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Projection operators for rearrangement collisions
ANNALS
OF
PHYSICS:
Proiection
36, 5346
(1965)
Operators
for
Rearrangement
RIARCEL Laboratory
for
Collisions*
cOZt
Nuclear Science and Physics Department, Massachusetts of Technology, Cambridge, Massachusetts
Inntitute
As in Mittleman’s recent paper we derive a system of equations which describes the scattering of a system into a rearranged channel. Instead of defining a projection operator into the open channel subspace, we define elementary projectors into each open channel; this definition leads to more sirnplified equations than those obtained by Mittleman. Discussion of the existence of these elementary projectors is given. I. INTRODUCTION
If we describe rearrangement collisions, we can write the total wave function as a linear combination of the different channels with unknown scattering functions in each. These unknown functions are called amplitudes ( 1). In order to obt’ain the coupled equations for these amplitudes it, is useful to construct appropriate projection operators (briefly projectors). A first step will be to define a projector P on the open channel subspace and its complement Q = 1 - P on the closed channel subspace. This has been essentially the work of Feshbach (2) and Mittleman (3). It can be seen that in the many channel case even with antisymmetrization these projectors P and Q exist and are Hermitian projectors, but it is not our purpose to show this. As Xttleman tries to do, we want to obtain coupled equations for the coupled amplitudes, but Mittleman’s t’echnique leads to equations for mixtures of these amplitudes. To avoid this difficulty, we define elementary projectors on each open channel and define P as the sum of these elementary projectors P, + Pd . In this paper we are rest,rict’ed to the nonantisymmetrized cases since arlt,isylllilletrization leads to technical difficulties we shall study in a second paper. We show here, that in physical cases such elementary projectors exist; they are not hermitian but Pi’ = Pi, P;Pi = 0. Their sum is a hermitian projector. With these elementary projectors we obtain coupled equations for t,he amplitudes; the coupling between these equations is only a potential coupling. Since * This work is supported in part through funds mission under Contract At(30-1) 2098. t On leave of absence from Facultb des Sciences 53
provided de Paris,
by the Group
Atomic 1, Orsay,
Energy France.
Com-
54
coz
we have equations for amplitudes for the uniqueness of solutions. In order to avoid explanatory (p, d) reaction with only the two of projectors (the second is only to which they lead. In this theory
it is easy to prescribe
the boundary
conditions
difficulties we restrict the construction to a (p and d) channels open. We give three kinds an intermediate one) and analyze the problems we consider functions
E(r2,7-3,
...
,ra
;ro,1'1
I7",
R)
(1.1)
with 1’0 - 7-l = r; which
can be written
in the form
E = +(Q,
r2,
. .. ,
rA)u(rO)
r. + rl = 2R
(1.2) +
#(r2,
r3
, . . . , rA)x(r)4R)
(1.2)
where t’he two choices (r. , rl), (r, R) in (1.1) express the possibility of rearrangcmerit. In (1.2), +(rl , r2 , . . . , rA) is the target wave function, fi(rz, . . . , ra) the residual nucleus wave function and x(r) the deuteron function. The methods used are those of refs. g-4. We begin in Sections II, III, and IV with some orthogonality assumptions, the existence of the operators will be proved under these assumptions. In the conclusion, we shall discuss the necessity of these restrictions. II.
PROJECTORS
OF
THE
FIRST
KIND
In Section II we consider two wave functions $(rl , r2 , . . . , ra), $( r2 , . . . , rA) which represents the ground state of the target nucleus A and the residual state of the nucleus (A - 1) after the p, d reaction has taken place. We assume that these two functions are not proportional. We assume also that if q is XP,,pen + Remainder, this remainder is orthogonal to the state +(rZ, . . . , rd) of the residual nucleus as well as to the core +(r2 , r3 , . . .rA) of the target nucleus. Two methods are proposed to build up the projections operators P, and Pd , the first is based on the Gram Schmidt orthogonalization process, the second breats the problem by solving integral equations. The two methods are identical. Integrations involved are over (A - 1) variables and the condition for existence of P, and Pd is exactly the nonproportionality between the two functions 4 and 9, i.e., that in (A, 2) +(rl,
where
g*(rl)
A. DEFINITION
7'2,
is some factor
,ra)
* *.
# #(r2,
... ,rA)g*(rl)
of proportionality.
OF PROJECTORS
The two projectors P, , Pd have to be such that they extract tion t the proton part and the deuteron part.
from every func-
REARRANGEMENT P,E P&t
= =
dn,r2, ti(r2,
A trivial j,*
.*.
of fp*(rl
clrz ) * . - ) dr&*(r1,?~Z,
calculation
,r.A)U(d
* * * ) 1./4)x(r)v(R)
TO obtain P, we need a function 4* and $* such that s
COLLISIONS
, . . . , TV), a linear combination
*.*,1..4)$(1’2,
.A.,ra)
of
= 0
gives
= f#l*(rl, . . ...).*. -
s
dr’z . . &‘A $5*(7-l) * * * ) 7.‘a)l//(r’2, . .
r’A)lCI*(r2,
. . . , r,( )
With the notations * /d,‘l) =
J
Cll~Z
. . clra 4*4;
(( 1.1) = /’ dr2
. dra 4*fi
we writ.e 1
P, = 4( )‘I ) 1‘2) . . . ) ‘?.A) Ph>
-
f*o.l)j*(I'l)
s
dY.2* * . &J4*
- f(h)+*1
(2.1)
In the same manner we obtain Pd = $(1’2, ‘.. ) r,)
1 P(l’l) - f’“h)j’(rJ
s
(jr2 .f. ckAp$b* - f*4*1
(2.2)
(2.1) and (2.2) we see that when p(,rl) - f*(rl)f(~J is different from zero (t,hen 4 and # are not proportional), the operators P, and Pd exist.
From
B.
