES OF
THE
GREENFUNCTION
T h e two symmetry properties (5.14) and (5.15) of the free Green function are inherited by the full Green function, i.e., for 5 $ SH (5.51)
G(r, r’; 5)
=
G*(r, r‘; <*),
(5.52)
G(r, r’; 5)
=
G(r’, r; 5)
holds almost everywhere in R6. T h e first of these two relations is a straightforward consequence of the formulas (5.39) and (5.40) for G(r, r’; c), resulting from the fact that V(r’) is real, and consequently K(r, r‘; 6)
=
Go(r7r‘; 5) V(r’) = Go*(r,r‘; [*) V(r’) = K*(r, r‘; 5*).
T o prove (5.52) we have to resort to a lengthier argument. First of all, we have to note that the function rdr, r’; 5)
=
1
R3
D(r, r”;-1) Go(r”,r‘; 5) dr“
is the kernel of a Hilbert-Schmidt operator. This follows from the fact that Go(r”,r’; 5 ) is the kernel of the bounded operator ( H , - C)-l, while D(r, r”; -1) is a kernel of the Hilbert-Schmidt type, since (see Exercise 5.4) it is the limit in the mean of the Hilbert-Schmidt kernels (see Exercise 5.5) N
C &(t, r”; --I), n=O
N
=
1,2, ...
Consequently, D(r, r”; - 1) is the kernel of a Hilbert-Schmidt operator
D(- I). Thus, rl(r, r’; l )is the kernel of the operator D( - l)(Ho - ()-I,
499
5. Green Functions in Potential Scattering
which is the product of a Hilbert-Schmidt operator and a bounded operator, and therefore is of Hilbert-Schmidt type (see Exercise 5.6). The function f*(r)g(r’) is square integrable in R6 whenever f,g €L2(R3).Since rl(r,r’; 5) is a kernel of Hilbert-Schmidt type, it is also square integrable in W. Hence, the functionf*(r) Tl(r, r‘; [)g(r’) is square integrable in R6. Consequently, Fubini’s theorem can be applied to infer that
Since G,,(r, r’; 5) is also square integrable in R6, a corresponding relation holds for it. Consequently, in view of (5.38), we have (5.53) =
j,,dr’g(r’) 1w3 dr G(r, r’; I)f*(r).
On the other hand, the relation
in conjunction with (5.51) implies that
=
=
J
R3
dr‘g(r’) J
1, dr’g(r’)
R3
w3
dr G*(r’, r; {*)f*(r) dr G(r‘, r; {)f*(r).
Comparing (5.53) and (5.54), we see that
j,, dr’g(r’) j w3 dr G(r, r‘; 5)f*(r’) =
I
dr’g(r’)
[w3
Iw3
dr G(r’, r; [)f*(r)
for all square-integrable functions g(r’). This is possible if and only if
1 G(r, r’; [)f*(r) dr R3
=
w3
G(r’, r ; [)f*(r) dr
500
V. Quantum Mechanical Scattering Theory
for almost all r' E [w3. Since the above equality holds for all squareintegrable functions f(r), we infer that (5.52) holds almost everywhere in R3 x R3 = R6. This completes the proof of (5.52). OF 5.7. THESPECTRUM
THE
SCHROEDINGER OPERATOR
We have seen already that if K(r, r'; 5 ) is a Hilbert-Schmidt kernel the full Green function exists as long as 5 $ SX. This is certainly the case if I m 5 # 0. T h e following theorem shows that this will be also the case if 5 is real but negative, with the possible exception of a finite or at most countably infinite number of values.
Theorem 5.8. Suppose the potential V(r) is locally square integrable and bounded at infinity. Then the continuous spectrum SE of the Schroedinger operator H 3 -(1/2m)A V(r) contains no negative values; moreover, there are only at most countably many negative eigenvalues, all of finite multiplicity, * and having no negative number as an accumulation point. I n order to prove the above theorem, we need a number of auxiliary results. Since these results are of interest in themselves, we shall present them as lemmas and theorems. We recall from 93 that if H and H , are any two operators (not necessarily self-adjoint) with identical domains of definition and 5 belongs to their resolvent sets, then the resolvent R ( ( ) of H can be expressed in terms of H , = H - Ho and of the resolvent I?,(<) of Ho by making use of the infinite series in (3.68), provided that this series converges in some sense. Lemma 5.2 represents, to some extent, a converse to this result, valid under the additional assumption that HlR0(5) has a bound less than one; in this context, it should be noted that when g H= BHo, the operator H,R0(5) is defined everywhere on iP since R,({) is defined everywhere on Z , and that it maps any f E Z into R , ( 5 ) f ~ 5 3= ~ ,g H 1so, that H,R,(<)f is defined.
+
Lemma 5.2. Suppose that H and H , are two operators (not necessarily self-adjoint) with identical domains of definition g X = gX,, , and that the complex number 5 belongs to the resolvent set of H, . If 11 HlRo(5)/1< I , H , = H - H , , then 5 belongs also to the resolvent set of H , and (5.55)
R(5) = RO(5) u-lim n++m
c (-1)k(KRo(5))k. n
k=O
Recall that the multiplicity of an eigenvalue is the number of linearly independent eigenvectors corresponding to that eigenvalue. #
501
5. Green Functions in Potential Scattering
Proof. Since I/ H,Ro([)/l < 1, the uniform limit in (5.55) exists (see Exercise 5.9). For any f E g Hwe have [RO(O
f
k=O
=
(-1)"(~1Ro(0)k]
(H - 5)f
RO(O(H0 - 5 ) f - Ro(5) fflR,(I)(HO - Of (-1)"Ro(5) ffl ... %(5)(Ho - I;)f
+ + Ro(5)Hlf + (-1)"Ro(5)
=f
Ro(5) HlRO(4-1fflf ffl *.. Ro(5) Hlf
-
+ (-1)"RoK)
+
.*.
+ ..*
Hl ... Ro(5) Hlf z f .
This shows that if A ( [ )denotes the operator on the right-hand side of the equality (5.55), then we have A({)(H - c) f = f for all f E g H . Since g HE gH0and Ro([) has the range g H 0 the , operator ( H - [) R,({) is defined everywhere on X . Consequently, we can write
Thus, we also have ( H - [) A ( [ ) = 1, i.e., A ( [ ) is identical to = I?({). Q.E.D.
( H - [)-'
If A is an unbounded self-adjoint operator, then its spectrum S A is an unbounded set. However, the set S A C [wl could be still bounded from above or from below, in which case we shall say that A is bounded from above or from below, respectively. I n the case where A is bounded from below, we denote by mA the greatest lower bound of the set SA: mA = inf A. A&
We note in passing that A is bounded from below if and only if the set
is bounded from below, with the same greatest lower bound (see Exercise 5.10).
502
V. Quantum Mechanical Scattering Theory
Theorem 5.9. Suppose H and Ho are two self-adjoint operators with identical domains of definition, that Ho is bounded from below, a < I , b 3 0, and that for some 0
<
/I Hlf /I < all H O f I l
(5.56)
for all f
E
+ bllfll,
Hl
=
H - Ho
3
gH0 = g H Then . H is also bounded from below, and
(5.57)
m, 3
mHo
b
- max 11 9 --a
+ a/
mHo
11.
Proof. We shall prove the main statement of the theorem by showing that for any real (5.58)
b
5 < m H o - max 11-a
9
f
mHo
1 1 9
we have 11 H1R,,(5)~~ < 1, and consequently, according to Lemma 5.2, such a 5 E R1 belongs to the resolvent set of H . This means that the spectrum of H contains only such points which do not satisfy (5.58); i.e., it is bounded from below and its greatest lower bound m, satisfies (5.57). T o prove that 11 H,R,(5)11 < 1, we use (5.56) to derive the inequality
Since S H o C [mHo,
+ co),and therefore for any f
we conclude that 11Ro(5)11 <
1 mHo -
5'
E
X and real
5 < mHo,
5. Green Functions in Potential Scattering
503
Combining the above two inequalities with (5.59), we obtain (5.60)
Since it is easily seen that the expression on the right-hand side of the inequality (5.60) is smaller than one when 5 satisfies (5.58), it follows from Lemma 5.2 that such 5 is in the resolvent set of H . Consequently, H is bounded from below by the expression on the right-hand side of (5.60). Q.E.D.
+
We recall from 97 in Chapter IV that when H = T, V, is the Schroedinger operator with a potential satisfying the conditions of Theorem 7.4 in Chapter IV, then BH = gH0 (Chapter IV, Theorem 7.5) if H , = T , is defined by (7.15) in Chapter IV, and also (5.56) is satisfied by any 0 < a < 1. Hence, we can state Theorem 5.10. Theorem 5.10.
T h e n-body Schroedinger operator H
=
H,
+ V,
corresponding to a potential V(rl ,..., r,) satisfying the conditions of Theorem 7.4 in Chapter IV, is bounded from below. As a by-product of the above theorem and (5.57), we can obtain estimates on the lower bound of an n-body Schroedinger operator H by computing the constants a and b, as was done in the proof of Theorem 7.4 in Chapter IV. A single-particle Schroedinger operator H with a potential satisfying the requirements of Theorem 5.8 also satisfies the conditions of Theorem 7.4 in Chapter IV. Hence, such a Hamiltonian H is bounded from below. T h e second resolvent equation (5.62)
( H - A)-1
=
( H o - A)-1
-
( H o - A)-lV(H - A)-1
holds for any value of h in the resolvent set, and therefore, in particular, for any real h < mH . T h e desired result about the spectrum of H , stated in Theorem 5.8, can be then obtained from Lemma 5.3, which will be derived with the help of the following important theorem. *Theorem 5.11 (Weyl's criterion for limit points). A real number A, is a limit point" of the spectrum of a self-adjoint operator A if and
* By definition, a limit point of the spectrum is either an accumulation point of the spectrum, or an eigenvalue of infinite multiplicity.
504
V. Quantum Mechanical Scattering Theory
only if there is a sequence fl ,f2 ,... E aAwhich converges weakly to the zero vector, and is such that Ilf,li = 1 and (5.63)
lim lI(A
n++m
-
h ) f n I1
0.
Proof. Suppose that 11 f, 11 = 1 , that w-limm++mf , = 0, and that (5.63) is true. For an arbitrary open interval (a, , b,) containing A, we can write II(A - An)fn
1/2 =
1
(A
R
~ IEiYn I 11~
-
2 (bn - An)2
jm ~ IEiYn I + (an 112
b0
== (bn
- An)2
IKI
- ER,>fnII
+
(an
- ~2
jao~ --a,
I Effn I
112
- ho12 II EtJn 1 1 ~ .
Now, the expression on the left-hand side of the above inequality approaches zero in the limit n + +a.Hence, we have
and consequently
=
lim 11 Eb”,fn fl+
W
1‘
+
lirn
n++m
11 Efofn /I2
=
1.
This implies that the subspace EA((ao,b n ] ) Z onto which EA((a,, b,]) projects cannot be finite dimensional; in fact, if this subspace were of finite dimension N , we could select an orthonormal basis {el ,..., eN> in it, and write
11 EA((an bol)fn /I2
= I(el
9
lfn>12 + . * *
+I(~N
lfn>l2
+
0,
since lim,,,, (el j f,) = = lirn,,,, (eN If,) = 0 due to the weak convergence to zero of fl ,f2 ,.... Thus, we conclude that An is either an accumulation point of S A or an eigenvalue with an infinite-dimensional characteristic subspace, i.e., that A, is a limit point of S A . Conversely, if A, is a limit point of the spectrum of A and I , 3 1, 3 are open intervals containing A,, then each EA(I,) is the projector on an infinite-dimensional subspace of X . Hence, we can choose an infinite orthonormal system {el , e2 ,...} such that EA(Im)en = e, , n = 1, 2,... . Since we have II(A - An)en
/12
=
1
In
-
An)2
~ IE,Aen I 11< ~ I I n I /I en 1 ’
+
0
5. Green Functions in Potential Scattering
505
when these intervals were so chosen that their lengths shrink to zero, and since limn++, (en I f ) = 0 for any f E 3 because of Bessel’s inequality, we see that the three conditions of the theorem are satisfied . Q.E.D. by this sequence el , e, ,... E 9A
*Lemma 5.3. If A and K are self-adjoint bounded operators on LP and K is also completely continuous, then the sets of the limit points K coincide. of the spectra of A and A
+
Proof. If A, is a limit point of the spectrum of A, according to Theorem 5.1 1 we can find a sequencef, ,f, ,... E LP such that 11 f, (1 = 1, w-lim,,+, f, = 0 and lirnnjfm ll(A - A,)f, 11 = 0. Since f , ,f,,... is a bounded sequence and K is completely continuous, it must contain a subsequence g , ,g , ,... such that Kg, , Kg, ,... converges strongly to some vector h E 2. However, since w-limnj+m g , = 0, we infer (K*h I g,) = 0. Hence, that 11 h 1, = limn,+, ( h 1 Kg,) = lirn,,,, lirn,,,, ll(A K - A,) g , 11 = 0, which in conjunction with 11 g , 11 = 1 and w-lim,++,g, = 0 implies, by virtue of Theorem 5.9, that A, is a K. limit point of the spectrum of A , = A Conversely, if A, is a limit point of the spectrum of A, , then, according to the same argument applied to ( - K ) , we deduce that A, is also a limit point of the spectrum of A = A , - K . Q.E.D.
+
+
We can return now to the proof of Theorem 5.8, and observe first the spectrum of ( H , - A)-,, that since the spectrum of H, is [0, +a), A < 0, is [0, -I/A], as seen from the relation EROyB)
= EHyF-yB)),
F(A’) = (A’
-
Ay.
T h e operator ( H , - A)-, V is an integral operator with the kernel K(r, r’; A) = G,(r, r‘;A) V(r’). Since this kernel is of Hilbert-Schmidt type, ( H , - A)-l V is a Hilbert-Schmidt operator, and therefore it is completely continuous. For A < m H ,the operator ( H - A)-l is bounded, and consequently, by Theorem 8.l(b) of Chapter IV,(H, - A)--, V ( H - A) is also a completely continuous operator. Hence, according to (5.53) and Lemma 5.3, the limit points of the spectrum of ( H - A)-, are the same as the limit points of the spectrum of ( H , - A)-,, i.e., they constitute the interval [0, - l/A]. This implies that the limit points of the spectrum of H constitute the interval [0, +a), i.e., H has no negative limit points. This means that the continuous spectrum of H is contained in [0, a), that any negative eigenvalues of H must be of finite multiplicity, and that these eigenvalues have no accumulation point on the negative real axis. Thus, Theorem 5.8 is established. .Theorem 5.8 deals with the point spectrum of the Schroedinger
+
506
V. Quantum Mechanical Scattering Theory
operator on the negative real axis. For the more restricted (but still very large) class of potentials V(r) for which there is an Ro > 0 such const, that for all r 3 R, the potential V(r) is continuous and I Y V(r)l it can be shown (see Kato [1959, Theorem 11) that the Schroedinger operator has no positive point spectrum, i.e., no positive eigenvalues. T h e reader should now recall that these results on the point spectrum of the Schroedinger operator are well illustrated by the case of the hydrogen atom treated in 97 of Chapter 11. I n that case we were dealing with a point spectrum of the form St = {-a/nz : n = 1, 2,...}, where a is a constant characteristic of the hydrogen atom.
<
5.8.
THERELATIONBETWEEN THE RESOLVENT AND THE SPECTRAL FUNCTION OF A SELF-ADJOINT OPERATOR
We know already that if the spectral function EA of any self-adjoint operator H is given, then its resolvent R(5) = ( H - c)-l can be computed by means of the formula (5.64)
It is not so obvious, however, that the converse of this statement is also true-namely, that the spectral function EAof H can be computed from the resolvent R(5) (i.e., from the Green function of H , if it exists). T o show that EA can always be computed from R(<),we need the following lemma. Lemma 5.4. If the function F(A) has a continuous derivative F’(A) on [a, 61, and is Riemann-Stieltjes integrable on [a, 61 with respect to cr(A), then (5.65)
lbF(h) a
&(A)
= F(b) u(b) - F(a) .(a)
1
b
-
a
F’(h) .(A)
dh,
where the integral on the right-hand side of the above relation is a Riemann integral. Proof. Taking A,‘ = A, in the defining formula (3.18) and recalling that A, = a and A, = b, we get
According to the mean value theorem of differential calculus, F(h,)
=q L 1 )
+
qJk-l)(&
-h l ) ,
4-1
<
Jk-1
< A,
*
507
5. Green Functions in Potential Scattering
Consequently, we have (5.66)
c
k=l
c n
n
F ( A k ) u( Ak)
-
F ( A k ) a( hk- l )
k=l
Since F’(A) is uniformly continuous on [a, b],
and the expression in (5.66) converges to the right-hand side of (5.65).
Q.E.D.
Let us apply the result of the above lemma to the integral in (5.64). For Im 5 # 0 this integral can be considered to be a Stieltjes integral, and since (A - 0 - l is continuously differentiable, we get
Letting a + - 00 and b -+
+
00,
we arrive at the result
from which we can derive Theorem 5.12.
Theorem 5.12. For any fixed f,g E 2, the function ( f I R ( 5 ) g ) is analytic at any complex 5 with I m 5 # 0, and for A, A,
<
where the above integrations should be carried out along any piecewise smooth curves which do not intersect the real axis.
Proof. According to (5.67) the complex derivative
508
V. Quantum Mechanical Scattering Theory
exists at any point 5 E C1 with I m 5 # 0 since the above improper Riemann integral converges absolutely and uniformly in any closed neighborhood of 5 which is disjoint from the real axis. Hence, ( f I R(5)g) is analytic in the upper and in the lower complex plain. Due to the analyticity of ( f I R ( 5 ) g ) the integrals in (5.68) are independent of path. If we carry out the integration along straight lines and use (5.64) and (5.67), we get
T h e integrand in the last of the above integrals is Lebesgue integrable in s and A on the set [A,, A], x [wl, since it is majorized by a function integrable on that set:
I
(s
(f I EAg) -
A
-+ i € ) 2 I
llfllllgll
(s - A)2
+
€2
.
Hence, the order of integration in s and h can be reversed and, after carrying out the integration in s, we obtain
By taking into consideration
from (5.69) we easily derive
509
5. Green Functions in Potential Scattering
It is quite obvious that the integrands of the above two integrals satisfy the conditions under which Lemma 4.2 is valid. Hence, by using that lemma, we arrive at the relation W+0 lim
-
(f I EAz-Og>
f
which is identical to (5.68).
(f I EAl-Og)
=
(f 1 E A z g )
-
(f I EAlg),
Q.E.D.
