quantal-classical correspondence impulse theory

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P HYSICAL R EVIEW LETTERS VOLUME 85 3 JULY 2000 NUMBER 1 Quantal-Classical Correspondence Impulse Theory M. R. Flannery and D. Vrinceanu School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332 (Received 10 January 2000) The quantal impulse cross section is derived in a novel form appropriate for direct classical corre- spondence. The classical impulse cross section is then uniquely defined and yields the first general classical expression for n 2 n 0 0 collisional transitions. The derived cross sections satisfy the optical theorem and detailed balance. Direct connection with the classical binary encounter approximation is also firmly established. The unified method introduced is general in its application to various collision and recombination processes and enables new directions of enquiry to be pursued quite succinctly. PACS numbers: 03.65.Sq, 31.15.Gy, 32.80.Cy, 34.60.+z The quantal impulse approximation (QIA) is an estab- lished and productive cornerstone in general scattering theory [1]. It is based on the assumption that an inelastic collision of incident particle i with a two-particle sub- system 2, 3 results from each binary i 2 j scattering under the two-body interaction V ij alone, leaving the “spectator” particle k unaffected. The internal interaction V jk between the “active” particle j and the spectator particle k is ignored during the i 2 j collision except insofar as it generates a distribution rp jk dp jk over the relative momentum p jk of particles j and k . This approach has been invaluable in nuclear physics for high-energy neutron scattering by complex nuclei [1] and in atomic physics for atom-Rydberg atom collisions at thermal energies [2–7] and for electron (ion)-atom and atom- atom collisions [8,9] at high energies. It is also valuable, for example, in the study of three-body recombination, Li 1 Li 1 Li ! Li 2 1 Li which limits the density and lifetimes of Bose-Einstein condensates at ultralow energy, and e 1 1 ¯ p 1 e 1 ! ¯ H 1 e 1 for the formation of antihydrogen. A classical binary-encounter approximation (BEA) has also been formulated for Rydberg collisions [3] and for high-energy collisional ionization [8–10]. Although QIA and BEA are conceptually connected, the formal interre- lationship between them is quite complex and has never been firmly established. Previous studies [6,8,11] have de- pended on fairly complicated theoretical analysis in an ef- fort to reduce QIA to BEA. In this Letter, the quantal impulse approximation is pre- sented in a new form which provides quite naturally “the royal road” to classical correspondence. A classical im- pulse approximation can then be uniquely defined. The novel expression yields the first general classical cross sec- tion for n 2 n 0 0 transitions and the standard result for n 2 n 0 transitions. The derived cross sections satisfy the optical theorem (which implies probability conserva- tion) and detailed balance. The method introduced then permits BEA formulas to be derived quite succinctly and in a unified way. There is substantial renewed [7,10–15] interest in the power of classical dynamics in almost all fields of modern physics, attributed to the desire [15] to ob- tain a more thorough understanding of the classical-quantal correspondence. This present development provides a pro- found and important quantal-classical connection in colli- sion physics. The new formulation also provides a natural origin for development of new semiclassical methods. The transition T-matrix for the free-free (p ij ! p 0 ij ) scattering under the two-particle (i 2 j ) potential V ij alone is T ij p ij , p 0 ij expi p 0 ij ? r ij ¯ h jV ij r ij jCp ij , r ij r ij , where Cp ij , r ij is the exact wave function for i 2 j scattering in the (i 2 j ) center-of-mass frame CMi , j . The momentum transferred from i to j is q j p i 2 p 0 i p 0 j 2 p j p ij 2 p 0 ij , where the momentum of particle i is p i in the fixed CMijk frame of all three particles, and is p ij in the 0031-9007 00 85(1) 1(5)$15.00 © 2000 The American Physical Society 1

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PHYSICAL REVIEW

LETTERS

VOLUME 85 3 JULY 2000 NUMBER 1

Quantal-Classical Correspondence Impulse Theory

M. R. Flannery and D. VrinceanuSchool of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332

(Received 10 January 2000)

The quantal impulse cross section is derived in a novel form appropriate for direct classical corre-spondence. The classical impulse cross section is then uniquely defined and yields the first generalclassical expression for n 2 n00 collisional transitions. The derived cross sections satisfy the opticaltheorem and detailed balance. Direct connection with the classical binary encounter approximation isalso firmly established. The unified method introduced is general in its application to various collisionand recombination processes and enables new directions of enquiry to be pursued quite succinctly.