PROJECTOR
PROPERTIES
It is easy to c,heck that P, and Pd are yroject,ors, Pd2 = Pd)
P,” = P,,
that is
PdP, = P,Pd = 0
(2.3)
Now we can define P = Pd + P, . Our claim is that P is an hermit’ian projector. Since P” = P, it remains to show that P = P*. Alt’hough it is possible to demonstrate hermitiality directly from Eqs. (2.1) and (2.2) it will be useful t’o show it by another derivat’ion of P because it introduces the integral equation method mentioned above. The existence of an hermitian project,or depends on a solution of the equation
56
coz
Sufficient
conditions s
for this equation
are
4*(r1 , rz , ... , rA) drl drz ... dra(\E (2.4)
s
dr2 ) --* ,rA)x*(r)
dr dr2 ... dra(\E
or also
J
4*(Q)
x*(r)
r2, . . . , rA) dr2 drs . . . drA(\k
- ,$) = 0 (2.5)
--a ,rA) dr2dr3 ...
J #*(r2,
drA(* - .$> = 0
With the notations
U(QI, rd = J 4%
dr2 . . . drA ;
V(r, R) = / $~*\kdr2 . . . drA
we have U(o,
4
= drdu(rd
V(r, R) = x(rMR)
+ f(rl)x(r)u(R) + f*(Odr~)
From these equations we obtain x(r)v(R), u(ro) if p(rI) - f*(rI)f(rI) is not zero, for some rl . Let operators P,’ and Pi be defined by P,‘\k = C$U,Pd)* = Qxv. When P,’ and Pi exist, i.e., when p(rl) - f*(rl)f(rl) # 0 we have P,’ = P, , Pd’ = Pd and their sum is an hermitian projector from Eq. (2.4). The operator P annihilates the remainder defined by Q - pk from the orthogonality assumpt,ions. C. EXPLICIT
EQUATIONS
FOR THE AMPLITUDES
u AND v
(a) Let I? = H + H&G&H with Q = 1 - P, G = Q(E - &II&)-‘& The effective equation for the open channels is P(E - B)p\k
= 0
(2.6)
where pk = 4u + #xv,
(2.6)’
from the orthogonality assumptions. We can write the hamiltonian H of a A (p, d)B reaction with (Y = A + p, /3 = B + d as follows H=T,+H,+Va,=
Td+Hg+V,vci.
(2.7)
REARRANGEMENT
From
(2.6) and the equality HP,%’
QHPp*
= (T, + e&u =
QV,,+u;
.i7
COLLISIOl’iS
P& = 0 we have + V&u; QHP&
HP& =
=
( Td
+
Q)#XV
+
~‘Bd#Xv
Qvi&x?.
(b) We use now the explicit expressions for P, and Pd , and mult’iply (2.6) give the coupled successively from the left by P, and Pd . These multiplications equations.
[(#J
= + (4 I 4) -II (4 I +I I2
( +)($‘I
Ved
14)
-
(9
1 +I(4
1 VBd
1 +)h
In (2.8) round brackets express integrations on ~2 , . . . , rA . The integro-differential equations for the amplitudes are obtained after multiplication by +[resp. +x] and integration on (ri , Q , . . . , r,) [resp. (r, r’2 , . . . , yA)J on the first [resp. second] equation of (2.8). III.
PROJECTORS
OF
THE
SECOND
KIND
In Section II we have assumed that the two functions 4 and # were not proportional to each other. Also we assumed a strong orthogonality condition. With these conditions we have been able to build two projection operat.ors P, and Pd using only (A - 1) variables. The two methods (orthogonalizat,ion procedure, and integral equation method), in order to obtain P, and Pd were equally easy to handle. In Sections III and IV we want to build more general projection II operators which can apply even in the case where 4 and \L are proportional t,o each other. Again we shall use the two methods recalled above. Now we must use int’egration over A variables. But only the last method (the integral equation method) can be used without difficulty. At the present stage of this work, we assume again an orthogonality condition which is, however, less strong than that of Section II. We assume that the remainder is orthogonal to the target nucleus and to the system residual nucleus-deuteron. Later we discuss the necessity of this assumption.
58
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In order to contruct the P, project in the orthogonalization method we define a function f,* = cw4*(?2!$ Y
) r2 , * ‘. ) rA) + p+*cr2 , -. - , LAX*(r)
such that
We obtain
Now the construction of P, needs the knowledge of the operator fi2, D,=
sdr,[p~~)-{SS(~)x(r)dr}f*~~)x(r)],
since we must write P, = +n,’
I
dr drp - . . dra f,*
The singularity appearing when the denominator of (2.1) is zero is replaced by the possibility &, to be a singular operator. If this singularity can happen, the operator P, cannot be defined. A similar method is necessary to obtain Pd . Let us assumenow that these P, and Pd exist and that we know their explicit forms. The integro-differential equations we are asking for, if we put Q = 0, are given by (E-E,-
T,)u(r,)
-
St;;’ 1 f,*V,,+u
dr dr2 . . . dr.., (3.1)
0;’
(E -
ED -
Td)v(r)
-
/ .fp*VB&
C&i’ / fd*VedJ/xv
= ($x(x+
(f&i’
dr dr2 . . . drA)
v(R)\
/
drl dr, . . . dr,
/sd*V~,Odrldrz
...
drA)u(n)>.
(3*2)
The inversions involved in fiil and flZ1 are difficult, so that these equations are not useful; however, in the alternative method (integral equation method), inversions of operators which replace 0, and fid are easier. At the same time singularities corresponding to the denominators of (2.1) and (2.2) can be understood.