We shall see in the next section that the above theorem provides an important means of computing the spectral measure of a self-adjoint operator from its Green function.
EXERCISES 5.1. Prove that the set DG0defined in (5.18) is a Bore1 set of Lebesgue measure zero in R6. 5.2. Assuming that JRa, I K(a, a ’ ) / 2&(a) dp(a’) < +a,show that = JOBS K(a, a ’ ) f ( a ’ ) dp(a’) on the adjoint K* of the operator (Kf)(a) L 2 ( W ,p ) is also an integral operator with the kernel (K*)(a,a‘) = K*(a‘, a ) , and that the kernel (K*K)(a,a’) of K*K is equal to JR3 K*(CY“,a ) K(a”,a’) +(a’’).
5.3, Prove that any integral operator K on L2(Rn,p), which has an L2 kernel K(a, a’), is a Hilbert-Schmidt operator, and that
5.4. Give the reason why the limit in the mean K(a, a’) of a sequence of Hilbert-Schmidt kernels Kn(a,a’),
is a kernel of Hilbert-Schmidt type.
5.5, Prove that the kernel D,(a, a’) in (5.31) is of Hilbert-Schmidt type when K(a, a‘) is of Hilbert-Schmidt type. 5.6. Show that if A is Hilbert-Schmidt operator in a separable Hilbert space and B is a bounded operator then B A is a Hilbert-Schmidt operator.
.5.7. Show that for I m 5 # 0 the function F(r, r‘; 5) defined in (5.50) is integrable in r and r’ on a domain of the form D x R3, where D is compact if V(r) is bounded at infinity.
510
5.8.
V. Quantum Mechanical Scattering Theory
Let A , , A,
,... be
a sequence of bounded operators for which
x,",11 A, 11 < + co. Prove that the uniform limit A
=
u-lim n++m
n
1 A,
k=l
exists and is a bounded operator with
k=l
5.9.
Use the results of Exercise 5.8 to show that n
u-lim n++m
exists if 11 A
/I < 1 and I ck 1
c,Ak k=l
< M for all k = 1, 2, ... .
5.10. Prove that the self-adjoint operator A is bounded from below i f a n d o n l y i f ( f 1 Af) > m , , I j f 1 1 2 f o r a l l f ~ 9 ~ .
6. Distorted Plane Waves in Potential Scattering 6.1.
THERELATIONOF DISTORTED PLANEWAVES TO THE GREEN FUNCTION
T h e distorted plane waves @++)(r) can be constructed from the full Green function G(r, r'; t;) by taking its Fourier-Plancherel transform (6.1)
G(r, k; 6) = ( 2 ~ ) -1.i.m. ~/~
G(r, r'; 6) ecikr'dr'
L
3
in the variable r', and then setting
T h e verification of (4.25) is then achieved by resorting to the fundamental formula
-
1
- lim Ti r++O
I,, ha
($ I [R(h
which was derived in Theorem 5.12.
+ ie)
-
R(h - ie)]#) dh,
511
6. Distorted Plane Waves in Potential Scattering
T h e main purpose of the present section is to establish that, under suitable conditions in the potential, the limit in (6.2) really exists, and that the functions defined by (6.2) really qualify as the advanced distorted plane waves introduced in 94. Theorem 6.1 represents the first step in this direction.
Theorem 6.1. If the potential V(r) is integrable and square integrable on R3, then for any complex number K with I m K > 0, the function (6.4) h(r, k; K )
k2 - K'
-2m eirk +
:
(24312
1.i.m.
(24312
s,
G (r, r';
&-)
eikr'
dr'
is well defined by the above expression for almost all" values of r, r' E R3; in addition, for any such K with K~ $ SH and almost all r E R3 and k E R3, this function satisfies the equation (6.5) h(r, k; K )
= g(r,
k; K )
-
m
-
2n
S 3,
exp(iK1 r - r' I r - rI I
in which (6.6) g(r, k; K )
=
m2
- ~-
4/8nb
SR3
exp(iK1 r
-
r'
I r - rt I
I)
I)
V(r') h(r', k; K ) dr',
V(r') exp(ikr') dr'.
Proof. By comparing (6.1) and (6.4) we see that (4.7)
Hence, the existence of h(r, k; K ) hinges on the existence of e(r, k; K2/2m), whose existence, in its turn, can be inferred from the square integrability of G(r, r'; K2/2m) in r' E R3. But we know already that for any fixed r' and K~ 4 SH, G(r, r'; K2/2m) is the square integrable in r E [w3 solution of (5.24). Since for any fixed r and K we have by (5.52) that G(r, r'; K2/2m) = G(r', r; K2/2m),we conclude that G(r, r'; ~'/2m) is square integrable in r' E R3. Therefore, h(r, k; K ) is well defined by (6.4) for almost all k E R3. Due to the existence of G(r, k; K2/2m), we have
* In this section all the measure theoretical concepts refer to the Lebesgue measure, except if otherwise stated.
512
V. Quantum Mechanical Scattering Theory
on account of the unitarity of the Fourier-Plancherel transform. Hence, the second resolvent equation (see Theorem 3.2)
can be written in the form
I t can be easily computed that for I m K > 0 the Fourier transform (with respect to the variable r’) of 1 r - r ’ 1-l exp(iK I r - r’ I) is equal to d G (k2 - K ~ ) -exp(-ikr), ~ so that
Now, it can be shown that the function
K (r, r‘;
c (r’; k; &)$(k)
$)-
is integrable in r’ and k on R6 (see Exercise 6.1). Hence, the order of integration in r’ and k can be inverted in (6.9) (by virtue of Fubini’s theorem), and we obtain R
c (r, k; $-) -
1
P3
$(k) dk
dk +(k)
1
R3
a)c
dr‘ K (r, r’;
(r‘, k; T). K2
6. Distorted Plane Waves in Potential Scattering
513
Since the above equation holds for arbitrary square-integrable functions &k), we infer that the following equation is satisfied for almost all k E R3, (6.11)
K2 G (r, k;-)2m
=
~
2m exp(-ikr) (277)”2 k2 - ~2 exp(iK1 r - r‘ I) _ _m V(r’) 2rS,3 I r - r’ I
e (r’, k; 2) dr’. 2m
After solving (6.7) for e ( r , k; ~ ~ / 2 r and n ) substituting the expression obtained for G(r, k; ~ ~ / 2 rinto n ) (6.11), we arrive at the integral equation (6.5). I n this equation g(r, k; K)is defined by the integral in (6.6), which is essentially the Fourier transform of the function
Ir
-
r‘ 1-l exp(iK1 r - r’ I) V(r’).
Since this function is the product of the two square-integrable functions V(r’) and I r - r ’ 1-l exp(iK I r - r’ I), it is integrable, and therefore its Fourier transform indeed exists. Q.E.D. Though the equation (6.11) has a kernel of Hilbert-Schmidt type, we cannot solve it by the Fredholm method because the inhomogeneous term 4277 I r - r ’ 1) - 1 / 2 exp(ikr) is not square integrable in r. The transition to the new function h(r, k; K) serves the purpose of providing the equation (6.5), which has an inhomogeneous term g(r, k ; K) square integrable in r. Hence, this last equation can be handled by the Fredholm method. 6.2.
EXTENSION OF G(r, k ;
5)
TO THE
REALAXIS
In the following lemma, we investigate the equation (6.5) in the case of I m k = 0.
Lemma 6,1, Suppose the potential V(r) is measurable and locally” square integrable on R3, that the integral (6.12)
* The function V(r) is locally square integrable if and only if it is square integrable on any compact set. 1 V(r)( = O(r-2-Co) if there are constantsR, > 0 and Co > 0 such that r 2 + Q V(r)l < C, for all r > Ro’. These two conditions when combined obviously imply that V(r) is square integrable on Rs.
V. Quantum Mechanical Scattering Theory
514
exists, and that (6.13)
+
for some E , > 0. Then, if H , 1 V 1 has no positive eigenvalues,* the integral equation (6.5) has a unique solution h(r, k; K ) for all complex values of K with Im K 3 0, except for those pure imaginary values of K for which ~ ~ / 2 rE nSf. We know already from Theorem 6.1 that the function K~ $ SH. We observe, however, that this integral equation does not have a kernel of Hilbert-Schmidt type when Im K = 0, since in this case, by Fubini's theorem, Proof.
h(r, k; K) satisfies the integral equation (6.5) for any K for which
exp(iK1 r - r'
I) V(r')
1
2
dr dr'
T o remedy this situation we introduce the auxiliary function
(6.14)
-dl V(r)l vdr)
>o
if
V(r)
if
V(r)
if
V(r)
= 0,
where vo(r) vanishes outside the set {r: V(r) = 0} and it is positive and continuous on that set; in addition, vo(r) is chosen as to make I V(r')l integrable on cW6 in the variables r and r'. vO2(r)lr - r' We rewrite the integral equation (6.5) in the form (6.15)
hl(r, k;K ) = gl(r, k; K )
+
w
&(r, r';
K)
Mr', k;K ) dr',
* The absence of positive eigenvalues of H can be established by studying the dependence of (1 - Kl(~))-'on K , considering in the process (1 - K ' ( K ) ) -as~an operator on a is suitable Banach space (see Ikebe [1960]).In fact, once the existence of (1 - K1(~))-l established by this method (even for the case of real K ) , then the absence of any positive eigenvalues of the Schroedinger operator can be derived as a consequence of the considerations in this section. Furthermore, we already mentioned at the end of $5 that due to a theorem of Kato, the Schroedinger operator has no positive eigenvalues when I V(r) I = O(r-l). Hence, later on, we shall not rentain this redundant condition on the spectrum of H.
6. Distorted Plane Waves in Potential Scattering
515
where
The newly introduced kernel Kl(r, r'; K ) is of Hilbert-Schmidt type whenever Im K 2 0. This is due to the fact that
and that the integral on the right-hand side of the above inequality exists by virtue of the requirement that (6.12) exists. The function g(r, k; K ) is well defined by (6.6) for almost all values of r even when Im K = 0. As a matter of fact, we have (6.17)
the existence of the integral (6.17) is a consequence of the restrictions imposed on V(r) (see Exercise 6.3). We note that the constant C can be chosen t o be the same for all the values of r, k, and K . Since the kernel Kl(r, r; K) is of Hilbert-Schmidt type, it can be solved by applying Theorem 5.4, provided that gl(r, k; K ) is square integrable in r E R3. But the square integrability of gl(r, k; K ) in r E R3 for almost all values of k E 083 and I m K 2 0 can be inferred easily from the existence of the integral
by using Fubini's theorem. Let us now write m
P(K =)C dn(l)(K), n=O
m
D("(r, r'; K )
=
1 Dn(l)(r,r';
K),
n=O
where d , ( l ) ( ~and ) Dn(l)(r,r'; K ) are obtained by taking Kl(r, r'; K ) to be the kernel in (5.30) and (5.311, respectively. According to Theorem 5.4, the equation (6.15) has the unique square-integrable (in r E R3) solution (6.19)
516
V. Quantum Mechanical Scattering Theory
provided that d ( l ) ( ~#) 0. Hence, if we knew that d ( l ) ( ~#) 0 also in the case when I m K = 0, the lemma would be established. ) if and only if the integral operator By Theorem 5.5, d ( l ) ( ~vanishes K,(K)with the kernel Kl(r, r’; K ) has the eigenvalue one. We will show, however, that even when K is real, Kl cannot have the eigenvalue one. The relations (5.6) and (6.16) indicate that for Im K > 0 the operator K l ( ~ )is identical to the operator -v(HO - ~~/2rn)-l I V 1 / 2 , where ZJ and I V 11/ 2 are the operators of multiplication by v(r) and I V(r)11/2, respectively. If K , were a real number for which K,(K,)#, = #,, 11 $, (1 > 0, then we would have (see Exercise 6.4)
#,
= w-lim K+KO
-)2m K2
K,(K)go = -w-lim v K+Ko
where in taking the limit we must keep Im K
$ E gIv, n 9 1 v , 1 / z , we could write (6.20)
(# I I V‘ I’”$Jo)
=
(I I‘P2#I # J
=
-(I
v I#
141) =
-(*
-1
I V‘
> 0.
3
Hence, for any
I I v l#l)>
where $1 appears as the weak limit of ( H , - ~~/2rn)-l I V /1/2#o ; this weak limit exists as a consequence of the existence of the advanced Green z,hl = function Gh+)(r,r’; ~2/2rn). Since, obviously, ( H , - ~,~/2rn) - I V 1 /2#o , we infer from (6.20) that ( H , - ~,~/2rn) = - I V I , i.e.,
P o
+ I V‘
IWl
= (Ko2/2m)h*
+
But, we have already stated that the operator H , 1 V 1 does not have positive eigenvalues. Hence we conclude that d ( l ) ( ~ ,.f ) 0. Q.E.D. The preceding lemma will be essential in deriving the rest of the results in the present section. Hence, it is desirable to understand very well the nature of the condition imposed in this lemma on the potential V(r). The lemma allows potentials which are singular on a set YvC R3, as long as the measure of this set is zero and (6.12) holds. For instance, in the case of the Coulomb potential V(r) = const I r I-l, 9’would consist of the single point r = 0. We note, however, that the Coulomb potential does not satisfy the condition (6.12) of the lemma. On the other hand, the Yukawa potential V(r) = const e-alrl 1 r 1-l satisfies these conditions, and it also has only one singularity which is at the origin.
517
6. Distorted Plane Waves in Potential Scattering
In general, an almost everywhere continuous potential V(r),which is at infinity, and has locally square integrable, vanishes faster than I r only a finite number of singularities r’ in whose neighborhood it diverges more slowly than 1 r - r’ fulfills all the conditions of the lemma. The last of these conditions guarantees, in conjunction with (6.13), the existence of the integral (6.12). It must be realized, however, that these are sufficient but not necessary conditions for Lemma 6.1 to hold true. On the other hand, they are satisfied, for instance, by any bounded and almost everywhere continuous potential, such as the square-well potential. Moreover, all the potentials commonly occurring in practice, with the exception of the Coulomb potential, satisfy such conditions. We shall refer to the set YVmany times in the future. Hence, it is desirable to be more specific and define YVas the set (of Lebesgue measure zero) which is such that V(r) is bounded on any closed subset of R3 which is disjoint from 9’:. This means that Yvcontains all the singularities of V(r), but not also all the points at which V(r) is discontinuous. Thus, for instance, SP, is empty when V(r) is the square-well potential, which has a two-dimensional set of points of discontinuity, but no singularities. 6.3.
c(r,
CONTINUITY PROPERTIES OF k ; 5) I N THE UPPER HALFPLANEOF 5
Our next task is to relate the functions G(r, k; 5) and h(r, k; K ) to the spectral measure EH(B) of the Schroedinger operator H. Such a relation can be established by means of the basic formula (6.3). First of all, let us relate the integrand in (6.3) to G(r, k; f ) . Using the first resolvent equation (3.62), we deduce that
(6.21)
(4I P ( 5 )
- R(5*)1$> = (5 - 5*K$ =
I R(5)R(5*)$)
(5 - 5*)(W*WI R(S*)$).
Since according to (5.52) the Green function G(r, r’; 5) is symmetric in r and r‘, we can write
If we choose $(r) to be continuous and of compact support, then we take the Fourier transform on both sides of the above equation, and thus obtain, after interchanging the order of integration in r and r’,
V. Quantum Mechanical Scattering Theory
518
This interchange of the order of integration can be easily satisfied when the domain of integration D is compact. As a matter of fact, according to Theorem 5.7,
I G*(r, r’; 5) +(r)l = I G(r’, r; 5) +(r)l< constF(r’, r; 511 +(r)l, where F(r’,r; 5)1 +(r)l is a function integrable in r’ and r on R3 x D when $(r) is continuous and of compact support (see Exercise 5.7); hence, G*(r, r’; 5 ) $(r) is integrable on D x R3 and Fubini’s theorem can be applied. Let us introduce the function
$(k; 5 )
(6.23)
= (2m5 - k2) 1.i.m.
1, e*(r,-k; 5)
+(r) dr.
Using this function and (6.22)’ we can write
Setting
5
=
h
+ ie, we easily obtain from (6.21) and the above relation 1
=
a
1,3
2k (k2/2m-
+ e2
I $(k; X
+ ;€)I2
dk.
Hence, in the present case, (6.3) assumes the following form: (6.25)
(+I (EE
+ EE-oM)
-- 1 -
4nim2
-
(4 I (EE + G-o)#)
!$IJ,, dXJ,s Aa
(A
-
+ c2 I $(k; X + ie)12 dk
2ie k2/2m)2
We observe that the reversal of the order of integration in k and h in the above integrals, taken at fixed e > 0, is a direct consequence of Tonelli’s lemma (which guarantees that the integrand is integrable in X and k on [A,, A,] x R3) and Fubini’s theorem. We are now faced with the problem of finding the limit E --f +O in (6.25). At first glance, this would seem to be a straightforward task since the factor .[(A - k2/2rn)2 €21-1 is strongly reminiscent of a similar
+
6. Distorted Plane Waves i n Potential Scattering
519
factor in (4.68). A more careful inspection reveals, however, that the k)12 situation is not quite that simple, since the other factor I $(k; X in the integrand also depends on E. Hence, the following lemma will be needed in computing the limit (6.25).
+
*Lemma 6.2. Suppose the potential V(r) is measurable and locally square integrable on R3, that the integral
exists, and that" (6.26)
Then for any compact domain 9 in the upper complex plane I m K 2 0}, containing no pure imaginary values of K for which ~ ~ / 2Er SF n , the following are true.
{K:
(a) For any (6.27)
E
> 0, there is a a(€, 9)such that I h(r, k;K
+ A K ) - h(r, k;.)I
< eFo(r),
+
for any I AK I < a(€, 9),K A K E 9, k E R3 and r E R3 - Y ; , where 9'"is the set of singularities of V(r), i.e., V(r) is bounded on any compact set disjoint from 9". (b) There is a constant C ( 9 ) such that (6.29)
I h(r, k;41 < C W F 0 ( r )
f o r a n y K E 9 , k ~ R ~ a n d r ~YV. R ~ -
+
* The behavior of V(r) for Y --+ a, is determined by the requirement that the integrals appearing in the definition (6.28) of F,(r) exist. In fact, since the integration in r' E Ra O(l/l r' Icltalla) involves products of three functions which behave as O(l/l r' and O(l/l r' la) (see Exercises 6.5 and 6.6) the integrand behaves for Y' --* +a, as O(l/l r' Icaa+al/z). For the convergence of the integral, we need g(5a 5 ) > 3, i.e., OL >
+
8.
