Time-independent nonrelativistic collision theory

ANNALS

OF

PHYSICS:

6,

58-93

Time-Independent

(19%)

Nonrelativistic 14:. GERJU~Y

University

of Pittsburgh,

Collision

Theory*

t

Pittsburgh,

Pennsylvania

Arbitrarily complicated rearrangement collisions describable by the timcindependent nonrelativistic Schrodinger equation are treated in a wholly timeindependent framework, without explicit reference to transition probabilities. Introducing a generalized probability current operator for many-particle systems, the current flow at large distances can be computed in complete analogy with one-particle scattering. This procedure leads to the same formal expressions for reaction rates as are obtained from more conventional formulations; it is an immediate consequence of t,he asymptotic behavior of the Gteen’s (-)*V&; = $,*V,*<(+) yield the scattering function that the matrix elementsqf amplitude A (i +f), and that*, (-) is a solution with incoming scattered waves. Moreover the time-independent formulation avoids perplexities associated with t,he necessity for computing transition probabilities from projections on sets of it turns nonorthogonal states. In the reaction a + b + c + d + e for instance out automatically, without projecting t,he solutions on plane wave states, that coincidences are observed in counters placed at large distances rc , rd , I’, only when re , rd , r, are in the rat,ios of the classical velocities of c, d, e; here c, d, e are a rearrangement of the particles contained in a, b. These procedures can be applied either in the laboratory or center-of-mass systems; eliminating the center-of-mass coordinates does not simplify the calculations for threebody breakup ab + cde. Reciprocity relations for two and three-body breakup are examined, and mathematical difficulties of this time-independent formulation are discussed. 1. INTRODUCTIOX

.4ND

YUMiLIARY

In recent years the formal theory of scattering has been the subject of numerous papers, wherein a variety of arguments have been employed to obtain the transition probabilities in a form presumably valid for a very wide class of processes, including particle creation and annihilation. The results and methods are well illustrated by the contributions of Lippmami and Schwinger (Ij, C;ellMann and Goldberger (2) and Belinfante and Moller (3). Related questions have been discussed by other authors (4)’ and further references to t,he voluminous literature c,an be found in 6he papers cited (r-4). * This work was supported in part by the Office of Naval Research. t Present address: John Jay Hopkins Laboratory for Pure and Applied Science, General Atomic Division of General Dynamics Corporation, San Diego, California. r These last four papers do not employ mathematically improper operator techniques, but pertain mainly to scattering without rearrangement. 5s

TIME-INDEPENDENT

COLLISION

THEORY

59

Cross sections for scattering of a single particle by a potential are more usually (and more directly) computed not from the probabilities for transition to new momentum states (5) but from the asymptotic form of the wave function at infinity (5). This observation has motivated our approach to many particle rearrangement collisions describable by solutions of the time-independent nonrelativistic (first-quantized) Schrijdinger equation. As we show in the following sections such collisions, however complicated, can be treated in a wholly timeindependent framework without explicit reference to transition probabilities. In this fashion we avoid the nonorthogonal state perplexities (6) which have been much discussed in the literature (7’,8), and bypass many of the subtle formal questions associated with the limits (as t -+ CQfor example) appearing in the time-dependent formulations (I-4). Use of a generalized probability current operator for many-particle systems is an important feature of our method;’ therewith the probability current flow at large distances can be computed in complete analogy with one-particle scattering. Subject to limitations explained below, our principal results are as follows: The scattered wave cp satisfies a probability conservation law involving J dS$(v*, VP) [see Eqs. (1.2) and (2.1)] integrated over the surface at infinity in the 3n-dimensional space of all the particles, where each surface element dS corresponds to a definite set of aggregates.3 Provided cpis ‘
(l.la)

with the arrow pointing right, coincidences are observed in counters placed at large distances re , rd , re only when rc , rd , re are in the ratio of the classical velocities of the aggregates; in Eq. (l.la) c, d, e are a rearrangement of the particles contained in the aggregates a, b. Moreover our procedures lead to the same formal expressions for reaction rates as are obtained from the more conventional formulations (1, 2) via transition probabilities. In particular for reactions a+bSc+d

(l.lb)

it is an immediate consequence of the asymptotic behavior of the Green’s func= $f*V,‘ki’+’ yield the scattering amplition that the matrix elements +f ‘-‘*V,$i is a solution to the Schrijdinger equation with intude A(i 4 f), and that *f’-’ coming scattered waves; these consequences pertain also to Eq. (l.la), although 2 Procedures resembling ours in some respects have been used by Morse and Feshbaeh (S)particle having no internal 3 By “particle” we mean always a spinless “fundamental” denotes either a single particle or a collection of degrees of freedom. The term “aggregate” particles; in the latter event the particles are presumed bound in a specified eigenstate of negative energy (in the rest system of the aggregate). The angular momentum of any such eigenstate is not restricted and may be different from zero.

60

GERJUOT

there are added complications when either the initial or final channel consists of more than two aggregates. The mathematical apparatus we employ in Chapter II is summarized in Section 1.1; to Chapter III which serves as an Appendix are relegated many of the mathematical details. The boundary condition specifying t’he solution has been examined previously (10) .4 By restricting this paper to spinless particles we have avoided analytic complications which arise when the (Hermitian) Hamiltonian has an imaginary part. Our results can be extended to particles possessing spin; details are left to a future communication. The parkles also are assumed distinguishable, which is not a serious limitation however; with indistinguishable particles, though the cross sections are given by squares of linear combinations of ordinary and exchange amplitudes, the amplitudes continue t.o be comput.ed as if the particles were distinguishable (6). Our methods have not yet been cxtended to time-dependent problems, nor to collisions involving particle creat,ion or annihilation. 1.1

hlATHEMATIC4L

PRELIMINARIES

Some aspects of the notation are described more fully in I (10). In the laboratory system the coordinates of the ‘n particles are rl , . . . , rn ; the entire set will be symbolized by r = rn which can be regarded as a vector in a 3n-dimensional space. The center of mass of all n particles is at R; ra is the location in t,he laboratory system of the center of mass of a; s, is the set of internal coordinates of a; r is any suitable set of coordinates in the center of mass system such t,hat the Jacobian of the transformation from r to R, r is unity. If *I and q2 both satisfy the same Schrodinger equation (H - E)* = (T + V - E)\k = 0, then ‘k,T\k, - q’lT\kz = 0, so that integration by parts (Green’s Theorem) in t,hc manydimensional r space yields (12a) where the integral is over the surface at infinity in r space; v is the outward drawn normal to the surface element dS; the components W,(j = 1, . . . , 71) of the 3~ dimensional vector W are defined by Wj = &

3

(\EzVj*l - 91Vj*J

(12h)

and pi = -ihVj is the momentum canonical to rj = rjnj . In direction n and therefore each surface element dS at infinity possible formation of a definite class of aggregates. For example ments dS, corresponding to possible formation of t aggregates a, 4 We make

frequent

reference

to this

source,

denoted

in the text

by I.

Eq. (1.2a) each corresponds to the surface eleb, . . . lie along

TIME-INDEPENDENT

COLLISION

61

THEORY

directions Il, for which as r --, m the distances ] raj - r,k ] . . . , 1 rbj - rbk ] , . . . between pairs of particles belonging to the same aggregate remain finite, but the separations Tab of each pair of aggregates became infinite. We assume explicitly (10) that whenever the reaction forming these t aggregates is energetically possible, the interaction vob between each pair a, b of these aggregates decreases more rapidly than r&-l as Tab -+ 03. Directions n are specified by the 2n angles in ordinary three-dimensional space determining the directions nl , . . . n, , together with the n - 1 ratios qj = rj/rl , j = 2, . . . , n; remains constant as r -+ 00 along given fixed n and it is presumed r1 --) ~0 with r (at least one of the ri does). The volume element in r space is Tj/Tl

dr = r?drldnlrtdrzdnz 2

dn = (1 +

422

J2

* * . r,,‘dr,,dn, 2 . . .

3

.“:

+

= drdS = ?‘-I

(1.3a)

drdn,

2 +

qn2)3n/2

dq2dq3dq4 * . . dq,dnldn2 . . . dn,.

(1.3b)

Unbarred symbols refer to the laboratory system. The corresponding quantities in the center-of-mass system usually are represented by barred symbols; where these are not available we use German letters or a degree sign superscript. Superscripts (-I-) and (-) distinguish, respectively, between solutions with outgoing and incoming scattered waves; subscripts i and f designate, respectively, initial and final reaction channels. Often we are not concerned with the direction of the arrow in Eqs. (1 .l) ; the prime is used where necessary to signify a reaction initiating with aggregates c, d or c, d, e and proceeding to the left. The incident wave $i satisfies (Hi - E)$i = 0; H = Hi + Vi = Hf + Vf . Eigenstates ua(s,) of a in its own center of mass system satisfy (H, - E&A, = (T, + Va - E&A, = 0. Usually we abbreviate the notation, denoting the combined set s, , sb by s& ; also the products u&b = u&5 ; however r,b has the customary meaning rb - r, . Integration over the internal variables so6 or scdeonly is indicated by brackets, e.g., [‘&b*J’i] is the projection of the incident wave on the eigenfunction U,,?.&, integrated over all s, , sb for fixed r,, , rb . In the reactions (1.1) with the arrow pointing to the right H - Hi = Vi = V,,b , $i

=

‘f&b

eXp

[i(

k-l-0

+

$i

=

‘&b

eXp

[i(%kb

kb. rb)] = $e exp

- Mbk,)

‘Tab/M]

(1.4a)

(iK. R) =

‘&b

eXp

[ik,b*r,b]

(1.4b)

and K = k, + kb ; specifically, if particles 1, 2 form a, particles 3, 4 form b, V, = VIZ, Vab = VI3 + VTB + VU + V2, . With the arrow pointing left in Eq. (l.la) H - Hi’ = Vi’ = V, = VcTed + V,, + Vdc, #i’ = ucaeexp [i(k,. rC + kd.rd + k,.r,)]

= $[ exp [iK* R],

(1.4c)

and K = k, + kd + k, . For the plane waves of Eqs. (1.4) E = i!? + h2K2/2M

= E, + &, + E,,b = EC + Ed + E, + Ecdd (1.5a)

62

GERJUOP

with Eab = h2h32M,

+ h2k:/2Mb

Ecde = ii2k,2/2M,

+ h’k,‘/2&

(1.5b) + ii’k,‘/2M,

.

