Use of the Wigner Representation in Scattering Problems.pdf

7/27/2019 Use of the Wigner Representation in Scattering Problems.pdf

1/41

ANNALS OF PHYSICS 95, 4555495 (1975)

Use of the Wigner Representation in Scattering Problems*E. A. REMLER

Department of Pkysics, College of William and Mary, Williamsburg, Virginia 23185Received May 5, 1975

The basic equations of quantum scattering are translated into the Wigner representa-tion. This puts quantum mechanics in the form of a stochastic process in phase space.Instead of complex valued wavefunctions and transition matrices, one now works withreal-valued probability distributions and source functions, objects more responsive tophysical intuition. Aside from writing out certain necessary basic expressions, the mainpurpose of this paper is to develop and stress the interpretive picture associated withthis representation and to derive results used in applications published elsewhere. Thequasiclassical guise assumed by the formalism lends itself particularly to approximationsof complex multiparticle scattering problems. We hope to be laying the foundationfor a systematic application of statistical approximations to such problems. The formof the integral equation for scattering as well as its multiple scattering expansion in thisrepresentation are derived. Since this formalism remains unchanged upon taking theclassical limit, these results also constitute a general treatment of classical multiparticlecollision theory. Quantum corrections to classical propogators are discussed briefly.The basic approximation used in the Monte Carlo method is derived in a fashion thatallows for future refinement and includes bound state production. The close connectionthat must exist between inclusive production of a bound state and of its constituents isbrought out in an especially graphic way by this formalism. In particular one can seehow comparisons between such cross sections yield direct physical insight into relevantproduction mechanisms. Finally, as a simple illustration of some of the formalism,we treat scattering by a bound two-body system. Simple expressions for single- anddouble-scattering contributions to total and differential cross sections, as well as for allnecessary shadow corrections thereto, are obtained. These are compared to previousresults of Glauber and Goldberger.

1. INTRODUCTIONThis paper developes the elements of what may be called the Wigner representa-tion of quantum scattering. It is well known that quantum mechanics can beformulated entirely in terms of density operators and linear operators (often calledsuperoperators in this context) that act upon the densities [l, 2, 91. The Wignerrepresentation of densities puts them in the form of real functions of the coordinates* Supported in Part by N.A.S.A. Contract NASI-11808.

455Copyright 0 1975 by Academic Press, Inc.Au rights of reproduction in my form reserved.


2/41

456 E. A. REMLERand momenta of the systems particles. In other words, density operator matrixelements are functions defined in the classical phase space of the system.Schroedingers equation, correspondingly transformed, is a linear equation forthe densitys time dependence. It looks like the equation governing a Markovianstochastic process in phase space. The integral form of Schroedingers equation,the Lippmann-Schwinger equation in the usual Hilbert space representation ofthe theory is transformed into an integral form of the equation of continuity.The equation of continuity relates densities, currents, and sources. Thus, upontranslation into the Wigner representation, the fundamental equations of quantumscattering appear in terms of these physically graphic and transparent objects.The main purpose of this paper is to develop and stress this picture and, inparticular, to indicate how its graphic intuitively appealing nature may beexploited in the formulation of approximations to complex multiparticle scatteringproblems.The formalism encompasses both classical and quantum scattering. In fact,the Wigner representation is especially constructed to go over smoothly to theclassical limit. In taking this limit, none of the equations change their form nordo any of the symbols appearing in them change in interpretation or role.Schroedingers equation goes over into the Liouville equation and its integralform similarly emerges as the classical integral equation of motion obtained herein its most general form [IO].Quantum mechanics assumes a classical guise in this formalism; in this, liesthe source of its intuitive power. Nevertheless, since we have here merely anotherrepresentation of ordinary quantum mechanics, all of the latters wave and inter-ference properties must be hidden within. What has happened is that this transla-tion of quantum mechanics necessarily gives birth to a nonclassical stochasticprocess. Densities are not positive definite and in addition exhibit long-rangeoscillatory behavior-just that needed to produce interference. Thus, althoughthe total structure of the formalism developed here and the various roles playedby its elements (densities, sources, etc.) are isomorphic to classical theory, theparticular functions needed to represent these elements will differ considerablyin some respects from those of classical stochastic theory.These qualitative remarks point to the circumstances in which it may or maynot be useful to employ this representation. The property being observed shouldnot depend critically on high-order interference effects or concomitantly on theexistence of certain long-range order or correlations. There also should be asignificant advantage in being able to visualize at least some of the main participantsin the process under consideration as having simultaneously defined positions andmomenta while moving along trajectories in classical phase space. We shall callsuch circumstances quasiclassical, to be distinguished from the more restrictiveterm, semiclassical. The latter requires that wavelengths be short compared to


3/41

WIGNER REPRESENTATION 457interaction length parameters in which case particles can be considered to moveas if on a continuous classical path both in the force field and out of it. Quasi-classical is a broader and less sharply defined category that depends on the propertybeing observed as well as on the state of the system. The most commonly knownquasiclassical problem is that of obtaining thermodynamic and transport propertiesof gases under ordinary external conditions. Maxwell-Boltzmann statistics andoften billiard ball models may be used even when the typical molecular wavelengthis comparable to molecular size-a condition that violates the semiclassicalcriterion. Generally, quasiclassical scattering problems have been difficult to treatin the usual Hilbert space representations and we may hope that the approxima-tions engendered by the Wigner phase space representation will be complimentaryto those previously established.An example satisfying these conditions occurs in medium and high-energycollisions between various projectiles and nuclei. Elastic scattering, most of itdiffractive, is treatable by the Glauber approximation, the multiple scatteringexpansion, the optical model, or various combinations of these. This is a highlycoherent process depending critically on high-order interference effects. Inelasticscattering, in which the target is left in one, or possibly a few, well-defined finalstates, also may be treated by such methods. This leaves about half of the possiblereactions unaccounted for-the nonelastic collisions-in which many nuclearspecies may be produced in assorted multiplicities and momenta. Since these finalstates are so complex, one generally observes averages over them such as inclusivecross sections and multiplicities. This washes out most high-order interferenceeffects. Since this formalism is written directly in terms of the density matrix, suchaveraging may be done automatically with ease. Finally, perhaps the most impor-tant attribute this formalism brings to such a problem is its quasiclassical guise.These collisions involve may particles, high orders of multiple scattering, andpossibly, collective motion. It is very difficult to see such phenomena in terms ofwaves in multidimensional position or momentum space. The minds eye isirresistably drawn to hydrodynamic, thermodynamic, or transport theory typepictures in which joint average distribution functions in position and momentumplay a central role. In contrast to previous ad hoc application of such concepts tosuch problems of scattering, they occur here as natural and systematic approxima-tion procedures of ordinary quantum mechanics.Another field that may be mentioned is chemical reactions. Here again com-plexity is most often the rule, statistical averaging is the natural ally, the importantsteric properties of the compounds are easily representable and, an added positivefactor, short wavelengths often made the problem semiclassical as well as quasi-classical. The Wigner representation is especially suited to the semiclassical limit.There are other potential areas of application [ll] that will not be discussedsince those already mentioned should serve to illustrate practical reasons and