REARRANGEMENT IV.
rZ. DEFINITION
PROJECTORS
OF
59
COLLISIONS THE
THIRD
KIND
OF PROJECTORS
Clearly the method given can be used for a general (p, d) reaction, but we consider only t,he case excluded in Section II. Let ,t(h, .a. , rd ; r0,rl 1r, R) = #(r2, . .. ,?*.~)(+(r~)u(h) + x(r)u(R)j
(4.1)
We use the notations U(r,)
=
/
#*(r2
) . *. ) r,)c$*(r,>~
V(R)
= f a+Q*(rg , . . . , r&*(r)l
dr1 drz
f..
dr,
dr d1.2. . . dr,
Using the orthogonalit,y assumption between the remainder and the target nucleus wave function and between the remainder and the wave function of the system residual nucleus-deuteron, as in (2.5) we obtain integral equations for u and v
U(ro> = u(d + / d~*(rdx(r)v(R) drl (4.2) V(R) = v(R) + / x*(r)~(rduhd
dr
Let us write (4.2) in a symbolic form
U(Q)
= U(Q) + KdR);
V(R)
= r(R) + K,u(d
(4.3)
The solution of (4.3) gives u(r0)
v(R) = l _ ;
=
We define now P,E = u( ro)$+; P& = v(R) = U(R)&; Pd exist they have the following form.
p, = 94
1 1 -
K,K,
1 1 -
K,K,
if the operators P, and
I)*+* drl dr2 . . . drA (4.4)
- / drl +*(n)x(r)
Pd = #x
K D’ - Kv VI. 21 u
/ do’ drz . . . drA#*x*(ro,
. . . ,rij]
J/*x* dr dr2 . . . drn - j dr x*(r)+*(rl)
/ dr’ dr2 . . . drpj?-+*
( >I R - ;
(4.5)
60
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B. PROPERTIES
OF THESE
OPERATORS
The formulas (4.4), (4.5) relate the existence of P, and Pd to the impossibility of eigenfunctions of eigenvalue 1 for both integral operators K,K , K,K, . If P, and Pd exist, they are projectors: P,’ = P, , Pd” = Pd ; P,Pd = PdP, = 0. Now we can define P = P, + Pd. The operator P is the projector II of Mittleman, which always exists (3) (even when separately P, and Pd have no existence) at last with our orthogonality assumption, and is hermitian. C. EXISTENCE
OF THE
ELEMENTARY
PROJECTORS
We come back to the different kernels involved and rewrite (4.2) in a more precise form
With trivial obtain
U(O)
=
U(‘~CI)
+
/” 2, (-‘\)
V(R)
=
v(R)
+
j
24 (R
~(7%
+
;)
4 (Ii
-
rl)$*(rl)
-
;)
drl
x*(r)
dr
changes of parameters and noting that 1’ and rl are vectors, we
U(f)
= u(E) + 8 / v(,t’)+*(2E - t’>x(2$
- 28 dE
v(t)
= v(t)
- 2{‘) dE
+ 8 / &‘>$(2[’
- t)x*(2t
We note t’hat the two kernels involved in these equations are adjoint one of another. From the homogeneous system udt)
= -8X /- dt’4*(25 - t’)x&
- 25’h(t’) (4.6)
v,(t)
= -8X
s
d&#OE - &*(2f’
- 2E)&‘)
we obtain m(t)
= 64X2N&, rh(1’);
VA(~) = 64X2Nv(F, rh(r)
with N,(E, r> = /- d&W
- r)9*(2E - &x(25
- 2C;‘)x*(2r - 2.5)
N,(E, r> = /- &‘d2 r - F/)4*(25 - .59x*(2{’ - 27-)x(2.$ - 2E’)
(4.7)
REARRANGEMENT
61
COLLISIONS
N, and N, are self-adjoint kernels, therefore X2 is real and positive. The critical eigenfunctions whose presence prevent’s the existence of P, and Pd are t’hose with X = 1 in (4.6). We are now asking what this means. Let us consider (4.6) with X = 1. Instead of (4.6) we can write
(4.8) o=
s
&*(2$
- 2{){x(2st' - 2.91(~)
+ qq2t: - $'ju&')
We multiply the first equation of (4.8) by u*(,t’), the second by v*(t’) ; in the second equation we permute t- and ,t’. Integrating now on [ and l’ we add these two equations; thus we obt,ain
or +(~‘l)u(cJ
+ x(r)v(R)
= 0
(4.10)
(4.9) or (4.10) are conditions of the same kind as (3.28) in (2). In other words when the critical eigenfunctions occur (4.10) is valid for some functions u( rO) and v(R) which are not identically zero. Conversely (4.10) implies the existence of these critical eigenfunctions. This means a strong functional dependence (Appendix II) between the functions 4 and X. This means also that the amplitudes u(,Q), v(R) cannot be uniquely determined: they have no physical meaning. In other words, the channels defined by 4 and x cannot be distinguished; they are not physical channels. The situation is very different from that encountered by Feshbach (2) in the antisymmetrized case. The direct and the exchange channels are mathemat’ical but not physical entities, therefore there is no contradiction when one asserts that ~(1,~) cannot be uniquely determined in
@ is the antisymmetrization operator. In conclusion (4.10) implies either the channels are not physical, or we have made an improper choice of the functions 4 and X; therefore we rule out the possibility that Eq. (4.10) is satisfied. We can therefore define P, and Pd and use them in the same manner as in Section II or Section III, t’o obtain the coupled equations of the problem.
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D. INVERSION OF THE Both operators K,K, tive eigenfunctions. We with eigenvalue C;equal
INTEGRAL OPERATORS INCLUDED IN THE PROJECTORS , K,K, are hermitian. Let ux and vx denote their respecadd to the two sets (UX) and (vh) the functions (ut), (zQ), to infinity. In this we obtain complete orthogonal sets.
l-K,,K,=l-&$u)(uk=~f$ (1 - KuKJ'
UA>( UA
= &ud{
ui
and similarly (1 -
K,K,)-'
= x&
= VA)(UA.