(1.k)

The solution to the SchrGdinger equation satisfying the outgoing boundary condition g(G”’ , cpi) = ih JmdX~(G’+‘, 07;) = 0 is (10) qiki’+’ = tii - G’“‘J~i,)L SE,bi + c,oi

(1.6)

where the Green’s function G(A) satisfies (H - A)G = I E 6(r - r’) for real or complex A, and the outgoing Green’s function G’+‘(E) is the limit, of G(X) as e -+ 0 with X = E + ie, E > 0. Generally we drop the superscript (+) on the Green’s function; G(E) = G”‘(E). The Green’s functions satisfy (10) G=G,-

GfVfG = Gf - GI)G,

G = GFcn)- ,‘&‘“‘vG

>

= GF(lz) _ GVG F(d ,

(1.7:1) (1.7hj

wherein GFtn) is the free space Green’s function for the system of n particles and (Hf - X)G, = I. Moreover Gadra

) rb

, Sab

; r,‘,

rbl,

sab’

; X)

= C GF(‘)(re , Gcde(rcde

, &de

; red:

, Sed:

rb

;

ra’,

rb’;

X’)?&,b(sab)uab*(sab’j

(l&t)

; A> =

c

GFc3)(fede

; rcde’

; X’)ll,de(Scde)‘Ucde*(s~d~).

(1%)

In Eq. (l.Sa) the asterisk denotes the complex conjugate; A’ = X - Ba - limb approaches Eabas X ---f E; the sum is over E:, , Eb and all other quantum numbers Pa , Pb necessary to specify the states 2La, &, ; integration Over 8, , i
GF(d(r,r’;$) = r$>“” . . . (+y”;

(g>” H.I’)/fl’;

where a(n) = (3n - 2)/2; 0 < arg X < 2a so that 0 < arg dx

; ’ I) (1.9) < a; H,(l) =

TIME-INDEPENDENT

COLLISION

63

THEORY

J, + iN, is the cylindrical function of the third kind (13) and the 3n-dimensional vector e E el , . . . , e,, is defined by ej = (2Mj/h*)“*rj . AS T -+ ~0 along fixed n, holding r’ constant, the asymptotic behavior of Gpcn) is given by (13) (l.lOa) eexp @j&K)

exp [-i&m-r:

+ ...

+ k,n,.r,‘)],

hj z2+fi”.

(l.lOb) P

We see the outgoing for E > 0

GF’“‘(E)

VjGp(%) (E) -

is exponentially

idE

‘2

decreasing at infinity

for E < 0;

!I!’ Gpcn) (E) = ikjnjGp@) (El. P

II. SCATTERING

AMPLITUDES

AND

CROSS SECTIONS

2.1 THE PROBABILITY CURRENT In this section we postulate the current operator representing, e.g., the scattered current of aggregates a, b in states u, , ub . This postulate, which relates the physical situation to the purely mathematically specified solution !~i(+’ of Eq. (1.6) cannot be “derived” but we shall attempt to make it plausible (Sections 3.1-3.2). As in I we concentrate initially on the laboratory system. For reactions (l.la) the current operator is algebraically simpler in the laboratory system than in the center-of-mass system; moreover the relationship between the laboratory and center-of-mass systems illuminates the reason that the centerof-mass matrix elements for reactions (l.la) sometimes are formally (though inconsequentially) divergent. We stress that in this and subsequent sections of this chapter the energy always is real, G = G(E). With spinless particles H has no imaginary part, H = H*. Consequently if \E is a solution of the Schrijdinger equation, so is q*. Substituting in Eq. (1.2a) qil = **‘+‘,\Ez = q*(+)* and noting that Eq. (1.2a) alsoholds substituting !!!I = $i , !P2 = #i*, we have / dSg(pai*, pi) = -i

m

Im /” dSv~W[#i*,

m

(oil

with the scattered wave vi = -GVi+i according to Eq. (1.6). Because of the bound state factors ?.&,b in $;, the only surface elements dX contributing to the right side of Eq. (2.1) are those along directions n corresponding to formation of the initial aggregates a, b. This observation suggeststhat Eq. (2.1) generalizes the theorem relating the total cross section to the forward scattering amplitude

64

GERJUOY

(1, 2, 14, 15), usually proved for reactions less complicated than (1.1). We conclude that the left side of Eq. (2.1) yields the total scattered current summed over all reaction channels. Assuming cpi behaves asymptotically like the Creen’s function G, it is possible to evaluate the contribution 5 to the left side of Eq. (2.1) from surface elements corresponding to formation of any given set of final aggregates. Namely for surface elements dS,b corresponding, e.g., to formation of the initial pair a, b we transform from r = rl , . . . , rn to r, , rb, s& , and integrate over the int)ernal variables sa6 (Section 3.2). It is found that (2.2) where Vab is the outward drawn six dimensional unit vector perpendicular to the surface element d&b bounding the sphere at infinity in ra , r!, space; the six dimensional vector Jab = ( Ja , Jb) has first three components Jo and last three components Jb (along the same unit vectors respectively as the components of r (I> rb) given by

Ja = & Jb

=

(&l,*va&b

-

&bVa&b*)

,

a

&

(2.3a) @ab*Vb&b

-

&bvb%b*);

b

for a given pair of bound states ZL, , ‘& . &bh

, rb)

=

/

dSabUab*(Sabh&a,

rb

, sab)

=

[%b*cPil

(2.3b)

and in Eq. (2.2) we sum over all J ob, i.e., over all pairs of bound stat’es ‘&b . Similarly the contribution to the left side of Eq. (2.1) from directions corresponding to formation of three aggregates c, d, e is (2.4) where d&d, bounds the surface at infinity in rC , vector Jede = ( JC , Jd , Je) has components

Jc = &

(Zede*Ve&de

-

rd

, re space; the nine dimensional

(2.5a)

Zedev&de*),

c

etc; and &dek

, rd

, &>

=

s

&de’&ie*~i

=

[&de*pi].

(2.5b)

TIME-INDEPENDENT

2.2

REACTION

COLLISION

65

THEORY

RATES

We proceed to apply the results of the previous section. To focus our attention, consider the reaction (l.la) with aggregates a, b incident, for which reaction we use cpifrom Eq. (1.6) in Eqs. (2.4)-(2.5). We require the asymptotic behavior of [‘&de*%] = - [&de*{ Gvi#i] 1 on a large sphere of radius rt in r, , rd , re space, whose surface elements are determined by the ratios qd = rd/r, , qe = re/rc and the directions n, , nd , n, along which rc , rd , re approach infinity. From Eq. (1.7a) with G, = Gede, Eq. (lBb), and Eqs. (1.9)-(1.10) with 72 = 3, we find lim dS’%d,*(S)Gcd& rpoo s lim lpf00 s

dS%de*(s)G(r;

r’; E> =

%debhb*(r’>,

(2.6a)

r’;

llede(P)*f(-)*(r’)9

(2.6b)

E)

=

(2.6~) where KP,(-)*(~‘) #f*(r’)

= #f*(r’)

- / dr”~f*(r”)V~(r~)G(r”;

= t&de*(d)

exp [-i(k,.r,’

+

kdd’

r’; E),

(2.7a)

+ kd)],

(2.7b)

with A’ = Ecde from Eq. (1.5a) ; h2p”/2 = M,r,’ + Mar,’ •k M,re2; and k, = [2M, fi rcnC]/h2p, etc. Thus from Eqs. (1.3b), (2.5) and (2.6) noting that Y& = (r,/r, , rd/rt , re/Tt), the term in Eq. (2.4) corresponding to this particular set of bound states is

(2.8b) In Eqs. (2.6~) and (2.8a) the notation \Ef (-)*vitii

=

s

for

dr\k,(+*(r)Vi(r)lti(r)

(2.9)

implies integration over the entire range of the suppressed intermediate variable, consistent with Eq. (1.6) and the usual operator formalism; in Eqs. (2.7a) and (2.9) *‘f(-)* and #f* are regarded as row matrices whereas *i(+’ and #i are column

66

GERJUOY

matrices in Eqs. (1.6) and (2.9). Because G is symmetric Eq. (2.7a) can be put in t,he form *I

(-1 = I++, - G*T’,&

(IO) and V is real,

(2.10)

where G* can be proved (II) identical with the incoming Green’s function G’-‘(E’) defined as the lim of G(E - ic) as t + 0, E > 0. Consequently (IO) ‘k/’ ) is a solution to the Schrddinger equation (H - E)!P = 0 differing from fi, = #?’ i lc h satisfies the “incoming” of Eq. (1.4~) by a function ‘pr (-I E Q/‘-’ - ,J, \-h’ boundary condition g(G’-‘, cpr(-)) = 0. According to our int,erpretstion of 5 the flow of scattered probability curreut into the nine-dimensional solid angle element dn,d, is given by the inbegrand of Eq. (2.8a); since k, , kd , k, , and rt/p are functions of yd , ye but not of r, = (rc2 + rdd2 + T:)“~, this integral depends only on the variables pd, qe, n, , n,f , n, . In other words the integrand of Eq. (2.8a) must represent the anticipated count.ing rate when three counters capable of detecting aggregate c in st.atc U, etc. are located at large distances along directions n, , nd , n, and so connect,ed in coincidence that a count signifies a rearrangement (l.la) sending the aggregates into the respective solid angles d n, , dnd , d n, , with t,he rat,ios r,Jr, , r,,,jrc of t,he aggregat’e distances found to lie between qd and @ + dqd , ypand qe + tip, , ;is it happens the quantities k, , kd , k, of Eq. (2.7h) have been so d&net1 that /id = ~~&&./~~c , k, = ~~~,f&/;l’, , while (2.11) These are precisely the values li, , kd , k, would have if k, had been defined in terms of the classical speed v, of c by hk, = M,v, , etc., and if we interpreted qd = ud/& , qe = &/b ; with these values of /i, , X.d, k, Eq. (1.5~) is satisfied. Hence it follows automatically from our purely quantum mechanical formulation that Eq. (2.8a), involving as it does integration over the two ratios qd, qerather than over all three distances rc , rd , re, is in agreement with the classical expectnt,ion that coincidences occur when rc , rd , re are in the ratio of the velocities v, , IQ , z’, , which velocities are not arbitrary but are restricted by the requirement that the total kinetic energy of the aggregates has t,o equal Ecde. Moreover, without project,ing the solution on any special plane wave state, the wave function

prescribing through *f(-’ * the probability current int.odn,d, turns out to represent a set of non-interacting aggregates c, d, E in bound states ‘&demoving with their expected classical speeds along directions n, , nd , n, . Conversely it is natural and consistent to interpret observed coincidences at, large distances r,. , rd , re as measuring scattering of the aggregates into n, , nd , n, nith speeds?:,:u~:P, =