520
V. Quantum Mechanical Scattering Theory
We base the proof of Lemma 6.2 on a detailed analysis of the relation (6.30)
h(r, k; K )
obtained by multiplying the right and left-hand side of (6.19) by (v(r))-l. . . .. First of all, we shall study g(r, k; K ) . If R, > 0 is sucl that r2+,oI V(r)l C, for all r > R, , then we have for any R, R,
I g(r, k;K
+
<
>
- g(r, k; .>I dr'
If R, is chosen large enough, the first integral on the right-hand side of the above inequalities can be made smaller then any given E, > 0 for all r E R3 (see Exercise 6.2). For any such fixed R, , the second
integral can be also made smaller then I AK I so small that
1 exp(i1 r
E,
for all r E [w3 by choosing
1I < exp(-2Ro Im A K ) { cos ~ 2R, 1 AK( -1 - r' 10,)
-
1 + 1 sin 2R0/A K 1 I}
where the constant a is such that
for all r (6.31)
E
R3 (see Exercise 6.3). Hence, we have
I g(r, k;K
+ A K ) - g(r, k;.)I
< 2e1
for all r E R3. Let us prove now that d ( l ) ( ~is)a uniformly continuous function of K in the upper half of the complex plane ( K : I m K O}. Specializing
>
521
6 . Distorted Plane Waves in Potential Scattering
(5.30) to the present case and expanding the determinant representing the integrand, we get
x
fq,...,u,(rl
,***,
rn;
K)
dr1 ... drn
-
In the above formula, ~ ( a,..., , a,) is zero for all even permutations ( a , ,..., a,) and one for all odd permutations. T h e summation extends over all permutations (a1,..., a,), except those in which aJ = j for some j = 1,..., n. T h e functionfal ,...,a,(rl,..., r, ; K ) is of the form
where the product has n factors and the sum has n terms, each vector ri occuring exactly twice. Hence, for given el > 0 and R , > 0 we have (6.33)
I f m l ....,Jr1 ,.-,r n ; K + AK) -fal ,...,a,(rl )..., r,; .)I
<
for all rl ,..., Y, R , and all sufficiently small I A K 1. Let us write the integral in (6.32) in the form
<
<
where Sn(R,) = {rl:rl R,} x ... x {r,: r, R,}. T h e second integral can be made smaller than any given E , > 0 by choosing R1 > R, sufficiently large, while the first integral is continuous in K on account of (6.33). This establishes that &')(K) is uniformly continuous in K in the region { K : I m K 3 O}. Since according to (5.43) (6.34)
Id3K)I
< {(e/41'zll
~l(K)ll2Y
< {(e/n)1'21/
~l(O)llZ)",
the series for P(K) converges uniformly in K. This fact and the uniform continuity of d L l ) ( ~ )for every n imply that &)(K) is uniformly continuous O}. in the complex domain { K : I m K
>
V. Quantum Mechanical Scattering Theory
522
Let us investigate now the terms in the sum of the integrand in = 0, we have
(6.30). For n (6.35)
m exp(-Kl r - r' IT-r'I
1 Do(1)(r, r'; K) = v(r) 2n
I) d -. I V(r'>l*
Let us introduce the function (6.36)
It is easily seen that for any E;,)
<
> 0 and A, > 0, we have
+
R, , I m K , Im(K A K ) 3 0 and all sufficiently small for all I, Y' values of I AK I. To derive a similar formula for Dil)(r,r; K ) , we specialize (5.44) to the present case, thus obtaining (6.38)
D!)(r,
r'; K ) = d:$(~)
Kl(r, r'; K )
+
R
D!(lr,
r,; K ) Kl(rl , r'; K ) dr,
Let us introduce the convenient notation K,"(r, r') =
1
R3"
&(r, rl) &(r1, rZ) Kz(r,, , r') dr,
dr,
.
6. Distorted Plane Waves in Potential Scattering
523
We specialize (6.39) to the case n = 1 and using the estimates (6.34),
(6.37), and (6.40)
I D?(r, r'; .)I
< K2(r,r'),
we easily obtain that for all r, r' (6.41)
(Dp)(r,r'; K
+ AK)
< $K2(r,
r')
-
I Kl(r, r'; K)I
< R, and
Dp)(r, r'; K ) )
+ EP)K:(r,
r')
< Kz(r,r'),
I m K , Im(K
+1
r+R1
+ A K ) 3 0,
Kz(r,rl) Kz(rl , r') drl ,
where for any given R, > 0, we can choose E ( ' ) , d 2 )> 0 small, with corresponding sufficiently small values for I AK 1. From (6.38) we also obtain, by using the estimates (6.34) and (6.40), (6.42)
1 Dp'(r, r'; .)I
< {(e/2)1'211K , ( ~ ) l l ~ } ~ Kr')~ (+r ,KZ2(r,r').
Proceeding by mathematical induction, and using the same type of arguments as in going from n = 0 to n = 1, it is straightforward to R, and I m K , Im(K A K ) 3 0 deduce that for all r, r'
+
<
where ck'), ..., ckn) are constants and EL'), ..., €in) > 0 can be chosen arbitrarily small for adequately small values of I AK 1; in this process of induction one must also generalize (6.42), by establishing that Dk')(r, r'; K ) = &K2(r, r') ... akn+')Kg+'(r,r'),where u;'), ..., a:) are constants. From (6.30) we get
+ +
(6.44)
I h(r, k; K + A K ) - h(r, k; K ) I < I g(r, k;K A K ) - g(r, k;.)I
+
I d(l)(K
f dK)
x I gl(r', k; K
- d(l)(K)I
+ A K ) dr' ~
~
D;)(r, r'; K
+
AK)
524
I
I I,.
+ d”1(K)
V. Quantum Mechanical Scattering Theory
fI
D!)(r, r‘; K
+ AK)
-
v(r)
n=O
X
A K ) dr’ ~
+
n=O
X
A K ) - gl(r‘, k; K ) I dr’
D!)(r, r‘; K )
II
I!
We have already established that the first term on the right-hand side of the inequality (6.44) can be made arbitrarily small, uniformly in r, k, and K . Since O ( K is )continuous and never vanishes for Im K 3 0, it must have a positive minimum in any compact region of the upper complex K plane. T o estimate the integral in this second term, we specialize (5.46) to the present case and easily obtain (6.45)
Since (see also Exercises 6.2 and 6.3)
where (6.46)
for rl for rl for rl
<1 > 1, > 1,
0 c0
< e0 < 1 > 1,
we see that the second term in (6.44) is majorized by EF,,(r), where E > 0 can be chosen arbitrarily small for adequately small values of I LIK I. To deal with the third term on the right-hand side of (6.44), we must split the sum into two parts, the first part containing only n = 1,..., N ,
F,(r) is defined in (6.28) and
525
6. Distorted Plane Waves in Potential Scattering
while in the second part N < n (6.45) and (6.46) that for given E we can achieve
x I gl(r’, k; K
+
< co. It is easy to establish by using > 0 and compact 3 C I m K 2 0}, {K:
+ O K ) ]dr‘ < EFo(r)
for all K , K AK E 9, k E R3, r E R3 - Y V by , choosing N sufficiently large. For any such N , it is then possible to make the integral
over the region S = {r’: r’ 3 R,} arbitrarily small for sufficiently large R, 3 R, ; this result easily follows from the estimates (6.45) and (6.46)
(see also Exercises 6.5 and 6.6). Finally, it follows from (6.43) and (6.46) that by choosing R, > 0 adequately large (to make the last term in (6.43) as small as desired) and I AK I sufficiently small, we can make the integral (6.47) over the region S = {r’: r’ I?,} smaller than E,F,(r) for any a priori given E , > 0. Thus, the third term in (6.44) also conforms to the pattern leading to the first statement (6.27) of the lemma. Moreover, it is evident upon an inspection carried out in the same vein that the last term in (6.44) also conforms to this pattern on account of (6.31) and (6.45). Hence, (6.27) is established. T h e proof of (6.29) proceeds along similar lines, and is left to the reader.
<
6.4.
OF THE GREENFUNCTION RELATION TO THE SPECTRAL MEASURE
By combining (6.7) and (6.23), we arrive at the relation
which is valid for any complex value of K with I m K # 0. The function F,(r) in (6.28) is bounded on any compact domain disjoint from YV(see Exercise 6.13). Hence, if we choose #(r)E %:([w3) to have a support disjoint from Yv , we can infer from Lemma 6.2 that, given E,, > 0, we have
V. Quantum Mechanical Scattering Theory
526
for all k E R3 and A E W, and for any sufficiently small value of This result, in conjunction with (6.25), implies that
E
> 0.
Let us take now A,, A, E:S with A, > A,. I n view of (6.48) and C, (where C, Lemma 6.2, $(k; A) is continuous in A and I $(k; A)/ is a constant) for all k E R3 and all A E [A, , A,]. Thus, we can infer that
<
< Clz (arctan
A,
-
k2/2m E
-
arctan
Hence, it follows from the properties of the function arctan x that for < E 6 and given 6 > 0 we can choose a constant C , such that
0
<
where the newly introduced function F,(k) is obviously integrable on R3. On the other hand, by using a slightly altered version of Lemma 4.3 (see Exercise 6.8) we infer that
6. Distorted Plane Waves in Potential Scattering
527
Hence, we can apply Lemma 3.1 of Chapter IV to deduce that the limit f O can be taken under the first integral sign in (6.49). Thus, we obtain E -+
after taking into consideration that A, , A, E Sf and therefore Eh,-o = EAi, i = 1, 2. We shall extend the relation (6.50) to more general functions a,h(r), thus arriving at the following theorem, in which we slightly reiax the restrictions imposed in Lemma 6.2 on the potential V(r) (see Exercise 6.9).
Theorem 6.2. Suppose that the potential V(r) is measurable and locally square integrable, that it is bounded on any compact set disjoint from a certain closed set YVof measure zero [which contains all the V(r') is locally" singularities of V(r)], that the function V(r)l r - rf integrable in r and rf in Iw6 and that I V(r)l = O(r--2--co)for some c0 > i . Then the function
= @Li)*(r),
exists for every k E [w3; in addition, for any a,h E L ~ J R ~=) EH(Sf)L2(Iw3) we have for any Bore1 set B C Sf C [0, 00)
+
where
By comparing (6.51)-(6.53) with (6.23), (6.49)) and (6.50) we see that we have essentially already proved Theorem 6.2 for the case when $(r)
is continuous and has compact support which is disjoint from SP,. Now, the set 9 0, of all functions which are continuous and of compact is dense in L2(Iw3); this easily follows from the support disjoint from Yv
* See Exercise 6.9.
V. Quantum Mechanical Scattering Theory
528
fact that %i(lw3) is dense in L2(R3),while any given function in VE(R3) can obviously be approximated arbitrarily well in the mean by an element of %; (see Exercise 6.14). Hence, if $(r) is any given squareE %'$ which integrable function, we can find a sequence $(l)(r),$(2)(r),... converges to $(r) in the mean. Now let us show that $y)(k), @)(k),... converges in the mean to a limit $+(k) which is the same for any sequence {$(")(r)} converging strongly to $(r). First of all, we note that $(i)(r)- $(j)(r) belongs to VO, when both $(i)(r)and $(j)(r) belong to Vb . Hence, (6.52) holds for yYt)(r)- $(j)(r),
Noting that
and then taking A, = 0 and letting Aa + fco, we obtain
11 #(i) - #(') 1 '
=
/ ,I $$'(k) R
- $y)(k)l2 dk.
This shows that i,&)(k), $F)(k),... is a Cauchy sequence in L2(R3). Consequently
$+(k) = 1.i.m. $f'(k) exists. Moreover, if {$in)(r)} is another sequence from %; converging to $(r) in the mean, then $(n)(r) - $in)(r)also belongs to V; . By the above procedure we easily obtain again
I1 #(n)(r)- #?)(r)Il'=
s ,I R
$p'(k)
- $k)(k)I2 d
k
and consequently {t,@(k)} converges to the same limit as {Jy)(k)}. By already standard methods (see Chapter 111, $5) we infer from (6.52) that for any Bore1 set B C [0, co)
+
Taking in both sides of the above relation the limit n + fa, we arrive at (6.52). Hence, Theorem 6.2 is established.
529
6. Distorted Plane Waves in Potential Scattering
6.5.
THELIPPMANN-SCHWINGER EQUATIONS FOR PLANE WAVES
We introduced the functions Ok+)(r) by means of the formula (6.2). Afterwards, we related these functions to the spectral measure of the Schroedinger operator. We have not yet established, however, whether @i+)(r)defined by (6.51) are indeed the outgoing distorted plane waves defined in 94. T h e first step in establishing the physical nature of Ok+)(r) is to show that these functions are indeed eigenfunctions of the Schroedinger operator. Let us introduce the functions wk(r) = [ g I 3 I 2 h ( r ,k; 1 k I)
(6.55)
for all k E R3. Combining (6.51) with (6.7) and using the above definition, we get
+
(6.56)
@F)(r)= ( 2 ~ ) - ~ / ~ ( e i k ' ok(r)).
According to (6.5) and Lemma 6.1, uk(r) satisfies the integral equation
where K (6.58)
I k I and
=
m exp(ik1 r - r' I) gk(r) = - V(r') exp(ikr')dr'. I r - r'I 277 S R 3
Expressing in (6.57) vk(r) in terms of result.
@i+)(r),we
obtain the following
Theorem 6.3. If the potential V(r) satisfies the conditions imposed on it in Theorem 6.2, then the functions @:*)(r) defined in (6.51) satisfy (almost everywhere in r E W) the integral equations (6.59) @k(*t)
(r) =
~
(243'2
exp(ikr) - m 2rr
I
R3
exp(+ikl r - r'
I r-r'I
I) V(r') @L*)(r')dr'
for all k E R3. T h e inhomogeneous term in the above integral equations is a free plane wave. Hence, after taking a look at (4.92), we easily recognize in (6.59) the type I Lippmann-Schwinger integral equation. Since we have already seen that the solution of (6.57) [and therefore also that of (6.59)]
V. Quantum Mechanical Scattering Theory
530
is unique, we can infer from Theorem 6.3 that if outgoing plane waves with the properties required in 94 exist, then they coincide with the functions @i+)(r)defined by (6.51). AND 6.6. THESCATTERING AMPLITUDE
THE
BORN APPROXIMATIONS
We can derive from the Lippmann-Schwinger equations (6.59) the asymptotic behavior of ok(r) which was predicted in (4.18).
Theorem 6.4. Suppose that the potential V(r) is measurable and locally square integrable in R3, that it is bounded on any compact set disjoint from a given closed set YV[which contains all the singularities V(r’) is locally integrable on R6, and of V(r)], that V(r)l r - r’ that for some e0 > 0 (6.60)
Then ok(r)satisfies* the following estimate for large values of r : (6.61)
where 9 and r/r and
rj5
are the two spherical coordinates of the unit vector
Proof. T h e conditions imposed on V(r) in the present theorem are such that all the conditions of Theorem 6.2 are also satisfied. Consequently, zlk(r)satisfies the integral equation (6.57). Let us introduce the function (6.63)
m 2n
wk(r) = - -(ezkr
+ nk(r)) V(r)
in terms of which (6.64) Strictly speaking, we should explicitely require that k2/2mE Sr. However, relying on the results of Kato [1959] (where Kato shows that, for his class of potentials, H has no positive eigenvalues), we can state that for potentials V(r) satisfying the conditions of Theorem 6.4, we have k2/2m4: Sp”.
531
6. Distorted Plane W a v e s in Potential Scattering
We note that zIk(r) is certainly uniformly bounded at infinity (see Exercise 6.10), so that for fixed k (6.65)
I zuk(r)I< const I V(r)l = O(l/r3+%).
Let us write now (6.64) in the form
According to (6.65), there is a constant M such that for suitably large R
Consequently, for such values of R
By selecting, for fixed r, the x1 coordinate axis in the direction of the vector r, and working in spherical coordinates, we arrive at the following estimate of the above integral:
Furthermore, using Taylor's formula at r,
1
Ir-rlI
1
=y
[l
=
+0( 3 1 . 2
0, we get
V. Quantum Mechanical Scattering Theory
532 Consequently, we can write (6.68)
exp(ik( r - rl Ir-rlI
1
Let us take in (6.66) R = ~ ( l - a ) / ~ with , some fixed 6 < 1, and then choose r so large that (6.67) becomes valid. Inserting (6.68) in IR’, we obtain, after noting that for the selected R we have YJY < ~ - ( l + ~ ) / ~ ,
Now, by virtue of (6.67), we also have
and consequently
On the other hand, for the chosen value of R,(6.67) yields IR” =
O
(
1 yl+[(1-6)/2]c0
) +O
1 (=)2
so that we obtain
+
Setting in the above estimate 6 = ~,,/(2 (6.61). Q.E.D.
< E,, , we
E ~ )
finally obtain
T h e main significance of the above theorem lies in the fact that it provides us with the explicit formula (6.62) for the scattering amplitude fk(e,#). This formula requires, however, a full knowledge of @L+)(r).
6. Distorted Plane Waves in Potential Scattering
533
On the other hand, even in the case where @)ljri)(r) is not known, we can still obtain an approximation of f k ( e , + ) by using (6.56) and neglecting the second term vk(r). Then we obtain the jirst Born approximation of the scattering amplitude: m f k ( 8 , c ) m - 2n
(6.69)
i,,exp
(-z%
7)r . r‘
v(r’)exp(ikr‘) dr’
Higher Born approximations can be obtained by an iteration method in which we start by setting, on the right-hand side of (6.59), @i+)(r’)w ( 2 ~ r ) - ~eikr’, / ~ thus obtaining the second Born approximation for @&+)(r): (6.70)
@k)(r)m
___
(243’2
exp(ikr) exp(ik1 r
-+IR. Ir-r’I
-
r‘ I)
V(r’) exp(ikr’) dr‘.
Inserting the above approximate expression for @$+)(r) in (6.62), we obtain the second Born approximation for the scattering amplitude
f (4 4).
can be inserted on the T h e second Born approximation for @)ll”(r) right-hand side of (6.59) to obtain the third Born approximation for @i+)(r),which in turn can be used in (6.62) to obtain the third Born approximation for fk(B, $), etc. For certain potentials, this iterative procedure converges, and provides a practical method” for the computation of fk(8, +). 6.7.
THEGENERALIZED PARSEVAL’S EQUALITY
We recall from $5.8 that under the restriction imposed in this section on the potential V(r), the Schroedinger operator H has a point spectrum SE only on the negative real axis. Moreover, by Theorem 5.8, all the eigenvalues of H are of finite multiplicity, and SF has no negative accumulation points. These results imply that we can choose an orthonormal basis (6.71)
of eigenvectors @ k , v of H , (6.72)
* See Massey [1956] for criteria of convergence of the above method, as well as for estimates on the degree of accuracy of the first Born approximation.