V. CONCLUSION
The existence of an operator P which annihilates \E - pk has been proved using orthogonality assumptions. We can ask if these assumptions were necessary. To give an answer we can look how P has been constructed. Let ((rz , . . . , ?-A); ro , r2 1r, R) be any function having good asymptotic behavior. How does P operate on .$? If for any such 4, PE is the sum of two terms, the first proportional to $(rl , r2 , . . . , ra), the proportionality factor depending only on r. , the second proportional to cp(rz , . . . , rA)x(r) the proportionality factor depending on R, we can choosef = P, and define a remainde asr \k - pk. This remainder will be automatically orthogonal both to #(rl, r2 , . . . , ?“A) and to dr2, - +. , rA)x(r). If it is not so for every E, although it can happen for some special .$,we need the existence of an orthogonality assumption between closedand open channels. Coming back to the operators P defined in Sections II, III, and IV, it is straightforward to verify that in Section II the orthogonality assumption is needed while in Sections III and IV it is useless.Therefore we have an operator P which can be constructed which annihilates XP- p\k. This is the general result announced in the introduction. APPENDICES
We are first concerned with a special case in which calculations discussedin Sections III and IV are simple, and afterward with the functional dependence between 4 and x which results when Eq. (4.10) is satisfied. I. APPROXIMATE
PROJECTORS
We assume that the following approximations discussed in Section I, D are valid.
/4*(hdr)u(R)
1 x(rMrdu(r0>
drl # u(R) / 4*(n)x(r) dr # u(r0)
1 x*(r>drd
ch dr
(A.1) (A.2)
REARRANGEMEKT
COLLISIONS
63
In this case we can define approximate projectors. A. COXSTRUCTION
OF THESE
OPERATORS
The approximate equations for (4.2) are written r:hl)
= ucro> + u(R) (4 / x)
(A.3)
I’(R)
= v(R) + 41.0) (XI $>
(A.4)
which gives t,he ‘Lproje~tor~” asymptotic in some sense
1 _ / (; , x> (1[of4 - (4 I XNXI
as. P, = ti(1.2 , . . . , r,,)+(r) as. Pd = +(Q,
-se , T.~)x(T)
Since / +“(R - r/2)x(r)
(A.5)
1
1 - I (4 I x> I2[44x - (x I ~)w?d.
cl?”= / c#l**(rl)x(ro - T-1)czr1
the denominator in (A.5) has been written as 1 - I(4 [ x)1’. The integration is to be taken on ~1[resp. r] when +[resp. x] goes after the symbol (. B. PROPERTIES (as. P,‘)
= as. P, ; (as. Pi)
= (as. Pd);
(as. P,)(as. Pd) = (as. Pd).(as. Pp) = 0 (as.P,+as.Pd)=P=P*=P2. C. EXPLICIT
EQUATIONS
In order to write shorter formulas we put & = 0 in the following formula L4.6)
(A.61
- (X / ‘+)(+#I VBd1+x))dR)
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D. USE OF THESE
OPERATORS
The validity of the approximations (A.l), the following integral
(A.2) can be discussedby studying
/ d~l~*o”l)x(r’MR) = / drl4*(rl)x(ro- rdv (,+)
(A.71
at large distances, i.e., when ro is big. Because of the asymptotic behavior of v we can write V
M v(r0) + r1v(ro) Fz (1 + k~?“~>V(1.0) w (1 + /&1.&(R)
The integral (A.7)
now becomes approximately r1 - f-1) dQ(1 + ikdrl)
Let r’ be the range of +*(rl).
If we write (A.7) as
v(R)
j
4*(rl>X(ro
-
r1)
dr1
the error committed is less than kdr’
s
9*(rl)x(ro
-
Tl)
(-h
.
Hence if kdrf << 1 we can use the approximate projector. The same argument gives the approximation of Pd , if i&r” << 1 where rn is the radius of the deuteron. II. MEANING
OF THE
EQUALITY
(4.10)
Suppose we have x(2
- 2b45’)
= at’
- IME)
(A.81
for some v and u nonidentically zero. With a trivial change of notations we write (A.8) as fWK&
- EM5’)
= x(2
- 20.
(A.91
We note x = 2.$’ - f and &/dx = 4’. The first number of (A.9) is a function of (2E - 2.5’); then we have the equality
$ w#Nl = --$
u?d
REARRANGEMENT
65
COLLISIONS
Explicitly
(A.10)
If we assume f and IJ nonconstant t,o (A.10) we have -g
5-J
and if we apply the condition
= $
[$I
= consl’ant
of integrability
= K
We obtain now
(All) x =
C4 exp - F [2,t’ -
2,t]”
In the above discussion we assumed that f and u were not constant. Let us now examine these cases. ((Y) f and u are constant: this implies x and 4 be constant,, i.e., llorlrlorlllalixable. It. is not a physical case (fl) f only is a constant; 11ow f’ = 0, and we obtain $J =
Cs exl) - C(2,t’ - ()
x = Cg exp (2) v only is a constant;
I~OW
(A.12)
(C/2) (a{’ - 20
v’ = 0, and we obtain
4 = CT exp - C(2t’
- E)
(A.13)
x = C, exp - C(2<’ - 20 From (A.ll), (A.12), particular situations.
and (A.13)
we see that (A.lO)
can happen only in very
ACKNOWLEDGMENTS
I would like to take the opportunity to thank Dr. C. Marty discussions. I am also very grateful to Professor Herman Feshbach ing of the manuscript, while I was a guest at M.I.T. The author
for many helpful critical for his advice, and readalso wishes to thank Dr.
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coz
Mittleman valid. RECEIVED:
for telling
APRIL
him
a condition
21,
1965
was necessary
to make
the construction
REFERENCES 1. 2. 3. 4.