TIME-INDEPENDENT

COLLISION

67

THEORY

We remark that our application and physical interpretation of 5 and J do seemto differ from those made in the past (16, 17).5 Equation (2.8a) becomes more recognizable and convenient when the wave numbers replace qd , qeas variables of integration. There results r,:rd:r,.

where Eq. (1.5~) determines k, as a function of the variables kd , k, and the known [from Eq. (1.5a)] energy EA ; the integrand of Eq. (2.12) must represent the rate at which particles c,.d, e are simultaneously scattered into solid angles d n, , dnd , d n, along n, , nd , n, , with the wave numbers of particles d, e lying between kd and kd + dkd , k, and k, + dk, . Multiplying Eq. (2.12) by 6(Ek - Ecde) and integrating over dEk , El, the function of k, , kd , k, on the right side of Eq. (1.5c), and then letting k, replace Ek as a variable of integration, the rate of scattering of c, cl, e into dk, , d& , dk, is seen to be w (ab -+ cde) = 2~ h9

1@f(-)*Vi#i

l’S(Ek - ECd,)dkdkddk,.

(2.13a)

In general, if the final channel contains t aggregates c, d, . * * in states uC, ua , * * * , it can be shown by the methods of this section that the rate of simultaneous scattering into dk, , dkd , . . - is w = ;

&

1gf(-)*vilc/i

I2 s(Ek - Eed...) &dkd

. ..

(2.13b)

wherein Eed... is the total kinetic energy in the channel; Ek = h2kz/2M, + . . 1 ; and q,(-)* is a time reversed solution given by equations like (2.6)-(2.7) in terms of the projection of the Green’s function on the set ucud 19. . 2.21

CONSISTENCY

OF FORMULATION

To be sure that our interpretation of Eq. (2.1) has not led us astray, in this section we shall verify that: (i) our methods are internally consistent, in the sensethat they lead to the same physical results whether formulated ab initio in the laboratory or center-of-mass systems, and (ii) our reaction rates agree with those deduced (1, 2) from the transition probabilities to specified final states. The reaction rates ti for any reaction can be computed directly in the centerof-mass system; the calculation (Section 3.3) is quite similar to the laboratory system calculation of this chapter. For instance in the reaction ab --f cd we readily find ti(ab

4

cd)

=

2

~ h3

5 As in J. J. Thomson’s Burhop (18).

) Gf(-)*Vi&

I2 6(K, - Ki)G(Ek - &)

theory of three-body

recombination,

dkdh

described

.

(2.14a)

by Massey and

68

GERJUOY

In the reaction

ab + cde

1 ziY(ab -+ cde) = ?I n (27r)G e1 +f(-)*V&

1’ 6(Kf - Ki)G(Z& -&de)

dk,dk&k,

.

(2.14h)

The algebra involved in computing Eq. (2.14b) is more tedious however than t.he algebra required for Eq. (2.13a), because in the c, d, e channel the current operator J”, and also the free space Green’s function C?‘,appearing in the analog of (l.Sbj, have more awkward forms than the corresponding quantities in the laboratory system. The center-of-mass system in effect fixes one of the initial aggregates at thr origin, e.g., aggregate a in Eq. (1.4b), aggregate c in Eq. (1.4~) if $l is expressed in terms of red, rCC. We presume as usual that unit amplitude plane waves (*orrespond to unit aggregate densities N, i.e., N, = 2\ib = 1 for Eq. (1.4aj, AT, = Nd = N, = 1 for Eq. (1.4~). Then with the incident wave $i of 1i;q. (1.4a) :I large S-dimensional volume r contains T of the aggregates a which were fixed at the origin in the center-of-mass system; simiIarly with the incident wave tii’ of Eq. (1.4~) the volume T contains 7 of those aggregates fixed at, the origin by $l. Thus if we suppose the interacting parts&es are confined to the volume 7, for any process (l.la) or (l.lb) the reaction rate in the laboratory system cornputed with unit amplitude incident wave #L = Gi exp (iK,.R) must be 7 timcls the reaction rate in the center-of-mass system computed wit,h the corresponding (necessarily unit amplitude) $I . The matrix elements of Eqs. (2.13) are formally divergent and yield Cfuncbion factors 6(K, - Ki) stemming from the relations (10) Pf(-’ = $,(-’ exp (iK,.R). The occurrence of these &functions, though merely expressing the expected requirement of conservation of momentum between initial and final st,ates, nonetheless necessitates the supposition that the particles are confined to a finit.c though large 3-dimensional volume 7; in an infinite volume the reaction or scattering rates computed from Eqs. (2.13) are infinite and therefore not comparable with experiment. Our reinterpretation of the integrals in Eqs. (2.13) is conventional, namely we write

z $,-hv&sTdRe-‘gf’ReiKi’R *f(-’*v&hi 1 qj,(-)*Vi#i

z

(2,$S(Kf

I2 g

/ @,(-)*Vi&

S 1 #-)*Vi$i

- I(&-)*Tii$i 12 j” d&e-i’Ki-K’).R’ r

, 1 dRei’K”-K”.R r

I2 (27r)36(K, - Ki) / dR 7 = (2~)~7 j @,(-)*Vi$i

(2.15b) 1’ 6(Kf - Ki).

TIME-INDEPENDENT

COLLISION

69

THEORY

In Eqs. (2.15) the matrix element 9, (-)*V&; integrated over all t presumably converges, and it is immaterial whether the limits of integration over r are infinite or large though finite; the same cannot be said concerning the divergent integral over R. From Eqs. (2.13) and (2.15) we infer that with incident wave #$ confined to a volume 7 the observed laboratory system reaction rate W, for any process (l-la) or (l.lb) will be given by wt = tir, where @ is found from Eqs. (2.14) using the corresponding incident wave SC. Referring to the preceding paragraphs we conclude that our methods do lead to the same physical results whether formulated ab initio in the center-of-mass or laboratory systems. We mention, perhaps superfluously, that this discussion shows the quantities ti of Eqs. (2.14) are the physical reaction coefficients, related to observation by equations such as those pertaining to Eq. (l.la), viz., (2.16a)

wl(ub + cde) = NaNb7@(ub + cde), zo,‘(cde -+ ab) = N,NdN,d(cde

--f ab).

(2.16b)

In Eq. (2.16a) wr is the observed number of reactions ab -+ cde, produced per unit time into specified eigenstates &de and into wave number ranges dk, , dkd , dk, in a volume r containing N, aggregates a per unit volume, Nb aggregates b per unit volume; similarly w: of Eq. (2.16b) is the observed number of reactions cde + ub per unit time. The reaction coefficient ti has dimensions cc/set and is related to the cross section (T by ti = 1v, - vb 1u. The reaction coefficient ti’ has dimensions cm6/sec and is interpretable as the rate at which a pair of aggregates collide within a sphere of influence of the third (18) ; it is not possible to define a cross section. The rule (1,2) for the rate of transition from an initial state #i to a set of final states $f labeled by quantum numbers Ef , /3 is w = 2

1~‘r(-)*V,&

1’ &)

= ‘;

1*,(-)*Vi#i

1’ /#ME,

- E) d-7$,

(2.17)

where \k,‘-’ is given in terms of #, by Eq. (2.10). However in Eq. (2.17) I/, are not the unit amplitude plane wave functions of Eqs. (1.4). Instead the wave functions #, of Eq. (2.17) are normalized on the same scale as is used to specify them, meaning the k, , kd , k, scale for the reaction ab + cde; the normalization of the set of states %P,(-’ defined by Eq. (2.10) is the same (11) as the normalization of $f. Furthermore, in comparing Eqs. (2.13) and (2.17) we must recognize that: by definition, using the k, , &, k, scale for example, p(p) dEf the “number” of final states in the energy range dEf about the total energy E, is found from p(p)dEf = dk,dbdk, ; E, = E, + Ed + E, Jr Ek so that, recalling Eq. (1.5a), is implicit. Ek - Ecde = E, - E; eventual integration over all infinitesimals These remarks make it evident that Eqs. (2.13) and (2.17) are identical. Similarly, transforming from k, , kd , k, to K, , kd, k,, , with hk,d = &d the mo-

70

GERJUOY

mentum canonical to red, and integrating over Kf , we seethat CJof Eq. (2. I-lb) is the transition rate in the center-of-mass system from the initial state qp to the set of final states e,(-‘/(2,)” which are normalized on the kcd , k,, scale. The reader is reminded that each of Eqs. (2.13) and (2.14) is derivable independently, without any reference to normalization in a finite volume; the large volume T was introduced solely to make plausible the connect’ion between U)and ,ti. It will be noted that our approach doesnot encounter any nonorthogonal state perplexities (6-8). There is no interference between the currents flowing in different channels, since these currents flow across entirely different, surfarc elements at infinity in r space. From our point of view therefore, no difficulty is presented by the fact that the current operators in different channels involve projections on non-orthogonal states. We have overlooked the following difficulty however. When I
as rr + m, whereas / [ucde *G] 1‘dS, is finite at infinity. In ot’her words, to put it bluntly, our derivation of Eqs. (2.l:$ was wrong. On the other hand the corresponding Eqs. (2.13) in t,he center-of-mass syst,rrn arc’ correctly derived, because with but two incident aggregates qi always is (10) everywhere outgoing, i.e., behaves like G. Since we have verified the assert,iolls (i) and (ii) with which we began this section, it appears that no errors of cons+ quence resulted from ignoring the plane wave factor exp (iK,.R) in cpi. Xcvertheless this discussion indicates that even in the center-of-mass syst,em formally (though very likely inconsequentially) divergent matrix elements are to be PSpected whenever the outgoing current is computed from expressions which ill actuality are not everywhere outgoing in the center-of-mass systtkm. In pnrticular such complications occur in reactions (1 .ln) (see Section 2.X2), l)ec:~usea!’ does not behave like 6 when three aggregates are incident (10). 2.3 RECIPROCITY The reciprocity relations for Eq. (1.1) are examined in Sections 231 and 2.32 below, but not with the primary intention of rederiving or even generalizing