V. Quantum Mechanical Scattering Theory
534 which is such that for any
+ €L2(jW3)we have
Hence, by combining the above formula with the relation (6.52) in which we take B = ,S: we obtain
II 4 112
(6.74)
=
I1 E H ( S ; ) $ 112
+ I1
EH(S:)+
112
T h e above equality is strongly reminiscent of (4.16) in Chapter I. Consequently, we can expect that other formal features of Theorem 4.6 in Chapter I are retained in the present case. This hope is realized in Theorem 6.5. Theorem 6.5. Suppose the potential V(r) satisfies the conditions in Theorem 6.2. Then for any $ €L2([W3)and any Bore1 set B C S : C [ O , +m)
&(k)
= 1.i.m.
@f)*(r) $(r) dr, L
and for any
3
z/A2) eL2([W3)
(6.76)
Proof. One can easily derive (6.76) from (6.52) by setting first + $(2) and then $ = $(I) + i$@), and using formula (5.13) in Chapter IV with f = +(I) and g = $(".
+
= +(l)
Let us select any +'(r) from the family %?; of all continuous functions with supports which are compact and disjoint from the set .SP, of singularities of V(r). From the last statement of Lemma 6.2 we infer that for such functions
(6.77)
I $+'(k)I =
15
R3
@p)*(r)$'(r) dr
I<
const
s,1
+'(r)ldr.
535
6. Distorted Plane Waves in Potential Scattering
We know from Theorem 6.2 that a function $+(k), related by (6.53) to an arbitrary square-integrable function $(r), is square integrable on R3. Hence, $(k) is integrable on any compact Borel set B' C OX3. This statement in conjunction with (6.77) implies that the function $'*(r) @L+)(r) $+(k) is integrable in the variables (k, r) on any set of the form {k: k2/2m EB,} x R3, if the Borel set B, C Sf is compact. Consequently, the reversal of the order of integration in the following derivation,
-
Sk2/2msB,,
=
dk $+(k)
1 dr $'*(r) R3
j
R3
kz12meBn
@L+)(r) $'*(r)dr
@F)(r) $+(k) dk,
can be justified by using Fubini's theorem. Since (6.78) holds for arbitrary $'E%?:, and since we already know that V: is dense in L2(R3),we conclude that (6.79)
whenever B, C S z , is compact. If the Borel set B C Sf is not bounded, we can introduce the sets B, = B n [0, n] which are bounded, and to which (6.79) can be applied. Since EH(B,)$ converges strongly to EH(B)$ when n ---t 00, the relation (6.75) immediately follows as a consequence of (6.79). Q.E.D.
+
6.8. DISTORTED PLANEWAVESAS EIGENFUNCTIONS OF THE SCHROEDINGER OPERATOR For the complete identification of @:+)(r)as outgoing distorted plane waves, we still have t o establish that @:+)(r)is an eigenfunction of the Schroedinger operator with the eigennumber k2/2m. Theorem 6.6 provides a set of precise criteria under which @:+)(r)is indeed an eigenfunction of the Schroedinger operator.
Theorem 6,6. Suppose that the potential V(r) is continuous in the neighborhood of a point r E R3, and that it satisfies the conditions
536
V. Quantum Mechanical Scattering Theory
imposed on it in Theorem 6.2. Then the distorted plane waves @:*)(r) satisfy the time-independent Schroedinger differential equation (6.80)
at that point r E R3. Before proceeding with the proof of the above theorem, let us make a few remarks. For the proof of this theorem we need, essentially, only to know that V(r) and @;*'(r) are continuous in some neighborhood of r, and that
T h e restrictions on @:*)(r)are fulfilled automatically if V(r) satisfies the conditions of Theorem 6.2 and the restrictions imposed on it in the present theorem. It can happen, however, that @:*)(r)satisfies these restrictions even when V(r) does not obey all the conditions of Theorem 6.2 (see Ikebe [1960]). Hence, the validity of Theorem 6.6 is somewhat independent of the validity of Theorem 6.2. T h e proof of Theorem 6.6 is straightforward, and it essentially consists in applying the Laplacian A to the right-hand side of equation (6.59) and computing the result. This computation is easily carried out with the help of Lemma 6.3.
Lemma 6.3, Suppose f (r) = O(r-2-eo) for some c0 > 0, and that there is a neighborhood A'" of the point r E R3in which f (r) is continuous and the function (6.81)
is well defined by the above integral. Then g(r) is twice differentiable
at that point in each of the variables x, y, and x, and
(6.82) Proof.
(-Ag)(r)
=
+ 47d(r).
k2g(r)
Let S be a neighborhood of r and write
We shall compute separately the two terms on the right-hand side of the above identity.
537
6. Distorted Plane Waves in Potential Scattering
Straightforward differentiation shows that
a exp(ik1 r - r' I) (6.83) ax Ir-r'I a2
(6.84) -
exp(ik1 r
Ir
ax2
-
-
~
(x
-
x')
exp(ik1 r - r' I)
Ir
-
r'
+
(ik
l2
r' I) r' I -
exp(ik1 r - r' I) Ir-r')
x (ik
+
1 - k2(x - x')2 - ik(x - x') Ir-r'I Ir-r'I2
-
Ir-r'I3
On any bounded domain which does not contain some closed neighborhood of r, we have
Now, the function on the right-hand side of the above inequality is integrable on account of the existence of the integral in (6.81) and the fact that if a function is integrable then it is also absolutely integrable. Hence, we conclude that f(r) is locally integrable. I t is easy to infer from (6.83) and (6.84) that for sufficiently large R, > 0 and all rl from some open neighborhood S C N of r const
an exp(ik1 r - r' I) I r' 13+ro ' ax. I rl - r' I If (r')l G constf(r'), for n
=
r' 2 R,
r' E R3 - S and r' R,
<
1, 2. Thus, we deduce that Ss,-
an exp(ik1 r - r' I) I r - r' I f (r') dr',
ax.
n
=
1,2,
exist for r E S. Consequently, a/ax and a2/ax2 can be carried under the integral sign when applied to the right-hand side of (6.81) (see Exercise 6.1 1). Since the same obviously holds for the variables y and x , and since
538
V. Quantum Mechanical Scattering Theory
we obtain (6.86) -A
exp(ik)r
Ir
SR3-,
-
rl
-
r'
I
I)
f(r') dr'
=
k2
exp(ik1 r JR3-s
Ir
-
-
r'
r' I)
I
f(r') dr'.
Let us choose
(6.87)
S = ( ~ 1 I:
x
-
I < a, I y1 - y I < a, I ~1 - x I < a } .
If a > 0 is sufficiently small, then the generalized mean-value theorem of the integral calculus can be applied to obtain
+a
exp(ik1 r - r' I)
-a
-
Ir
f(r')
I
r'
I
x,=x+-a
,
x =x
dy'dz' dy' dz'
I
x'=x+a
x =x--a
,
where / y o- y 1 < a and j x,, - z j < a. Using (6.83) and the meanvalue theorem of integral calculus, we get
La +a
+a
8 exp(ik1 r
a~
1r
-
-
r'
r' I)
1
1
x'=a
dy' dz'
x'=--a
where j 7 I < a and 1 f I < a. This shows that, due to the continuity o f f ( r ) in M , by taking the limit a + 0 we shall get in (6.88) a contribution of the form (6.89) f ( r ) lim
a-t+O
i_,.La +a
fa
-2a exp(ik[a2 (a2
+ ( y - y')2 + (Z
+ (y
-
y')2
-
z')2]1/2)
+ (z - z')2)3/2
dy' dz'.
To compute the above limit, we must compute the following integral explicitly: +-a exp[ili(a2 y12 z12)1/7 dY1 dz, -a (a2 y12 212)3/2
:s
i
+ +
+ +
539
6 , Distorted Plane Waves in Potential Scattering
where rav is obtained by applying again the generalized mean-value theorem to the first of the above two integrals, and it satisfies the inequality 1 rav 1 < a&'. Hence, the limit (6.89) is equal to -(477/3)f(r). Due to the symmetry of the considered expressions under the exchange of x, y , and z, when S is the box in (6.87), we have
=
3 lim a+o
a2
~
ax2
1
exp(ik1 r - r' I r - r' I
I) f(r') dr' = 47~f(r).
Hence, the desired result in (6.82) is obtained by combining (6.90) fO. Q.E.D. with the result (6.86) in which we let a ---f
In order to be able to apply the above lemma to derive (6.80) from (6.59), we have to establish that (6.91)
and that the above function is continuous in some neighborhood of the considered point r E R3. Since I @i*)(r)l= O(1) (see Exercise 6.10), it immediately follows that (6.91) is true. T h e continuity of Oi*)(r) is an easy consequence of (6.59) (see Exercise 6.12). Hence, in computing the effect of applying the Laplacian A to the right-hand side of (6.59), we can use Lemma 6.3; then (6.80) easily follows. EXERCISES 6.1 +
Show that if the potential V(r) is square integrable, the integral
J,. I K(r, r'; 5) e(r', k; 5) &k)l dr' dk
exists for every r E [w3 when I m
drml > 0.
6.2. Prove that the integral in r' of the function 1 r' ) 2 + E o 1 r - r' I over the region {r': I r' 1 3 R,} exists when R, > 0 and c0 > 0, and 1, and as O(l/r) that when r + co, it behaves as O(l/fo) if 0 < c0 if co > 1.
+
<
6.3. Prove that if the potential V(r) obeys the restriction of Lemma 6.1, then
where the constant C is independent of r.
540
V. Quantum Mechanical Scatering Theory
6.4. Show that K ( K )is a weakly continuous vector-valued function ) = s-limAK+,K ( K A K ) +’ of K in the upper half plane, i.e., K ( K +’ whenever Im K , 2 0 and +’ E for all K in some neighborhood
+
Of K O .
6.5. Show that if the potential V(r) satisfies the conditions of Lemma 6.1, then
and that the above integral defines a locally integrable function of r.
6.6, Show that if the potential V(r) satisfies the conditions of Lemma 6.1, then for any fixed r I$ SP,
and the above function is locally square integrable in r l . 6.7. Prove that if the potential V(r) satisfies the conditions of Lemma 6.2, then the integrals appearing on the right-hand side of (6.28) exist.
6.8. lim
Show that iff(x) is continuous on [a, b], then
1
l b
r++O.X
a
E
x2
+
€2
f(x) dx
if a t O < b if a > O or b < O , a < b .
=
6.9. Show that if the potential V(r) is locally square integrable, Co~-2-fo,c0 > 0, for almost everywhere continuous, and I V(r)l Y 2 R o , then the function V(r) I r - r’ V(r) is integrable on R6 if it is integrable on the set {(r, r’):Y R, , Y‘ Ro}.
<
< <
6.10. Prove that when the potential V(r) satisfies the conditions of Lemma 6.2 then the function vk(r) defined by (6.55) is uniformly const) for Y 3 Ro and k assuming values from bounded (i.e., 1 uk(r)l a compact set D, containing no points for which k2/2m E.:S
<
7. Wave and Scattering Operators in Potential Scattering
54 1
6.11. Let h(r, u ) be a function with a continuous (in r and u ) partial derivative hu(r,u ) = ah(r, u)/au, and assume that for r 3 R,
where p(r) is integrable on {r: r 3 Ro}. Show that if hu(r,u)f(r) and f ( r ) are integrable over the sphere {r: r R,},then
<
6.12.
Show that the function
is continuous in r if If(r)i
=
O ( T - ~ - ~for O )some
e0
> 0.
6.13. Prove that the function Fo(r) defined in (6.28) is bounded on any closed set X , disjoint from Yv[provided that the potential V(r) satisfies the conditions of Lemma 6.21. 6.14. Show that the family YF of all functions in %?;([w2) with supports disjoint from the set YV(mentioned in Theorem 6.2) is dense in L2(R3). 7.
Wave and Scattering Operators in Potential Scattering
7.1. THEEXISTENCE OF ASYMPTOTIC STATES IN
POTENTIAL SCATTERING
In the study of a particular scattering model it is crucial to determine from the start the magnitude of the family of asymptotic states. I n other words, we have to establish the size of the initial domain M, of 52, . I n potential scattering, under very mild restrictions imposed on the potential, this question is answered by Theorem 7.1. Theorem 7.1. Suppose the potential V(r) is locally square integrable and that for some E > 0 V(r)
=
0
1 . L+J -
542
V. Quantum Mechanical Scattering Theory
Then the initial domain M, of the wave operators 52, corresponding V(r) coincides with the to H,,> -(1/2m)A and H 3 -(1/2m)A entire Hilbert space L2(R3). T h e proof of this theorem is based on Theorem 2.8, and it will be carried out in several stages. It is important, however, to verify before starting with the proof that in the present case H is indeed self-adjoint. This self-adjointness of N is a consequence of the restrictions imposed on V(r), which has to be locally square integrable and bounded at infinity to satisfy these restrictions. Consequently, -( 1/2m)A V(r) determines a unique self-adjoint operator by virtue of Theorem 7.5 in Chapter IV. The starting point of the proof of the above theorem consists in choosing an adequate domain g1 which satisfies the conditions of Theorem 2.8. We shall prove now that we can choose for g1the linear manifold spanned by all functions
+
+
qp(k) = k,k,k, exp[-(1/2m) k2 - ikp],
(7.2)
pE
R3,
where p varies over all vectors in R3 and k, , k, , and k, are the Cartesian components of the vector k E R3. As a straightforward consequence of (5.2) we obtain
= exp (-it
--k2I
&(k)
2m
= k,k,k,
exp
( -1 2m- i t
k2
- ikp).
Hence, by virtue of Theorems 7.3 and 7.5 in Chapter IV we have" n g H and , consequently exp(-itH,) 9, C g Hn 0gH. exp(-itHo) $, E gH0 The second condition imposed on g1in Theorem 2.8 is that when t , -+ - co and t, + 00, the strong limits of Wl(tl , t,) $, exist. We have already pointed out, at the end of $2, that the existence of the integrals (2.41) and (2.42) is sufficient for the fulfillment of this condition. I n view of this fact, we shall settle the question by showing that
+
(7.4)
for all
pE
R3.
* Note that Ts = Ho and Ts +
Vs = H.
7. Wave and Scattering Operators in Potential Scattering
543
By taking the Fourier transform of both sides of the relation (7.3), we obtain for any real 6 (7.5)
I(exp( -itffo)#P)(r)l
=
const
If we now take 0 < 6 < 712, so that 712 - 6 > 0, we get, by multiplying both sides of the above inequality by V(r),
where C is a constant which majorizes the function
and is obviously independent of r, p, and t. If we choose 6 so that in addition 6 < E , then by virtue of (7.1) the function on the right-hand side of (7.5) is square integrable. Consequently, we can write
where Cp is constant with respect to the variable t E R1. Thus, (7.4) holds and the second condition of Theorem 2.8 is fulfilled. This establishes that the closure g1of the linear manifold 9,spanned by all functions (7.2) is contained in M, .
544
7.2.
V. Quantum Mechanical Scattering Theory
WIENER'STHEOREM ON THE CLOSURE OF TRANSLATIONS
T o conclude the proof of Theorem 7.1, we still have to show that Gf, = L2(R3), and thus establish that M, = L2(R3).T o arrive at this result we need the following lemma, which is a straightforward generalization, to arbitrarily many dimensions, of a theorem by Wiener.
*Lemma 7.1. Let us denote by Jv; the set on which the Lebesgue square-integrable function f ( x ) , x E Rn, vanishes. T h e Lebesgue measure of Nfis zero if and only if the linear manifold (Ff) spanned by the family gf of all functions exp[ip x ] f ( x ) , obtained by varying p over all vectors p E Rn, is dense in L2(R"). Proof. We set out to prove that, given any g E L2(Rn) and E > 0, we can find vectors p , ,...,p , E Rn and complex numbers a, ,..., a, such that
1,. I
g(x) - f ( x )
We note first that for any (7.7)
s
(I dnx < 166.
uyeiz'9v "=l
E
> 0 we can satisfy the inequality
J" B , I g(x)I2 dnx <
6,
B'
=
R"
-
B,
with a compact Bore1 set B , by choosing that set sufficiently large. Let us define the functions
for any positive integer M ; note that h,(x) is almost everywhere well defined, sincef(x) vanishes only on a set of measure zero. I t is easy to see that by virtue of Theorem 3.10 in Chapter I1 g(x)I2dnx = 0.
Consequently, we can find an M for which
7. Wave and Scattering Operators in Potential Scattering
545
Since (7.7) obviously implies that
we easily obtain from this inequality and (7.8), by using in the process the triangle inequality in L2(Rn),
Let L be a box L
=
{x: -1
< xi < I, i = 1, ...,n}
which contains B, and let g l ( x ) be a function equal to xB(x) hM(x)in L and periodic in each of the variables x, ,..., x, , with the period 21. For sufficiently large 1 we have
where L' = Rn - L, and the prime on the summation sign indicates = k, = 0. Comthat the sum does not include the term with k, = bining (7.9) and the above inequality we obtain (7.10)
by making use of the triangle inequality. Let us expand g l ( x ) in a Fourier series,
c.=--J
where k
=
1 (21)"
L
. k-) * x
gz(x)exp (-in
1
dnx,
(k, ,..., k,) and the convergence of the above series is in
546
V. Quantum Mechanical Scattering Theory
the mean. A straightforward computation yields, for any positive integers N and N o < N ,
where the summation C” extends over all K = ( K , ,..., K,) in which 1 k, I > N o for at least one value o f j = I , ..., n. Due to the convergence of G j ck 12, the second sum on the right-hand side of the above inequality can be made arbitrarily small by choosing No large enough, while the first sum can be then made arbitrarily small by taking N sufficiently larger than No . Hence, the sequence of functions
N
=
1 , 2,...,
+
converges in the mean to zero when N + co. Thus, it follows from , Theorem 4.5 of Chapter I1 that there is a subsequence f,,(x), f N 2 ( x )..., which converges to zero for almost all values of x E Rn. Now we can apply Lemma 3.1 of Chapter I V to infer that (7.11)
after noting that the use of this lemma is justified by virtue of the inequality I.fN(x)f(x)I
< {I gL(x)l + M)i.f(x)l’
I n fact, the above inequality follows from the estimate
547
7. Wave and Scattering Operators in Potential Scattering
the estimation of the last integral in the above relation could be carried out due to the formula
which can be proven by mathematical induction in N . Since (7.1 1) is true, we can choose one N, = N' for which
x (1 -
9)
k.x
exp (inT)l
dnx
< 6.
