Quasicausal expansion of the quantum Liouville propagator G. W. Bund* Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, rua Pamplona 45, Sa˜o Paulo, Sa˜o Paulo 01405-900, Brazil S. S. Mizrahi† Departamento de Fı´sica, Universidade Federal de Sa˜o Carlos, Rodovia Washington Luiz Km 235,13565-905 São Carlos, Sa˜o Paulo, Brazil M. C. Tijero‡ Instituto de Fı´sica, Universidade Estadual Paulista and Pontifı´cia Universidade Cato´lica (PUC), São Paulo, Sa˜o Paulo, Brazil ~Received 14 July 1995! The quasicausal expansion of the quantum Liouville propagator is introduced into the Weyl-Wigner picture. The zeroth-order term is shown to lead to the statistical quasiclassical method of Lee and Scully@J. Chem. Phys.73, 2238~1980!#. PACS number~s!: 34.10.1x, 03.65.Sq I. INTRODUCTION Several works were devoted to the development of semi- classical methods in order to describe collision phenomena in the molecular, atomic, and nuclear heavy-ion domains @1–3#. Among these, some have adopted the quantum Weyl- Wigner ~WW! picture @4,5#, since it possesses a well suited structure that enables the development of\ power-series ex- pansions. In Refs.@4,6# the reader can find a detailed discus- sion about the problems and difficulties involving these ex- pansions. Lee and Scully@2# have developed a method, first suggested by Heller@7#, that they called the statistical quasi- classical method and they applied it to calculate transition probabilities of colinear molecular collisions. However, some of their calculated transitions present discrepancies with the exact quantum-mechanical values@8#. The aim of this paper is to show that the approach of Lee and Scully corresponds to the zeroth order of a general expansion of the full quantum Liouvillian written in the WW picture; there- fore their calculated transition probabilities accept higher- order corrections, and here we derive the analytical expres- sion of the first-order term. As will be made clear later, we find it more suitable to call the zeroth-order term the causal approximation~CA! and the first-order term as the quasicausal approximation~QCA!. But first it will be necessary to clarify the meaning of taking the classical limit, which in general leads to annoying singulari- ties @7# in the Wigner distribution function~WDF!. We un- derstand that Planck’s constant arises in the quantum Liou- ville equation from two sources: one of these being the WDF at initial time and the other the presence of\ in the quantum evolution operator exp@2iLQ(t2t0)#, through the Liouville operatorLQ . In the next section we show how to handle the limit \→0. II. CAUSAL AND QUASICAUSAL APPROXIMATIONS Let us consider the quantum Liouville equation in the WW picture @4#, ]W~pW ,qW ,t ! ]t 52 iLQW~pW ,qW ,t !, ~1! whereW(pW ,qW ,t) is the WDF and (pW ,qW ) is a point of phase space in one or several degrees of freedom. The quantum Liouvillian is LQ5H~pW ,qW !F i 2\ sin \ 2 LJ G , ~2! H(pW ,qW ) being the Hamiltonian of the system. The operator LJ5 ]Q ]qW • ]W ]pW 2 ]Q ]pW • ]W ]qW ~3! is the Poisson bracket, where the arrows indicate on which side the derivatives operate. In order to develop a\ power-series expansion and to analyze the classical limit, we introduce the dimensionless parametera in LQ , LQ ~a!5H~pW ,qW !F i 2 a\ sin a\ 2 LJ G . ~4! The formal solution of Eq.~1! is given by W~a!