TIME-INDEPENDENT

COLLISION

THEORY

71

previously known results. Rather we wish to demonstrate the connection between the reciprocity relations and the boundary conditions on Qi(+’ and \kf(-), discussedin I. In these immediately following sections we concentrate anew on the laboratory system and do not refer specifically to the center-of-mass system unless there is some special point to be made, thereby avoiding the use of barred quantities. As Eq. (2.15) illustrates, corresponding expressions in the laboratory and center-of-mass systems usually are related by simple d-function factors; thus it generally is neither necessary nor advantageous to distinguish between formulations in the two systems. We note particularly that if +1 and % are any two functions such that%(r) = exp (iK1.R)&(r), cPz= exp (-iK?.R)& , then as in Eqs. (1.2), and using Green’s theorem in r space6

= (27r)36(K1- K2) 1 &%‘.W”[&, m

&]

The surface integral on the right side of Eq. (2.18) vanishes by definition whenever &I and @zare everywhere outgoing in the center-of-mass system. We conclude that scattered waves known to be everywhere outgoing in the center-ofmass system can be manipulated as if everywhere outgoing in the laboratory system. Henceforth the unqualified phrase “everywhere outgoing” shall mean “everywhere outgoing in the center-of-mass system.” 2.31 TWO-BODY

REACTIONS

The discussion of this section pertains to reactions (l.lb) only; the more complicated reactions (l.la) are deferred to Section 2.32. With this restriction the initial scattered wave cpi= (pi’+’ = -GV$i is everywhere outgoing, as is cp/‘-‘* = -GV&f*, the scattered part of PI’-‘* according to Eq. (2.7a). If the f = c, d channel is a rearrangement of the i = a, b channel (IO), *ki(+’ = +%+ (pi(+) = - G,Vf4?i (+‘; under these circumstances moreover lim [Ucd*+i] = 0 as rC, rd + a~, so that lim [u,d*(pJ = lim - [u,d*(G,V~XPi(+))]. Proceeding as in Section 2.2, Eq. (2.6~) evidently is replaced by lim [u,a*(aic+‘] = -qhJp)$f*Vf’Ei’+ rpc.3 and correspondingly #f*VfFi(+)

replaces *f(-)*V&i

(2.19)

in Eqs. (2.13) for w(i --+ f).

6 The components of W” are defined explicitly in Eqs. (2.9) of I; they are equals (A)-'W" when the system generalization of Eqs. (3.31) for JCL, which three particles of mass M, , Ma , M, .

an obvious comprises

72

GERJUOY

E’urthermore, since [uCd*(pc(+)] has the same limit as rC , rd become infinite, whet#her computed from -GVirLi or from -GfVjilXt’+), we infer that +,*v,\Iri(+)

= \Il,(-)*v*#i

.

(2.20)

The above line of reasoning would furnish a perfectly valid proof of Eq. (2.20) were it not for the fact that -GfVf~i’+ cannot be everywhere outgoing because it contains the incoming wave #i ; the derivation of Eq. (2.19) has ignored the strictures at the end of Section 2.21. Eq. (2.20) also can he proved however by first demonstrating each of J/.f*vf+i J/r*vfcP;(+)

= +r*vz+i = (&-)*vt&

(2.21a,!

+ g(+r*, tit), - g[&)*, pi(+)]

(22lhi

and then adding the two Eqs. (2.21), noting that S(#,*, #J = 0 because $i and #,* propagate in different channels, while $(cp,(-I*, vi(+)) = 0 because (pi(+) and outgoing. Each of Eqs. (2.21) is demonstrated using T’:(l. cPf‘-I* are everywhere (2.18), which is applicable after substituting: in Eq. (2.21aj Kki = (I-I - Z$j#; , l’f#,* = (H - E)$,*; in Eq. (2.21b) vtJ/i = -(H

- E)$p,

r&j*

= -(N

- &Jr(-)*.

As may be anticipated from the last chapter of I, the vanishing of g[p,(--‘*, (pL’+‘] is equivalent to the possibility of invert’ing the order of a repeated int,egral. Namely, recalling G is symmetric, ICq. (2.21b) can he rewritt,en in t)he form -$,*V,(GVi~i}

= - {fif*VfGJ

V&j

- &,‘-‘*,

&+‘I

(2.22,

The vanishing of $(J/f*, #J is equivalent to the well-known (6) equality of Born approximation matrix elements computed using “post” Vi and “prior” T’i int’eractions. The function JF~“* which actually appears in the expression (2.13) for the reaction coefficient, and which actually results from evaluating (for aggregates C, d out) the limit on the left side of Eq. (B.Cih), has an everywhere outgoing scattered part of (-I* . From the time-independent standpoint of this paper, it seems that our attention is directed to solut,ions with incoming scattered waves solely because of our insistence on writing makix elements in the form q,,.*l’J/, , thereby prejudicing us to regard \kf not 9 f* as t’he significant funct)ion. In similar fashion when the theory is reformulated so as to replace t,he incoming plane waves # of Eqs. (1.4) by nonplsne wave functions x the functions xi (+I :t11cl appearing in the matrix elements have outgoing scaattlctrcd parts. X/ (--)* actually We suppose ,

(2.23:1)

Vcd = urrz + CC;&)

(2.231,)

vab

=

[Tab

+

u’ab

TIME-INDEPENDENT

COLLISION

73

THEORY

where U& , %b are Sufficiently rapidly decreasing as r,b ---f co, and similarly ud , uCd . Also we presume that in the reaction a5 -+ cd, the solutions to

for

(Hi + uab - E)x;

= (Xi - E)x;

= 0,

(2.24a)

(Hf +

= (x,

= 0

(2.24b)

- E)xf

Ucd

- E)xf

are known, so that vob, VCd now are thought of as the interaction potentials; in most cases of practical interest u,l, = U&r&) only, UCd &pen& only on red . Since but two aggregates are incident, there exist solutions xi(+)&, b) to Eq. (2.24a) differing from #i( k, , b) of Eq. (1.4a) by functions which are everywhere outgoing. Thus just as in Eqs. (1.6) and in I, Xi

(+) = #i - siu,+i

,

(2.25a)

where Ui = u,%b, (Xi - x)$j’i = S(r - r’), and the outgoing Green’s function S,(E) in Eq. (2.25) is the limit of Si(X) as E --f 0 with X = E + ic, e > 0. Moreover JJ?~(+’ and xi(+) differ from each other by a function which is everywhere outgoing, because they each differ from #i by an everywhere outgoing function. Consequently, again just as in Eq. (1.6) and in I *.(+I E Equations

(2.25) imply

- xi(+) - Gz,~~~(+) E xi(+) + (Pi.

(2.2513)

cpi(+) of Eq. (1.6) also is given by (+) = -$U,$i - Gvixi(+). Pi

(2.26)

When xi(+) does not propagate in the f = c, d channel, meaning the projection [&d*s;u$+i] is negligibly small as re , rd approach infinity, we infer immediately from Eq. (2.26). in complete analogy with Eqs. (2.6) and (2.19), that lim [‘&d*$&(+)] = -Qd(P)*‘/(-)*‘udXd(+)

(2.27)

Tt+cc

and therefore

that Eq. (2.20) must be supplemented +,*y,lpi(+)

by

= gp*Qxi(+).

(2.28a)

In general, #f*vpPi(+) Equation

= %p*wixi’+’

+ s(l//,*, xi(+)).

(2.28b)

(2.28b) is proved substituting

$‘,I//,* = -(H

WiXi(+) zz -(H

- E)&)*,

using Green’s theorem; recalling Eq. (1.2a) and letting g((pf(-)*, +i(+‘) = 0 since a;(+’ vanishes when xi(+) does not propagate in Eq. (2.27), but need not vanish otherwise. (fJ - JW

= V&f,

- E)@i+‘;

holds with *l = ?P,i(+), \kz = \kf(-)*; is everywhere outgoing. g(tif*, xi(+)) the j channel as assumed in deriving Furthermore (H - @xi (+) = =(Jixi(+)

74

GERJUOY

and Green’s theorem imply iq$f*,

xii+))

= l)f”(V,

- w;)p,

(2.29)

which generalizes Eq. (2.21a). If Uab = Uab(rab) only, in which circumstances Uu6 can scatter but cannot rearrange or excite a, b then: (i) the matrix element on the right side of Eq. (2.29) vanishes and Eq. (2.28a) holds for final channels corresponding to rearrangement or excitation of a, b, but (ii) for f = i corresponding to elastic scattering +,*vqp’

= q/,‘-‘*‘uxi(+)

+ #,*[yx;(+’

(2.30)

wherein unnecessary subscripts have been dropped. By the methods of I it can be shown that *i’+’

= -$fvf*i’+’

(2.3la)

+ S($jf y #i)

and that, as in Eqs. (1.7) sf = Gf - GfUfS/

(2.31h)

,

G = s/ - ~fwfG. If g($, yield

$J is neglected

in Eqs.

(2.31), the procedures

lim [u,d*(pi’+‘] rpm permitting

(,2.31(@) of Eqs. (2.19)-(2.20)

= --~~(p)~,(-)*u,*~(+)

(2.322)

the inference ip*viJ/i

where, referring

(2.32h)

to Eqs. (2.7a) and (2.25) Xf

(-I*

= $I* - $/f*L:,s,

\k/(-h Using Green’s Eq. (2.32b)

= Xf(-)*q\Ei(+),

theorem

= x,(-)*

however,

\E,‘-‘*v,+, 4(x,(-)*,

#i)

)

(x3:3:1)

_ x,(-)*~,Ge

(2.:Nb)

as in Eqs. (2.28b)-(2.29),

=

p*Vf\k;‘+’

=

Xf(-)*(U.,

-

Vi)&

$(xf(-)*, .

we find instead of lj/<),

(2.34n) (2.:341,‘)

Equation (2.34), the analog of Eq. (2.28b), is precisely the result we would have obtained directly from Eq. (2.31a) had the t,erm g(sj, tii) not been neglected. The analog of Eq. (2.30) is qr,‘-‘*v#i

z xf(-)*qJ\k,“’

+ x,(-)*cT+i

(2.351

TIME-INDEPENDENT

2.32 THREE-BODY

COLLISION

THEORY

75

REACTIONS

we now turn our attention to reactions (l.la), supposing first that the arrow points right, i.e., a, b are incident. Then (pi+’ is everywhere outgoing and, as shown in I, gQ+, = -GrVf!k’+’ =WJ/ ,1Li) = $i -

+ s(Gf , fii),

G,(H, - -O&i.