Combining (7.10) and (7.12), we immediately obtain, by means of the triangle inequality,
I
N'
g(x) - f ( x )
k,.
... k,=-N
, (I
-
$!)!
**.
(1 -
!$!)
I
k * x T) dnx < 166.
exp (in
I
Since the above inequality becomes identical to (7.6) after an appropriate change of symbols, we have proven the statement that the linear manifold (gj) spanned by Fj is dense in L2(R3). Conversely, if (Fj) is dense in L2(R3),then the measure of JV; is zero. In fact, if this were not so, then by choosing g(x) = xX,(x) we would have
.s, I
C a,eiPY'" S
g(x) - f ( x )
"=l
so that (7.6) would not be satisfied for all
andp, ,..., p s E R n . Q.E.D.
E
> 0 by
any a, ,..., a, E C1
548
V. Quantum Mechanical Scattering Theory
I f f ( p ) is the Fourier-Plancherel transform off(x), then it is easy to Psee that the Fourier-Plancherel transform of f(x)eiPu.z is T h e function f(p - p”) is usually called the translation of f ( p ) by the amount p , . By taking the Fourier-Plancherel transform of the integrand in (7.6) we obtain
PC
P Y ) ’
Hence, we conclude that the linear manifold spanned by all translations of 3 ( p ) is dense in L2(Rn).This statement constitutes Wiener’s theorem on the closure of translations. Let us return now to the proof of Theorem 7.1. If we set
then (7.2) assumes the form
$,(k)
= f(k) e c i k p .
Sincef(k) vanishes only on a set of Lebesgue measure zero, we infer from Lemma 7.1 that the linear manifold g1spanned by all function (7.2) is dense in L2(R3). Consequently, 6,= L2(R3),and since we already know that M, 3 g1, we finally obtain Ma = L2(R3). This concludes the proof of Theorem 7.1. We note that this theorem guarantees that Ma = L2(R3) for all almost everywhere continuous potentials V(r) which decrease at infinity faster than the Coulomb potential. However, at this stage we cannot say anything about the unitarity of the S operator in potential scattering. Only when additional restrictions on the potential are imposed, as required in the theorems of the preceding section, will we be able to prove that in potential scattering the scattering operator S is unitary. ON THE POTENTIAL FOR PHYSICALLY 7.3. CONDITIONS SATISFACTORY POTENTIAL SCATTERING THEORIES
T h e next theorem will relate wave operators to distorted plane waves, providing at the same time the answers to all the basic questions regarding R,, M a , and the unitarity of the scattering operator S in potential scattering.
Theorem 7.2. Suppose the potential V(r) is measurable and locally square integrable, that it is bounded on any compact set disjoint
I . Wave and Scattering Operators in Potential Scattering
549
from the closed set 9“ of measure zero [containing all the singularities of V(r)], that V(r) I r - r’ V(r’) is locally integrable in R6, and that (7.13) for some > 6. Then the initial domain Ma of 52, is L2(R3),the final ) the domains R, of 52, are both identical to Lic(R3)= E H ( S 3 L 2 ( R 3and S operator is unitary on R = R, = R- ; furthermore, (7.14)
(Q*+)(r)= 1.i.m. J.3
@f)(r) &k) dk,
where (7.15)
$(k)
=
1.i.m.
I.,
+(r) ecikrdr,
and @L+)(r) are the outgoing distorted plane waves, while @L-)(r)= @L+k)*(r) are the incoming distorted plane waves. T h e validity of statements which are essentially equivalent to the assertions made in the above theorem about M a , R, , S, and 52, has been taken for granted by physicists from the very beginning of quantum scattering theory. In proving the above theorem, we confirm that this faith was justified only to the extent that the potential satisfies the conditions imposed on it at the beginning of the theorem. Consequently, it is important to realize that these conditions are essential only to the extent of making applicable all the theorems on distorted plane waves in the preceding section. We discussed these conditions in $6, mentioning at the same time that those theorems are also valid under Ikebe’s condition requiring that V(r) is locally Holder continuous and locally square integrable, and that (7.16) for some P > 0. Thus, the assertions of Theorem 7.2 are also valid under Ikebe’s conditions, which are in some respects more stringent than the conditions of Theorem 7.2, but are slightly more relaxed in their restriction of the behavior of V(r) at infinity (by the 6 power in r). As a matter of fact, the proof of Theorem 7.2, which will follow, uses the same method which Ikebe devised to prove that the above assertions about M a , R,, S , and 52, are true when the potential fulfills his conditions (see Ikebe [1960, $10, $111).
550
V. Quantum Mechanical Scattering Theory
I n the case when the potential is spherically symmetric, the above conditions can be further relaxed by replacing (7.16) with
and making the additional requirement that r V ( r ) is locally integrable on [0, +a)(see Green and Lanford [1960]). Since the assertions of Theorem 7.2 are not true for the Coulomb potential, it is clear that for spherically symmetric potentials one cannot hope to do any better by further relaxing (7.17). However, in the general case, it is an open question whether one could relax (7.16) any further. 7.4.
DISTORTED PLANEWAVESAS KERNELS OF INTEGRAL OPERATORS
We shall prove Theorem 7.2 in a few stages. T h e first step consists in verifying that the integral operator in the right-hand side of the equation (7.14) is a well-defined mathematical entity.
*Lemma 7.2. If @:*)(r) are distorted plane waves belonging to a Schroedinger operator H with a potential satisfying the conditions of Theorem 7.2, then the operators W+ and W- are defined by the relations (7.18)
(W,+)(r)
=
1.i.m.
on all I)€L2([W3), and satisfy the relations
(7.20)
W**E"(B) = EHo(B) W**,
B E B1.
Proof. We shall prove the lemma only for W - , since the corresponding results for W+ easily follow from the relation @:-)(r) = *(r). First of all, we note that under the present restrictions on the potential, Lemma 6.2 holds. I t follows that for any given bounded closed set X,, which is disjoint from the set Yvof singularities of V(r) we can find a constant C such that
@s)
(7.21)
1
I @F'(r)l= ( 2 ~ ) -eikr ~ /+ ~ h(r, k; I k I) 2m 1
~
I<
C
7. Wave and Scattering Operators in Potential Scattering
551
as long as r E X,, and k assumes values from a bounded set (see Exercise 6.13). Let us consider the bilinear form
(7.23)
$(k)
=
( 2 ~ ) - ~ / ~ecikr#(r)dr, JR3
(7.24)
$+’(k) =
1 @F)*(r)+’(r)dr. R3
With the help of Theorem 6.2 we show that this bilinear form is bounded:
Consequently (see Chapter 111, Exercise 2.5), there is a unique bounded linear operator W- satisfying the relation
W+#I *’) = (* I $7
(7.25)
for all t,h, $’ €L2([W3). Let us choose now $(k)E$?i([W3)and $ ’ ( r ) E $ ? t , where VO, is the family of all functions in $?;([w3) with supports disjoint from Y;.For such $(k), “1.i.m.” in (7.18) can be dropped, and (7.22) can be written in the form
Since for such functions $’(r), (7.21) is satisfied, we immediately find $’(r) is integrable in R6. Consequently, by Fubini’s that $*(k) @L+)*(r) theorem, we can interchange the order of integration in (7.26), thus arriving at the result (7.27)
(W+* I $7 =
J
R
dr P
W j JR3 @W) $(k) dkl*.
Since 2‘ 7; is dense in L2(R3)(see Exercise 6.14), equality (7.27) holds for all $’(r) E 2‘ 7; if and only if (7.28)
V. Quantum Mechanical Scattering Theory
552
almost everywhere. Hence, (7.18) is established for $ E %?:((w3). Moreover, due to the fact that qE([w3) is dense in L2((w3),the general validity of (7.18) for W, and arbitrary $ €L2(R3)is an immediate consequence of (7.28). I n order to express W+* as an integral operator, let us take in (7.22) +(r) E %?;((w3) and $+'(k) G V;([w3),and write
(# I W+*#') =
=
( 2 ~ ) - J~ / ~dk$+'(k) R3
)I R3
e-ikr#(r)
&I*.
Since, under the present conditions, the function $*(r) eikr$+'(k) is integrable in r and k on [wB, we can invoke Fubini's theorem to invert the order of integration in k and r:
(# 1 W+*#')= ( 2 ~ ) - ~ / dr ~ #*(r) R3
lj eikr$+'(k)dkl. R3
Now, the type of reasoning earlier applied to (7.27) leads us to the conclusion that (7.29)
(W+*#)(r) = ( 2 ~ ) - 1.i.m. ~ / ~J
R
eikr$+(k) dk
for arbitrary $ E ~ 2 ( ~ 3 ) . Combining (7.18) and (7.29), we obtain (W+W+*#)(r) = [W+(W+*#)](r) = 1.i.m. =
1.i.m.
JR3
1., &)(r) $+(k) dk
@Pk(+'(r)(W+*#)" (k) dk
=
(EH(SF)#)(r),
where the last step follows from Theorem 6.5, and in particular from (6.75). Hence (7.19) is established. Finally, combining (6.75) with (7.29), we easily derive ( W+*EH(B))(r) = ( 2 ~ ) - 1.i.m. ~/~
eikr(EH(B)#);(k) dk L
=
( 2 ~ ) - 1.i.m. ~/~
3
eikr$+(k) dk, 'kZ/2mEB
= ( 2 ~ ) -1.i.m. ~/~
eikr$+(k) dk. ' kZ/ ZmE B
This establishes (7.20).
Q.E.D.
553
7. Wave and Scattering Operators in Potential Scattering
7.5.
THERELATION OF WAVEOPERATORS TO PLANEWAVES
We are now ready to carry through the proof of Theorem 7.2 to its completion. Let us adapt the relation (2.37) to the present case, setting t, = 0, t, = T, and f = $ to obtain (7.30)
((Q(7) -
I)# I g )
=i
j T((eKiHot $ I He-iHtg) - (HoediHot I e - i H t g ) ) dt )I
0
for any g E B H. Since the restrictions imposed at present on the potential meet the demands of Theorem 7.5 in Chapter IV, we infer that if # E g H, then th, E g Hn 9fH0 = 9". Hence, for E gH0( H - Ho)e-iHot # = Ve-a*ot O . *, and the relation (7.30) assumes the form )
(7.31)
(Q(,)*
I g> = (+ I g> + i
1' 0
(eiHtVe--iHot
*I
g> dt.
The above relation can be extended to arbitrary g E L2(R3).Setting g = W-4, €L2([W3),and using the relation W-*eiH1 = eiHol W- *, which is a direct consequence of (7.20) (see also Exercise 2.1)) we obtain
+
Since it can be readily shown (see Exercise 7.1) that (7.32)
we easily arrive at the result
It should be emphasized that, in this context, all the integrals over the infinite interval [0, +m) are improper Riemann integrals, and not Lebesgue integrals.
V. Quantum Mechanical Scattering Theory
554
We can calculate the integrand in the above relation by using (7.29) and (4.8),
We resort now to the relation (6.53), which for +(r) E Wz(R3) yields
We note that the limit in the mean in (6.53) could be immediately taken when deriving the above relation, since the integrand in (7.35) is integrable on R3. T o arrive at this conclusion, we combine (6.29), (6.55), and (6.56) to derive the estimate
SinceF,,(r) 1 V(r)/ is locally integrable (see Exercise 7.2)while ( c t H o t +)(r) is uniformly bounded for all t E R1 (see Exercise 7.3), we conclude that F,(r) V(r)(eriHot$)(r) is locally integrable. Furthermore, since IFo(.)] const when r 2 R and R is sufficiently large (see Exercise 6.13), we obtain for such values of r E R3
<
T h e function on the right-hand side of the above inequality is the product of two square-integrable functions and therefore is integrable. Thus, the integrand in (7.35) is integrable for r 3 R, and consequently it is integrable on R3. By combining (7.34) and (7.35) we obtain (7.38)
(4 I exp(iH,,t) =
.s,
W-*V exp(--iH,,t)+)
dk $*(k)
s
R3
dr &'*(r)V(r)(exp[-i(H,
Let us show now that when +(r) E %;(R3),
we have
- k2/2m)t]$)(r).
555
7. Wave and Scattering Operators in Potential Scattering
where C , is a constant independent of k E R3 and t E R1. T o see that, consider separately the above integral over B, and B,‘, where B, is a closed set disjoint from SP,, and its complement B,’ is bounded. Since the argument leading to (7.37) can be applied when r varies over B, (see Exercise 6.13), we have
LoI
&)*(r) V(r)(exp[-i(Ho
~
k2/2m)t]+)(r)ldr
where 11 V I/ and 11 # 11 denote the L2 norms of the square integrable functions V(r) and $(r), respectively. Furthermore, since (exp[-i(H, - k2/2m)t]#)(r) is uniformly bounded in k E [w3 and t E R1 (see Exercise 7.3), by using (7.36) we get
f
Bo’
I &’*(r)
V(r)(exp[-i(H,
- k2/2m)t]+)(r)ldr
< const
thus establishing the validity of (7.39). Let us insert (7.38) in (7.33) and choose &k)
where (7.41)
&(+, +)
=
is+“ dt eft 0
x
s,
E
1,.
%:(R3),
(1
+ Fo(r))dr,
thus obtaining
dk$*(k)
jR3 dr @i-’*(r) V(r)(exp[-i(H,
- k2/2m)t]#)(r).
Using (7.39), we easily infer that the above integrand is integrable in t E [0, CO) and k E R3. Consequently, the reversal of the order of integration in t and k is justified by Fubini’s theorem. Hence, for closed B disjoint from Y Vwe , have
+
(7.42)
M,4 4) = I?(#, 4)
+ i I,, dk4*(k)
dt (F: I exp[-i(Ho - k2Pm - ic)tl$>,
556
V. Quantum Mechanical Scattering Theory
where (7.43)
I;(#,
4) = i j dk$*(k) Iw
x (7.44)
Is,
j+m dt ecGt 0
dr &)*(r) V(r)(exp[-i(Ho
-
k2/2m)t]#)(r),
Ff(r) = XB(r)@(i)(r) V(r).
We note that the inner product under the integral sign in (7.42) is indeed defined since Ff €L2(R3),due to the fact that I @L-)(r)l const on B and that V(r) is square integrable. Using Theorem 3.1 of Chapter IV, we infer that
<
Consequently, we easily compute
= i lim r++m
=
-i (F'
(Ff 1 (exp [-i
1
R,
(H,
-
k2 (= + i c ) #)
k2
- - ic
2m
=
-i
k2 (Ho(- 2m
-k)Ff
If we combine (5.5) and (5.6) with the above result, we get
On the other hand, using the fact that the function
1 #).
557
7. Wave and Scattering Operators in Potential Scattering
is uniformly bounded with respect to following estimate:
k E R3 and t E R1, we obtain the
This estimate shows that for any sequence B, C B, C we have
*-.,(J, B,
= R3,
Let us combine this last result with (7.45) in order to take the limit B, --t R3 on the right-hand side of (7.42). Since the integrand on the right-hand side of (7.45) is integrable on R9 in r , r’, k E R3, the limit B, --t R3 can be carried under the integral sign. Hence, we have
L(A4) = $J
R
IJ
R3
exp[i(k2 - 2 r n i ~ ) lI /r~ - r‘ Ir-r’)
x V(r’) @i-)(r’)dr‘
1* $*(k)
11
+(r)d r dk.
Inserting this result in (7.40), and noting that exp[i(k2- 2 r n i ~ ) l1 /r~ - r‘ I] Ir-r’I
=
V(r’) @L-)(r’)dr‘
I r - r’ I) V(r’) @L-)(r‘)dr‘, r --‘I
I,exp(-ikI
and that the convergence in the above limits is uniform for r f SUPP and k E supp $ (see Exercise 7.4), we arrive at the following result: exp(-ik I r - r‘ I) x V(r’) @A-)(r’)dr‘
I*
$*(k) +(r)dr dk.
From (7.18) we obtain (7.48)
1 dr +(r)1 dk @i-)*(r)$*(k) R
J,
R
@;-)*(r) $*(k) +(r) dr dk,
*
V. Quantum Mechanical Scattering Theory
558
where the second equality follows from Fubini’s theorem; this theorem can be applied, since” by (7.36)
where C , and C , are two constants and consequently the integrand in (7.48) is integrable on Re by Tonelli’s theorem. Substituting the expression for (W-4 I $) from (7.48) in (7.47), we obtain
x =(
i R 3
exp(-ik I r - r’ I) V(r’) @i-)(r)dr I r - r’I
1
2 ~ ) - ~ /dk$*(k) ~ R3
1
R3
dr cikr $(r)
I*
=
by using in the process the Lippmann-Schwinger equation (6.59) (r). for @i-) I n the above relation $(r) and the Fourier transform $(k) of d(r) can assume any value in YC(R3). Since Y:(R3) is dense in L2(R3),it follows that W-*SZ- = 1. Consequently, by virtue of (7.19) and the fact that E,EH(Sf),we conclude that
<
w-= w_w-*sz-= EJys;)Q-
= Q-
.
Completely analogous considerations lead to the conclusion that
W+ = SZ, . Hence, we have thus established the representation (7.14)
for the wave operators SZ, in potential two-body scattering for arbitrary $ eL2(R3).This shows that in the present case M, = L2(R3).Moreover, in view of (7.19), we have also established that R, = R- = Ltc(R3). Consequently, the proof of Theorem 7.2 has been completed.
7.6. THERELATIONOF TO THE
THE S MATRIX SCATTERING AMPLITUDE
In scattering theory, the object of principal interest to the physicist is the scattering matrix, since it is related in a relatively straightforward
* Note that according to (6.29) the constant in (7.36) can be chosen to be independent of k on the compact support of &k).
559
7. Wave and Scattering Operators in Potential Scattering
manner to the differential scattering cross section, which is directly measurable. T h e computation of the scattering matrix is made comparatively easy by the following theorem which relates it to the distorted these in their turn can be computed by solving plane waves @J+)(r); the integral equation (6.59) or the differential equation (6.80).
Theorem 7.3. Suppose the potential V(r) satisfies all the conditions imposed on it in Theorem 6.4. Then the scattering operator S can be expressed by the following integral formula: (7.49)
( S $ ) - ( k ) = $(k)
-
r* 1"
i d/z;J d+ 0
0
dB sin ef-k(B,
4) $1(k,
8, d),
wherefk(B, #) is the scattering amplitude defined in (6.62), and (7.50)
yT1(k, B, 4)
= $(K
cos 4 sin B, k sin 4 sin B, K cos 8).