4/41

458 E. A. REMLERrequisite criteria for use of the representation. However, it should be noted finallythat the Wigner representation has had a long history of use in, especially, transporttheory [12]. What is being developed here is a version suited to collision phenomenaon complex yet microscopic systems.Section 2 gives elementary definitions, examples, and theorems connected tothe Wigner representation. A bra-ket notation is introduced to designate elementsof the vector space of density operators. The mapping of operators from theirusual representation into the new one is given. Schroedingers equation is thentransformed into the new representation.In Section 3, the semiclassical limit is discussed very briefly. The propagatorfor finite time translations on densities (not pure states) is obtained as a perturba-tion series in terms of its classical, not free, value. Thus, just as one thinks of wavespropagating freely between successive scatterings by a potential in the usualrepresentation of perturbation theory, here a particle moves on its classicaltrajectory between successive quantum jumps. These ideas lead to practicalformalas for computing quantum corrections to semiclassical processes, but thisbranch of the subject will not be developed in this paper.Section 4 introduces the most basic concepts used in scattering in the contextof potential scattering. The time-independent integral equation for scattering isderived. The effect of the potential on the incoming stationary flow of particlesfrom the accelerator is expressed by a source distribution function in phase space.This gives the net rate of production of particles being produced by the potentialwith a certain position and momentum according to the heuristic quasiclassicalinterpertation of the symbols in this formalism. Source functions are in manyways analogous to transition matrices, but here they occupy an even more centralrole in the theory. The relation between the solution to the integral equation andthe observed scattering cross section is established by noting that the spatialintegral of the local production rate of particles of a certain momentum is thetotal production rate, which in turn, is the incident flux times the differentialcross section.

In Section 5, the results of the previous section are extended to the multiparticleproblem where, it is expected, the most effective use of the formalism is to befound. Formulas continue to be analogous to standard results of scattering theoryin multiparticle Hilbert space.In Section 6, the multiple scattering expansion is derived. The result is quitesimilar to that of Watson. Here the role of the fully off-shell transition matrix istaken by a jump operator.The general properties of the jump operator for two particles interacting viaa phenomenological potential are fully as mysterious as those of its counterpart,the transition matrix. Much energy has been expended but little physicalintuition can illuminate the off-shell properties of the latter complex function. In


5/41

WIGNER REPRESENTATION 459Section 7, two limiting forms of the jump operator are examined, the classical andthe dilute. The relation between a particle moving on a classical trajectory and theconcept of sources previously developed is explained. In the dilute limit, in whichall particles are very far from each other compared to all other length parametersof the problem, we obtain the basis of the Monte Carlo method [4]. However,having obtained this not as an ad hoc procedure but as a well-defined quantummechanical approximation, one sees mmediately how its range of application maybe properly extended to include bound final states [5] and, in addition, howsystematic improvements in the approximation may be made for less dilutesystems.In Section 8, the multiple scattering formalism is developed further and thenapplied to find expressions for various simple but important production crosssections. In additions, the close relation between inclusive production of abound state and of its unbound constituents is discussed. It is pointed out howcomparisons between these measurements can provide information on productionmechanisms.In Section 9, as a final illustration of the formalism we look in some detail at thesimplest problem to which it might usefully be applied, scattering of an elementaryprojectile by a two body bound state in the dilute limit. The lowest-order term inthe multiple scattering expansion gives immediately both the differential crosssection formula first derived by Goldberger for quasifree scattering by a boundparticle [6] as well as a corresponding expressing for the total cross section. Thesecond-order term gives an obvious shadow correction to Goldbergers formula(the effects of which may be seen quite dramatically in reducing the backward peakin proton-deuteron eIastic scattering at intermediate energies) [13], a shadowcorrection to the total cross section, which is compared to that of Glauber [7] and,the expected double scattering contribution to the differential crosssection. Allthese results are straightforward to obtain and transparent in physical meaning.They are written in terms of observed cross sections. They are extendable ina straightforward fashion to more complex systems and also to nondilutesystems [8].

2. ELEMENTS OF THE WIGNER REPRESENTATIONWe begin with a resume of previous results concerning the Wigner representa-tion. Details of certain standard derivations that may be found in the earlierliterature are omitted. Initially, discussion will be confined to the case of onespinless particle. The extension to more than one spinless particle is straightforwardand will be used in later sections. Throughout the paper, we ignore spin and:in-

distinguishability of particles.


6/41

460 E. A. REMLERLet 0 be any operator on single-particle Hilbert space. Its Wigner representativeis defined to be [l]:

Ou(x, p) = 1 dy eiP.y x - :Y I 0 I x + i!Y)s

(1)= dq eiq.x(p + &q j 0 1p - +q).We will give some simple examples. Let X and P denote the ordinary position andmomentum operators and let 0(X, P) denote some functions of these operators;then

(mm& 69 P) = O(x)and (w%u (XTP) = 060. (2)When 0 depends on products of conjugate operators, then their ordering will havedue effect. Another instructive example is the density operator of a normalizedGaussian wave packet state in Hilbert space. The Wigner representative of thisoperator is

(3)where 52and @ are averages and dx, Apa = +.For any operator 0,

= j 4 CW3 Owtx, P),(4)

= 1 dx (2~)-~ OttAx, P),j 0 I = j- dx (x I 0 I x)

(5)= s dx dp (27r)-3 Ow(x, p).In particular, if 0 is a density operator p, then the normalization condition is

IPI==l. (6)It can be seen that pw appears to be a sort of joint probability distribution func-tion in phase space, the volume element of which is given by

dx dp (27~)-~ = dx dp h-3


7/41

WIGNER REPRESENTATION 461(since h = 1). We find it convenient to denote a point in phase space by a singlesymbol

v = 1% P>and correspondingly a volume element as

drp = dx dp (2~)-~.It is straight forward to show in general that for any two operators 0 and 0,

I 00 I = j- dg, O,(T) O,(T). (7)In particular, the expectation value of any 0 in the state described by p is given by

I PO I = s 6 pdv) ow(~).

This is consistent with a probability density interpretation of pw . What is incon-sistent with this interpretation, however, at least in the ordinary sense, is the factthat although pro s necessarily real, it is not necessarily positive everywhere. Thus,Jde dp, pw(y) cannot refer to the probability of a realizable measurement (measuringwhether the particle is in dv) for arbitrary regions, LI v, of phase space. This isconsistent with the quantum mechanical fact that to every region of phase spacethere does not necessarily correspond a physically realizable measurement. Inparticular, dp, might violate the uncertainty principle. As we shall see, this factdoes not appear to affect the heuristic value in thinking of pw as a joint probabilitydistribution function in the quantum scattering formalism to be developed.Equation (1) maps Hilbert space operators onto phase space functions. Theinverse mapping may be performed with the help of the operatorR(x, P> = 1 4 eiaex I p - .$q)(p + $q 1 (9)

which has Wigner representative(ax, PNW w, P) = cw3 6(x - x> %P - P) (10)or

(~(FNtlJ (94 = SC% v>* (11)Thus, Z?(F) maps into the Dirac delta density in phase space. Although it is not atrue quantum mechanical density operator, since it does not describe a physicallyrealizable state of the system, it will be a convenient heuristic device to speak of it asa density.


8/41

462 E. A. REMLERWe not note some useful properties of R(q);

s 4 R(v) = 1, (12)I W9J)l = 1, (13)Ow(v) = I OR(v (14)

I R(v) R(F)l = V% 90. (15)The above equations imply

In fact, it is easy to see that this integral has the same Wigner representative as 0.Therefore, it must equal 0 as long as distinct Hilbert space operators map intonecessarily distinct phase space functions. The latter statement is true because theWigner representation is accomplished via a Fourier transformation on the matrixelements of 0.Another bit of notation is now introduced which serves, among other things, toenhance the analogy between the usual quantum scattering formalism and therepresentation of it to be developed here. 0, is considered as an element of a linearvector space [2] and written in braket notation as I 0).