H. FESHBACH, H. FESHBACH, M. MITTLEMAN, M. Coz, Paris
Ann. Whys. (N.Y.) 6, 357 (1958). Ann. Phys. (N.Y.) 19, 287 (1962). Ann. Phys. (N.Y.) 28, 430 (1964). Congr. Phys. Nucl., July, 1964 (Ed. C.N.K.S,
Paris).
of Section
II
OF
PHYSICS:
Proiection
36, 5346
(1965)
Operators
for
Rearrangement
RIARCEL Laboratory
for
Collisions*
cOZt
Nuclear Science and Physics Department, Massachusetts of Technology, Cambridge, Massachusetts
Inntitute
As in Mittleman’s recent paper we derive a system of equations which describes the scattering of a system into a rearranged channel. Instead of defining a projection operator into the open channel subspace, we define elementary projectors into each open channel; this definition leads to more sirnplified equations than those obtained by Mittleman. Discussion of the existence of these elementary projectors is given. I. INTRODUCTION
If we describe rearrangement collisions, we can write the total wave function as a linear combination of the different channels with unknown scattering functions in each. These unknown functions are called amplitudes ( 1). In order to obt’ain the coupled equations for these amplitudes it, is useful to construct appropriate projection operators (briefly projectors). A first step will be to define a projector P on the open channel subspace and its complement Q = 1 - P on the closed channel subspace. This has been essentially the work of Feshbach (2) and Mittleman (3). It can be seen that in the many channel case even with antisymmetrization these projectors P and Q exist and are Hermitian projectors, but it is not our purpose to show this. As Xttleman tries to do, we want to obtain coupled equations for the coupled amplitudes, but Mittleman’s t’echnique leads to equations for mixtures of these amplitudes. To avoid this difficulty, we define elementary projectors on each open channel and define P as the sum of these elementary projectors P, + Pd . In this paper we are rest,rict’ed to the nonantisymmetrized cases since arlt,isylllilletrization leads to technical difficulties we shall study in a second paper. We show here, that in physical cases such elementary projectors exist; they are not hermitian but Pi’ = Pi, P;Pi = 0. Their sum is a hermitian projector. With these elementary projectors we obtain coupled equations for t,he amplitudes; the coupling between these equations is only a potential coupling. Since * This work is supported in part through funds mission under Contract At(30-1) 2098. t On leave of absence from Facultb des Sciences 53
provided de Paris,
by the Group
Atomic 1, Orsay,
Energy France.
Com-
54
coz
we have equations for amplitudes for the uniqueness of solutions. In order to avoid explanatory (p, d) reaction with only the two of projectors (the second is only to which they lead. In this theory
it is easy to prescribe
the boundary
conditions
difficulties we restrict the construction to a (p and d) channels open. We give three kinds an intermediate one) and analyze the problems we consider functions
E(r2,7-3,
...
,ra
;ro,1'1
I7",
R)
(1.1)
with 1’0 - 7-l = r; which
can be written
in the form
E = +(Q,
r2,
. .. ,
rA)u(rO)
r. + rl = 2R
(1.2) +
#(r2,
r3
, . . . , rA)x(r)4R)
(1.2)
where t’he two choices (r. , rl), (r, R) in (1.1) express the possibility of rearrangcmerit. In (1.2), +(rl , r2 , . . . , rA) is the target wave function, fi(rz, . . . , ra) the residual nucleus wave function and x(r) the deuteron function. The methods used are those of refs. g-4. We begin in Sections II, III, and IV with some orthogonality assumptions, the existence of the operators will be proved under these assumptions. In the conclusion, we shall discuss the necessity of these restrictions. II.
PROJECTORS
OF
THE
FIRST
KIND
In Section II we consider two wave functions $(rl , r2 , . . . , ra), $( r2 , . . . , rA) which represents the ground state of the target nucleus A and the residual state of the nucleus (A - 1) after the p, d reaction has taken place. We assume that these two functions are not proportional. We assume also that if q is XP,,pen + Remainder, this remainder is orthogonal to the state +(rZ, . . . , rd) of the residual nucleus as well as to the core +(r2 , r3 , . . .rA) of the target nucleus. Two methods are proposed to build up the projections operators P, and Pd , the first is based on the Gram Schmidt orthogonalization process, the second breats the problem by solving integral equations. The two methods are identical. Integrations involved are over (A - 1) variables and the condition for existence of P, and Pd is exactly the nonproportionality between the two functions 4 and 9, i.e., that in (A, 2) +(rl,
where
g*(rl)
A. DEFINITION
7'2,
is some factor
,ra)
* *.
# #(r2,
... ,rA)g*(rl)
of proportionality.
OF PROJECTORS
The two projectors P, , Pd have to be such that they extract tion t the proton part and the deuteron part.
from every func-
REARRANGEMENT P,E P&t
= =
dn,r2, ti(r2,
A trivial j,*
.*.
of fp*(rl
clrz ) * . - ) dr&*(r1,?~Z,
calculation
,r.A)U(d
* * * ) 1./4)x(r)v(R)
TO obtain P, we need a function 4* and $* such that s
COLLISIONS
, . . . , TV), a linear combination
*.*,1..4)$(1’2,
.A.,ra)
of
= 0
gives
= f#l*(rl, . . ...).*. -
s
dr’z . . &‘A $5*(7-l) * * * ) 7.‘a)l//(r’2, . .
r’A)lCI*(r2,
. . . , r,( )
With the notations * /d,‘l) =
J
Cll~Z
. . clra 4*4;
(( 1.1) = /’ dr2
. dra 4*fi
we writ.e 1
P, = 4( )‘I ) 1‘2) . . . ) ‘?.A) Ph>
-
f*o.l)j*(I'l)
s
dY.2* * . &J4*
- f(h)+*1
(2.1)
In the same manner we obtain Pd = $(1’2, ‘.. ) r,)
1 P(l’l) - f’“h)j’(rJ
s
(jr2 .f. ckAp$b* - f*4*1
(2.2)
(2.1) and (2.2) we see that when p(,rl) - f*(rl)f(~J is different from zero (t,hen 4 and # are not proportional), the operators P, and Pd exist.
From
B.