As a starting point for the derivation of (7.49) we take formula (7.31) O g Hand g = 52-4) at T = t , and T = t , , (with V = H , , # E g H = and subtract the corresponding sides of the two equations, thus obtaining
By letting t, -+
-
co and t,
--f
+ 00 we arrive at the relation
where the above integral is an improper Riemann integral. On the other hand, we have
since in the present case Q-*Q-= EMo= 1 . Combining the above relation with (7.51), and using the intertwining property (2.3), we arrive at the result
By making use of (7.14), we easily derive (Q- exp( -iHot)$)(r) = 1.i.m.
I,.
@k;)(r)exp[ -i(k12/2m)t]$($) dk, .
560
V. Quantum Mechanical Scattering Theory
Inserting the above formula and the relation (V eXP(-~KJt)W) =
exp[ --i(k,2/2m)t] exp(ik,r) $(k,) dk,
V(r) 1.i.m. JR3
in (7.52), and taking &k), $(k) E %‘g(R3) (so that the integration extends, effectively, over compact domains, and therefore 1.i.m. becomes Lebesgue integration over R3), we obtain
(7.53)
(4 I s+>- (4 I *)
[ (- kZ2 - K)t] exp(ik,r) +*(k,)$(k,).
x exp --i
2m
2m
A potential V(r) obeying the restrictions imposed on it in Theorem 6.4 is obviously integrable on R3. Since I @i-)(r)lis uniformly bounded at k, E supp $, we infer infinity (see Exercise 6.10) for all” k, E supp from the inequality
6,
that the integrand in (7.45) is integrable on R9 in r, k, , and k, . Hence, the order of integration in (7.53) in the variables r, k, , and k, can be interchanged on account of Fubini’s theorem. Therefore, we conclude that
x $*(k,) $(k,)
J
Iw
dr exp(ik,r) V(r) @i;)*(r).
* Here we take it for granted that H does not have any positive eigenvalues. However, even if H had positive eigenvalues, the proof would have to be altered only slightly by choosing &kl) E Vr(R3) and &kz) E V?(R3) with supports disjoint from the sets {kl : k12/2mE S,”} and {kz : kZ2/2mE S,”}. Since these sets are closed sets of measure zero, the totality of such functions is still dense in L2(R3),as can be proven by the method employed in solving Exercise 6.14. T h e reader can then easily make the slight alterations which are necessary t o carry the proof of the present theorem under these conditions.
56 1
7. Wave a n d Scattering Operators i n Potential Scattering
For fixed 7 > 0, the above function is obviously integrable in t , k, , and k, on [-r, +TI x [ws. Hence, we can interchange in the above integral the order of integration in t and (k,, k,), thus arriving at the relation
(4 I S+) - (4 I+) = 2i(2~)-3/2lim
r++m
k2'/2m)7 1 dk, dk, sin(k12/2m k12/2m k,2/2m -
x $*(k,) J(k2) J dr eikzrV(r) @i;)*(r). RS
The above relation can be written in the form
r" 1"
where (k, , Oz , C#Lare J the spherical coordinates of k, and (7.55)
h(k2 , k 1)
= 2 i ( 2 ~ ) - ~ / ~ $ * ( k ~d4, ) 0
x sin 8, &k2) The above limit for r -+
do2
0
1 dr eikzrV(r) @i;)*(r). R3
+ co can be calculated by using Lemma 7.3."
*Lemma 7.3. If the function f ( u ) is integrable and Holder continuous in [wl, i.e., if for some O > 0, (7.56)
I f(u +4
- f(4l < CO(4 I flu
for sufficiently small values of 1 d u (7.57)
1 f ( u ) = - lim T
r++m
+m
IS
I, then sin T(U - V )
f (4dv.
* At first glance, it would seem that this lemma states that lim,,,, in the sense of distribution theory, i.e.,
(1 /?r)(sini&)/.% = 8(x)
However, the usual proofs of the above relation require not only that f ( x ) be Holder continuous, but that it be at least once continuously differentiable.
562
V. Quantum Mechanical Scattering Theory
Proof. We note that 1 dv=-[ 77
fm --m
sin w dw w
___
=
1,
and consequently (7.58)
Let us split up the domain R1 of integration into the three parts 61, and (u 6, +a), and study each part separately. Using the Holder continuity condition (7.56), we obtain for any given E > 0
(-a, u - a), [u - 6, u
+
+
for all T 2 0 and sufficiently small 6 > 0. The integral over the interval (u 6, CO) is equal to
+ +
+m
When
T
+ f co ,
T(U
- v)
dv
+-
f ( v ) dv.
the first of the above integrals converges to zero, i.e.,
for sufficiently large T o see that, write (7.61)
sin
T
> 0. The second integral f ( v ) dv =
+"
[
rL4
also converges to zero.
sin TW
f
(u - w> dw. W
The right-hand side of (7.61) converges to zero according to the Riemann-Lebesgue lemma (see Chapter 111, Lemma 4.1). Thus, the 3 sufficiently large T. expression in (7.60) can be made smaller than ~ / for
563
7. Wave and Scattering Operators in Potential Scattering
A completely analogous argument leads to the conclusion that the u - 8) to the contribution of the integration over the interval (-a, integral on the right-hand side of (7.58) is smaller than r / 3 if 7 is sufficiently large. Consequently, (7.57) is true. Q.E.D. Let us show now that the above lemma can be applied to the computation of the limit in (754). By setting u =
kZ2/2m,
we obtain
(7.62)
(4 I S 4 ) - (d 14) = lim
dk,
-
-03
du
sin(k12/2m- U ) T k12/2m- u
f (u, kl),
where we have extended the integration in u to the entire real line by introducing the function m
(7.63)
u A(&,
k)
for u
>0
for u
< 0.
It can be proved (see Exercise 7.5) that h(k, , k,) is Holder continuous min(1, E,,}: in k, for any 0 < 6
<
I t follows that f ( u , k,) is also Holder continuous for those values of B which are also smaller than one. This can be seen from the following inequality:
If(u + Au, k d - f ( ~ ,ki)l < n- 4 2 \(I u + AU 1 /’ ~ 3 1 2
-
I h ( f i K , k1)l
1 1’2)
+
in which u1 is some value between u and u Au obtained by applying the mean-value theorem of differential calciilus to
564
V. Quantum Mechanical Scattering Theory
I u + du 1 / 2
- Iu
11/ 2 .
Consequently, Lemma 7.3 can be applied to
derive the fact that
1 lim -
(7.64)
7++m
T
j
+m
sin(k12/2m- U ) T kI2/2m- u f (%kl) du
However, we cannot insert this result immediately in (7.62), since we do not know yet whether it is permitted to invert the order of integration in k, and of taking the limit T -+ co. T o justify the taking of the limit T -+ f c o under the integral sign, we employ (7.58) to derive
+
<
Then, using the fact that I f ( u , kl)l C, , where C, does not depend 1, by on k E supp and u E [wl (see Exercise 7.6), we obtain for T means of (7.59)-(7.61),
6
Since the function on the right-hand side of the above inequality is integrable in k, over supp6, and is independent of T, we can apply Lemma 3.1 in Chapter IV and infer that the expression in (7.65) converges to zero when T -+ +a.Hence, the limit T -+ +co can be taken under the integral sign in (7.62). Thus, in view of (7.64), we arrive at the result (7.66)
(4 I S*) =
-
(9 I *)
6j iWz dk, k16*(k,) j2nd+2
i 25
0
0
do, sin O2 &(kl , w2)
565
7. Wave and Scattering Operators in Potential Scattering
where SZ, = [0, 771 x [0, 27~1,dw, = sin 8, do, db2 and w, is the unit vector in R3 with the spherical coordinates O2 and b 2 , while f k ( w z ) is the scattering amplitude defined in (6.62). and %?;(R3) is dense in L2(R3),the relation Since &k), $(k) E g;([w3), (7.66) can be immediately extended to the entire Hilbert space L2(R3). Thus, we arrive at (7.49), and at the conclusion that Theorem 7.3 is valid. 7.7.
RELATIONOF THE SCATTERING AMPLITUDE TO THE SCATTERING CROSSSECTION
T h e formula (7.49) assumes a simpler form when the S operator is replaced by the T operator, (7.67)
(T$)(k)
=
1-s ( T ) ( k )
1
= -=
d277
4
kf-k(w’) y$(k, w‘) dw‘.
Comparing this result with (1.31), we see that the T matrix T ( l ) ( kw, ; is related to the scattering amplitude f k ( w ’ ) by the formula (7.68)
T‘l’(k;W,
w’) =
w‘)
1 kf-k(O’), 2/27
where w = k / k . Formula (1.47) for the differential scattering cross section is valid in a frame of reference with its x axis pointing in the direction w’ of the relative momentum of the incident particles. I n such a frame of reference the relative momentum k after the collision points in the direction w. Thus, (1.47) can be written in the form (7.69)
u(k2/2m,w)
=
8a3 l f k ( e J 2 ,
where e, is the unit vector pointing in the direction of the z axis. T h e above relation (7.69) is not the familiar formula for the differential cross section. T o recast it in the familiar form we have to use the following symmetry property of f k ( 0 ’ ) : (7.70)
f ( k w , ( w ’ ) = focw‘dw).
This property follows from the obvious formal symmetry in of the right-hand side of the relation
w’
(7.71)
m fk(0‘)= - 27
I,,
exp[--ik(w’
- r) V(r) @g(r)dr
w
and
566
V. Quantum Mechanical Scattering Theory
This relation is obtained by using (6.59) to eliminate exp[--ik(w’ * r)] from the integral in the first line of (7.71). T h e usual procedures establish that
is integrable in r and r’ on [w6 (see Exercise 7.7). Hence, Fubini’s theorem can be applied to infer that in (7.71) the iterated integration in r, r’ E [w3 can be reduced to an integration on R6 in (r,r’). Thus, the formal symmetry in w and w’ of the right-hand side of (7.71) is an actual symmetry, in the sense that the expression on the right-hand side of (7.71) remains unchanged under the intercharge of the vectors w and w‘. From (7.69) and (7.70) we immediately obtain
where obviously (7.73)
Formula (7.72) is the very familiar relation expressing the differential cross section in terms of the scattering amplitude. T h e only unusual feature in (7.72) is the presence of the factor 87r3, which usually does not appear in physical literature. Its presence is due to a different “normalization” of the distorted plane waves (6.56), which in physical literature are chosen to be
I n fact, the scattering amplitude fk’(r) obtained by taking in (6.62) Ok(+)(r)instead of @$+)(r) is related to fk(r) by
In terms of fk’(r)(7.72) assumes the form
567
7. Wave and Scattering Operators in Potential Scattering
7.8.
THEPHASE-SHIFT FORMULA FOR
THE
SCATTERING OPERATOR
vs) defined in $7 of Chapter I1 Let us introduce in the space L2(Qn,, [see (7.16) of Chapter 111 the operators
(7.74)
S ( k ) = 1 - 2m' T(k),
(7.75) (T(k)u)(w)=
k
dzJ,
~
f(k",)(-w)
u(w') dw',
u EL2(-%
, PSJ.
The unitarity of the S operator imposes the condition that S ( k ) should be unitary in L2(Qn, , ps) for all k 3 0: (7.76)
S*(k) S ( k ) = S ( k ) S*(k) = 1.
As a matter of fact, for arbitrary u E L ~ (, Q ps)~ and normalized
p eL2(Qn,, p r ) [see (7.16) of Chapter 111 it easily follows from (7.67),
(7.70), and (7.75) that
(S*Sp * u)(k, W )
= p(k)(S*(k)S(k)u)(w).
Consequently, we have for arbitrary u., u €L2(QS, p s ) v"(w)(S*(k)S(k)u)(w)dw =
JrnI p(k)I2k2 dk J",
v*(w)(S*(k)S(k)u)(w)dw
0
= =
J, J"
(p
. ~ ) * ( kw)(S*Sp , . u ) ( w ) k2 dk dw
(p
. v)*(k,w ) ( p . u)(k,w ) dw
=
QP
This means that S*(k) S ( k ) = 1 . A similar argument establishes that S ( k ) S*(k) = 1. The unitarity of the operators S ( k ) , 0 k < +a,enable us to derive Theorem 7.4.
<
Theorem 7.4. Suppose the potential V(r) satisfies all the conditions imposed on it in Theorem 6.1. Then for each k E [0, co), there is an orthonormal basis {w,(k),w2(k),...} in L2(Qs, p s ) such that
+
568
V. Quantum Mechanical Scattering Theory
where the numbers 6,(k), 6,(k),... are real and such that (7.78)
Tr[T*(k) T ( k ) ] =
m
1 I sin S,(k)I2 < +a.
”=l
Proof. T h e kernel T(l)(R; w, w’) of the operator T ( k ) is square integrable in (w, w’) on fin x Q S with respect to the measure ps x pa (see Exercise 7.8). Hence, the operator T ( k ) on L2(fis,pa)is of the Hilbert-Schmidt type” and therefore completely continuous. Since S(k) is unitary, we can apply Theorem 6.1 of Chapter I11 and write (7.79)
S(k) =
2a
eiAdE,(k),
0
where the above integral converges in the uniform sense of (6.1) in Chapter 111. We shall prove now that the complete continuity of the operator T ( k ) implies that the spectral function E,(k) is constant in A except for an at most countable number of discontinuities Aio’(k), Aio)(k),..., where EAF)(k)# EAy)-,,(k). Suppose to the contrary that there is a point A, E (0, 27r) such that EAo(k)= EAO-,,(k)but Ea(k) # E,(k) for any a < A, < b such that a, b E (0, 27r). Then we can choose a sequence A, , A, ,... + A, which is # EA,+,(K). monotonically increasing or decreasing, and such that EAm(k) T o be specific, let us say that A1 < A, < < A,, and let us choose - EAn(k))f,= f , vectors fl ,f2 ,... E A? which are such that (EAn+l(k) and 11 f , 11 = 1. We shall prove that no subsequence of T ( k )fl , T ( k )f , ,... is convergent, thus contradicting the complete continuity of T ( k ) . I n fact, it is easily seen that T ( k ) f , 1 T ( k )f , for m # n (see Exercise 7.9), and consequently
I1 W
) f m- W f n
On the other hand,
II W
f
Il2
=
I1 W
) f W ,
/I2
+ /I W ) f n l 1 2.
11 T ( k )f , 11 3 C > 0 for all n n
/I2
=
=
1 , 2,...,
1
p ((W- 1)fn I ( S ( 4 - l)fn>
1 2inf I sin A, l2 > 0. 4772 n=1,2....
* This can be easily inferred by setting p s ( B ) = 0 for B C [w2 - Q, , thus extending p s to RZ,and then applying the result of Exercise 5.3 and Theorem 8.6 of Chapter IV.
7. Wave and Scattering Operators in Potential Scattering
569
Thus, no subsequence of T ( k ) f , , T ( k ) f , ,... is a Cauchy sequence, which is impossible if T ( k ) is of Hilbert-Schmidt type. Hence, E,(k) can increase only in a discontinuous manner. , p s ) is separable, there can be at most countably many Since L2(Qn, such points A:O)(k), Aio)(k),... of discontinuity. Thus, (7.79) becomes
where it is easily seen that the convergence of the series is in the strong sense, but not necessarily in the uniform sense (think of the case S ( k ) = 1, when the convergence is not uniform). If we introduce an orthonormal basis in each one of the mutually orthogonal subspaces on which E,,;)(k)(k)- E,;)(k)-o(k), n = 1,2,..., project, we obtain an orthonormal basis in L2(Qs,,us) by taking the union of all these bases. After adequately labeling the elements of these bases, and replacing Ac)(k)with the adequately labeled 6,(k) = iAc)(k), we obtain (7.77). T h e corresponding expression for T ( k ) is (7.80)
1 T ( k ) = -( S ( k )- 1) 2ni
1 "
=-
1 I w,(k)) ei'u(k)
u=l
sin S,(k) (w,(k)l.
Thus, we have
and since T ( k ) is a Hilbert-Schmidt operator, (7.78) follows.
Q.E.D.
When the potential V(r) is spherically symmetric, physicists call the numbers 6,(k) phase shifts. We will apply this terminology (see also Ikebe [1965]) to the general case, when V(r) is not necessarily spherically symmetric. Such a name for a,,(,+) seems appropriate, since the absence of interaction is characterized by vanishing phase shifts 6,(k) = 0; in fact, when there is no interaction we have S = 1, and therefore
cI m
S(4 = 1=
W"(k))(WU(k)l.
u=l
If we expand f ( k e , ) ( w )in the orthonormal basis {wl, w p,...} and compare the Fourier coefficients of this expansion with (7.80), we get (7.81)
570
V. Quantum Mechanical Scattering Theory
Hence, the total scattering cross section a(k2/2m) at a given energy k2/2m can be expressed conveniently in terms of phase shifts:
Naturally, both formulas (7.81) and (7.82) are unambiguous only if T ( k ;w', w) and w y ( k ;w') are continuous functions of w', so that (1.47) is valid.
7.9. PARTIAL-WAVE ANALYSIS FOR SPHERICALLY SYMMETRIC POTENTIALS If the potential V(r) is spherically symmetric,
W) = Vok),
(7.83)
r
=
I r I,
the functions w,(k;w) can be easily computed. Using the well-known Bauer's formula (see Exercise 7.10), m
exp(ikx) = exp(ikr cos 19) - C i1(2Z l)j,(kr) P,(cos e),
(7.84)
z=o
+
and combining it with the addition theorem for spherical harmonics, (7.85)
we obtain (7.86)
exp[--ikr(w . w')]
=
1 1 iz(2Z+ l)jz(kr) yzm(w)(ytm(wl))*,
1=0 m=-1
where w = (0,d) and w1 = (el,dl) are spherical coordinates which correspond to the unit vectors w E R3 and w1 E R3, respectively. Let us insert the above series in the integral in (6.62) defining fk(w). Due to the uniform convergence of the series for w, w' E Q8 (see Exercise 7.10) we can integrate term by term, thus arriving at the following relation: (7.87)
fk(w) =
-2nm
m
1
1
1=0 ni=-Z
x
J0s
dw,(
m
izYzm(w) dr rVo(y) 0
Ylm(wl))*@k+)(rl).
7. Wave and Scattering Operators in Potential Scattering
571
From the Lippmann-Schwinger equations (6.59) we immediately see that when V(r) is spherically symmetric @L+)(r) is a function of only Y, k, and cos 0 = r k/rk: (7.88)
=
@(I.)
@ P ) ( r ,cos 0).