(2.36a) (2.36b)

But (Hf - E)#i = (H - Vf - E)$i = (Vi - V,)#; , so that *.(+) = #i - Gf(Vi - V,)I& - G,Vf!Pi(+’ *

(237a)

implying, substituting Pi(+) = tii + pi(+), (+) 4% = -G,V&

- Gfv,qi(+).

(237b)

Equation (2.37b), the right side of which is everywhere outgoing and does not contain the incoming plane wave #i , is a rigorous starting point for evaluating hm [u&*pi(+)] as r, , rd , r, --f 00. Using our by now familiar procedures, we find Eqs. (2.19)-(2.20) are replaced by (2.38) (2.39a) or qjyvi

- Vj)~~ + Jl,*vfPi’+’

= 9z’-‘*vi$c.

(2.39b)

At this point we remark that for reactions involving dissociation of one of the incident aggregates without additional rearrangement, the matrix element #f*v,‘ki(+) is formally divergent, even in the center-of-mass system. Consider for example deuterium breakup d(pn) in the field of a massive nucleus a. For simplicity we assume: (i) Coulomb forces are negligible, (ii) a has no internal structure, and (iii) a is infinitely heavy, so that the center-of-mass and laboratory systems coincide and a can be replaced by a center of force located at the origin. Then Vi = V, + VP , VI = V, + V, + V,, , and the term #f*Va&i in $f*VfQi(+) is +f*~,p+i

= / dr,dr, exp [-

i(k,.r,

+ -V,

k,-dl exp [ikd. (m + rP)121~(rP - rJ

(2.40a)

$6 where

GERJUOY

IL is the ground

state of the deuteron.

s=r,-r

11,

Introducing

rd = (f, + ml/2

-u(s)

exp i(kd - k, - k,) .rd

zz (2n)36(kd - k, - k,) / dsV,&)u(s) Because Ed is negative

exp [k ;(k,

(2.4Ob)

- k,) .s]

while

= h”(k, the h-function the deuteron.

new c*oordinates

+ k,)‘/‘JM

+ h2( k, -

k,)‘,Ml,

in Eq. (2.40b) always is zero; the force ITnp alone cannot dissociate Similarly, in the more general reaction a + b -+ n + d + e, $Q*vl&

= $.f*(v,

-

Vi)&

is the formally divergent part of #f*V#i(+) and can be interpreted to vanish. Thus the difference between Eqs. (2.19) and (2.38) is not. significant. Nonetheless it is gratifying that the more rigorous treatment leads directly to t,he expression (2.38) in which no formally divergent terms appear. Consider now the inverse reaction ade - nb. In this event, as ant,icipated at the end of section 2.21, the mat,rix element *i;/“--)*l~~$~ is divergent, u-jt,h tii = ‘GURU. exp [i( 1. r, + kd. rd + k,. r,)]. The divergence corresponds physically to the fact that in the center-of-mass system, with particle II for example fixed at the origin, particles d and c may react with each other anywhere in a large volume 7, not merely in the vicinity of a. When these two-body ctollision terms-which cannot produce the reaction a& + a&are subtracted from &(+), and the reaction coefficient ti’(ade 4 ab) is computed from t’he remainder, no divergent terms appear. We shall demon&ate this assertion, recognizing however that the matrix element ‘kl ‘“*r’lJ/,’ remains correct; because t,he two-body collisions cannot contribute to #(ade - ab) the formally divergent terms vanish, as they do in the reciprocal matrix element, #,*Y,,~lC+’ for t,he illverse reaction ab --+ ade. The two-body collisions of d and e are those which can result when 1/‘,, and V,, are zero. Thus, with the incident wave $i, the t,wo-body collisions of d and e are described by the wave function *de(+), satisfying (Hd, - E)%e(+)

= 0,

Hde = H -

(2.41x) Vod -

I’,, .

(2.41 h)

TIME-INDEPENDENT

COLLISION

THEORY

77

Since Vat, = Va/;ld+ V,, = Vf’, Hde is identical with H,’ which in turn is identical with Hi when a, b are incident. In terms of the outgoing Green’s function G,’ E Gi therefore,

%e(+) = t,bi’ - GI’V,&’

.

(2.42)

In the center-of-mass system the scattered wave pdeCi) = -G,‘V,&[ of Eq. in rd , r, , but contains the plane wave factor exp (ik,-rJ. The wave function *,ir(+) = $P’ + (pd(+) is

(2.42) is outgoing

q,irf+) = h’ - G(V,d + v,, + vi&ii = #i’ - GV,$k’

- GV&i

(2.43)

.

Substituting G = G, - G,‘V/G for G in Eq. (2.43)) we see that cpi(+) - pde(+) = ‘(+)a, which we term cp’(+) subtracted, is given by cp (#)8

= -GV,‘$i

+ {G/V,‘GJ

Pi ‘(+)’ does not have a factor exp (ik,.r,). lim [~~*~il(+)~] rm +b+m

= -~~~b(p)(!Pfl(-)*VflJ/~

V&i’.

(2.44)

From Eq. (2.44) we find

-

{#;*Vf’G)

V,&~).

(2.45a)

Recalling Eq. (2.7a), -$f’*V,‘G = (pfl(-)* in Eq. (2.45a). Moreover letting the incident wave propagate along -k, , - kd , - k, , so that #i’ = rClr* of Section 2.31, q/(-j* F *,;(+I, v,’ z Vi, Vf = Vi + Vde, Eq. (2.45a) takes the form lim [u=i,*(pi‘(+)81 = -?dp)($,*Vi?I/if+) T.. vJ’4) = -

tlab(P)(!b*vi+i

+ ~j*v&?(P) (2.4513) + It/*vjcd+9

as required by time reversibility comparing with Eq. (2.38). Actually we should to two-body have subtracted additionally from cpil(+) the terms corresponding collisions of a, d and of a, e; this more careful procedure also leads to Eqs. (2.45). Furthermore, starting with the primed analogue of Eq. (2.37a), and subtracting (+I as above, leads to the non-divergent matrix element $f’*Vf’\kl(+) = (pd’e’*Vs4i again in accord with time reversibility. Qf As a final illustration of our methods, we derive the matrix elements for d(pn) reactions, retaining assumptions (ii) and (iii) above, but no longer assuming Coulomb forces are negligible. The most obvious approach is to choose the Coulomb interaction U(r,) = U,a = UCd = U, in Eqs. (2.23). Then vi = up + wi ;

Wi=

Vn+

Vp; v.f = up + w, ;

‘of = vn + v, + v,, )

78

GERJUOT

with I’, the nuclear non-Coulomb cleus a. In Eq. (2.26) Si

interaction = S/ -

of the proton with

the heavy nu(2.46)

SfVnpSi

implying (12.47:1) lim (pi(+) = -dP)(Er TP’i-,-m (-)* = Xf (-) * Ef

(-*T-J& Xf (-I

-

+ q/,(-)*2)iXi(+)),

* vnpsi

(2.47b) (‘2.47d

.

The function [f’-‘* satisfies (Xi - E)&-)* = 0; recollecting Eq. (2.33a) t;,‘--‘* is seen to represent the collision of a neut’ron and proton interacting with each other through V,, and with the center of force through U(r,) only. The term ,&-‘*U&i in Eq. (2.47b) is the matrix element for the disintegration of a deuteron in the Coulomb field U(r,) . Such Coulomb disintegration can occur because rP does not coincide with the center-of-mass of the deuteron; in the not’ation of I
=

-s,vi+i

-

S,Vf$Q

so that (-)*v&

lim (pi’+’ = --rl(P)(Xf Tp,v,-m Various (+I =

Xi

other formulas l)i

+

can be found.

+ xf’-)*vfqj(+))

For instance,

as in Eq. (2.37bj,

mrit,ing

Tit+), r .(+) *

=

-G/U&i

lim Tic+’ +p*r,-roo

=

-dP)(#f*wi

-

G/(t-,

+

(2.49a)

T’,,p)T,‘+’

so that

yielding plying

also the rate of deuteron 5.t’-‘*up**

= *f*U&

+

#,*(I’,

disintegration + */*cup

+

Vnpm(+f)

in the Coulomb + &Jt(+‘.

(2.49b) field and im(2.50)

Terms such as #f*U,#i above do not converge unless me are willing to make the approximation that the Coulomb forces are screened. This difficulb, stemming from failure of our original assumption that v& decreases more rapidly then r,b -I, is avoided if we introduce Uab = lri = V(r,) in Eqs. (2.23)-(2.24); U(rd) = Ud is the Coulomb interaction between the deuteron and the nucleus when the charge is located at the deuteron center of mass. Then Eq. (2.2:&l) is

TIME-INDEPENDENT

COLLISION

THEORY

79

modified: Vi = Ud + Ui”; ‘ui’” = V, + VP + UP - Ud ; Vf and U, are as before. xi(+jn, the solution to Eq. (2.24a) with Ui = Ud , has no probability of disintegration, and the matrix element for the entire reaction is therefore lp+QQ+)y as in Eq. (2.27). The reciprocal matrix element is best found from qk,i(+, = .pm + @;+)m, (H - E)&+)” wr - wsf

= -QmX*(+)m,

(2.51a)

= 1,

(2.51b)

yielding *i(+jm = -s&mxi(+)m

- sy(J@i(+)”

(2.52)

so that Eq. (2.4813)is replaced by limpi Tg*+n+*

=

lim #Jam = -~(~)(~(-)*u~~x~(+)~ r*. *n+-

+ x,(-)*~,.+~‘+‘“).

(2.53)

Sections 2.31 and 2.32 elaborate, and to someextent generalize, relations given by Gell-Mann and Goldberger (2). III.