PROJECTOR
PROPERTIES
It is easy to c,heck that P, and Pd are yroject,ors, Pd2 = Pd)
P,” = P,,
that is
PdP, = P,Pd = 0
(2.3)
Now we can define P = Pd + P, . Our claim is that P is an hermit’ian projector. Since P” = P, it remains to show that P = P*. Alt’hough it is possible to demonstrate hermitiality directly from Eqs. (2.1) and (2.2) it will be useful t’o show it by another derivat’ion of P because it introduces the integral equation method mentioned above. The existence of an hermitian project,or depends on a solution of the equation
56
coz
Sufficient
conditions s
for this equation
are
4*(r1 , rz , ... , rA) drl drz ... dra(\E (2.4)
s
dr2 ) --* ,rA)x*(r)
dr dr2 ... dra(\E
or also
J
4*(Q)
x*(r)
r2, . . . , rA) dr2 drs . . . drA(\k
- ,$) = 0 (2.5)
--a ,rA) dr2dr3 ...
J #*(r2,
drA(* - .$> = 0
With the notations
U(QI, rd = J 4%
dr2 . . . drA ;
V(r, R) = / $~*\kdr2 . . . drA
we have U(o,
4
= drdu(rd
V(r, R) = x(rMR)
+ f(rl)x(r)u(R) + f*(Odr~)
From these equations we obtain x(r)v(R), u(ro) if p(rI) - f*(rI)f(rI) is not zero, for some rl . Let operators P,’ and Pi be defined by P,‘\k = C$U,Pd)* = Qxv. When P,’ and Pi exist, i.e., when p(rl) - f*(rl)f(rl) # 0 we have P,’ = P, , Pd’ = Pd and their sum is an hermitian projector from Eq. (2.4). The operator P annihilates the remainder defined by Q - pk from the orthogonality assumpt,ions. C. EXPLICIT
EQUATIONS
FOR THE AMPLITUDES
u AND v
(a) Let I? = H + H&G&H with Q = 1 - P, G = Q(E - &II&)-‘& The effective equation for the open channels is P(E - B)p\k
= 0
(2.6)
where pk = 4u + #xv,
(2.6)’
from the orthogonality assumptions. We can write the hamiltonian H of a A (p, d)B reaction with (Y = A + p, /3 = B + d as follows H=T,+H,+Va,=
Td+Hg+V,vci.
(2.7)
REARRANGEMENT
From
(2.6) and the equality HP,%’
QHPp*
= (T, + e&u =
QV,,+u;
.i7
COLLISIOl’iS
P& = 0 we have + V&u; QHP&
HP& =
=
( Td
+
Q)#XV
+
~‘Bd#Xv
Qvi&x?.
(b) We use now the explicit expressions for P, and Pd , and mult’iply (2.6) give the coupled successively from the left by P, and Pd . These multiplications equations.
[(#J
= + (4 I 4) -II (4 I +I I2
( +)($‘I
Ved
14)
-
(9
1 +I(4
1 VBd
1 +)h
In (2.8) round brackets express integrations on ~2 , . . . , rA . The integro-differential equations for the amplitudes are obtained after multiplication by +[resp. +x] and integration on (ri , Q , . . . , r,) [resp. (r, r’2 , . . . , yA)J on the first [resp. second] equation of (2.8). III.
PROJECTORS
OF
THE
SECOND
KIND
In Section II we have assumed that the two functions 4 and # were not proportional to each other. Also we assumed a strong orthogonality condition. With these conditions we have been able to build two projection operat.ors P, and Pd using only (A - 1) variables. The two methods (orthogonalizat,ion procedure, and integral equation method), in order to obtain P, and Pd were equally easy to handle. In Sections III and IV we want to build more general projection II operators which can apply even in the case where 4 and \L are proportional t,o each other. Again we shall use the two methods recalled above. Now we must use int’egration over A variables. But only the last method (the integral equation method) can be used without difficulty. At the present stage of this work, we assume again an orthogonality condition which is, however, less strong than that of Section II. We assume that the remainder is orthogonal to the target nucleus and to the system residual nucleus-deuteron. Later we discuss the necessity of this assumption.
58
co2
In order to contruct the P, project in the orthogonalization method we define a function f,* = cw4*(?2!$ Y
) r2 , * ‘. ) rA) + p+*cr2 , -. - , LAX*(r)
such that
We obtain
Now the construction of P, needs the knowledge of the operator fi2, D,=
sdr,[p~~)-{SS(~)x(r)dr}f*~~)x(r)],
since we must write P, = +n,’
I
dr drp - . . dra f,*
The singularity appearing when the denominator of (2.1) is zero is replaced by the possibility &, to be a singular operator. If this singularity can happen, the operator P, cannot be defined. A similar method is necessary to obtain Pd . Let us assumenow that these P, and Pd exist and that we know their explicit forms. The integro-differential equations we are asking for, if we put Q = 0, are given by (E-E,-
T,)u(r,)
-
St;;’ 1 f,*V,,+u
dr dr2 . . . dr.., (3.1)
0;’
(E -
ED -
Td)v(r)
-
/ .fp*VB&
C&i’ / fd*VedJ/xv
= ($x(x+
(f&i’
dr dr2 . . . drA)
v(R)\
/
drl dr, . . . dr,
/sd*V~,Odrldrz
...
drA)u(n)>.
(3*2)
The inversions involved in fiil and flZ1 are difficult, so that these equations are not useful; however, in the alternative method (integral equation method), inversions of operators which replace 0, and fid are easier. At the same time singularities corresponding to the denominators of (2.1) and (2.2) can be understood.