PACS numbers: 03.65.Sq, 31.15.Gy, 32.80.Cy, 34.60.+z

The quantal impulse approximation (QIA) is an estab-lished and productive cornerstone in general scatteringtheory [1]. It is based on the assumption that an inelasticcollision of incident particle i with a two-particle sub-system 2, 3 results from each binary i 2 j scatteringunder the two-body interaction Vij alone, leaving the“spectator” particle k unaffected. The internal interactionVjk between the “active” particle j and the spectatorparticle k is ignored during the i 2 j collision exceptinsofar as it generates a distribution rpjkdpjk over therelative momentum pjk of particles j and k. This approachhas been invaluable in nuclear physics for high-energyneutron scattering by complex nuclei [1] and in atomicphysics for atom-Rydberg atom collisions at thermalenergies [2–7] and for electron (ion)-atom and atom-atom collisions [8,9] at high energies. It is also valuable,for example, in the study of three-body recombination,Li 1 Li 1 Li ! Li2 1 Li which limits the density andlifetimes of Bose-Einstein condensates at ultralow energy,and e1 1 p 1 e1 ! H 1 e1 for the formation ofantihydrogen.

A classical binary-encounter approximation (BEA) hasalso been formulated for Rydberg collisions [3] and forhigh-energy collisional ionization [8–10]. Although QIAand BEA are conceptually connected, the formal interre-lationship between them is quite complex and has neverbeen firmly established. Previous studies [6,8,11] have de-pended on fairly complicated theoretical analysis in an ef-fort to reduce QIA to BEA.

0031-90070085(1)1(5)$15.00

In this Letter, the quantal impulse approximation is pre-sented in a new form which provides quite naturally “theroyal road” to classical correspondence. A classical im-pulse approximation can then be uniquely defined. Thenovel expression yields the first general classical cross sec-tion for n 2 n00 transitions and the standard result forn 2 n0 transitions. The derived cross sections satisfythe optical theorem (which implies probability conserva-tion) and detailed balance. The method introduced thenpermits BEA formulas to be derived quite succinctly andin a unified way. There is substantial renewed [7,10–15]interest in the power of classical dynamics in almost allfields of modern physics, attributed to the desire [15] to ob-tain a more thorough understanding of the classical-quantalcorrespondence. This present development provides a pro-found and important quantal-classical connection in colli-sion physics. The new formulation also provides a naturalorigin for development of new semiclassical methods.

The transition T-matrix for the free-free (pij ! p0ij)

scattering under the two-particle (i 2 j) potential Vij

alone is

Tijpij , p0ij expip0

ij ? rijh jVijrijjCpij , rijrij ,

where Cpij , rij is the exact wave function for i 2 jscattering in the (i 2 j) center-of-mass frame CMi, j.The momentum transferred from i to j is

qj pi 2 p0i p0

j 2 pj pij 2 p0ij ,

where the momentum of particle i is pi in the fixedCMijk frame of all three particles, and is pij in the

© 2000 The American Physical Society 1

VOLUME 85, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 3 JULY 2000

CMi, j frame. The primed 0 and unprimed momentadenote postcollision and precollision values.

The quantal impulse T-matrix for i 2 j, k scatteringdecomposes as the sum Tif T

i2if 1 T

i3if of binary

collision matrices Tijif , with j 2 or 3. Each T

ijif in the

momentum-space representation (MS) is the overlap[1,6,7],

MSTijif pi , p0

i ffpjk 1 qjk

3 jTijpij , p0ijjfipjkpjk , (1)

of the free-free Tij matrix with the initial and final mo-mentum wave functions

fpjk 2p h232Z

crjk exp2ipjk ? rjkh drjk ,

for the j, k subsystem. The momentum transferred toj 2 k relative motion is

qjk p0jk 2 pjk

Mk

Mj 1 Mkqj ,

where pjk is the relative momentum in the initial CM j, kframe. After the (i 2 j) collision, CM j, k moves withthe residual momentum MjqjMj 1 Mk. The bi-

nary Tijif matrix has also the configuration space (CS)

representation

CSTijif pi , p0

i expip0i ? rihcfrjk

3 jVijrijjC1i pi ; ri , rjkri ,rjk . (2)

The impulse approximation [6] for the initial total systemwave function in (2) is

C1i Cimppi; ri , rjk

2p h232Z

fipjkFpij , pk; rij , rk dpjk ,

where the active and spectator particles are j and k, re-spectively. The free-free (i 2 j) scattering wave functionis Cpij , rij in the CMi, j frame and is

Fpij , pk; rij , rk expipk ? rkhCpij , rij

in the fixed CMijk frame. With the aid of the(rjk , ri)–(rij , rk) transformation equations for the inte-gration variables in Eq. (2), of the identity pi ? ri 1

pjk ? rjk pk ? rk 1 pij ? rij , and of crjk 2p h232

Rfpjk expipjk ? rjkh dpjk , it can be

shown that the momentum-space and configuration-spacerepresentations (1) and (2) of Tif are equivalent.