MATHEMATICAL

APPENDIX

We now amplify some of the assertions made in this and the preceding paper (10). It is to be understood that this chapter aims merely to illuminate our procedures. We do not pretend that our treatment of rearrangement collisions is rigorous from a strict mathematical standpoint. 3.1 ASYMPTOTIC

BEHAVIOR

OF THE GREEN'S

FUNCTION

If the interaction Vjk between each pair of particles decreasesmore rapidly than rik-‘, the particles when infinitely separated should propagate as if free. In other words, if r -+ co along directions n such that every rjk becomesinfinite, then along such directions, corresponding to complete dissociation of all the aggregates, G should behave asymptotically like GpCn).This expectation is borne out by an argument like the one used to find the asymptotic form of the radial wave equation in one-particle potential scattering (19). Let B(r; r’; A) = G( r; r’; A)/G,‘“‘(r;

r’; A).

For fixed r’ and sufficiently large r, 6(r - r’) = 0 and

-G,~~,~~B-~~,~jGI.VJB+ 3

3

B satisfies VGFB=O.

(3.1)

Divide Eq. (3.1) by GF ; neglect the terms in Vj’B, to be justified a posteriori; and use Eq. (1.11) (with E = A). There results -

2i4+VB+VB=O

(3.2)

80

GERJUOT

where we have written Cj rj. Vi = r. V, V the Xn-dimensional gradient operator. and integrating from r. to r along a path Multiplying Eq. (3.2) by p&r-‘B-l, of constant n, we have B(m;

r’; X) = B(ron; r’; X) exp

with r. some constant lower (1 .I1 ) t,o be valid when r > P -z

r

0

2 @

1’2

TO

(5&h:)

limit which is large enough for Eqs. (1 .lO) and . In the integral on the right side of IQ. (i3.Z)

(Ml

+

a!r,g;

+

.

.

.

+

M,y,“)“”

(1 + y2? + . . . + y,“)“’

is constant for fixed n; I’ = CV,(r, - rr;); rjh decreases more rapidly t,han -1 = rlpl / yj2 + q1;2- 2qjqkIlj’ nk l-l, mhereyl = 1; r1 = (1 + y?” i- ..’ + YYi2~ r. Hence the integral in Eq. (3.3) converges as 7 --+ CL,,provided n is so chosen that for each pair of particles qj = yk and nj = nk are not simultaneously true. For such n the upper limit of the integral in Ey. (3.3) can be replaced by infinity when 7 is sufficiently large, making the right side of Eq. (3.Z) indepentlent of r, though not of n or r’. It now can be seen that the first term in l?q. (3.1) is NT+ compared to the second term in gradB, ergo negligible. We conclude t,hat for n chosen as above rjk

G(r; r’; X) -

A (n; r’; X) ‘Gp n

_

describes the asymptotic behavior of G at lsrge r, -4 the product of R and the remaining factors in Eq. (l.lOa). The anomalous behavior of G occurs along directions n for which one or more nj = nk , i.e. directions for which 1’ does not go to zero as r - x . ,Supposefor simplicity our system consists of two particles only. The anomalous surface elements at infinity center on the intersection of the infinite sphere r = (r12+ r2’)“’ + 4r with the plane r, = r2 . Consider that part X2of the spherical surface lying within the planes r, - rz = &s, s a given fixed vector. As long as s is finite, St is an infinitesimal fraction of the ent,ire surface of t,he sphere, within which Eq. (3.5) fails. The terms modifying Eq. (3.5) must vanish however as s --, 00, since s infinite corresponds to portions of the sphere for which Eq. (3.5) was demonstrated. We expect, and have consistently assumed, t.hnt these modifying terms represent propagation in bound states of 1 and 2, jvhen such bound states exist. Since the bound state eigenfunct,ions u(rl - r2j = r!(s) are exponentially decreasing, the bound stat’e current across S, , which should be computed holding s finite though large, can be evaluated letting s approach infinity. In this fashion (see Section 3.2) we are led to evaluate the bound st.ate current from the project’ion [u*G] integrated over all s, and conclude t,hat the of

qj

=

qk

7

TIME-INDEPENDENT

COLLISION

81

THEORY

total current across the sphere is the bound state current evaluated as described, plus the two-particle current from Eq. (3.5) integrated over the entire sphere at infinity. Eq. (3.5) is not valid within X1 of course, but the p5” dependence of G (for two particles) guarantees that for any finite s the current across St from Eq. (3.5) is negligible, as is the current across S, from the cross product of Eq. (3.5) and U(S); in the phase space of possible k, , kZ the particles can move to infinity together without being bound, but only with negligible probability. In a many-particle system we assume similarly that the current along directions nL corresponding to possible formation of aggregates c, d, e is computed from projections on the bound states Undo. Other terms in the asymptotic limit presumably either make no contribution to the current along n, of G or (pi’+’ because they represent propagation in fragments of c,, d, e, or else contribute only along a subset of nt because they represent agglomerations of c, d, e. The contentions of this paragraph are made plausible by the discussion of this and other sections, and by the overall consistency of our results; a general proof would be most welcome however, since these contentions are basic to our methods. For complex X G, like any other quadratically integrable function, can be expanded in terms of the complete set of functions U&S& for instance, including the continuum functions. Guided by the expansion (1.8b) we write

Gk from which,

r’; V = CB(rede ; r’; X’)Gpc3)(rcde ; red:; x’)zL~~&~~)

substituting

r&‘3’TB

- $

in the differential c

V,GF’3’V,B

-

equation

(3.6)

for G, we infer

.. .

+ (Vcd + Vc, + VdGk3’B

1

= 0

The potential V,d in Eq. (3.7) is a function of sc, sd , but is assumed to decrease more rapidly than red -’ when rC , rd , red -+ cc holding SC , sd finite. If v were independent of &de , the bracketed expression in Eq. (3.7) would equal zero for every A’, and the argument following Eq. (3.1) would be applicable. Thus we immediately could conclude that the coefficient of ucda in the expansion of G(r; r’; X) behaves like lim GFC3)(rcde ; f6de; X’), i.e., the aggregates propagate as if free, when rc , rd ,r e + 00 along directions such that each of red , rCe , rde become infinite. Since it seems clear that the fundamental asymptotic behavior of the Green’s function is determined by the rate at which V decreases as r, , rd , r, + 00, our conclusion about the behavior of B should not be altered by the fact that in Eq. (3.7) V actually depends explicitly on a& . We infer, in agreement with the previous paragraph and with expectation, that the bound UC& terms in Eq. (3.6), exponentially decreasing with increasing a& and therefore negligible except

82

GERJUOY

in the immediate neighborhood of nt at infinity, make a finite contribution the current. In fact, referring to Eqs. (1.11) and (2.4)-(2.5), if propagation thCdeis energetically possible, i.e., if E ede= E - E, - Ed - E, > 0,

Vc[ucde*Gl -

to in

ik, n,[u,de*Gl

and / [u,de*G] 1’ dS cdeis finite and independent, of r. According to the foregoing discussion the continuum functions in Eq. (3.6j must represent propagation in fragments of c, d, e and therefore should not make a finite contribution to the current in the neighborhood of n, . On the other hand these continuum functions seemingly do make a finite contribution since, like the bound state functions, they are multiplied by Gpo’(X’). The apparent discrepancy arises because the coordinate system r&e, scde is deceptive at’ infinite to directions n, , while all other dire&ions n are rCd,finite sede corresponds included in the points s& infinite. As the continuum functions are not exponentially decreasing, the current from the continuum terms in Eq. (3.6), when computed from the prescription Eqs. (2.4)-(2.5), includes contributions from infinite s&e , and therefore includes contributions from directions n rorresponding to partial or complete dissociation of c, d, e. Thus the prescription (2.4)~(2.5) applied to the continuum terms of Eq. (3.6) must yield the total current in fragments of c, d, e, which current also can be found we know, perhaps less perhaps more conveniently, by projecting on all possible bound states of these fragments. We shall illustrate this contention for d(p) reactions neglecting Coulomb forces [see assumptions preceding Eq. (2.40a)l. A convenient set, of deuteron continuum functions, normalized on the k-scale, are

u*(s, k) = vf* - $Vnpvf*,

(3.&l)

ik.s v.f=k2e

,

$j”

=

6(s

-

(3.8c)

s’),

with p the reduced mass of the deuteron and s = rP - r, . When the total Green’s function G is expanded in terms of this set, the projection on u behaves like GFC1)(rd ; rd’; X’). Hence, computing the current in each u(k) by our usual procedures (assumed legitimate for these continuum states), then integrating over allowed k, the total rate of scattering into continuum states such that the center of mass of the neutron and proton moves with momentum between hkd and h(kd + dkd) is dk 1\k/(-)*’

\kf’-‘*“(ka,

k) = u* exp (-ikd.rd)

V&i

- G(V,

I2 dk&Ef

- &I,

+ Vp)u* exp (-ikd.rd).

(3.9b)

TIME-INDEPENDENT

COLLISION

THEORY

83

Equation (3.9a) is integrated over values of k for which kd is real, where Ef = h2k2/2p + h2ki/4M. The rate of scattering of protons and neutrons into dk,dk, is, according to Eqs. (2.13)

’ [!I!{(-)*V& *,(-)*

I* dkpdknS(Ef

(3.10a) (3.10b)

= #f* - G(V, + Vp + Vn,k*,

#f = exp [i&-r,

- Ei),

(3.1Oc)

+ k,.r,)l.

Introducing kd = k, + k, , k = s(k, - k,) it can be shown by the methods of sections 2.31 and 2.32 that qf(-)* = (27r)3’%?f(-)*c,and therefore that WC above is identical with the integral of w over k. 3.11 ASYMPTOTIC

BEHAVIOR

OF INTEGRALS

In the remainder of this chapter, as in chapter II, we are concerned with projections on bound state eigenfunctions only. Our results in chapter II for the asymptotic behavior of [uf*(pi] frequently employ Eqs. (1.7), as in Eqs. (2.6) for instance. Although results so obtained are self-consistent and consistent with the preceding Section 3.1, our use of Eqs. (1.7) merits additional discussion. We note first that Eq. (2.6b) defines 9f(-)*, as Eq. (2.6a) defines tir*, and then remark that whereas the first equality of Eq. (1.7a) yields the formula (2.7a) for !Ff(-)* in terms of #f *, the second equality of Eq. (1.7a) yields qf(-)*

= tif* - @)*VfGf

.