IO) = 0, (17)

We also write

from which one getsI R(v)) = I v>, (19)

CT I 0) = Odd (20)(Y I 9) = XT, T? (21)s dp, I v>(v I = 1. (22)

Quantum operators appear in this formalism in two distinct ways: as kets, of thetype just introduced, or as operators on kets. Generally, density operators are


9/41

WIGNER REPRESENTATION 463mapped into kets. Now let 0 and p be any Hilbert space operators (p generallybeing a density operator), we write

OLl p) = lop),ORIP) = IPO). (23)

Previous authors have shown [3], in effect, that the matrix elements of theseoperators are given by

where fl is the Poisson Bracket operator,

Alternatively, we note(Y I OL I F> = = I R(#)OR(y)l.Thus, they are related by transposition

(27)

.

We are now prepared to consider the form that dynamics takes in the Wignerrepresentation. Schroedingers equation isa/atp = -i(Hp - pH) (29)

as written in the usual Hilbert space. In terms of Wigner representatives, thisbecomes a/at I p) = -i 1Hp - pH)= -WL - HR)I P>-DIP>.

(30)


10/41

464 E. A. REMLERIt is easy to see from the preceding that the hermeticity of H implies

H,* = H R (31)(complex conjugation taken with respect to the q~ representation), from which isobtainedIn addition, we note D = D. (32)D = -DT. (33)The antisymmetry of the time evolution operator is sufficient to prove conservationof probability and time reversal invariance.

If (p [ p) were a true positive semidefinite probability density, then p(t) wouldcorrespond to a true stochastic process. One aim of this paper is to demonstratethat the analogy to statistical physics implied by a formal probability densityinterpretation of (g 1p) may be extended to all the basic equations of quantumscattering theory, which thereupon assumes most naturally the guise of a sto-chastic process in classical phase space. The situation encountered here with p issomewhat reminiscent of that of the Dirac delta function. The latter is not a truefunction (nor can the various commonly used distributions containing it correspondto true quantum states). Nevertheless, it is extremely useful to treat it formally as afunction (and to use nonnormalizable eigenfunctions) during intermediate stagesof a calculation. One knows that such distributions have meaning when finallymultiplied by test functions and integrated. Similarly, (p 1p), while not a trueprobability density may be formally treated as such, and usefully treated as suchduring intermediate stages of a calculation. At the end, one must not ask of theformalism quantum mechanically disallowed questions such as: What is the proba-bility for finding a particle infinitely sharply localized in phase space? Providedonly that sensible questions are asked, quantum mechanically correct and sensibleanswers will be obtained. Mathematically, this again will entail intergration withtest functions of a certain class.

3. QUANTUM CORRECTIONS TO CLASSICAL DYNAMICSThe dynamical equation can be reexpressed as

alat = H(d(--2 sin WWF I P>, (34)where H(v) is just the classical Hamiltonian. The first term in the expansion of thesine gives the classical Liouville equation

alat = --H($n CT I P>- (35)


11/41

WIGNER REPRESENTATION 465Let us call D, the operator that generates classical time evolution;

(36)It is clear that the solution to the classical problem is given formally by

eDct 9J>= I d% t>>, (37)where vC is the function that gives the coordinates of the phase point of the classicaltrajectory at time t such that y&y, 0) = y.Let A(t) be the operator that corrects the classical propagator,

It satisfies e Dt = d(t) eDCt.a/at n(t) = eDt(D - DC) eeDCt

= deDCt(D - DC) eCDCf.Its perturbative solution with initial condition d(0) = 1, is

(38)

(39)

d(t) = 1 + 1 dt eDCt(D - DC) epDCt0

+ Iot dt Jot dt eDct(D - DC) eeDCt. eDct(D _ DC) e-Dct + . . . . (40)

The first quantum correction to the matrix element of the time evolution operatoris thus given bycv' 1 ,Dt - e%t I v> = Iot dt' (w(F', -0 I D - DC I wAy, t - t')> + a*- (41)

using Eq. (37).The interpretation of this equation is that a particle may travel from q~ o v in atime t via an infinite variety of paths in phase space. Each path is itself made up ofsome number of classical path segments, the first beginning at q~and the last endingat q (Fig. 1). The particle moves classically along the first segment, performs aquantum jump through phase space to the second when it reaches the end of thefirst, and proceeds in like manner until the end. Each quantum jump, say, thatgoing from point y1 to vz, occurs with a probability (vz / D - D, 1 c&. Thisprobability may be negative. In fact, due to the antisymmetry of both D and D, ,the time-reversed jump always occurs with a sign opposite that of the time-directjump.


12/41

466 E. A. REMLER

T = t-t'

FIG. 1. This illustrates the first-order correction to the classical propogator given in Eq. (41).A particle is known to be at p initially and one asks for the probability of it being at p at a timet later. In addition to the classical value of this probability there is a contribution correspondingto the particle traveling a time I - t along its initial classical trajectory and then performing aquantum jump to a new classical trajectory which brings it to I in the required time 1. Thetrajectories are pictured in position space emphasizing the fact that for a local potential there isa jump in momentum only as with a classical stochastic impulse. The probability that the particlemakes the jump is given by Eq. (41).It is interesting at this point to exhibit in more detail the jump probabilityoperators matrix elements for potential scattering. If H = X2/2&fis easy to see that only V can contribute to D - D, . We have

(9) I VL I 90 = I NT) VR(p,)l

so that

= s dy df ,+~Y+ip~Y-Ilx+~y)(x-~yIvIx+~y)(x-

= 6(x - x) j dy e+~-~)v(x - +y)

t V(P) then it

$Yf II(42)

(p' I D - DC I q~'>= S(x - x'> Ij dy ei(p--p)T(V(x + $y) - V(x - iy))- (21T)3(a/ax V(x) * a/ap S(p - p.))/. (43)

This shows that a particle may experience a finite jump in momentum but not inposition, which is what is expected for a particle subject to a random force. Onecan show that the average of this random force is just the classical force.

4. CURRENTS, OURCES, ND CROSSSECTIONSN POTENTIAL SCATTERINGIn this section, we wish to develop the essential ideas of this paper in the simpli-fied context of potential scattering. These results will be extended subsequently tothe general multiparticle multichannel case. The plan here is first to derive theintegral equation for scattering-the Lippmann-Schwinger equation in the Wigner

representation. Next, we want to relate the solution of this equation to the cross


13/41

WIGNER REPRESENTATION 467section, and finally, to interpret the symbols appearing in these results in a waythat brings out clearly the analogy to statistical physics.Let D, = -i[(P/2M)L - (P2/2M),] (44)be the time evolution operator in the absence of interaction. Explicitly, using theexpansion in terms of Poisson bracket operators, it is

(9 I D, = --v * a/ax (F I , (45)where v = p/M. Let 1p(t)) represent any density of freely moving particles,

alat 1p(t)> = 4 1 (t)> (46)and consider a density I p(+)(t)) satisfying/ ,.4+(t)> = / p(t)) + J:, dt eD@)(D - D,) / p+(t)). (47)

Then, clearly a/at 1p(+yt)j = D I p(+yt)) (48)andpm (I P+(t)> - I p(t)>> = 0. (49)

Thus, 1p(+)(t)) is that solution to the potential scattering problem with theboundary condition that it approach a specific free density 1p(t)) as t + - cc.One already knows that solutions to this integral equation exist in which thedensities correspond to wavepackets. More generally, here they can be mixtures ofpackets. We now assume that an exponential convergence factor will suffice toallow limits in which p(t) is stationary to be taken in the integral equation. Thisleads to the definition1 p(+)(O)) = I p) + [j-4, dt e-(Do-Z)] (D - D,) ! p+(O))

= 1 p) - (D, - z)- (D - D,) 1 p-(O))with

D, I p) = 0. (51)If, in addition, the limit as z - 0 of the solution of this time independent equationexists, algebra suffices to derive

whereD / pf) = 0,

li+i / p+(o)) SE 1PC+).