The quantal differential cross section is

dsif

dp0i

y0i

yi

µ1

4p

2Mi

h2

∂2

jTifpi , p0ij

2, (3)

where Mi MiMj 1 MkMi 1 Mj 1 Mk is the re-duced mass of the i 2 j 1 k collision system and p

0i

2

Miv0i are the momenta of i with v

0i measured with re-

spect to initial CM j, k. Although QIA assumes that eachscattering is a separate event, the cross section (3) involves

jTif j2 jT

i2if 1 T

i3if j2 which exhibits interference be-

tween the two-body (i 2 j) scattering amplitudes. Forthree identical particles jTif j

2 4jTijif j2. The classical

representation of Eq. (3), however, involves only jTif j2

jTi2if j2 1 jT

i3if j2 which is valid for binary and indepen-

dent scattering (as, e.g., in Rydberg collisions).The key idea of this paper is to find a phase-space rep-

resentation for jTijif j2. This is accomplished by writing

jTijif pi , p0

ij2 CST

ijif MST

ijif (4)

as the product of the momentum-space (1) and the con-figuration-space (2) representations. It can be shown thatEq. (4) with Eqs. (1) and (2) for (i 2 j) scattering alonereduces to

jTijif pi , p0

ij2 2p h3

Zdp

Zdr r

fr, p 1 q

3 jTijpij , p0ijj2rir, p , (5)

where rjk , pjk , qjk are now denoted by r, p, q. Theinternal j 2 k quantal distribution in phase space is

rr, p 2p h232cre2ip?r hfp , (6)

in agreement with that [12] previously defined. Thestandard probability densities rr

Rrr, p dp

jcrj2 R2nrjYmrj2 in configuration space and

rp R

rr, p dr jfpj2 f2npjYmpj2 in

momentum space are recovered directly from (6), whichis the standard ordered version [12] of the Wigner distri-bution [16]. In terms of the phase-space distributions forthe initial and final internal j 2 k states, the differentialcross section (3), with Eq. (5) for independent scattering,separates into (i-2) and (i-3) components, each given by

dsijif

dp0i

y0i

yi

µMi

Mij

∂2

2p h3Z

dpZ

dr

3 rfr, p 1 q j fijpij , p0

ijj2rir, p ,

(7)

where the i 2 j and i 2 j 1 k reduced masses areMij and Mi , respectively. The scattering amplitude,fijpij , p

0ij fijgp, qj, for free-free i 2 j colli-

sions is a general function of momentum change qj andof the i 2 j relative velocity g vi 2 vj . It thereforedepends implicitly via g on the j 2 k relative momentump. Thus, (5) and (7) are the free-free transition probabili-ties/differential cross sections, respectively, averaged overthe initial and final phase space distributions.

The above novel expression (7) for the quantal impulsecross section (3) has the phase-space representation ap-propriate for direct classical correspondence, obtained by

VOLUME 85, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 3 JULY 2000

replacing the quantal densities ri,f of (6) by the corre-sponding classical phase-space distributions [12]:

rcE, L; r, pdrdp dp22m 1 V r 2 EdE

3 djr 3 pj 2 LdLdrdp

2p h3 ,

(8)

for a j 2 k symmetric interaction V r. Theinternal Hamiltonian Hr , p p22m 1 V r of thej 2 k subsystem (with relative momentum pjk pand reduced mass Mjk m) and the internal angularmomentum Lr, p r 3 p are conserved quantities inphase space. The distribution (8) holds also for boundstates (n, ) but with dE and dL replaced [12] by hnnl andh, respectively, where the frequency (or inverse period)for bounded radial motion is nnl t

21nl . The distribution

(8) is normalized toR

rcn, dr dp 2 1 1 states over

all of phase space.(a) n ! n00 transitions.—The angular rur , fr in-

tegration, resulting from Eq. (8) in Eq. (7), involves theangular momentum overlap integral,

Rifr , q 2p h23

4p

Z 1

21drp sinur 2 Ldcosur

3Z 2p

0drp0 sinur 2 L0 dfr , (9)

where cosur p ? r and cosur p0 ? r with p alongthe Z axis and with p0 p 1 q fixed in the XZ plane.Integration via the techniques outlined in [12] yields

Rifr , qdE dL dE0 dL0 h3

m2 rcnrrc

n00r

3 G1ifr , q 1 G2

ifr , q .