(3.11)

This Lippmann-Schwinger integral equation for XPf(-)* is consistent with Eq. (2.7a), as is to be expected since we merely have used alternative forms of the identity GfVfG = GVfGf . Suppose now f = c, d or c, d, e of Eqs. (1.1) is a rearrangement of i = a, b. We know from G = Gi - GiV&’ = Gi - GViGi , that defining #i*, iP:-‘* by lim [ui*Gi] = r]i(p)#i*, 7,.q--=2

(3.12a)

lim [ui*G] = qib)*i(-)*, I,, 16-00

(3.12b)

pi(-)* = ,&* _ ,&*F/‘&’ = $,;* _ qi(-)*ViGi

(3.13)

leads to

as in Eqs. (2.6)-(2.7). On the other hand assuming as always that lim [ui*Gf] -+ 0 when r,, rb + * 00, i.e., assuming that G, does not propagate in the ith channel, the first equality in Eq. (1.7a) seemingly implies [ui*G] -+ 0 in this limit, whereas the second equality implies q.(-)* = --\k *?‘*V f Gf. (3.14) z

81

GERJUOY

Equation (3.14) is (10) the correct integral equation for \ki(-)*. We wish to understand why the identity GfVfG = GV,G, has led to an inconsistency when projecting on ui . The inconsistency stems from an assumption similar to the assumption pointed out at the end of Section 2.21; in the preceding paragraph we have assumed lim [ui*( GfVfG)] = lim { [ui*Gf]VfG}. We observe that this assumption proves false even though GfVfG = GVfGf = G, - G surely is everywhere out’going, in the laboratory as well as the center-of-mass system-there are no incident plane waves in Gf or G to complicate the behavior of GfVfG. As r - r= along n the integral over r”G,(r; r”)Vf(r”)G(r”; r’) appears to converge rapidly as r” + to along all directions n” except for n” parallel t.o n. In such directions there are values of r” near r no matter how large r is, and Gf is not part,icularly small when r” is close to r. With this in mind the difference in the behavior of Gf(r; r”)Vf(r”)G(r”; r’) and G(r; r”)Vlf(r”)Gj(r”, r’) can be understood. Suppose r + a, along a direction corresponding to possible formation of aggregates i. Because G, does not propagate in the ith channel and G does, Gr(rN; r’) decreases more rapidly than G(r”; r’) as r” becomes large in a direction parallel to r; G,(r; r”) and G(r; r”) are of the same order of magnitude when r” .-’ r. Thus, without attempting a detailed proof, we see that the integral Gll’,,G (*an he espetted to converge less rapidly than the integral GVfGf when r - m along directions in which G, does not propagate. Consist’ent, results are obtained howin ever from [.uf*{ GVfGf)] and [uf*(GfVfG} I, i.e., when r - 00 along dire&ons which both G and Gf propagate, and along which T’f decreases ‘more rapidly than (r”-’ (Section 3.1). We infer that (i) Gfl’,.G is almost, sufficiently rapidly converging to behave like G/ along all directions of r, failing to do so only because V,(f) remains finite (10) as r” + m along directions corresponding to renrrangrments of J’; (ii) judging by Eq. (3.14), even though Vr(r”) rem:tins finite, the more rapid decrease of G/(r”; r’) compared to G(r” ; r’) is sufficient to guarantet~ GVfG, does behave like G along all directious of r. We infer further that for an integral like GiVicp7(+’ to behave like 6i it is necessary but not sufficient that GiVicpi(+’ be outgoing; before projecting on :L set of bound states the convergence of the integral must be examined. In particular, with two incident aggregates gi(+’ = -G(r ; r’)vi(r’)$i(r’) obviously behaves like G, because UC = ‘~,b is exponentially decreasing unless r’ + = in dir&ions corresponding to the incident channel, and IYi decreases more rapidly t,han (r’)-1 along such directions. Similarly, dropping the bars, Pi (+) = -G.V.+.‘+’ z EI

= -Gl[‘ill,i

- GJ$+)

behaves like Gi along directions corresponding to the incident channel. On the other hand we know (pi(+) = -GiV$i(* behaves like G not Gi , the anomalous behavior in directions corresponding to rearrangement stemming not from

TIME-INDEPENDENT

COLLISION

THEORY

85

GiV,qi but from GiVipai(+); since cpi(+) behaves like G, GaVipe(+’ behaves like GiViG of the previous paragraph. It is for these reasons that we have to obtain lim [uf*‘pi(+)] of Eq. (2.19) from Pi(+) = -GfVf’k/+’ rather than from pi(+) = -GiV$Pi(+). AS a last illustration, it is legitimate to project on ui the integral equation

for G/ in terms of Gi

and thereby

G, = Gi - Gi(Vi

- V,)G,

(3.15)

0 = I/&* - $i*(V,

- V,)G,

(3.16)

to infer

consistent with Eqs. (2.36)-(2.37) of i. On the other hand Gi(Vf Gi nor GI but like G; assuming

bi*~Gr(V,

-

when s(G, , #i) = 0, i.e., f a rearrangement VJG, = Gf(Vf - Vi)Gi behaves neither like

J’JGill

= {[ui*Gzl(V,

-

ViPil

evidently leads to an erroneous result. The manipulations of chapter II have been kept consistent. with the considerations of this section. 3.2 THE CURRENT OPERATOR This section supplements Section 2.1. We seek to evaluate the contribution to the left side of Eq. (2.1) from surface elements dS corresponding to possible formation of aggregates a, b. We introduce the coordinates f, , rb , saz = ra2- ral , sa3 = ra3 - ral , . . . , sbz = rb2 - rbl , * . *; let r, , rb approach infinity along n a , nb ; and, as in Section 3.1, seek the amount of current crossing that portion of the sphere at infinity subtended between planes of constant s, , sb symmetrically placed about the region %b = 0. On this part of the spherical surface the distance from the origin of each particle in a is approximately r, , similarly for the particles in b, i.e.,

Val = v,z = . . * = ra/r E v. ;

Vbl z Vb2= . . . z

rb/r

=

Vb ;

r

=

(fara’

+

2 l/2 .fb?“b > ,

where fa , fb are, respectively, the numbers of partideS in a, b; fa -/- fb = n. Referring to Eqs. (1.2), $(qi*, pi) in the new coordinate system is seen to equal

jh%*, cpi)= &-p-y

&9*v,$9- cpv&*) + &

fbvb’ (cP*vb@

-

qvb‘P*)

(3.17)

86

GERJUOY

where Vaz signifies differentiation with respect to ~~2 and the + . . . signifies similar terms in v, . Vae, . * * and in Vb’ vb2 , - * a. cp*= cpabove is assumed to behave asymptotically like G, i.e., t’o be expandnble in terms of t&b for large r, , rb ; moreover we presume that the continuum terms make a negligible contribution to the current within plmw of finite sa, sb . Thus within this region we approximate cpin Eq. (i3.17) by vi

A

c

&b(ra

, rb)&b(%b)

(X18)

summed over bound ‘&b only, with z& given by Eq. (2.3b). lcor small At-, recalling Eq. (1.3a), the integral we seek equals (AT)-’ S 3((pz*, pi)d7 integrated over the volume between the planes of constant sab , and between t’wo spheres of radius r and r + Ar. Transforming to I-, , rb , sab space this volume, as ra , 7’b+ m for any finite s,b , becomes identical with t’he volume between t,he same planes of constant s&,, and between two 3+dimensional cylinders whose equations in ra , rb , s,b space are fara2 + jbrb’ = r2 and f,r,’ + jirb’ = (r + Ar)‘; we remember dr = dr,drbdS,b . The limits of integration over s,b in this caylindrical region (*an be extended to infinity however, because &b(&b) are bound states and vanish exponentially with increasing &b ; the necessity for excluding cont,inuum functions from Eq. (3.18), discussed in Section 3.1, is apparent at this poiut. Substituting Eq. (3.18) in Eq. (3.17); performing the indicated manipulat,ions; integrating over all %b ; and making use of the ort’hogonality properties of &,b , we seethe current at infinity from surface elements (*orresponding to form:lt,ion of a, 11is

summed over all ‘&+,, with Ja , Jb defined by Eq. (Ma). The volume of integration in Eq. (3.19) is the six-dimensional volume in r, , rb space between what are now ellipsoids j,r,’ + fbrb’ = r’, far,? + fbrb’ = (r + Ar)“. We note that only the C, , Vb terms of Eq. (3.17) contribut,e t’o Eq. (3.19). After substit,uting Jlq. (3.18) into (3.17), but before integrating over s& , the V,? t)ermsfor example arcA proportional to J”’ = ~a.2 Za*Z&*V,suB

- z@,,?u,*),

(3.20)

where CYand p are each summed over all possible bound ‘&b . When [la and 1~~ have different energies, the corresponding product .Z,*Zb in Eq. (3.20) is oscillat,orv at large r and makes no net contribution to the current. Excluding aceidental degeneracy, when 1~~and ug have the same energy they have t,he same parity, i.e., (Yand p index levels of different j,, , jbz belonging to the same family of eigenstates u,(ja), ?lb(jb); for such Q, p the term in %I. (3.20) vanishes bec~aus~~

TIME-INDEPENDENT

COLLISION

83

THEORY

Vaz is an odd operator, after integrating over all sab or over a finite reflectioninvariant range of sob . normal to the SUrface (r(r, , rb) = fara + At given r, , rb , the six-dimensional r2 is V&,’ = 2 I grad g I-‘(jdk , fbrb), w h ere grad u = (Vau, VbU) ; the perjbrb2 = pendicular distance between the ellipsoids is Ah = 2r ] grad u ]-‘Ar; the volume between the ellipsoids is the integral over d&Ah where dS, is the surface element on the ellipsoid u(ra , rb) = r2. Thus Eq. (3.19) becomes saab= c

1 dSev.b’ ‘Jab

with Jab as in Eq. (2.2). In the asymptotic region, using Eq. (1.11) and the a, b analogs of Eqs. (2.6)-(2.7), Eq. (3.21) is equivalent to Eq. (2.2). Suppose A(r), not necessarily equal to \ki(+), is a wave function describing the system of particles taking part in the reactions (1.1). We customarily assume the corresponding probability amplitude for finding the system in eigenstates u, , Ub of a, b is the projection [?&b*ii] ; the probability of finding the system in these eigenstates with centers of mass ra , rb lying within limits prescribing a six dimensional volume r is proportional to P,b = Directly

sr

from the time-dependent

dpab 1s7 dt=iFi dr&{

habA*]

dr,drb ) [?&*A] j2.