~pW ,qW ,t !5e2 iLQ ~a! ~ t2t0!W0~pW ,qW !, ~5! whereW0(pW ,qW ) is the WDF at the initial timet0 . In the limit a→0,LQ (a) goes over into the classical Liouvillian Lcl5 iH ~pW ,qW !LJ , ~6! which becomes\ independent, and Eq.~5! is written as@9# *Electronic address: bund@vax.ift.unesp.br †Electronic address: dsmi@power.ufscar.br ‡Electronic address: maria@vax.ift.unesp.br PHYSICAL REVIEW A FEBRUARY 1996VOLUME 53, NUMBER 2 531050-2947/96/53~2!/1191~3!/$06.00 1191 © 1996 The American Physical Society W~0!~pW ,qW ,t !5e2 iLcl~ t2t0!W0~pW ,qW ! 5W0„pW ~ t02t !,qW ~ t02t !…. ~7! Each point of the phase space of the initial WDF will evolve classically~although it will be reversed in time! following a trajectory according to Hamilton equations; however, quan- tum ingredients may still be present through the choice of W0(pW ,qW ) at timet0 . This description constitutes the CA and it is the same approximation originally adopted in Ref.@2#. It is important to realize that the introduction of the parameter a avoids the confusion between the different origins of\ in the WDF when we consider the limit\→0. Thus, to obtain the CA this limit has to be considered in the Liouvillian only and not in the initial time WDF, where quantum ingredients are present and must remain. In order to get the first-order approximation we have to perform, initially, the expansion of LQ (a) in a power series of (a\)2, LQ ~a!5 ( n50 ` ~a\!2nL2n , ~8! where L2n5H~pW ,qW !F i ~21!n 22n~2n11!! ~LJ !2n11G ~9! ~note thatL05Lcl), which permits us to write Eq.~1! as ]W~a!~pW ,qW ,t ! ]t 1 iL0W ~a!~pW ,qW ,t ! 52 i( n51 ` ~a\!2nL2nW ~a!~pW ,qW ,t !. ~10! Using the classical propagatore2 iL0(t2t0), the integral equa- tion corresponding to Eq.~10! is W~a!~pW ,qW ,t !5e2 iL0~ t2t0!W0~pW ,qW !2 i( n51 ` ~a\!2n 3E t0 t dt8e2 iL0~ t2t8!L2nW ~a!~pW ,qW ,t8!. ~11! This equation may be solved iteratively to any order in (a\)2; at first order we establish the QCA WQCA ~a! ~pW ,qW ,t !5e2 iL0~ t2t0!W0~pW ,qW !2 i ~a\!2 3E t0 t dt8e2 iL0~ t2t8!L2e 2 iL0~ t82t0!W0~pW ,qW !. ~12! It can be verified that fora50 the CA is obtained, but for aÞ0 the terms on the right-hand side of Eq.~10! are respon- sible for the breaking of the causal evolution of the WDF phase-space points. Moreover, it should be noted that in Eq. ~12! the only evolution operator connecting different times is e2 iL0(t2t0), which gives a causal character to the evolution. If we wish to analyze the behavior of the classical limit we first have to make a50 and thereafter \→0 in WQCA (a) (pW ,qW ,t), since the operatorL2 may still bring down \21 factors. If such a limit exists the classical distribution function d„pW 2PW (t)…d„qW 2QW (t)… is to be obtained, where PW (t) andQW (t) are the solutions of Hamilton equations. Oth- erwise, if the classical limit is of no interest, it is sufficient to takea51 after Eq.~11! has been iterated up to the desired order in (a\)2. III. TRANSITION PROBABILITIES FOR A COLLISIONAL PROCESS We are now going to derive the expression of the transi- tion probabilities up to the QCA, corresponding to the colli- sional process treated in Refs.@2# and@8#, where a molecule with a known internal structure suffers a collision from a pointlike projectile; the molecule is transferred from the ini- tial discrete energy levelum& to the final levelun&. The den- sity matrix for the target plus projectile system is r(RW ,RW 8;xW ,xW 8,t)5^RW ,xW ur̂(t)uRW 8,xW 8&, where the coordinates RW , RW 8 specify the relative distance between the projectile and the c.m. of the target, whilexW , xW 8 refer to one or more internal coordinates of the target. The density matrix is re- lated to the WDF by@1# r~RW ,RW 8;xW ,xW 8,t !5E dPW dpWe i \P W •~RW 2RW 8!e i \p W •~xW2xW8! 