14/41

468 E. A. REMLERAnother representation of the time-independent equation that will be seen tohave a very interesting and important interpretation is obtained by noting that

eDOt x, p) = I x + vt, P> (52)from which, by transposition, one obtains

(x, p I emDot = (x + vt, p [. (53)This, inserted into the integral equation yields (limit notation implicit)

(x, P I P+) = + sf, dt (x + vt, p I D - D, I p+). (54)One can now verify by direct substitution that when p represents a plane wave of,e.g., momentum pE and unit density

I p> - I 0 = j- dx I x, PE)= Gw3 ICIPEXPE I)>

(I pE), a Hilbert space ket), then in factI E+) = G-I3 ICIP%(+PE ID (56)

satisfies the time-independent equation in the limit z -+ 0, 1pk:) being the out-going wave solution to the Schroedinger equation. Thus, the limit of this equationis not vacuous and possesses the physically expected solutions.We next show how to obtain the correct formula for the cross section based onthe assumption that (p I EC+)) may be treated as a density. The purpose in doingthis is to develop and emphasize the heuristic power of the symbols in the forma-lism. In addition, the resulting procedure is remarkably simple. A rigorous deriva-tion for potential scattering, leading to the same result, has been given elsewhere [5].Under this assumption then, the current density of particles of momentum p is

andJE(x,PI = v


15/41

WIGNER REPRESENTATION 469that an incident particle of momentum pE be scattered by the potential into thecorresponding interval. Thus,

Q(p, PE) ZE = s JE . dy (27r)-3.sApplying Gausss theorem, we get

G(P, PE) VE = (27F j. dx SE(X, P), (59)where the integral extends over all space and the source function SE is defined by

f&(X, P> = a/ax 'JE(K P)= v . a/ax (x. p 1 EC+))= -(x,plD,/E+)= (x, p / D - D, j Et+).

The equation of continuity when applied to a stationary distribution shows thatthe divergence of the current, the source function as defined here, does indeed givethe local rate of production of particles within the element ds, = dx dp (2~)-~.Therefore, our equation for the cross section merely states that the total rate ofproduction of particles in interval dp is the integral over all space of the localproduction rate. It is easy to see that the usual differential cross section is given by

U(P, ~4 = I P I- %I P I - I PE I) d4dQ. (61)The equation for the source function in terms of the density function can berewritten as

SE(q) = s dq (g, 1D - D, 1$j(p;' ! Et+)\. (62)Now (y / D 1 IJJ) is the total probability per unit time that a particle at y will jumpto cp, while (40 I D,, / cp'j is the contribution to that probability due to simple freeparticle streaming. Thus, by the above equation, since dq( F j E(+j is the numberof particles in steady state in dy, S,(v) dg, is the rate at which particles are jumpinginto dg, due to the potential. This is equal and opposite to the rate at which particlesare jumping into dzp due to streaming. The net rate at which particles are jumpinginto dp, is zero since we are in steady state, i.e.,

f dcp (p, I D I y)( y I EC+) = 0. (63)


16/41

470 E. A. REMLERThe Wigner representation of the Lippmann-Schwinger equation for the timeindependent wavefunction of the scattering problem can be rewritten as

=


17/41

WIGNER REPRESENTATION 471We find it useful to define, as before,

RD(yD) = s dq eiqxD I PO + k>(PD - h I. (68)This is now a two-body operator, but it has much the same properties as before. Inparticular,

I dw, RD(VD) = SD (69)I RD(wJI = 1, (70)

I RD(TD) RD(FD)~ = a(~, > TO), (71)where pD = xD , PO, and gD is the projection operator for D. Using an obviousextension of the expansion theorem written in terms of the elementary particledensity operator R(v) that was discussed previously, we have

Carrying through, the algebra yieldsRD(YD) = j- dg, wD(?) R,(R) RAoA (73)

where x1 , p1 , xz , pz are the linear combinations of Xn , Pn , x, p appropriate to thecenter-of-mass transformation andwD(x, p) = j dq eiq (P - $q I D)@ I P + iq> (74)

is just the Wigner representative of the internal structure wavefunction of D.The set of internal phase coordinates of a bound system will be given the genericsymbol [. Thus, in the simple case we have discussingbD > PD ; x, p> = i?D ; $Dh

(2~)-~ dXD dpD (27~-~ dx dp = dcfD df, ,etc. If the system D were to be instead, a three-body system, the set 50 woulddenote two internal phase points and so on. In all cases of N elementary particlesbound into a state D we write

(75)(76)


18/41

472 E. A. RJZMJLERConsider next the integral equation of the stationary density describing a beamof particles B scattered by a target A in the laboratory frame. Both A and B ingeneral may be bound states. Assume the whole system to be made up of N elemen-tary particles. The incoming channel will be denoted by the subscript AB. Theincoming channel Hamiltonian is HAB . In the absence of any interaction between

A and B, the incoming density operator satisfies the Schroedinger equation

or a/atP,m(t)= -~(HABPAB(~) PAB@HAB)alat 1 pAB(t)) = DAB 1 PABct)). (77)

Proceeding as before, we obtain(78)

for the stationary scattering state.To obtain an expression for the cross section, we again construct a standarddensity describing the unperturbed system1 PAB) = I AB) = 1 RA@ 0) J ~XB &(XB 3 PB)). (79)

At this point, the objection might well be raised that since this operator does notreally represent a physically realizable quantum mechanical state of the system, anerror may be made in its use as the unperturbed initial state in a quantum scatteringproblem. A more detailed derivation, given in the Appendix, shows that this is notthe case. It is yet another example of a phenomenon noted earlier; the formalism isindifferent to the quantum nature of the problem. This density describes a targetparticle A localized at the origin with zero momentum, while at the same time asteady stream of bombarding particles B flows in with momentum pB and fluxUB = 1 VB / = 1 PB/MB / .

Due to the interaction, this distribution is altered to become1AB+) = 1AB) + [J:m dt eMDAB] (D - &) ] AB+). (80)

Suppose one were interested in the inclusive cross section, A + B -+ C + anything,where C is some, in general, bound state of the system. Asymptotically far fromthe origin, the components of C, if it is a bound state, will have a negligable


19/41

WIGNER REPRESENTATION 473probability of being close together unless in fact they are in their bound state. Thus,at large distances (yc / AI++)), where

I rpc) = I M~cD (81)is the probability density for finding C within &I,. Note that this inner product,since it is in fact the Hilbert space trace of R,(yc) with the stationary densitydescribing the collision, includes an integration over unobserved final state particlesand an integration over [c , the internal coordinates of C. The current of particlesC far from the region of interaction is

JAB = VC(FC AB?. (82)Proceeding exactly as in potential scattering yields, now for the inclusive productionrate

where Sc:,a(~)c) = a/& . JAB(~= vc . a/ax, (qJc I ABf).

Note next that if Hc is the Hilbert space Hamiltonian acting on the subspace ofpracticles in C such thatHc I PC) = (P~/~~c)I PC), (64)

where / p,-) is the Hilbert space eigenket of momentum pc, then it is straight-forward to prove that--iVWc(q~c) - Rc(w) Hc) = Vc . a/a% &(yc). (85)

This implies DC yc) = vc . a/h I vc,cqc I DC= -Vc . a/a% (FC !/Ct 1& = j e-HCt&(vc)e-=Ct)

= I R&c + vd, PC))= I xc + vet, PC).

