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The Journal of Chemical Physics

Phase space interpretation of semiclassical theory

Eric J. Heller

Published Oct 1, 1977• 144 citations
Publisher

Abstract

A useful and intuitive phase space picture of common semiclassical approximations and procedures is proposed and developed. Underlying the intuitive pictures is the semiclassical, C→0 limit of the Wigner distribution, which is discussed in an Appendix. Simple classical analog phase space diagrams are suggested which represent quantum amplitudes, completeness relations, stationary phase points and integrations, caustic singularities, uniformizations, and the interesting short and long time behavior of the semiclassical propagator. Semiclassical wave packet amplitudes, now known to be on equal footing with the more usual amplitudes between delocalized eigenstates of Hermitian operators, are included in the phase space pictures. It becomes apparent why the primitive semiclassical wave packet amplitudes are to some extent uniformizing, and numerical results are presented to support this conclusion. The power of the phase space picture as an aid in formulating new approximations is illustrated.

DOI
10.1063/1.435296

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A useful and intuitive phase space picture of common semiclassical approximations and procedures is proposed and developed. Underlying the intuitive pictures is the semiclassical, C→0 limit of the Wigner distribution, which is discussed in an Appendix. Simple classical analog phase space diagrams are suggested which represent quantum amplitudes, completeness relations, stationary phase points and integrations, caustic singularities, uniformizations, and the interesting short and long time behavior of the semiclassical propagator. Semiclassical wave packet amplitudes, now known to be on equal footing with the more usual amplitudes between delocalized eigenstates of Hermitian operators, are included in the phase space pictures. It becomes apparent why the primitive semiclassical wave packet amplitudes are to some extent uniformizing, and numerical results are presented to support this conclusion. The power of the phase space picture as an aid in formulating new approximations is illustrated.
We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends Gaussian wave packet dynamics into complex phase space. Although we were able to give an accurate formulation of the method, we had at the time of writing that paper only an intuitive, heuristic understanding of the deeper causes which make the method work. A more mathematical understanding was needed. To close this gap we show in this paper the equivalence of the new method with a first order expansion of ℏ of the Schrödinger equation. We further prove that the new method is equivalent to the stationary phase approximation, using the usual WKB formula for the propagator. The latter equivalence enables us to show that all the symmetry properties of time-dependent quantum mechanics also hold in the new semiclassical theory. Finally, we provide some elaboration of the method, and clarify several issues that were not discussed before. With this new insight we are able to formulate a simple rule for the calculation of semiclassical wave functions that contain contributions from more than one branch. This corrects for the divergence of semiclassical wave functions near caustics, a problem that we encountered in the preceding paper.
The theory of quantum amplitudes in the semiclassical, h/→0 limit is considerably extended by including overlaps between Gaussian coherent states (wave packets) in the primitive ’’semiclassical algebra.’’ Traditionally, semiclassical formulations have exclusively involved overlaps between eigenstates of Hermitian operators; these are delocalized in most representations. Paradoxically, the states which are most localized and particlelike in the classical limit (wave packets) are not eigenfunctions of Hermitian observables, and have been omitted from asymptotic, formal theories of the semiclassical limit. When incorporated into a generalized theory, the Gaussian wave packet states apppear to alleviate many of the practical difficulties encountered in implementation of the Hermitian, delocalized state formulation. We present the generalized theory here, and show how it contains and reduces to the Hermitian semiclassical theory in various limits.
The Gaussian wave-packet phase-space representation is used to show that the expansion in powers of \ensuremath{\Elzxh} of the quantum Liouville propagator leads, in the zeroth-order term, to results close to those obtained in the statistical quasiclassical method of Lee and Scully in the Weyl-Wigner picture. It is also verified that, propagating the Wigner distribution along the classical trajectories, the amount of error is less than that coming from propagating the Gaussian distribution along classical trajectories.