3W„PW , 12 ~RW 1RW 8!,pW , 12 ~xW1xW 8!,t…, ~13! and the matrix element that is appropriate to a final state un& then becomes rn~RW ,t !5^nRW ur̂~ t !unRW & 5E dxWdxW 8wn* ~xW !wn~xW 8!r~RW ,RW 8;xW ,xW 8,t !, ~14! where thewn(xW )’s are the eigenfunctions of the Hamiltonian for the internal degrees of freedom. Introducing the WDF Wn corresponding to the internal motion, defined through wn* ~xW !wn~xW 8!5E dpW 8e2 i \p W 8.~xW2xW8!Wn„pW 8, 12 ~xW1xW 8!…, ~15! into Eq. ~14!, making further the change of variables jW5xW2xW 8, qW 5 1 2(xW1xW 8) and integrating overjW andpW 8 in this order, we obtain rn~RW ,t !5~2p\!3E dPW dpWdqWW~PW ,RW ;pW ,qW ,t !Wn~pW ,qW ! 5~2p\!3E dPW dpWdqWWn~pW ,qW !e2 iLQ~ t2t0! 3W0~PW ,RW ;pW ,qW !, ~16! 1192 53BRIEF REPORTS LQ being the complete Liouvillian of the system and W0(PW ,RW ,pW ,qW ) the WDF at timet0 . The total transition prob- ability is then fort→` and t0→2` Pm→n5E dRW dPW dpWdqWW0~PW ,RW ;pW ,qW !eiLQ~ t2t0!Wm~pW ,qW !, ~17! where the unitarity property of the evolution operator was utilized. The WDF at timet0 is chosen such that the projec- tile is at positionRW 0 (R0→` or outside the interaction range!, while the target is in the stateum&. As no initial correlations between target and projectile are assumed we write W0~PW ,RW ;pW ,qW !5d~RW 2RW 0!d~PW 2PW 0!Wm~pW ,qW !, ~18! which substituted into Eq.~17! yields for t→` and t0→2` Pm→n5~2p\!3E dpW 0dqW 0Wm~pW 0 ,qW 0!e iLQ~ t2t0!Wn~pW 0 ,qW 0!. ~19! Up to this point the above result is exact, and now, on the basis of what was discussed in Sec. II, we introduce the QCA that consists of replacing the exact by the quasicausal propa- gator, and this leads in the limit fort→` and t0→2` to Pm→n QCA5~2p\!3H E dpW 0dqW 0Wm~pW 0 ,qW 0!Wn„pW ~ t !,qW ~ t !… 1 i ~a\!2E dpW 0dqW 0Wm~pW 0 ,qW 0!E t0 t dt8eiL0~ t2t8!L2 3Wn„pW ~ t82t0!,qW ~ t82t0!…J , ~20! wherepW (t), qW (t), pW (t2t8), qW (t2t8) depend on the initial valuespW 0 , qW 0 , RW 0 , PW 0 but neverthelessPm→n QCA shall be inde- pendent on the coordinateRW 0 . The first term in Eq.~20! corresponds to the CA, while the second term characterizes the QCA. Quantitative evaluations ofPm→n QCA have been car- ried out @10#, and they will be presented in due course. ACKNOWLEDGMENT S.S. Mizrahi thanks CNPq, Brazil, for partial financial support. @1# P. Carruthers and F. Zachariasen, Rev. Mod. Phys.55, 245 ~1983!, and references therein. @2# H. W. Lee and M. O. Scully, J. Chem. Phys.73, 2238~1980!. @3# H. Esbensen,Nuclear Structure and Heavy Ion Collisions, Pro- ceedings of the International School of Physics ‘‘Enrico Fermi,’’ Course LXXVII, Varenna on Lake Como, 1979, edited by R. A. Broglia, R. A. Ricci, and C. H. Dasso~North-Holland, Amsterdam, 1981!, pp. 572–591. @4# S. R. de Groot and L. G. Suttorp, inFoundations of Electro- dynamics~North-Holland, Amsterdam, 1972!, Chap. VI. @5# M. Gadella, Fortschr. Phys.43, 229 ~1995!. @6# G. W. Bund, J. Phys. A28, 3709~1995!. @7# E. J. Heller, J. Chem. Phys.65, 1289~1976!. @8# D. Secrest and B. R. Johnson, J. Chem. Phys.45, 4556~1966!. @9# D. N. Zubarev, inNonequilibrium Statistical Thermodynamics, Studies in Soviet Science~Consultants Bureau, New York, 1974!. @10# M. C. Tijero, Ph.D. thesis, Instituto de Fı´sica Teorica, 1994 ~unpublished!. 53 1193BRIEF REPORTS