REARRANGEMENT IV.
rZ. DEFINITION
PROJECTORS
OF
59
COLLISIONS THE
THIRD
KIND
OF PROJECTORS
Clearly the method given can be used for a general (p, d) reaction, but we consider only t,he case excluded in Section II. Let ,t(h, .a. , rd ; r0,rl 1r, R) = #(r2, . .. ,?*.~)(+(r~)u(h) + x(r)u(R)j
(4.1)
We use the notations U(r,)
=
/
#*(r2
) . *. ) r,)c$*(r,>~
V(R)
= f a+Q*(rg , . . . , r&*(r)l
dr1 drz
f..
dr,
dr d1.2. . . dr,
Using the orthogonalit,y assumption between the remainder and the target nucleus wave function and between the remainder and the wave function of the system residual nucleus-deuteron, as in (2.5) we obtain integral equations for u and v
U(ro> = u(d + / d~*(rdx(r)v(R) drl (4.2) V(R) = v(R) + / x*(r)~(rduhd
dr
Let us write (4.2) in a symbolic form
U(Q)
= U(Q) + KdR);
V(R)
= r(R) + K,u(d
(4.3)
The solution of (4.3) gives u(r0)
v(R) = l _ ;
=
We define now P,E = u( ro)$+; P& = v(R) = U(R)&; Pd exist they have the following form.
p, = 94
1 1 -
K,K,
1 1 -
K,K,
if the operators P, and
I)*+* drl dr2 . . . drA (4.4)
- / drl +*(n)x(r)
Pd = #x
K D’ - Kv VI. 21 u
/ do’ drz . . . drA#*x*(ro,
. . . ,rij]
J/*x* dr dr2 . . . drn - j dr x*(r)+*(rl)
/ dr’ dr2 . . . drpj?-+*
( >I R - ;
(4.5)
60
coz
B. PROPERTIES
OF THESE
OPERATORS
The formulas (4.4), (4.5) relate the existence of P, and Pd to the impossibility of eigenfunctions of eigenvalue 1 for both integral operators K,K , K,K, . If P, and Pd exist, they are projectors: P,’ = P, , Pd” = Pd ; P,Pd = PdP, = 0. Now we can define P = P, + Pd. The operator P is the projector II of Mittleman, which always exists (3) (even when separately P, and Pd have no existence) at last with our orthogonality assumption, and is hermitian. C. EXISTENCE
OF THE
ELEMENTARY
PROJECTORS
We come back to the different kernels involved and rewrite (4.2) in a more precise form
With trivial obtain
U(O)
=
U(‘~CI)
+
/” 2, (-‘\)
V(R)
=
v(R)
+
j
24 (R
~(7%
+
;)
4 (Ii
-
rl)$*(rl)
-
;)
drl
x*(r)
dr
changes of parameters and noting that 1’ and rl are vectors, we
U(f)
= u(E) + 8 / v(,t’)+*(2E - t’>x(2$
- 28 dE
v(t)
= v(t)
- 2{‘) dE
+ 8 / &‘>$(2[’
- t)x*(2t
We note t’hat the two kernels involved in these equations are adjoint one of another. From the homogeneous system udt)
= -8X /- dt’4*(25 - t’)x&
- 25’h(t’) (4.6)
v,(t)
= -8X
s
d&#OE - &*(2f’
- 2E)&‘)
we obtain m(t)
= 64X2N&, rh(1’);
VA(~) = 64X2Nv(F, rh(r)
with N,(E, r> = /- d&W
- r)9*(2E - &x(25
- 2C;‘)x*(2r - 2.5)
N,(E, r> = /- &‘d2 r - F/)4*(25 - .59x*(2{’ - 27-)x(2.$ - 2E’)
(4.7)
REARRANGEMENT
61
COLLISIONS
N, and N, are self-adjoint kernels, therefore X2 is real and positive. The critical eigenfunctions whose presence prevent’s the existence of P, and Pd are t’hose with X = 1 in (4.6). We are now asking what this means. Let us consider (4.6) with X = 1. Instead of (4.6) we can write
(4.8) o=
s
&*(2$
- 2{){x(2st' - 2.91(~)
+ qq2t: - $'ju&')
We multiply the first equation of (4.8) by u*(,t’), the second by v*(t’) ; in the second equation we permute t- and ,t’. Integrating now on [ and l’ we add these two equations; thus we obt,ain
or +(~‘l)u(cJ
+ x(r)v(R)
= 0
(4.10)
(4.9) or (4.10) are conditions of the same kind as (3.28) in (2). In other words when the critical eigenfunctions occur (4.10) is valid for some functions u( rO) and v(R) which are not identically zero. Conversely (4.10) implies the existence of these critical eigenfunctions. This means a strong functional dependence (Appendix II) between the functions 4 and X. This means also that the amplitudes u(,Q), v(R) cannot be uniquely determined: they have no physical meaning. In other words, the channels defined by 4 and x cannot be distinguished; they are not physical channels. The situation is very different from that encountered by Feshbach (2) in the antisymmetrized case. The direct and the exchange channels are mathemat’ical but not physical entities, therefore there is no contradiction when one asserts that ~(1,~) cannot be uniquely determined in
@ is the antisymmetrization operator. In conclusion (4.10) implies either the channels are not physical, or we have made an improper choice of the functions 4 and X; therefore we rule out the possibility that Eq. (4.10) is satisfied. We can therefore define P, and Pd and use them in the same manner as in Section II or Section III, t’o obtain the coupled equations of the problem.
62
co2
D. INVERSION OF THE Both operators K,K, tive eigenfunctions. We with eigenvalue C;equal
INTEGRAL OPERATORS INCLUDED IN THE PROJECTORS , K,K, are hermitian. Let ux and vx denote their respecadd to the two sets (UX) and (vh) the functions (ut), (zQ), to infinity. In this we obtain complete orthogonal sets.
l-K,,K,=l-&$u)(uk=~f$ (1 - KuKJ'
UA>( UA
= &ud{
ui
and similarly (1 -
K,K,)-'
= x&
= VA)(UA.