A detailed framework for a new formalism of phase-space path integral is presented, the outline of the theory of which has been reported elsewhere [K. Takatsuka, Phys. Rev. Lett. 61, 503 (1988)]. This path integral is described in terms of a phase-space distribution function, which was proposed earlier by the present author under the name of the dynamical characteristic function (DCF). The DCF is characterized by two independent wave functions and with two phase spaces. In the present paper, we extend the DCF by utilizing the Feynman kernel in place of the two independent wave functions. This is called the identity DCF. The dynamics of the DCF for general wave packets is reduced to that of the identity DCF. Some characteristics of the identity DCF are presented. For example, the quantum-mechanical time-evolution operator is represented in terms of the identity DCF and of a complete set of quantum q (coordinate) and p (momentum) operators. A semiclassical theory for the identity DCF is also developed, its primary aim being the study of heavy-particle dynamics such as molecular collisions.It is shown that the propagation of the identity DCF is represented by classical trajectories, each of which is associated with an amplitude factor and with the action integral as the quantum phase factor. This amplitude factor is free from singularity, unlike the primitive semiclassical expression of the Feynman kernel, which is well known to diverge at caustics. The physical significance of the identity DCF is analyzed in the context of chaotic dynamics. In particular, it is shown that its amplitude factor is an explicit function of the eigenvalues of a Jacobian matrix [\ensuremath{\partial}(qf{\mathit{q}}_{\mathit{f}}qf​,pf{\mathit{p}}_{\mathit{f}}pf​)/\ensuremath{\partial}(qi{\mathit{q}}_{\mathit{i}}qi​,pi{\mathit{p}}_{\mathit{i}}pi​)], where (qi{\mathit{q}}_{\mathit{i}}qi​,p){\mathit{p}}_{)}p)​ and (qf{q}_{f}qf​,pf{p}_{f}pf​) are the initial and final points, respectively, of a classical trajectory. These eigenvalues are well known in the study of the classical chaos of Hamilton systems and have been used to construct a measure of the linear (and local) stability of an orbit, the Liapounov characteristic exponents, and so on. Further, if the identity DCF is applied to a periodic orbit, the amplitude factor becomes equivalent to the square root of the determinant of a matrix, which is the so-called monodromy matrix minus the unit matrix. This matrix is one of the standard quantities used in bifurcation theory to investigate the stability of fixed points. In fact, the amplitude factor is closely related to the so-called residue, which was proposed by Greene [J. Math. Phys. 20, 1183 (1979)] in order to predict the occurrence of global chaos in classical systems. It is emphasized that all these classical quantities arise naturally in our representation of quantum mechanics.
Initial-value phase-space integral representations for a time-dependent propagator are obtained in the coordinate and momentum representations. To do so we first derive nonuniform semiclassical propagators for the various representations, obtaining the global-time semiclassical phase indices (Maslov indices) due to caustics. Results include readily implementable general phase index formulas for any type of caustic, including cases where the Morse index theorem is inapplicable. The method of obtaining the indices is general and based simply on concatenating uniform short-time propagators which also gives rise to alternative path-integral forms. Initial-value integral representations are then derived by introducing a method of extending short-time initial-value propagator formulas to global times via a simple stationary-phase asympotoic-equivalence approach. The integrals reduce to the nonuniform semiclassical propagators within the stationary-phase approximation, are uniform about caustics, and have integrand phases which properly account for the global-time phases in terms of appropriate Maslov indices. The initial-value integrals are also consistently derived via a canonical mapping procedure on the coordinate-space path integral. Initial-value integrals for time-dependent wave-function propgation are also given. Evaluation of the initial-value integral expressions do not require trajectory root searches for propagation.
Semiclassical mechanics allows for a description of quantum systems which preserves their phase information, and thus interference effects, while using only the system's classical dynamics as an input. In particular one of the strengths of a semiclassical description is to present a coherent picture which (to negligible higher-order ℏ corrections) is independent of the particular canonical coordinates used. However, this coherence relies heavily on the use of the stationary phase approximation. It turns out, however, that in some important cases, a brutal application of stationary phase approximation washes out all interference, and thus quantum, effects. In this paper, we address this issue in detail in one of its simplest instantiations, namely the evaluation of the time evolution of the expectation value of an operator. We explain why it is necessary to include contributions which are not in the neighborhood of stationary points and provide new semiclassical expressions for the evolution of the expectation values. The efficiency of our approach is based on the fact that we treat analytically all the integrals that can be performed within the stationary phase approximation, implying that the remaining integrals are simple integrals, in the sense that the integrand has no significant variations on the quantum scale (and thus they are very easy to perform numerically). This to be contrasted with other approaches such as the ones based on initial value representation, popular in chemical and molecular physics, which avoid a root search for the classical dynamics, but at the cost of performing numerically integrals whose evaluation requires a sampling on the quantum scale, and which are therefore not well designed to address the deep semiclassical regime. Along the way, we get a deeper understanding of the origin of these interference effects and an intuitive geometric picture associated with them.