(86)

(87)

Further, note that


20/41

474 E. A. REMLERwhere DC is the generator for all particles not included in the cluster C. This is truebecause the symbol (vc 1 as used here contains an implicit integration over all thephase coordinates of these unobserved particles; this fact, in conjunction withEq. (12), yields the result.Combining Eqs. (87), (88), and (83) yields

&:AB(w)


21/41

WIGNER REPRESENTATION 475form is important. In fact, were the order to be reversed, the integral would vanishsince for fixed xD , (vcyD 1A?(+)) must rapidly vanish as 1xc 1+ co, and viceversa, due to the necessarily correlated nature of the particles emissions. We cantake advantage of this fact and add to the above integrand the perfect differential

VD * w&l


22/41

476 E. A. REMLERwhere

GAB = f dt eeDABtJ-03 (99)= -(DAB z)-1; z-++oandlAB = D - DA; (lfw

are the Greens function and interaction operators, respectively. As in the case ofthe Hilbert space representation of the problem, there is an alternative form for thisequation:where 1 ABC+) = 1AB) + GZAB / AB) w w

G = -(D - z)-.The full Greens function G may be expanded in the usual manner. We write theinteraction as a sum over pairwise parts

D=D,+Z (102)z= CL, (103)awhere 01 uns over all pairs of particles in the system A + B. Now define a jumpoperator by

Ja= I, + I,G,J, (1WwhereGo = -(Do - z)-. (10%

Thus, the two-body jump operators take the place of the two-body transitionoperators in the usual development. However, there is one interesting differencebetween the two representations. Neither the Greens function nor, consequently,the jump operators used here have the energy of the A + B system as a parameteras is the case with the transition matrices in the Watson expansion. Thus, there isno possibility of an off-shell or half off-shell type jump operator and, in fact,the same jump operator is valid for any energy and any system in which the cor-responding two particles appear. This fact does not effect the algebraic structure ofthe equations, which is the same here as in the usual analysis so that we get finally

G = Go + G,,JGo , (106)J = Z + ZG,J, (107)

Jo = I + JG Zc J(a) = ;

0 L1, (108)(109)OL

Jcol) = J, + c JbG,J, (110)B# m


23/41

WIGNER REPRESENTATION 477GL = Gda -k GoJGd,

= G& + Go[J(a) - Im]= G,,J(o). (111)The last result inserted into Eq. (101) yields

where the prime on the summation indicates that a is restricted to all pairs notinteracting in the AB channel. The perturbation expansion of Eq. (110) insertedinto Eq. (112) is the multiple scattering equation:I AB+> = I AB) + c GoJ,, I AB) + c c G,,J,G,J, / AB) + ... . (113)a 0: Bfcr

7. PROPERTIES OF THE JUMP OPERATORBefore extending the multiple scattering theory begun in the previous sectionfurther, it will be useful to examine the properties of the elementary operatorsappearing in the expansion given in Eq. (113).Consider first potential scattering and the matrix element

(Y I 0 I cp> = =s ODt v . a/ax (p 1eDt 1F>.0

(114)

According to the probability interpretation of the symbols developed in this paper,CT I eDt 1 v> is the probability density of a particle to be at v at a time t after itwas localized at v. The integrand in (114) is thus the divergence of the current setup at t at v. Applying the equation of continuity we get

= j-= dt S(v, t> + CT I v>,where S(y, t) describes the sources due to the interaction and p(y, t), the particleprobability density at any time t. Since S(v, t) is a time rate of production, its timeintegral equals the net production. We can now describe the situation as follows:An external source pumps a net of one particle into the system at 9). The system


24/41

478 E. A. REMLERover a period of time reacts and redistributes this source distribution from ( y [ y)to (v 1 u 1 CJJ).Since u is in the nature of a probability one must have

(115)This is in fact satisfied by the theory already since, using Eq. (106).

and

identically.

u = 1 + JG,, (116)s d# (F I J = 0 (117)

The relation between the operator CJand the function SE(~) already introducedin the discussion of time-independent potential scattering (Eq. (60)) isSE(TJ) = j db (9, I u I m) UB, (118)

where xE = (bE , E ,) and bE is the impact parameter two-vector perpendicular tothe entering momentum pE , and zE -+ - co, along the beam line, where the accele-rator is situated. The equation says that the accelerator is a source distributedrandomly over the plane zE = -cc with an average density of one per unit area.The system reacts to this external probe, by redistributing this density. Thus aparticle introduced at TE will reappear at F with probability density (p ( u I TE).The cross section is

O(P, PE> = j dx W7F3 db


25/41

WIGNER REPRESENTATION 479To see what this says, imagine a numerical integration of the classical equation ofmotion. Time is segmented; the first interval is 0 < t < t, ; the next is tI S t < tz ,etc. During the ith interval, the trajectory is approximated by a straight line

xi(t) = x,(O) + v*(t - tg)X,(G) = Xi+,(O) (122)

while the momentum pi is constant. The momentum changes discontinuously, asif by an impulsive force, at the instants t, , tz ,. . . Thus,

(123)= < qJ' I fp> + 1 [-(4 I xi+1 , Pi> + CT I xi+1 9 Pi+21i=l

having used,-DoeDot = -(d/dt) eDot,(Xl 9 PI) = (5 P) = V-

Thus, the reaction of the system in the classical limit to an imposed unit sourceat C+Is a line distribution along the classical trajectory of sources and sinks. At eachspatial point subsequent to x, there is a unit sink for the old momentum going intothe point and a unit source for the new mementum leaving the point (Fig. 2).

P ttlb,t1/.:.I 2 3 4

x=x, PI x, Pz x, Ps x, p4

FIG. 2. This illustrates the classical limit of scattering in terms of source functions as givenin Eq. (123) and discussion subsequent to it. A circled portion of the spatial trajectory vo isenlarged. The continuous force is approximated by a sequence of inpulses. A particle starts atx1 with momentum pl. Thus at x1 there is a unit source of such particles. At x8 it receives animpulse, changing its momentum from p1 to pz . Thus, at x2 there is a unit sink for p1 and sourcefor pz , and so on. In this picture, classical motion is a consequence of the potentials creation of aline of sources and sinks in response to the externally imposed source at x1, pI .


26/41

480 E. A. REMLERAnother limit of considerable practical and conceptual interest that can bediscussed in some generality may be termed dilute. This will be applied to amultiple scattering problem when the distance between successive collisions is largecompared to other characteristic lengths of the problem, wavelengths and rangesof potentials of particles. In particular this limit implies no necessary orderingbetween wavelengths and ranges, whereas the classical limit requires wavelengthsto be smaller than all other characteristic lengths.Consider, for simplicity, scattering by a system of fixed scatters centered atpoints x, . The multiple scattering series is still given by Eq. (113); the subscriptsnow range over fixed scatterers indices. In any one of the terms of the series inwhich the orth jump operator J, appears it operates on some particle density

function; call it 1p). In the dilute limit, 1p) is set up by sources centered far awayfrom x, . This leads to the conjecture that any such density must be slowly varyingover the scale of distance set by the size of the olth scatterer, the range of its poten-tial. We could then, expand I p) about x, and, in the most favorable case, keeponly the lowest order term; thus,

(124)