Schrodinger t&b*ffA]

(3.22)

equation -

[%b*dh&zb(~~)

*] ).

(3.23)

H = T + v = Ha + Hb + Tab + vab, where T,b = p,2/2M, + pt/2Mb, p. = -ihV, ; since uab are bound states and H is real, [uab*{ Ha -/- Hb) A] = [((Ha i- Hb]u&*A]. Thus H in Eq. (3.23) can be replaced by Tab -I- Vab. Next suppose the volume (more precisely a sequence of volumes) 7 is so chosen that for all points in 7 ra , rb and rab become infinite; then because v& decreases more rapidly than r~’ , v,b can be neglected in Eq. (3.23), assuming A behaves like cpi(+) . Integrating by parts (Green’s theorem) in r,, , rb space yields therefore dpaa -=dt

s 7 d&a%* Jab

(3.24)

integrated over the surface elements dSti = dSabvabbounding 7, with Jab’ defined by Eqs. (2.3) substituting A for cpi . Equations (2.4)-(2.5) and the c, d, e analog of Eq. (3.24) are obtained similarly. The result (3.24) and our derivation of Eq. (2.2) starting with Eq. (3.17) support the beliefs that: (i) the left side of Eq. (2.1) represents the total scattered current; (ii) $((pi*, cp<)dS represents the scattered current of particles into the

GERJUOY

88

directions in physical space corresponding to t’he &z-dimensional surface element dS, whether these directions correspond to complete dissociat’ion or to possible propagation in bound states; (iii) Eqs. (2.2)-(2.3) yield the scattered current in bound states of the aggregates a, b. 3.3 THE

CENTER-OF-MASS

SYSTEM

The solution to the Schrijdinger equation in the center-of-mass syst,em is ,(+) 2

= St - fp~‘&

= gi + pi ,

(3.25)

with (A - @%i(+) = 0; (A, - ,!?)$E= 0; (II - EjG = fY(r - r’j; H = A + ~,~/2Yil; Hi = ITi + ~,~/21cf; E = E + h’K’/2dl; PR E -iihVR = hK. Pl(1. (3.25) is identical with the result of barring R-dependent quantities in E;q. (1.6) ; the same operation yields the center-of-mass syst,em analogues of l
- h)6r’2’(h)

= 6(rub - mbl),

- x C;rF(3)= S(r,d - rCd’)C4rra- rCfl’), >

(:3.%ib)

with pcd = -iihV,d the momentum canonical to red = rd - rC, and &d the reduced mass of particles c, d. cFc2)is identical with t,he usual one-particle C;reen’s function [substituting ~(~6for M, in Eq. (l.(3)], 1 2p,+,exp [i(&bh/h”)“2 dp(‘) (rab; f,bl; Xl = r nl, 3r 1 &b -

1rab - rab’ 1] rab’

(3.27)

1

Because of the p&‘&e term in Eq. (3.26131,(;F(3) is more complicated in structure than GF@),namely

with M the total massas usual, and M*$ = M,Md 1red- red’ 1’ + MdM, j ree- rrp’ 1’ + L11,1f71, I rde- r,l,’ I”.

(S.28h)

Equation (3.28b) is obtained via a canonical transformation diagonalizing the kinetic energy operator in Eq. (3.2613);employing Eq. (1.9) with n = 2; and t,hen transforming back to the red, rcecoordinates. In Section 3.2 the argument following Eq. (3.22) shows integration by part.s, using the kinetic energy operat,or in ra , rb space yields the laboratory system current operator for scattering into bound st’at,esu,,, . This simple procedure,

TIME-INDEPENDENT

COLLISION

89

THEORY

consistent with regarding [u,b*(p] as the probability amplitude for observing ?&b in the scattered wave cpi, hardly can fail in the center-of-mass system-there is no need to go through another derivation like that following Eq. (3.17) starting with Green’s theorem in the (3n - 3)-dimensional F space. Thus defining Z,b = [%&*@i], we obtain the center of mass system analogs of Eqs. (2.2)-(2.5) (3.29a)

with dficdevcdeothe surface element on the sphere at infinity rce , and similarly for Eq. (329a) in rab space. Of course

Jabo

=

&-j a

@~b*V,b$b

-

in the space of red ,

zabVabZab*)

With the more complicated kinetic energy operator )Tcde of Eq. (3.26b) it is somewhat less obvious that -(iii)-‘(Z*TZ - ZTZ*) is the six-dimensional divergence in red , rce space of a vector Jcdeo = (Jedo, JCeo) whose first three components JcdO (along the same unit vectors as the components of red) and last three components JeP are -

zv,dz*)

+

&

(Z*v,,Z

-

ZvceZ*),

c

(3.31) - Zv,,Z*)

+ &

c

(Z*vJ

- zv,dz*).

Starting with Eqs. (3.25), using Eqs. (3.27) and (3.30), and proceeding as in Section (2.2), it is apparent that for the reaction ab -+ cd the rate of scattering in the center-of-mass system into dk,d is ti(ab+cd)

=?I! n h3

1%f’-‘*Vi$i

1’ 6(I% - i%) dk,d ,

(3.32)

with B, the function h%d2/2pCd; I!?,~the knownenergyE - h2K2/2M - E, - Ed ; and the other functions as in Eq. (2.14a). Equation (3.32) is identical with Eq. (214a), noting dk,ddKf = dk&kd and I!?, - Eed = Ek - Ecd . For the reaction ab --+ cde, when redand rde-+ a, Eqs. (3.28) make CF(3)

N

(M,MaMJ312 TX~‘~ (-3~414exp (‘J&2,2)7/4 M1/4 e

ti(2h/Mh2)"2Q1

QS/2 *exp [ - i(ked.rcd’ + kce.ree’)]

(3.33a)

90 where Q” =

GERJUOY

M&Iered2@,

Q = rcr/rcd , and we define k,d , k,, by the equations

(8.34)

With x = Ecde the quantities

k,d and k,, of Eq. (3.34) satisfy

Moreover if r, , rCz, r, are supposed proportional (3.3-l) imply

t.o their classical velocities I+.

(3.36)

which are precisely the classical values of pCdand pee . In other words the matrix element giving the current in the center-of-mass system across that surface clcment specified by large red , r,, involves a plane wave state

$, = exp[i( k,d. red + k,, . r,,) I wherein k,d , k,, have their expected classical values; there is complete analogy and consistency with the discussion following Eq. (2.11). Applying the operator of Eq. (3.31) to (3.33a), the current corresponding to the particular set of bound states zdcdeis 5 cde

=

s

dqdncddnee

(X.374

Bringing Eq. (3.37a) into the form (214b) is a little complicated. First multiply Eq. (3.37a) by I@, - $&) and integrate over dl?k , replacing the factor ECdcL’ by Ek2; ultimately 2, will be identified with the function on the left side of E:q. (3.35), but for the present l?‘k simply is an auxiliary variable of integration. Next for constant n,d , rice introduce new variables iz = J!?~:“‘/Q, B = q~‘?n-“‘jQ; then dI?kdq = 2$2dAdB and Eq. (3.37a) takes the form

TIME-INDEPENDENT

COLLISION

91

THEORY

But A2B2dAdBdncddn,, can be regarded as the volume element dAdB in the six dimensional space of A = AnCd and B = Bnce . In terms of these variables, letting X = 8, in Eq. (3.34).

(3.38)

The transformation from A, B to kcd, k,, now is trivial; Eq. (2.14b) results after inserting the 6(K, - K;) factor and substituting dk,&k,JK, = dk,dbdk,. It will be noted that the current (3.37a) in the center of mass system involves no divergent or otherwise singular expressions; the d-function factors are introduced solely to make manifest the identity We = zi~r discussed in Section 2.21. This general relation, and the above arduously obtained expression for a in the particular reaction ab --f cde, also are consistent with

The notation and proof of Eq. (3.39) are as in Eqs. (2.15) and (2.18). The barred analog of Eq. (2.1), obtainable by direct application of Green’s theorem to solutions $ .(+), @i(+)* of (w - ~??)!i! = 0, also is consistent with Eq. (3.39). With two incideni aggregates a, b the left side of Eq. (2.1) is ~TZZ = T 1V, - vb 1 U, u the total cross section. Using Green’s theorem in c space, and (ai - E’)@< = - V$i’+‘. the right side of Eq. (2.1) is 2 7 Im dr(&*(Ri s ii

The elastic scattering

- i!?)@i - pi(Bi

- @$i*) (3.40)

amplitude

A(abni + abttf) is defined by (3.41)

f& -+ 00 along n/; in Eq. (1.4b) the incident and 9; = -G’iVi%<(+) imply the expected

as

A(abtti + ~bttf) = - $ ‘$ n-

kab = k,b&

$f*Vi%i(+),

Equation

(3.27)

(3 42)

92

GERJUOT

Thus we infer that Eq. (2.1) does yield the cross section theorem c = ‘e Im A (ablti +

&Ii).

(3.43)

ab

Analogously to Eqs. (1.8) G(r; r’; X) = c C?(r; t’; X’)U(R; K)u*(R’; K)

(3.44)

in which ej(R; K) are plane waves normalized on the K scale, u = (2?r)-3”esp(iK. R) and X’ + B as X -+ E’. Direct integrat’ion, using Eq. (:LHj,

shows t,hat

whenever the functions A(r) and x(r) are related by A = ;iesp (iK.R). Equation (3.45) establishes the equivalence of the laboratory and center-of-mass system Eqs. (1.6) and (3.25). The equivalence of t,he outgoing boundary caonditions (10) is established by

s

dSv.W[G(r; co

r”), A(r)]

= eiK’R“

ds”“~W”[G(r, Im

tN), ii(t)]

(3.46)

which shows $(G, cp) = 0 implies s”(C?,9) = 0 and vice versa. Equation (3.46) is proved employing Green’s theorem in r space and r space, then integrating over R, using Eq. (3.44) and T = T + p.‘/2il/. This section supplements mainly Sections 2.1 and 2.21 and explains some assertions made in I (10). ACKNOWLEDGMENTS

I should like to thank the Physics Department at the University of Washington, Seattle for its hospitality during the summer of 1957, when much of this paper was written. I am much indebted to Mr. Benjamin Lee for his painstaking assistance in preparing and checking the manuscript. RECEIVED:

May 8, 1958 REFERENCES

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