Hence, by carrying out a change of variables in which w1 is replaced by spherical coordinates of r with respect to k , we obtain
+ 1) s'
for m # 0
-1
@(+) k ( r , 24) P l ( U )
for m = 0.
du
Consequently, (7.87) can be written in the form (7.90)
T"'(k; w',
W)
=
k
6
-fk0+--w)
=
1 47+7 1 a,@) P,(-W
z=o 21 ~
where the convergence is in the mean on SZ, x (7.91)
az(k)= -kmiz(l
*
w')
a,,and
+ 4)3/21 dr rzVo(r)/" du @ P ) ( r ,u ) Pz(u). co
0
-1
Thus, we have arrived at the following expansion for S(k): (7.92)
m
z
S ( k ) = 1 - 2 ~ -T(k) i = 1C
Z=O m=-2
1 Y,")(l
-
2 n i ~ l ( k ) ) ( Y1. z ~
Comparison with (7.77) yields the following explicit expression for the phase shifts: (7.93)
eZiS,(lC) = I - 2niac(k).
T h e reader can compare the above results with the considerations in 94.3, and easily arrive at the conclusion that (7.94)
S,(k)
= e2iS,(k)
is the S matrix in spherical coordinates.
572
V. Quantum Mechanical Scattering Theory
The great importance of the above formula for the S matrix lies in the fact that in conjunction with (7.91) and (7.93) it enables us to analytically continue S,(k) in the complex planes of the variables I and k. T h e properties of the resulting complex S matrix S ( k , I) provide the foundations of a very fruitful theory of the S matrix, which has become the fundamental stepping-stone of much of the theoretical physics in the past two decades. T h e reader is urged to consult the References for literature dealing with this very important subject."
EXERCISES 7.1. Show that if f ( t ) is continuous and the improper Riemann integral I - lim r f ( t ) dt - oi-tm
=
/
+m
f ( t ) dt
0
exists, then the improper Riemann integral Ic =
+m
ecftf(t) dt
0
exists for every e
> 0, and lirnE++,Ie= 1, .
7.2, Prove that the function F,(r) j V(r)j is locally integrable on R3, where F,(r) is the function given in (6.28).
7.3. Show that for any given $(r) E V;([w3) there is a constant C , such that I(e-iHot $)(r)l C, for all r E R3 and all t E R1.
<
7.4. Prove that the convergence of the limit in (7.46) is uniform with respect to k E Do and r E D if Do and D are any two compact sets in [w3.
7.5. Show that when V(r) satisfies the conditions of Theorem 6.4, the function h(k, , k,) defined in (7.55) is Holder continuous in the sense that
I h(kz for any given 0
+ 4 , ki)
<8
-
h ( 4 , ki)l
< min{ 1, eo>.
< Cdkz
ki) I 4 Is
<
C, 7.6. Establish the existence of a constant C, such that I f ( u , k,)l for all u E R1 and k, E R3, where f ( u , k,) is the function introduced in (7.63) in the course of proving Theorem 7.2.
* A mathematically rigorous treatment of analyticity properties of the S matrix in potential scattering is given in the work by de Alfaro and Regge [1965].
573
I . Wave and Scattering Operators in Potential Scattering
7.7.
Show that the function h(r, r’) = @2)(r)V(r) I r - r‘ 1-1 V(r’)@k’(rr)
is integrable in (r, r’) on R6. 7.8.
Prove that for fixed K
exists, where T ( l ) ( k ;w, 7.9,
w’)
E
[0, + co) the integral
is defined in (7.68).
Suppose [EAir(k) - EAj(K)]fi = fi, i
(4 > A,’] n (A2 ,A,]’
=
1 , 2 for
= 0,
where EA(k)is the spectral function of S(K),and A,, Prove that T(K)f, T(K)f,.
Al’, A,‘
A,,
Prove that
7.10.
and that the convergence is uniform in u E [- 1, Show that for any
7.11.
(exp(-iHOt) =
+ I].
> 0,
+ko)(r)
(2p2
+ -)mit
-312
[
t exp i rko - -b2)] exp -
[(
2m
if y5ko(r)is a wave packet with a Fourier transform of the form $ko(k)
= exp[--P2(k
-
b)21*
Use this result to show that for any bounded Bore1 set B
= B
I(e-iHot$ko)(r)12 dr < #(B)
[4 (isz
+
-&)]-3’2
E
(0, 2 ~ ) .
574
V. Quantum Mechanical Scattering Theory
7.12.
Any free state $:, $$(.I
= (e-i'fot
d)(r)9
corresponding to E L2([w3),displays a tendency of "spreading out" in time. This conclusion can be reached by establishing the evanescence of the wave packet $i(r) from any bounded Borel set B :
Combine the result of Exercise 7.1 1 with Lemma 1.2 and Lemma 7.1 to prove that the above limit is indeed equal to zero. 7.13. packet,
Prove that any interacting state &(r) = (ciH1 1Crn)(r),
$t
represented by a wave
$n E L ~ ( R ~ ) ~
which has an incoming (outgoing) asymptotic state $f($SUt) evanesces from any bounded Borel set B in the configuration space when t + - 00 (t +a). +
8.
Fundamental Concepts in Multichannel Scattering Theory
8.1. THECONCEPTOF CHANNEL T h e outcome of the mutual interaction of three or more particles can lead to results which are qualitively different from the possibilities encountered in two-particle interactions. Indeed, a system of two particles can either be found after interaction in a bound state, or the two particles become eventually free. However, in the case of three particles P , , P , , and P , , there already are other possibilities in addition to the two alternatives of P I , P , , P , becoming eventually free or constituting a bound system (Pl-P2-P3). For example, it is possible that PI and P, will stay bound forever, thus building a new system P l - P 2 , while P, becomes free, etc. Each one of these possibilities determines a particular outgoing arrangement channel of the scattering process between P I , P , , and P , . I n general, an arrangement channel in the scattering process of n particles P, , P , ,..., P , is a particular partitioning of the set { P I ,P , ,..., P,} into a number of subsets, called fragments, where all
8. F u n d a m e n t a l Concepts i n Multichannel S c a t t e r i n g T h e o r y
575
the particles in a fragment are bound together, and where no interaction takes place between particles in different fragments. Clearly, we can talk about incoming channels and outgoing channels, depending on whether the particular channel we are considering refers to the system before or after the scattering had taken place. I n the case where some of the particles in the system { P , , P2 ,..., P,} are identical, not all partitions of the set { P , , P2,..., P,} will represent distinct arrangement channels. This is due to the presence of BoseEinstein or Fermi-Dirac statistics, respectively (see Chapter IV, 94) which rules that identical particles are indistinguishable. For example, if Pl , P , , and P , are identical, then the channel ( P l - P z , P 3 ) in which Pl and P2 are bound, while P , is free, is identical to the channels ( P , , P,-P,) in which P , is free and P, is bound to P , . As a matter of fact, in this example both above arrangement channels are also identical to ( P 2, P,-P,) so that we have only three distinct arrangement channels: (P1-p2-p3), ( P I , p2-p3), and (P1 , P2 , P3). If each fragment 9is in an eigenstate of a set 9, of commuting observables which are constants of motion in that particular arrangement channel, then the system is said to be in the channel determined by eigenvalues of the observables in the family Lo,. I n practice, it is very convenient to take an 0, containing the internal (binding) energy of the fragment, its internal angular momentum (called the spin of the fragment), its angular momentum with respect to the reference system, etc. If 9 , contains only the internal energy of then we call a channel determined by the given internal the fragment 9, energy eigenvalues an energy channel.*
u,
8.2. ELASTIC AND INELASTIC SCATTERING I n the case of many-body (three or more particles) scattering we can distinguish between elastic and inelastic scattering+ by looking at the distribution of energy among fragments before and after the scattering
* We depart somewhat from conventional terminology, in which a channel with respect to some of the above-specified observables is simply called a channel. However, since there does not seem to be complete agreement among different authors o n the precise contents of the set 03 when a channel is defined, the above general approach seems quite desirable. t T h e distinction can be made also in the case of two-body scattering in an external field. In fact, the external field describes the interaction of the two particles with one or more additional particles which are not explicitly included in the system, but are effectively represented in an approximative manner by the “external” field. I n that case, each one of the particles in the system can be bound to the field. However, it is clear that two-body scattering in an external field is, in fact, many-body scattering.
576
V. Quantum Mechanical Scattering Theory
has taken place. Elastic scattering is, by definition, the scattering process in which all the fragments and their internal energies are preserved; for example, if the initial energy channel consists of a particle Pl being scattered from a bound system P2-P, of two particles, then the scattering is elastic if after the scattering has taken place the particles P, and P, are still in a bound state of the same energy as the initial bound state, i.e., if no energy has been transferred from Pl to P,-P,. Naturally, any scattering which is not elastic is called inelastic. Suppose all the fragments in an energy channel are in their internal energy ground states and that A E is the smallest energy gap between the internal energies of these fragments and the energetically nearest eigenstate of their respective internal energies. Then, obviously, no inelastic scattering can take place as long as there is not sufficient kinetic energy to be transfered to the k,th fragment to raise it to an A E . This amount of energy “excited” state of internal energy E T is then called the threshold of the inelastic process in that particular energy channel. If in an inelastic scattering the fragments themselves are decomposed and new fragments result as an outcome of the scattering process, then we talk of a rearrangement collision; for instance, in the above example this is the case when a bound state of PI and P, is formed, while P, goes free after the interaction has taken place. I t should be immediately realized that a rearrangement collision cannot take place at any distribution of energy between the different fragments. For example, in an energy channel of (P,, P2-P,) a minimal energy equal to the internal binding energy of P2-P, is necessary for the decomposition of the fragment P,-P, occurring in the initial state. This energy is then the threshold energy for a rearrangement collision.
+
8.3. CHANNEL HAMILTONIANS I N THREE-BODY SCATTERING We shall illustrate the mathematical counterpart of some of the physical concepts introduced above with the case of three-particle potential scattering. For this purpose we consider three distinct particles which interact with one another via two-body forces. Here, the statement that we are dealing with two-body force means the total potential can be written in the form
and therefore each one of the particles interacts with each other separately. I n addition, we simplify our considerations by assuming that these forces ptj . are of finite ranges pij , i.e., Vij(r), i < j = 1, 2,3, vanishes for r
<
577
8. Fundamental Concepts in Multichannel Scattering Theory
T h e total Hamiltonian of the above system is given by the Schroedinger operator (8.2)
+ +
= HO
‘12
f
‘23
v13
7
where Ho is the free Schroedinger operator
and Vij , i < j , is the potential energy operator between the ith and the j t h particle, (8.4)
, rz rs)
(vij#)(rl
9
=
Vik-i - rj) w . 1 rz 9
9
r3).
Naturally these operators act in the Hilbert space L2(R9). T h e arrangement channel (Pl , P, , P3) in which all three particles are free, is characterized by the fact that all these particles are so distant from one another as to be outside the ranges of the forces with which they interact on one another. Hence, for such distances V(rl , r, , r3)vanishes and the Hamiltonian H(1,2,3)of the system is given effectively by the operator H o , which is said to be the Hamiltonian of that particular arrangement channel. In the arrangement channel (Pl, P2-P,), the particles P, and P3 are within the range p23 of their mutual interaction forces, while PI is outside the range plz from P, , and outside the range p13 from P, . Consequently, in this case V(rl , r 2 ,r3) becomes equal to V , 3 ( r 2 - r3), and the Hamiltonian of the arrangement channel is H(1,2-3)= H, Vz3. We see that in the present situation we are dealing with a much more intricate framework then in two-body scattering: instead of a single “free” Hamiltonian, we have a different Hamiltonian for each one of the five arrangement channels. We can write the Hamiltonian of each one of the arrangement channels as the sum of the Hamiltonians of each one of the fragments in that channel. Thus, for example,
+
(8.5)
H(1.2-3)
= H(l)
+
H(2-3)
7
where, in general, H ( i j denotes the kinetic energy operator of the ith particle, (8.6)
W(i)W(P1
3
Pz > P 3 )
= Pi2/2mi$(Pl
Y
P 2 > P3),
i = 1,2,3,
and H ( 2 - 3 ) describes the system (P2-P3);we can write H(2-3) in the form (8.7)
H(2-3)
= N(2.3)
$-
‘23
3
578
V. Quantum Mechanical Scattering Theory
where H(2,3)is the kinetic energy of the fragment (P2-P,),
and V,, describes the interaction between the two particles in the fragment. When the system, in some arrangement a: and there is no interaction or negligible interaction between the fragments, then the state Y ( t ) = e-iHt Y(0) of the system is described quite well by the asymptotic state YF(t) = p
(8.9)
a
t
YF(0)
in the sense that (8.10)
lim Ij Y(t)- YF(t))I
t+Fm
= 0,
+
where “ex” stands for “in” ( t ---f - CO) or “out” ( t -+ CO), depending on whether the channel is an incoming or outgoing channel, respectively. I n two-body scattering we could formulate a general consistent timedependent scattering theory for any two self-adjoint operators H a n d H,, . However, in multichannel scattering one has to be careful and establish from the very beginning whether the channel description is indeed consistent, since it is not a priori clear that some interacting state might not have two or more distinct asymptotic states satisfying (8.10). I n order to be able to find the conditions under which such ambiguities in the asymptotic description do not occur, we have to understand more fully the structure the arrangement channel Hamiltonian Ha . 8.4. THEINTERNAL ENERGY OF FRAGMENTS IN THREE-BODY SCATTERING
Let us consider the arrangement channel ( P I ,P,-P,) in the scattering of the three particles PI , P2 , and P, interacting via the two-body potential (8.1). T h e fragment P2-P, is asymptotically in a bound state (8.1 I )
Y(2-3)(t) = exp( -iN(2-.3)t) Y(2-3)(0) E L2((wB)
of P, and P, . Let us assume that Y(2-3)(0) is represented by the wave function (8.12)
where according to 95 of Chapter 11, $’ describes the state of the center of mass of the system P2-P, , while $“(r3- r,) describes the state of the
8. Fundamental Concepts in Multichannel Scattering Theory
579
two particles in relation to one another. The two particles are in a bound state when #”(r, - r2)is an eigenvector (8.13)
Ht2n43)#”
E(2-3)
E(2-3)#“2
< O,
of the internal energy operator
where A is the Laplacian in the variable r = r3 - r 2 .Naturally, the most general Y(2-,)(0)has to be an element of the closed subspace of L2(Rs) spanned by all such vectors (8.12), corresponding to all eigenvectors $’’ of H&,”_t,) . When P2-P, is asymptotically in the state Y(2-3)(t) E?(W) and PI is asymptotically in the state Y(l)(t), then the entire system is in the state (8.15)
! F ( t ) = Y(,)(t) @
Y(2-&)
ELZ(R9).
If Y(2-,)(0) is of the form (8.12) and #” is represented by the eigenvector in (8.13), then it is easily seen that
(8.16)
H(1,2-3)
yex(0) = ( H ( l ) = (ff(1)
+ f
HEf3)
f
Hf,n43))
Htjf3)
f
E(2-3))
yex(o)
yex(0),
where H(tk;) is the kinetic energy operator of the fragment P2-Pa,
The quantity
P,-P,.
1 E(2-3)1 is called the internal energy of the fragment
Naturally, it is often desirable to consider energy channels with respect to such internal energy operators. This is possible since in general, as well as in the above particular case, these internal energy operators commute with the arrangement channel Hamiltonian, and ther’efore they are constants of motion. Let us assume that (Pl, P2-P,) is an incoming energy channel, and that the kinetic energy part $’(pZ3)of #(r2,r3) has a sharp distribution of momentum, i.e., that the incoming momentum of P2-P3 is prepared very accurately. We can choose the system of reference in such a manner that p,, m 0, and consequently
580
V. Quantum Mechanical Scattering Theory
Hence, we see that in this case the incoming particle P, must have an amount of kinetic energy at least equal to I E(,-,) I in order to be able to impart it to P,-P, in the collision process and break up that fragment, thus giving rise to a rearrangement collision. Consequently, in this case -E(2-3)represents the threshold energy for a rearrangement collision of the system {P, , P, , P3}in the given incoming energy channel. T h e reader can easily verify that the Hamiltonian H , of any of the above arrangement channels of the system {Pl , P, , P3} is the sum of a kinetic energy part HFinof the center of mass of the fragment, and of an internal energy part Hpt. For example, in the arrangement channel (P, , P,-P,),
in the arrangement channel (P, , P, , P,),
in the arrangement channel (Pl-P2-P3), H&!:-,) is the kinetic energy operator of the center of mass of the entire system, and (8.21)
int
H(1-2-3)
‘12
f- ‘23
+
‘13
It is obvious that the kinetic energy operators of all channels commute with one another since they are all functions of the compatible momentum observables of the three particles. We also know from $5 of Chapter I1 that each center-of-mass kinetic energy operator commutes with the corresponding internal energy operator. However, in general, the Hamiltonians for different arrangement channels do not commute among themselves. This is easily seen to be so in the case of (P, , P, , P3) and (Pl-P2-P,) when the channel Hamiltonians are H,, and H = Ho V,, V,, V13, respectively.
+
+
+
8.5. WAVEOPERATORS IN MULTICHANNEL SCATTERING Following the guidelines set up by the above example, we postulate that in any multichannel scattering process a unique channel Hamiltonian H, is attached to every arrangement channel 01. Let us denote by MZ) the sets of all vectorsf which are not eigenvectors of H , and for which the respective strong limits
8. Fundamental Concepts in Multichannel Scattering Theory
581
exist. By applying Lemma 1.2, we immediately conclude that M$) are closed subspaces of Z which might contain only the zero vector, as often happens in practice (see Exercises 8.1 and 8.2). Hence, we can introduce the projectors EM:) onto MF) and define the wave operators ):?L for the channel a by the formula (8.22)
The above formulas are obviously straightforward generalizations of (2.1). Hence, as mathematical objects,):?L will have for each channel 01 the same properties as Q,. I n particular are partially isometric operators with initial domains M): and final domains R$), which coincide with the ranges of these operators. T h e intertwining properties (2.3)-(2.5)
Qz)
~k)
(8.23)
eitHa - e i t H
(8.24)
Q).!
(8.25)
Q!.',
tER',
E ~ ~ (= B E) ~ ( BQ).! , Q'.!
H, f
=
HQk)f,
f
B E @, E
BH,,
and the relations (2.10) and (2.11)
Qy*Qy= EM ( d
(8.26)
Qn'd
(8.27)
f
Qy*= ER(d
I
Y
will hold in the present case. Theorem 3.1 is also still valid, so that
Moreover, the above two formulas can be taken to be the starting point of the time-independent approach to multichannel scattering. 8.6.