However, quantum mechanics implies the existence, in principle, of substantialoscillations in 1p) associated with interference. These would vary on the scale of awavelength and contradict the conjecture. This question is resolved in the furtherquantum mechanical requirement that sources be distributed over a certainminimum region in phase space. Oscillations due to distributed sources wash outincreasingly with distance; in the dilute limit they disappear.When the condition of diluteness is not strong enough to allay all worriesassociated with the approximation of Eq. (124) other factors may assist.Randomness is often invoked to accomplish phase averaging. A relatively densitypacked medium such as a liquid or a nucleus will possess only short-range order.The approximation in Eq. (124) might be good provided only that I p> contains nocontribution in which a near neighbor of cywas the last scatterer. Terms in themultiple scattering series that describe successive scatterings by near neighborsmight be treated separately, leading to an expansion in powers of correlationfunctions of the medium.Another escape hatch opens if the wavelength of the particles (or particles) beingscattered is short compared to a long-range component of the potentials doing thescattering. It is straightforward to show that the multiple scattering series can berewritten so that the classical propagator Gc replaces G, . Concurrently, the jump


27/41

WIGNER REPRESENTATION 481operator is replaced by one that represents the stochastic quantum jump correctionto the classical motion, as discussed in Section 3. This correction is due mainly tothe short-range component of the potential. Such cores may be spaced widelyenough apart for the diluteness condition to apply.J, 1p) is the source distribution created by the olth scatterer when 1p} impinges.This must always be integrated over another density, say (p 1 , to calculate anobservable probability (p j J, / p). But the arguments applied to J, 1p), alsoapply to ip 1J, , using time-reversal invariance. Thus, one can write

= s dg, cs %x - x,) 6(x - XJ j dx dx (y / J, I cp). (126)In contrast with the general expression for u given in Eq. (120), this approxima-tion for the closely related jump operator is easy to evaluate and interpert. Notethat sdx I x, p) is the density for a unit incoming plane wave state. Therefore, it

can be written ascw3 ICIPX P ID (127)

where / p> is a Hilbert space ket. Next, note thatJ, = G;GI, , (128)

which, in combination with our integral equation for scattering, Eq. (101) yieldsJ, j dx I xpl = (27d3 G% P+)= (2d3 j dx (~7 P I GMl P+%P+ I)> - I(1 P>(P I)>1= (2~)~ j dx v - a/ax KP+ I W, P) I P+) -

I>

(129)

(130)


28/41

482 E. A. REMLERusing Eqs. (5), (7), (18), and (19). Inserting an expansion for R(x, p) such as thatgiven in Eq. (9) and writing

(P + &I I p(+> = (P + &I I P>+ (C(P) + io - 4 + $q))-l TdP + $a; p), (131)


29/41

WIGNER REPRESENTATION 483Hence we obtain in the dilute limit,Xv I J, I F> = 86 - x,)G7d66(x - x,1

{a(~ - P) 2 Im TAP; P) + %e(P) - c(P)) 2n I T,(P; P)l>. (139)Using the optical theorem and the relation between the transition matrix and thedifferential cross section

VU,oTAL(p) = -2(277)3 T(p; p) (140)WJp; P) = (27? &C(P) - C(P)) I T(p; P)12, (141)

where a,(~; p) is related to the differential cross section by Eq. (61) this result canbe rewritten as

. (243 {-6(p - p) Uo:oTAL(p) + YAP; PH. (142)This equation (142) says something that could have been written down right awayusing elementary considerations. In particular, a density ( p) impinging upon thec&h fixed scatterer gives rise to a source distribution function

(y I J, I P> = (2n13 w - 4 J dP {(257)-3 L.&Y 9 P I P>I. W(P - P) zoTAL(P) + %(PG PH. (143)

The first bracket is the incoming flux per unit area of particles between p andp + dp at x, . This is multiplied by two factors in the second bracket; the first givesthe number produced per unit dp at p, a positive source term; the second gives thenumber lost to the incoming beam, a negative sink term.

This result can be immediately generalized to include nonfixed scatterers. Let 01now refer to a pair of particles, e.g., particles 1 and 2 Then we have

= PI6 m2 - X12) WY2 - P12) W2) ~(x12)

. {- I Vl - v, I S(PL - Pl2) GYTAL(PIZ) + I Vl - vz I (J12cp;2 9 PI,)), (144)where X,, , P,, refer to phase coordinates of the center of mass while xl1 , p12 applyin the center of mass. This result may be obtained from the formalism but can bewritten down directly on physical grounds.The advantage gained in having derived this approximation via this formalismis that it is now part of quantum mechanical scattering theory. The possibility of595/9512-v


30/41

484 E. A. REMLERsystematic refinements to any approximation is thereby created. We have alreadydiscussed quantum corrections to the classical limit in Section 3. The possibility ofcombining the classical approximation (for motion through the long-range part ofa potential) with the dilute approximation (for transitions induced by short-rangepotential cores) has been mentioned. In addition, finite size corrections to thedilute approximation, arising when the range of the potential is not negligably smallcompared to the distance between scatterers, have been caiculated [8]. The relationbetween the dilute approximation and Monte Carlo calculations has been discussedelsewhere [ 151.

8. MULTIPLE SCATTERING EXPANSIONS OF INCLUSIVE CROSS SECTIONSThe previous section developed certain limiting cases that are useful for futureapplications and serve to illustrate and clarify the meaning of the basic jumpoperators that appear in the multiple scattering expansion. We now return to moregeneral developments of the theory. Formulas for the simplest inclusive productionprocesses are rewritten in a multiple scattering form, which facilitates their inter-pertations and computation in many cases of interest.It is useful to define a symbol (@).V for the sum of all scatterings beginning withthe 01and ending with the ,f3pair. Simple expansion in powers of the jump operators

will verify that it satisfies,(B)P) = .I$,, + c JB(l - 6,,) G,,(Y)J(OL)

= J,&, + ;. @J(l - a,,,) G,,J, . (145)Furthermore, using Eqs. (106) and (107), and some strictly algebraic manipulations,one can show that cO)J = IaS,, + I,GZ, . (146)

Particle B impinges on a composite target A. Consider first the inclusive sourcedistribution for B observed in the final state. Application of Eq. (89) shows&: AB(P) = (w I 1~ I ABC+)

= (9)~ I b(1 + Gfs-m) A&. (147)Since IB = IAB , the sum of all interactions between B and constituents of A, oneimmediately obtains

&;&cpB) = c (Q)B I cBa)JBa I AB) (148)a*a


31/41

WIGNER REPRESENTATION 485where a and a run over al l constituents of A. If, for example, A were a two-bodybound state composed of particles labeled 1 and 2, the multiple scattering expansionof Eq. (148) yields

~B;AB(TB') = (9~' IJBI + Jm + JB&,JB~ + JBI '&JB~ t . ' . I A@ (149)These expressions will be evaluated in the dilute l imit in the next section.

Expressions for the inclusive production of a single particle other than theincident one (knockout), follow in a similar manner. These source functions are,of course, to be integrated over space to give the total production rate or, equi-valently, the cross section.

Several examples of practical interest involve two or more particles observed inthe fina l state. They may be in bound or unbound condition. Our intuition wouldlike to relate, for example, a two-particle inclusive production rate to that of abound state of the same two particles. The present formalism brings this out in anespecially graphic manner.