(10)

The classical radial density and the classical radial proba-bility for continuum states of energy E are

rcEr

Zr

cnr, p dp 4pL dE dL2p h3r2 r ,

and 4prcEr2dr 2LdLh2 2dEdth, respectively.

The bound state distribution is rcn 2 1

14p 2rtn. Each of the contributions,

G6ifr , q q2 2 A2

6B26 2 q2212 , (11)

must be real, so that q must then lie within the classicallyaccessible range given by

A26r m2r 6 r 02 1 L 2 L02r2 # q2 # B2

6r

m2r 6 r 02 1 L 1 L02r2.

For a given momentum transfer q, the r integration pro-ceeds over the various [12] radial ranges Rq withinwhich the G6

if are real. The initial and final radial speedsr and r 0 determined by

12m r 02 E0 2 V r 2 L022mr2

are functions only of r for given initial nl and final n0l0

states. Hence, G6ifr , q and the angular integral Rif r, q

depend only on r and q and not on p. The remaining pintegration in Eq. (7) therefore involves only the overlap,Z

dHr, p 2 E j fijgp, qjj2

3 dHr, p 1 q 2 E0 dp ,

of the i 2 j free-free differential cross section with theinitial and final energy distributions at a fixed r . Whenintegrated [17] and combined with the angular momentumoverlap integral (9), the cross section for n 2 n00 tran-sitions resulting from i 2 j collisions is

sijn,n00pi

2p

M2ijy

2i

Z q1

q2

qj dqj

ZRq

Pn,n00r , q dr

(1p

Z g21r ,q

g22r ,q

j fijg, qjj2dg2

g2 2 g22 g2

1 2 g212

). (12)

The classical probability that the i 2 j impulsive transition transfers momentum q to the j 2 k system with internalseparation r in the interval r , r 1 dr is

Pn,n00r , qdr

µ2pm2

q

∂4pRifr2dr dE dL dE0 dL0 4p2r

cnrrc

n00rr2 h3

2qG1

ifr , q 1 G2ifr , qQr , q .

(13)

The classical limits to the g2 integral in Eq. (12) for fixed(r , q) are

g26r , q y2

i 1 y2j r 2 2yiyjr

3 cosur , q 6 uiq ,

where, for given transfers E E0 2 E and q in energyand momentum, the angles that pi and p each make withq are, respectively, determined by

cosuiq,E 2MiE 1 q2j 2piqj ,

cosuq,E 2mE 2 q22pq ,

in accord with energy and momentum conservation,implicit in the d function and impulse conditions.Since j cosuj # 1, the j, k relative momentump . p0 j2mE 2 q2j2qj so that the step func-tion Q in Pn,n00 has value 1 for pr $ p0 and zero

3

VOLUME 85, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 3 JULY 2000

otherwise. Since j cosuij # 1, then q2 jpi 2 p0ij #

qj # pi 1 p0i q1, as expected.

The integrated probability for the impulsive (n 2 n00)transition,

Pnl,n0l0q ZR

Pn,n00r , q dr , (14)

is simply the classical form factor as defined in Ref. [12].Equation (12) therefore illustrates the interesting mannerin which Pnl,n0l0 becomes deconvoluted over r whenthe general i 2 j scattering cross section sg, qjdepends additionally on the (i 2 j) relative speed g. Forg-independent s, (12) reduces to the standard impulseresult [1,6] involving the full form factor (14).

Since the integration limits and Pi,f are symmetrical ini and f, the basic cross section (12) satisfies the detailedbalance relation p2

i sn,n00pi p02i sn,n00p0

i. “Semi-

4

quantal” probabilities may now be defined by replacingeither or both of the classical radial densities r

cn in (13)

by their r-independent quantal equivalents,

rqnr

Xm

jcnmrj2 2 1 1R2nr4p ,

for both bound n and continuum e states.(b) n ! n0 transitions.—On integrating Eq. (9) over

all final angular momenta L0, then

R0ifdEdL

(12

Z 1

21drp sinur 2 L dcosur

)dE dL2p h3

1

4pmpr

cnr . (15)

The classical probability for the n 2 E0, E0 1 dE0 im-pulsive transitions in the j, k subsystem with internalmomenta in the interval p, p 1 dp is then