V. CONCLUSION
The existence of an operator P which annihilates \E - pk has been proved using orthogonality assumptions. We can ask if these assumptions were necessary. To give an answer we can look how P has been constructed. Let ((rz , . . . , ?-A); ro , r2 1r, R) be any function having good asymptotic behavior. How does P operate on .$? If for any such 4, PE is the sum of two terms, the first proportional to $(rl , r2 , . . . , ra), the proportionality factor depending only on r. , the second proportional to cp(rz , . . . , rA)x(r) the proportionality factor depending on R, we can choosef = P, and define a remainde asr \k - pk. This remainder will be automatically orthogonal both to #(rl, r2 , . . . , ?“A) and to dr2, - +. , rA)x(r). If it is not so for every E, although it can happen for some special .$,we need the existence of an orthogonality assumption between closedand open channels. Coming back to the operators P defined in Sections II, III, and IV, it is straightforward to verify that in Section II the orthogonality assumption is needed while in Sections III and IV it is useless.Therefore we have an operator P which can be constructed which annihilates XP- p\k. This is the general result announced in the introduction. APPENDICES
We are first concerned with a special case in which calculations discussedin Sections III and IV are simple, and afterward with the functional dependence between 4 and x which results when Eq. (4.10) is satisfied. I. APPROXIMATE
PROJECTORS
We assume that the following approximations discussed in Section I, D are valid.
/4*(hdr)u(R)
1 x(rMrdu(r0>
drl # u(R) / 4*(n)x(r) dr # u(r0)
1 x*(r>drd
ch dr
(A.1) (A.2)
REARRANGEMEKT
COLLISIONS
63
In this case we can define approximate projectors. A. COXSTRUCTION
OF THESE
OPERATORS
The approximate equations for (4.2) are written r:hl)
= ucro> + u(R) (4 / x)
(A.3)
I’(R)
= v(R) + 41.0) (XI $>
(A.4)
which gives t,he ‘Lproje~tor~” asymptotic in some sense
1 _ / (; , x> (1[of4 - (4 I XNXI
as. P, = ti(1.2 , . . . , r,,)+(r) as. Pd = +(Q,
-se , T.~)x(T)
Since / +“(R - r/2)x(r)
(A.5)
1
1 - I (4 I x> I2[44x - (x I ~)w?d.
cl?”= / c#l**(rl)x(ro - T-1)czr1
the denominator in (A.5) has been written as 1 - I(4 [ x)1’. The integration is to be taken on ~1[resp. r] when +[resp. x] goes after the symbol (. B. PROPERTIES (as. P,‘)
= as. P, ; (as. Pi)
= (as. Pd);
(as. P,)(as. Pd) = (as. Pd).(as. Pp) = 0 (as.P,+as.Pd)=P=P*=P2. C. EXPLICIT
EQUATIONS
In order to write shorter formulas we put & = 0 in the following formula L4.6)
(A.61
- (X / ‘+)(+#I VBd1+x))dR)
64
coz
D. USE OF THESE
OPERATORS
The validity of the approximations (A.l), the following integral
(A.2) can be discussedby studying
/ d~l~*o”l)x(r’MR) = / drl4*(rl)x(ro- rdv (,+)
(A.71
at large distances, i.e., when ro is big. Because of the asymptotic behavior of v we can write V
M v(r0) + r1v(ro) Fz (1 + k~?“~>V(1.0) w (1 + /&1.&(R)
The integral (A.7)
now becomes approximately r1 - f-1) dQ(1 + ikdrl)
Let r’ be the range of +*(rl).
If we write (A.7) as
v(R)
j
4*(rl>X(ro
-
r1)
dr1
the error committed is less than kdr’
s
9*(rl)x(ro
-
Tl)
(-h
.
Hence if kdrf << 1 we can use the approximate projector. The same argument gives the approximation of Pd , if i&r” << 1 where rn is the radius of the deuteron. II. MEANING
OF THE
EQUALITY
(4.10)
Suppose we have x(2
- 2b45’)
= at’
- IME)
(A.81
for some v and u nonidentically zero. With a trivial change of notations we write (A.8) as fWK&
- EM5’)
= x(2
- 20.
(A.91
We note x = 2.$’ - f and &/dx = 4’. The first number of (A.9) is a function of (2E - 2.5’); then we have the equality
$ w#Nl = --$
u?d
REARRANGEMENT
65
COLLISIONS
Explicitly
(A.10)
If we assume f and IJ nonconstant t,o (A.10) we have -g
5-J
and if we apply the condition
= $
[$I
= consl’ant
of integrability
= K
We obtain now
(All) x =
C4 exp - F [2,t’ -
2,t]”
In the above discussion we assumed that f and u were not constant. Let us now examine these cases. ((Y) f and u are constant: this implies x and 4 be constant,, i.e., llorlrlorlllalixable. It. is not a physical case (fl) f only is a constant; 11ow f’ = 0, and we obtain $J =
Cs exl) - C(2,t’ - ()
x = Cg exp (2) v only is a constant;
I~OW
(A.12)
(C/2) (a{’ - 20
v’ = 0, and we obtain
4 = CT exp - C(2t’
- E)
(A.13)
x = C, exp - C(2<’ - 20 From (A.ll), (A.12), particular situations.
and (A.13)
we see that (A.lO)
can happen only in very
ACKNOWLEDGMENTS
I would like to take the opportunity to thank Dr. C. Marty discussions. I am also very grateful to Professor Herman Feshbach ing of the manuscript, while I was a guest at M.I.T. The author
for many helpful critical for his advice, and readalso wishes to thank Dr.
66
coz
Mittleman valid. RECEIVED:
for telling
APRIL
him
a condition
21,
1965
was necessary
to make
the construction
REFERENCES 1. 2. 3. 4.
H. FESHBACH, H. FESHBACH, M. MITTLEMAN, M. Coz, Paris
Ann. Whys. (N.Y.) 6, 357 (1958). Ann. Phys. (N.Y.) 19, 287 (1962). Ann. Phys. (N.Y.) 28, 430 (1964). Congr. Phys. Nucl., July, 1964 (Ed. C.N.K.S,
Paris).
of Section
II