The inclusive source function for the bound state production of D, made up ofconstituents 1 and 2 originally in A is given by Eq. (89):

S,,~P),,) = (qD I ZD 1ABC+:,.That for 1 and 2 unbound is given by Eq. (97):

(150)

whileID = 4, + 1 4, + 12, + 2 12, (152)

ail.2 Q#l,Z

z;, = Zo + Z12 (153)Thus, we see that the basic source functions required to calculate bound andunbound production differ only in the term (~i~~ 1 ,, 1AB(+)). This termrepresents final state interaction between the observed constituents. It is the sumof al l contributions to the multiple scattering series in which the very last inter-action is between the constituents of the observed final state. This final stateinteraction must be deleted in the computation of bound state production-included in the unbound.

Except for kinematics that especially emphasize the role of the fina l state inter-action (e.g., at low energies and low relative momenta pi - p2 of the observedpair), this term is unimportant. It is useful to define a source function that neglectsthis term,

%:AB~+&) = (91q32 I ID I AB+j. (154)


32/41

486 E. A. REMLERThen

uD;AB(pD) ZB= i/ dxD (27~)-~ f dx dp (27~)-~ WD@, p) S12;AB(&&) (155)

= s dx, (2?~)-~ dx, (2~r))-~ &AB(Q)&$), (156)where 6jlzzAB s the cross section less final state interaction contribution. In theabove formulas we have used


33/41

WIGNER REPRESENTATION 487in one for a factor of (27~)~~ n the other. Thus, the ratio

~D:AB(PD)/u12:AB(PDI2, PD/2)is related to the volume from which particles 1 and 2 emanate, and hence, themechanism of their production.

9. MULTIPLE SCATTERING BY A TWO-BODY BOUND STATE IN THE DILUTE LIMITThe multiple scattering expression for inclusive scattering of B by a two-body

bound state A is now evaluated in the dilute limit. The methods and results of thisformalism may be connected with others in the context of this well known example.The contribution first order in JB, to this process is given by (see Eq. (149)),s~XBW3 (YBJB~AB)Since we are in the laboratory system, p1 + pz = 0. As before

are the relative coordinates and momenta of 1 and 2.The dilute expression for JB1 can be written in the present notation, usingEq. (144), as(fPlP29)B I JBl I 9)lwPB)

= CR I ~zx@;ll I @Bl) GBl) ~(xd (277)-3 VB - vl 1 {-%&?l - PSI) &PB 3 PI) + QdIb 'P,'; PBkb (162)

@,, = X,, , PB1 are the phase coordinates of the Bl center of mass; FBI = xB1,PB1 refer to relative coordinates, specifically xB1 = xB - x1 andPSI = (MIPB - MBk)/(Ml + MB);

&(PB,P1), which really only depends on psi, is the total cross section for B on 1;UB~(PB, PI; PB , Pi), which really on depends on PBl and p;ll, is the differentialcross section for scattering into dp;, . Its relation to the usual center-of-mass


34/41

488 E. A. REMLERdifferential cross section is given by Eq. (61). Combining these equations andintegrating over spatial functions yields

s dPl dP dx @71r)-3 w4(x, P) w%, - Pm). 1 vB - v1 1 {-6(Pil - PSI) &pB 2 PI) + (TB1(P B', PI'; PB , PI))

= --6(I),' - PB) j & 1 &4(p)i2 / VB - v1 j &pB, PI)

+ j dP I #A( I VB - vl 1 (TB1(PB', PI'; PB, PI) (163)where now

PI = P pl' = PB - PB' + p.The first term gives the contribution to the total cross section due to collisions

of B with 1 only. If we add to this the corresponding contribution from 2, thefirst-order estimate of the total cross section is

The second term in Eq. (163) is the well-known single scattering term originallyobtained by Goldberger [6],

j dp I $A(p)I" 17 1 UBdpB', PI'; PB, PI), (165)to which one should add the corresponding term for scattering of B on 2.

The contributions second order in the jump operators provide shadow correc-tions to the first order results as well as double scattering. The contribution fromthe JBzGJBl term in Eq. (149) is

f dXB'(2r)-3 &'1'&'2'& dxB wx4(9)) (%'~'z'~B' I JB,G ,JB, 19)1P)2P)B). (166)Expanding over intermediate states and using

Go 1~1, 932, .-a> = jam dt I ~)o(vl, t), ~(972, 0, . ..> (167)where

wl(% t) = x + vt, PY


35/41

WIGNER REPRESENTATION 489we get

(plyz~a I JB&~JB~ I RRFB>=.r f; dt dy; dy; d& (P)~P)~P)B I JB2 I Y&V; 3 t) . ..>

kmi I JBl I 9)lWB)=.c scdt dy$ (F~F~ I JB2 I v,,(Y~ 7 t>> R,(v; > t)>

*!~ob,~ -11, f?J; JBl I g, Q. (168)Inserting this result into Eq. (166) and integrating over the spatial functions, we get

s Idt dpl dpz dp dpz, dx (27r-3 wA(x, p)0* 6(x - v,t + v;;t>{l 6 - vz I %PB + Pz - P;; - Pz)* F&P;; , Pz) %P& - P&J + %B(PB, Pz; P;; 3 PAI I VB - VI 1 a(& + PI - PB - PI) b&B 9 PI) s(& - PBd + aBl(pii 2 Pl'; PB 3 Pd). (169)

The spatial 6 function requires that at some time t after collision of B with 1, theposition of B, x1 + vit, coincides with the position of 2, x, + v,t. The only newnotation used in Eq. (169) is p;lz for the relative momentum of pi and pz , pbz forthat of PB and p2, piI for that of pi and pl and, PSI for that of PB and p1 .The term quadratic in Us represents a shadow correction to the total crosssection; the terms linear in uT represent shadow corrections to single scattering;the term quadratic in the differential cross sections represent double scattering. Weconsider each in turn.The shadow correction to the total cross section given by this expression is

- j dp (h--3 Irn dt / VB + I I 1i A(-(vB + V) f, p) d&B , VB - v-P) 1 7 1 &pB 9 Ph (170)

which must be added to a similar term coming from collisions in the reverse orderto get the total shadow correction to second order. This expression simplifiessomewhat in the high-energy limit, where it can be compared to the correspondingcorrection due to Glauber. Using1 VB f v 1 = LB

UB dt = dzs dp (277Y WA@, P) = I &&)I:


36/41

490 E. A. REMLERfor the bound state spatial density, we get,

Taking into acount that the contribution to the shadow correction coming fromthe reverse order of collision, one finds that the total shadow correction in thedilute limit is twice that obtained by Glauber [7] in his simple approximation toblack sphere scattering. The reason for the difference in these results may be tracedto the fact that they each hold true in different nonoverlapping regimes. Our resultis accurate in the dilute limit in which the Glauber approximation is invalid. Itshould be reemphasized at this point that the basic formalism, the Wigner represen-tation of scattering, is not limited to the dilute limit but is susceptible to otherapproximations that may lead to simple formulas valid in other regimes.The shadow correction to single scattering is composed of two parts. Theparticle casting the shadow may lie either between the accelerator or between thedetector and the single scatterer. In either case one gets a reduction in the singlescattering, a unitarity correction since it is merely a manifestation of probabilityconservation. The shadow correction to single scattering by 1, to be added to theuncorrected term given in Eq. (164) is obtained by straightforward application ofthe equations just discussed yielding again in the dilute limit,

- Jamt J4 CW-3 ] VI? + v 1 w4(-(VB + v) t, p) &PA -p)

+ 1 VB - v 1 w&B - v> & P> oB1(PB, PI ; PB > P>

(172)where pl is again given as in Eq. (164). A similar term reduces single scattering onthe other particle. The physical origin of these terms is transparent. See Fig. 3.We might note that the effect of such corrections is quite striking in reducingthe backward peak observed in proton-deuteron elastic scattering (~50 % correc-tion) [13]. It may be that a simple formula, such as that derived here (perhapssomewhat refined to take into account the not completely dilute nature of nuclearmatter), will be quite accurate in calculating such shadow corrections for manynuclear targets, including the deuteron. This would be especially helpful at those