Pn,E0p, qdp µ

2pm2

q

∂4pR0

ifr2dr dE dL dE0 mdE0

2pq4pr

cnpp2dp ,

since 4prnpp2dp 4prnrr2dr under the energy constraint p2r 2mE 2 V r for the initial classical dis-tribution. Note that mq Mjqj . The classical impulse cross section for n 2 E0, E0 1 dE0 transitions is therefore

sijn,E0pi

2p

M2ijy

2i

Z q1

q2

qj dqj

Z `

p0qPn,E0p, q dp

(1p

Z g21 p,q

g22 p,q

j fijg, qjj2dg2

g2 2 g22 g2

1 2 g212

). (16)

The cross section for n 2 n0 transitions is obtained by substituting hnn0 for dE0 in Pn,E0 . By adopting the quantaldistribution 2 1 1f2

np for 4prcnp, then Eq. (16) provides semiquantal cross sections [6].

(c) Relationship with Optical Theorem and Classical Binary Encounter Theory.—Integration of Eq. (16) over all pos-sible final E0 states is facilitated by recognizing that the required integral,Z g2

1

g22

2gqjdE0

g2 2 g22 g2

1 2 g212

Z E1

E2

dEE 2 E2 E1 2 E 12 ,

is simply p . The classical limits E6qj , vi ? vj; yi , yj tothe energy transfer E , for the prescribed fixed arguments,need not then be specified. The cross section for all elasticand inelastic transitions is then

sijn yi

Xn0

sn,n0yi 1Z `

0sn,eyi de

1yi

Zr

cnp dp

Z Ωg

dsij

dp0ij

ædp0

ij . (17)

i.e., the rate yisn for transitions to all states is thereforethe rate for all i 2 j binary encounters, averaged over theinitial momentum function for j in the field of k. Expres-sion (17) simply restates the optical theorem, which, whenapplied to (1), provides (17) directly.

Expression (17) is the original basis [8] of the stan-dard classical binary-encounter approximation (BEA) forenergy-changing collisions, where the p0

ij- region of in-tegration is constrained by the required energy change.Previous BEA semiquantal results [3,6] for n 2 n0 col-lisional transitions can then be shown [17] to be identicalwith (16). Generalization [17] of BEA to cover the an-

gular momentum changes in n 2 n00 collisional transi-tions is made simply by replacing

Rrnp dp in (17) by

the integralR

rnp, r dp dr over phase space. See alsoRefs. [13,14].

In summary, the quantal impulse cross section (3)has been presented in a valuable new form (7) which isthe appropriate representation for direct classical corre-spondence. The classical impulse cross section has beendefined by (7) with (8) which yields, in quite a succinctfashion, the first general expression (12) for the classicalimpulse cross section for n 2 n00 and n 2 e0 elec-tronic transitions. The cross section satisfies the opticaltheorem and detailed balance. Direct connection with theclassical binary encounter approximation (17) has beenestablished and the derived n 2 n0 and n 2 e crosssections (16) reproduce the standard BEA cross sections[3,7]. The present unified method can also furnishsemiclassical impulse cross sections, obtained simplyby adopting semiclassical phase-space distributions (atpresent, unknown) within Eq. (7). Although appliedhere to a time-independent formulation of collisional

VOLUME 85, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 3 JULY 2000

electronic transitions in an atom, the prescribed methodis general in that it can be applied, where appropriate,to atom-molecule rovibrational collisions, to three-bodyrecombination, and to explicit time-dependent problemsas laser-pulse excitation [18]. The method presented alsohelps elucidate, quite succinctly, the role played by thequantal-classical correspondence in collision dynamics.

This work is supported by grants from AFOSR: F49620-99-1-0277 and NSF: 98-02622.

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(1973).[4] Rydberg States of Atoms and Molecules, edited by R. F.

Stebbings and F. B. Dunning (Cambridge University Press,New York, 1983).

[5] T. F. Gallagher, Rydberg Atoms (Cambridge UniversityPress, Cambridge, England, 1994).

[6] M. R. Flannery, Phys. Rev. A 22, 2408 (1980).[7] V. S. Lebedev and I. L. Beigman, Physics of Highly Exited

Atoms and Ions (Springer-Verlag, Berlin, 1998).[8] L. Vriens, Case Studies in Atomic Collision Physics I,

edited by E. W. McDaniel and M. R. C. McDowell (North-Holland, Amsterdam, 1969), p. 364.

[9] M. R. Flannery, J. Phys. B 4, 892 (1971).[10] M. Gryzinski and J. A. Kunc, J. Phys. B 32, 5789 (1999).[11] V. S. Lebedev, J. Phys. B 24, 1977 (1991).[12] D. Vrinceanu and M. R. Flannery, Phys. Rev. Lett. 82, 3412

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