THEUNIQUENESS OF ASYMPTOTIC STATES AND THE
GENERAL STRUCTURE OF CHANNEL HAMILTONIANS
T a k e f E Mg) = MY) n ME) and writef,
=
Qzy.Since
582
V. Quantum Mechanical Scattering Theory
we see that e r i * m 1 f is the incoming asymptotic state of the interacting state e c i H t f ):, and the outgoing asymptotic state of the interacting state e - i H 1 f 2). T he questions now arises, however, whether is the only incoming(outgoing) asymptotic state of e c i H 1 f $ ()e r i H t f ? ) ) ; namely, there certainly cannot be another incoming asymptotic state e+ff=t g of e c i H 1 f $ ) in the same arrangement channel, but, in general, there could be some other incoming asymptotic state e - - i H @ g of e c i H t f : ) in some other arrangement channel /3 # 01. I t was pointed out earlier (in the three-body case) that we cannot expect that such a uniqueness of asymptotic states would hold for any arrangement channel Hamiltonians H , picked u p at random from the family of self-adjoint operators in A?.T o prove such a uniqueness of asymptotic states we have to restrict the families of candidates for arrangement channel Hamiltonians, by requiring that such operators obey certain conditions, which are dictated by the physical situation at hand. Extrapolating from the example considered earlier of three-body scattering, it seems sensible to require that every channel Hamiltonian H , on MY) and on Me) be the sum eriHWt
(8.30)
H,x = Htin + H P t
of a center-of-mass kinetic energy part Hfin and an internal energy part H F t ; here Hakin is the operator which represents the sum of the kinetic energies of the centers of mass of all fragments, while the operator HFt represents the sum of the internal energies of all fragments in the arrangement channel a. Judging from the wave mechanical three-body problem, it is also reasonable to postulate that HFt has a pure point spectrum and commutes with Hakin,while Hnkin has only a continuous spectrum, and that all kinetic energy operators Hfin commute among themselves. When the channel Hamiltonians have the above indicated structure we can prove that incoming and outgoing asymptotic states of any interacting state are uniquely determined by that interacting state, and consequently the description of the scattering experiment in terms of the asymptotic behavior of the system is completely unambiguous. Let us first understand better the problem of uniqueness in mathematical terms. We have seen in (8.19) that if Y(t) = e c i H t Yois an interacting state, and if Yo = L?y)Yp,then Yin(t) = e - i H o l Yin is an incoming asymptotic state of Y(t). If Y y ( t ) = exp(-iH,,t) !Pin were another incoming asymptotic state of Y(t),then we would have Yo= l2:’)Y.P. Thus, Yin(t) could be a state different from Yp(t)only if 01 # 01’ and lj7 - Q(a)Yin= f p+ ’ ) y i n , i.e., . if Yobelonged to both ranges Rf) and 0 + 0
8. Fundamental Concepts in Multichannel Scattering Theory
583
R(a’) + of ace) + and a$’), respectively. Consequently, the necessary and sufficient condition which has to be fulfilled in order to have a unique asymptotic state corresponding to an interacting state is that the different ranges RY) should have no state vectors in common. Naturally, a similar conclusion holds for the ranges RE), which are also required to have no state vectors in common. 8.7.
A THEOREM ON UNIQUENESS OF ASYMPTOTIC STATES
T h e two ranges R’: and RY‘), 01 # a‘, will have no state vectors in common if, in particular, they are orthogonal to one another; in fact, in that case R!,? and RY’) have in common only the zero vector which is not a state vector. T he following theorem shows that if the channel Hamiltonians have the earlier mentioned structure, which is reflected in (8.30), then R’: 1 RY’) for all a # a’ and RY) 1- RF‘)for all p # p’. Consequently, under those conditions every interacting state has at most one incoming and one outgoing asymptotic state.
Theorem 8.1. Suppose the arrangement-channel Hamiltonians H , , acting in the separable Hilbert space H,are of the form H,
=
Hahin-1- H,nt,
where all Htin commute among themselves, while each HFt has a pure point spectrum when restricted to MF) and it commutes with the corresponding H:in. If 01 and p are two distinct arrangement channels and Hakin- Hfinhas no point spectrum, then RY) 1 RY) and R(a) 1 R?.
Proof. It follows from the basic definitions of wave operators that (8.31)
Since H p t has a pure point spectrum on M:), the integral in the spectral decomposition of Hint can be written as a sum: (8.32)
584
V. Quantum Mechanical Scattering Theory
here the summation extends over all eigenvalues of HLnt corresponding to eigenvectors in Mg), and Ea({A}) is the projector in the eigenspace corresponding to the eigenvalue A. It should be observed that in this context it is indeed possible to sum over all eigenvalues X on account of the fact that there can be at most countably many eigenvalues of a selfadjoint operator in a separable Hilbert space. We immediately get from (8.32), by using in the process the commutativity of H:in and HEt,
where the above sum, if infinite, converges in the strong sense. Since a similar relation can be derived for EM?)eiHgt, and since HFin and HFin commute, we can write for any f E Mf) and g E MF)
wheref,, = E(@)({A})fandg,= E(=)({X}) g. Let us take any two vectorsf E MY) and g E M Y which have only a finite number of nonzero components%, andg, , respectively. Then, in view of the fact that (f 1 eiHPte-i H a ' g ) is continuous and therefore integrable in t on any finite interval, we have (8.34) dt
=
c 1 J" A'
A
T
( f A , I exp[i(H;ln
0
+ A' - ~t~~ ~ ) tgA') ] dt.
+
-
Since Hiin A' - H:in - X is a self-adjoint operator, we can apply the mean ergodic theorem (see Theorem 8.3 in Appendix 8.10 to this section) to each term in the sum of (8.34). According to this theorem
=
(fA,
kin
I EH8
kin
({A - x } ) g A )= 0,
8. Fundamental Concepts in Multichannel Scattering Theory
585
where the above inner product is zero because the projector
on the characteristic subspace of Htin- H:in corresponding to the eigenvalue h - A' has to be zero on account of the assumption that Hfin- HFin has no eigenvalues. By using (8.31), we obtain (see also Exercise 8.3)
(f
If f E M Y and g E ME) have only a finite number of nonvanishing components fA* and gA, respectively, then in view of (8.34) and (8.35), we conclude that
for all suchf and g. Since the set of all such vectorsf€ MLB) and 6 E M Y is dense in Mf) and MY), respectively, it follows that
for all f,g E A?.This establishes the assertion that RP) 1 Rf) for 01 # /!I. Q.E.D. A similar argument yields that RY) 1 Ri!) for 01 # p. T h e above theorem is valid under the restriction that Hakin- HFin should have no point spectrum when 01 # p. It should be noted that this condition is certainly satisfied if HFinand Htinare two distinct functions of the momentum operators of the n particles partaking in the scattering process. T h e reader can easily convince himself that such is the case, for example, in potential scattering.
8.8. INTERCHANNEL SCATTERING OPERATORS In multichannel scattering we are faced with the possibility that a system prepared in some given arrangement channel does not have to end up in the same arrangement channel. Hence, obviously, no single scattering operator S could describe all the transitions which the system can undergo. Instead, to any pair (p, a ) consisting of an incoming
586
V. Quantum Mechanical Scattering Theory
arrangement channel 01 and an outgoing arrangement channel to attach a different scattering operator, (8.36)
we have
= R -(8)*9,(a),
I s ,
describing the transitions between these two channels. In fact, if Y(t)has Yfi(O), then the transition amplitude the incoming state YE(t) = to the free state Yyt(t) = e-iHd Y:ut(0) is eWiHUt
(8.37)
lim (Yyt(t) I Y-(t))== t++m lim (e-iHat YY'(0)I eiHt Y-(O))
t++m
-
(Qyy o u t (0) I Y-(O))= (98' YFt(0) I 9:) Y!!yO))
=
< Y y ( o I) s,, Y'"(0)).
+
From (8.23) we easily obtain for all t E R1
s,, eiHet
(8.38)
* Q?) p
=~i_") - eiH8t -
a t =Q )!
* eiHt 92)
= eiHst
Q!)*
s,,
*
We can also derive in similar manner (8.39)
S,, EHa(B)= EHB(B)S,,
,
B
E
a',
One of the most important properties of the single-channel scattering operator is its unitarity. T o establish the presence of some analogous property for multichannel scattering we have to consider simultaneously the entire array of all interchannel scattering operators. Then, using (8.27) we obtain (8.40)
c Y
Sy*S,,, =
c (Q~'*Q!?'')("?''"Q~') Y
I n the case of single channel scattering the identity of R, and R- is a necessary and sufficient condition for the unitarity of S on M, (see Theorem 2.5). A reasonable generalization of that condition to the multichannel case would be to require that (8.41)
@ Rt) Y
= @ R?" = Y
R,
8. Fundamental Concepts in Multichannel Scattering Theory
587
where we tacitly assume that the conditions of Theorem 8.1 are satisfied, and therefore RY) 1 RY’ and Re) 1 R1_8)for a # p. T h e assumption (8.41) implies that
since QY) is a partial isometry with final domain RY). Hence, (8.40) yields
2 K3,,
(8.42)
Y
=
Qi Q, a)*
(8)
-8 E - a5 R$ ? l)
where a,, = 1 for 01 = /3 and a,, = 0 for a # /I. A similar procedure leads to the conclusion that
when (8.41) is satisfied. 8.9.
THEEXISTENCE OF ASYMPTOTIC STATES IN
PARTICLE WAVEMECHANICS
Let us consider the wave mechanical case of n particles without spin interacting via two-body forces, so that
where
Let us choose an arrangement channel a containing the fragments
FL1),...,F:n,). T h e free motion of each one of these fragments FLk’is described by the Hamiltonian
(8.45)
588
V. Quantum Mechanical Scattering Theory
where the summation is over the indices of the particles in that fragment. T h e channel Hamiltonian is (8.46)
H,
==
H:)
+ ... + H?).
Take R, to be the center-of-mass position vector of the lzth fragment, and let pk stand for the rest of the internal motion coordinates, so that (Rk, p k ) completely describes the positions of all the particles in the fragment. Then proceeding in the manner indicated in $5 of Chapter 11, we can write (8.47)
Hik) = -1/2Mk ARk
+ Ha(lc)int,
where M k is the mass of the fragment 9Lk and ) HLk)int is the Hamiltonian of the internal motion of the particles in that fragment. If +:(pk), k = I , ..., n, , represent eigenvectors of Hik)int,then e--iHat Y(O), with Y(0) given by (8.48)
k=l
yt(Rk) #:(fk),
y : )...,Vl,". EL2(R3),
is a "free state" in the channel a. Obviously, the closed linear manifold N(,) spanned by all the functions of the form (8.48) coincides with the set of all free state vectors in the channel 01.
Theorem 8.2. Suppose the functions Vij(r), i < j , i, j = 1,..., n, are Lebesgue square integrable on R3. Then the initial sets M;) of the channel wave operators Sz?) coincide with the set N'") of all free state vectors in the arrangement channel a. I n order to avoid very involved notation, we shall prove Theorem 8.2 for the special case of the arrangement channel {PI , P2-P,} in three-body scattering. I n this case
where m2, is the reduced mass of the system P2-P, and (8.50)
T he space N ( n )of free state vectors in this channel is the closed linear manifold spanned by all the functions (8.51)
d(r1 r2 r) = Ydr1) FAR) 1Cro(r23> 9
3
8. Fundamental Concepts in Multichannel Scattering Theory
589
corresponding to all Yl , Y2€L2([W3)and all eigenvectors # of
On account of Wiener's theorem (see Lemma 7.1) the family of all functions (8.51) with Yk, R = 1, 2, represented by functions which are Fourier transforms of Gaussian wave packets exp[-(p - P ~ ) ~k] ,= 1 , 2, is dense in N(a)when p1 and pz are allowed to vary over OX3. For such functions we can easily compute
Hence, according to Theorem 2.8, the strong limits
E ( ~if) the improper Riemann integral for all exist for all Y Y N
is convergent for all 4 of the above indicated form. Now, using (8.52) we get the following kind of estimate:
11 vlZepiHwt # 112
x exp[ --a,(R
-
bJ2#02(r23)dr,, dr2, dR t
< const E4 (1 + (2m,)l
2 -312 *
590
V. Quantum Mechanical Scattering Theory
Hence, we conclude that (8.53) converges, and consequently N ( a )= M(*)= M(n) Mk).
+
~
8.10. APPENDIX: VON NEUMANN’S MEANERCODIC THEOREM T h e following theorem was first derived by von Neumann [I9321 in order to prove the quasi ergodic hypothesis of classical Hamiltonian mechanics, and it was used by us to obtain (8.35).
Theorem 8.3. space Z and
Suppose A is a self-adjoint operator in the Hilbert
(8.55)
irrespective of the mode in which t , - t , tends to infinity. We note that there is a unique bounded linear operator B(t,, t,) (a Bochner integral) for which the relation (8.56)
holds for all f,g E A? (see Exercise 8.7). Hence, (8.55) states that E({O)) = tw;lim B(t, , tz). 2-
p + m
I n order to prove Theorem 8.3, we write (8.57)
( f I B(t, > t 2 ) E ) =
< f I B(t,
From (8.54) we get
3
tz)
E({O))g)
+ ( f I B(t,
3
tz)
E(R1- {OHg).
59 1
8. Fundamental Concepts in Multichannel Scattering Theory
Thus, we see from (8.57) and (8.58) that (8.55) is true if and only if (8.59)
lim
t,-t,++m
(fl
B(t, , t2)h) = 0,
h
=
E(R1 - (0))g.
We shall prove that (8.59) holds by showing that
converges to zero when t2 - t, , we get Since U,I* =
-+
+
00.
After introducing in the above integral the new variables s and t = t’ t”, we arrive at the following relations:
+
=
t” - t’
Since the integral
obviously exists, we can apply Fubini’s theorem to interchange the order of integration in t and A. Thus, we obtain, after carrying out the integration in 1,
=
s,
(
sin 3(t2 - tl)A 3(tz - t l ) A
)
11 EAh 112*
592
V. Quantum Mechanical Scattering Theory
Let us split the domain R1 of integration in three parts (-00, -71, m), and majorize the integrand by 1 on (-7, +7] and by ( * ( t z - t,)h)-, on the other two intervals. Then we arrive at the estimate (-7, +7], and (7,
+
By choosing 7 small enough, we can make the first term on the right-hand 2 any a priori given E > 0; side of the above inequality smaller than ~ / for this is due to the fact that
Then, for such a value of 7, the second term in (8.60) can be made smaller than ~ / by 2 choosing I t , - t , 1 sufficiently large. Hence lim
t2--t,++m
11 B(t, , t,)h /I
= 0.
Thus, (8.59) holds, and therefore (8.55) is ’ true.
EXERCISES 8.1. (a) How many arrangement channels can there be in the scattering process of three distinct particles ? (b) How many arrangement channels have actually been observed in the scattering of a protonp, a neutron n, and an electron e, i.e., for which channels 01 do we have Mk’ = {O}? (c) Give the conventional names of the fragments in all experimentally realizable arrangement channels in the scattering of n, p , and e interacting by means of the nuclear and the Coulomb force.
8.2. (a) Count the arrangement channels in four-particle scattering in which there are only two pairs of distinct particles. (b) How many of these channels are not empty in the scattering of two protrons and two neutrons (as far as our present experimental knowledge extends) ? 8.3, Suppose that f ( t ) is a continuous function for t and that limt++mf(t)= a exists. Show that
E
[0, +a),
593
References for Further Study
8.4. Consider multichannel scattering with a finite number of arrangement channels a. Show that the operator
commutes with e i H t and with EH(B). Remark. This operator was introduced by Jauch [1958b] as a candidate for a scattering operator in multichannel scattering. However, its inadequacy for this role is reflected in the fact that its knowledge is not sufficient to compute transition probabilities for scattering processes in which we have transitions between distinct arrangement channels (see also Exercise 8.6). 8.5. Show that if RY' 1 RY' and R!? addition (8.41) holds, then S'*S'
=
S'S'*
=
1 RE'
for a #
6, and
if in
ER.
8.6. Show that if Y J t ) and Y+(t)have the respective asymptotic states Y?(t) = e-iHal YE(0)and Yyt(t) = e--iHf16YOut + (0),then the transi= (YYt(0) I S,,Yp(O)) can be written tion amplitude (YJO)I Y+(O)) in the form
(Y+(O) I Y-(O))= (YYt(0) I QJB'*S~Qnj.'Y!n(o)) =
(YYt(0) I Ql"'*SfQ'.'!P
(O)),
provided that the conditions stipulated in the preceding exercise are satisfied. 8.7, Prove that for any family U , of unitary operators defined by (8.54) there is a unique bounded operator B(t, , t z ) which satisfies (8.57)
for all f,g E X .
References for Further Study The basic ideas of two-body time-dependent scattering theory in Hilbert space are contained in the work of Jauch [1958a]. These basic definitions have to be somewhat modified [PrugoveEki, 1971al in order to be applicable to potential scattering with longrange potentials or other more general cases. This has been pointed out by Dollard [1964, 19691, Jauch et al. [1968], and others. The time-independent approach to scattering theory in Hilbert space has been discussed on a general level by Jordan [1962a]. Examples of nonunitary S operators are given by
594
V. Q u a n t u m Mechanical Scattering T h e o r y
Kato and Kuroda [1959]. T h e Lippmann-Schwinger equations in Hilbert space are derived by Prugoverki [1969b, 1971bl. Some further papers on mathematical questions related to two-body scattering theory are by Jauch and Zinnes [1959], Birman and Krein [1962], Lirnid [1963], and Belinfante [ 19641. A subject closely related to scattering theory is the perturbation theory of linear operators. A general survey can be found in the work of Kato [1967]. A reference book with an extensive bibliography is the one by Kato [1966]. Some other papers on the subject are by Kuroda [1959b, c, 19671, Wilcox [1966], and Howland [1967]. On the subject of Green functions and eigenfunction expansions the reader is referred to the works of Titchmarsh [1962] and Berezanskii [I9681 as general reference books. T h e last one also contains an extensive list of references. Representative papers dealing with the quantum mechanical aspects of this subject are by Kodaira [1949], Povzner [1953, 19553, Green and Lanford [1960], and van Winter [1964]. Much of the treatment of potential scattering given in $56-7 has been inspired by Ikebe [1960, 19651. This reference adopts, however, a Banach space approach in solving the basic integral equation for the full Green function rather than the more restrictive Hilbert space approach. For a general physicist’s treatment of the subject the reader is referred to Newton [1966], which also contains a very extensive bibliography. A mathematically rigorous exposition of complex analysis methods in potential scattering can be found in the book by de Alfaro and Regge [1965]. T h e reader desiring a better insight in the mathematical problems of multichannel scattering theory is referred to the following sample of articles: Jauch [1958b], Hack [1959], Zhislin [1960], Jordan [1962b], Hunziker [1964, 19661, Fadeev [1965], van Winter [1965], and van Winter and Brascamp [1968].