37/41

WIGNER REPRESENTATION 491

FIG. 3. This illustrates two phenomena generated by the second-order term of the multiplescattering expansion for scattering of projectile B by a two particle bound state A (see Eq. (149)).When B hits constituent 1 it happens, in this instance, to have momentum x and be displacedfrom constituent 2 by x. The solid line shows the trajectory of B. Between collisions B, havinggotten intermediate momentum pi, displaces itsel f along vi t while constituent 2, since A wasinitally at rest in the laboratory frame, has momentum -p and has moved -pt. The collisionbetween B and 2 at a time t after the first collision has two consequences: First it reduces thenumber of (single scattering) events into pi, i.e., particle 2 shadow events shining from particle 1.This phenomenon is described by the first bracketed term in Eq. (172). Second, it increases thenumber of (double scattering) events into p B. This is described by Eq. (173). Note that the smallcircles denote the particles positions and the small arrows through them denote their momenta,not spin.intermediate energiesat which the Glauber approximation begins to break down.We also note that, as before, this formula simplifies considerable when vg is muchgreater than Y, the Fermi velocity.Finally, the double scattering term obtained from Eq. (169) is

j dp (27~)-~dp;; Jrn dt ~)B~w~(-(v;; + v) t, p)0

. / 6 + v 1aB2(pBP2;;; , -p) 1VB v 1 o,,(p;;p,; PB p), (173)where now momentum conservation gives

PI = ps + p - p;; Pi?= p;; - p - PB.To this must be added the corresponding term from the reversed order of collisionto get the total double scattering.We have here a five-dimensional integral since there are two energy conservinga-functions implied in the elastic cross-sections.When uB s large compared to v,we can again effect a considerable simplification by integrating analytically overdp. One is then left with a two-dimensional integral. In either case his integral. Ineither case his integral is especially suited to Monte Carlo integration techniques.


38/41

492 E. A. REMLERAPPENDIX

The intent here is the derivation of the general relation between densities andcross sections. Particular cases were derived in Section 5 based on the assumptionthat the Wigner representative of the density could be formally considered as aprobability. Here, the formula is derived along conventional lines. The fact thatthe same results are obtained in both ways is then further evidence that the afore-mentioned formal identification of the density is both safe and effective.Let y index possible subsystems and C index possible channels of y. Thus, wewriteY=y10yzOf, (4.1)c = (n 7 yz Y), (A.3

where each yi is a set containing either a single elementary particle or a boundstate of more than one. The possibility of multiple bound states of a yi involves asimple extension of this notation and will be ignored. Let jj denote the complimentof y so that y @ jj = the whole system in question.The channel Hamiltonian for C is denoted as Kc. This is the same notation asin [6]. The total Hamiltonian for y is denoted as H, . The channel and totalHamiltonians are not identical except when C is comprised of a single fragment, i.e.,when C = (y). We now define

andHc = Kc + HP (A.3)H = Hc + Vc (4.4)=K+V

where His the total Hamiltonian of the system. Thus, Hc contains the sum of allthe kinetic energy operators of the entire system (- K), all interactions amongstelements of 7 (- V,), and all interactions necessary to hold the fragments yi in Ctogether:

v, = c V,< . (A.5)The residuum is

V,= v- v,- v,. 64.6)The direct product of the plane wave channel eigenstates of C is denoted by

I W = I k,,) 0 I k,) 0 ... (A.7)


39/41

WIGNER REPRESENTATION 493and the projection operator onto these is

wk) = I k&kc I - (A.8)Evidently then[Hc ) sykc)] = 0. (A.9)

Denoting again the target and projectile by A and B respectively and, a unitnormal packet centered about zero momentum by g we obtain

/ gA , PB) = j dpA gcpA) / PA, PB) (A.lO)(gig) = 1 (A.1 1)

[0B(27T)-3] oc;AB(kC) = [a/a? (@)gA , pB / eiHt8(kc) epiH 1gA , p$?))]t=o (A.12)In this expression, the cross section uC;AB has the meaning given in the text withthe proviso that C may more generally denote an arbitrary channel with

kc = k,, , k,, ,...being the set of channel momenta. The flux factor is z1~(27r-~. Finally,

I g, > P2) = -Q,B I g,4 2 PB), (A.13)where fiAB is the Moller operator for incoming channel AB.

Using Eq. (A.4) we getwC:/&c) = (27~)~ i((+)gA > PB 1 vc'gP(kc) 1 g, , I&+') (A.14)

+ complex conjugate. One may now expand over intermediate states using theset j kc, k,), which is complete in 7. Taking the limi t in which g is inf initely narrow,one obtains,

%?~c;AB(kC) = (2~)~ i 1 dk, &K, + Kc - PB)x 7-c*& , kc ; 0, PB) 4, , k, ; 0, PB) + c.c., (A.15)

where&= CL Kc = 1 hiasp i

kp = @a); aejj(k, , kc 1PA, Pl;t)) = %K, + KC - PA - pa) w(k, , kc ; PA, PB) (A.16)

(ki, , kc I vc I PA , P% = W, + Kc - PA - ps) T,(k, , kc ; PA , 14. (A.1 7)


40/41

494 E. A. REMLEROn the other hand, the integrand in the expression for the flux can be rewritten as

which leads to

= CW3 i j- 4, I ~(k,Wc, 52~~1 i& > PB)

(A.20)

= (243 j- 4x4 1&A , PB)(- $9, , PB1.(A.21)

Eq. (A.19) is the result to have been demonstrated.

ACKNOWLEDGMENTSI wish to thank Dr. J. Hornstein and Dr. A. P. Sathe for several useful discussions and for acareful reading of an earlier version of this work.

REFERENCES1. E. WIGNER, Phys. Rev. 40 (1932), 49.2. SEWELL, Lectures n TheoreticalPhysics,Vol. 10, p. 289, nterscience, ew York, 1967.3. K. IMRE, E. OZIZMIR, M. ROSENBAUM, AND P. F. ZWEIFEL, J. Math. Phys. 8, 1097.4. H. W. BERTINI, Phys. Rev. C6 (1972),631.5. E. A. REMLER AND A. P. SATHE , Ann. Phys. (N.Y.) 91 (197%295.6. M. L. GOLDBERGER AND K. M. WATSO N, CollisionTheory, Wiley, New York, 1964.


41/41

WIGNER REPRESENTATION 4957. R. J. GLAUBER, Lectures in Theoretical Physics, Vol. 1, p. 315, Interscience, New York,1959.8. E. A. REMLER, unpublished.9. U. FANO, Rev. Mod. Phys. 29 (1957), 74.10. J. R. N. MILES AND J. S. DAHLER, J. Chem. Phys. 52 (1970), 616.11. P. CARRUTHERSAND F. ZACHARISEN, Transport Equation Approach to Multiparticle Produc-tion Processes, A. I. P. Conf . Proc., Vol. 23, p. 481, American Institute of Physics, Williams-burg, Va., 1974.12. H. Mom, I. OPPENHEIM,AND J. Ross, The Wigner Function and Transport Theory, Studiesin Statistical Mechanics, Vol. 1, p. 217, Interscience, New York, 1962.13. E. A. REMLER AND R. A. MILLER, Ann. Phys. (N.Y.) 82 (1974), 189.

Use of the Wigner Representation in Scattering Problems.pdf

Documents