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Journal of Mathematical Physics

On the derivation of the semiclassical approximation to the quantum propagator

S. Fischer, Andreas Buchleitner

Published Jul 1, 2015• 2 citations
Publisher

Abstract

In order to rigorously derive the amplitude factor of the semiclassical approximation to the quantum propagator, we extend an existing method originally devised to evaluate Gaussian path-integral expressions. Using a result which relates the determinant of symmetric block-tridiagonal matrices to the determinants of their blocks, two difference equations are obtained. The first one allows to establish the connection of the amplitude factor to Jacobi’s accessory equations in the continuous-time limit, while the second one leads to an additional factor which, however, contributes to the final result only in exceptional cases. In order to demonstrate the wide applicability of these difference equations, we treat explicitly the case where the time-sliced Lagrangian is written in generalized coordinates, for which a general derivation has so far been unavailable.

DOI
10.1063/1.4926353

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In order to rigorously derive the amplitude factor of the semiclassical approximation to the quantum propagator, we extend an existing method originally devised to evaluate Gaussian path-integral expressions. Using a result which relates the determinant of symmetric block-tridiagonal matrices to the determinants of their blocks, two difference equations are obtained. The first one allows to establish the connection of the amplitude factor to Jacobi’s accessory equations in the continuous-time limit, while the second one leads to an additional factor which, however, contributes to the final result only in exceptional cases. In order to demonstrate the wide applicability of these difference equations, we treat explicitly the case where the time-sliced Lagrangian is written in generalized coordinates, for which a general derivation has so far been unavailable.
We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.
The coherent path integral is generalized such that the identity operator represented in a complete (actually overcomplete) set of the coherent states with the ``time-variable'' exponents are inserted between two consecutive short-time propagators. Since such a complete set of any given exponent can constitute the identity operator, the exponent may be varied from time to time without loss of generality as long as it is set common to all the Gaussians. However, a finite truncation of the coherent state expansion should result in different values of the propagator depending on the choice of the exponents. Furthermore, approximation methodology to treat with the exact propagator can also depend on this choice, and thereby many different semiclassical propagators may emerge from these combinations. Indeed, we show that the well-known semiclassical propagators such as those of Van Vleck, Herman-Kluk, Heller's thawed Gaussian, and many others can be derived in a systematic manner, which enables one to comprehend these semiclassical propagators from a unified point of view. We are particularly interested in our generalized form of the Herman-Kluk propagator, since the relative accuracy of this propagator has been well established by Kay, and since, nevertheless, its derivation was not necessarily clear. Thus our generalized Herman-Kluk propagator replaces the classical Hamiltonian with a Gaussian averaged quantum Hamiltonian, generating non-Newtonian trajectories. We perform a numerical test to assess the quality of such a family of generalized Herman-Kluk propagators and find that the original Herman-Kluk gives an accurate result. The reason why this has come about is also discussed.
Rather general expressions are derived which represent the semiclassical time-dependent propagator as an integral over initial conditions for classical trajectories. These allow one to propagate time-dependent wave functions without searching for special trajectories that satisfy two-time boundary conditions. In many circumstances, the integral expressions are free of singularities and provide globally valid uniform asymptotic approximations. In special cases, the expressions for the propagators are related to existing semiclassical wave function propagation techniques. More generally, the present expressions suggest a large class of other, potentially useful methods. The behavior of the integral expressions in certain limiting cases is analyzed to obtain simple formulas for the Maslov index that may be used to compute the Van Vleck propagator in a variety of representations.
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.
The extremal paths that arise in the stationary phase evaluation of coherent state path integrals do not seem to have a simple physical interpretation, in contrast to the extremal paths that occur in the conventional path integrals. On the other hand, a recently derived semiclassical formula for the coherent state propagator involves a path that is determined in exactly the same manner as the extremal paths of the conventional path integrals. Since both the semiclassical and the stationary phase analyses yield asymptotic (h(cross) to 0) approximations, the stationary phase and the semiclassical expressions for the propagator should be identical. The author presents a simple and direct proof that, in spite of the apparent differences, this is indeed the case. The simplification in the semiclassical formula is due to the utilisation of an appropriate set of canonical variables to describe the classical dynamics. In order to illustrate the usefulness of the semiclassical formula they present an application to the problem of the degenerate parametric amplifier, which had been treated before by operator ordering and path integral methods. The semiclassical approach has a simple classical interpretation that is absent in the alternative treatments.
Gaussian quantum channels are well understood and have many applications, e.g., in Quantum Information Theory and in Quantum Optics. For more general quantum channels one can in general use semiclassical approximations or perturbation theory, but it is not easy to judge the accuracy of such methods. We study a relatively simple model case, where the quantum channel is generated by a Lindblad equation where one of the Lindblad operators is a multiple of the internal Hamiltonian, and therefore the channel is not Gaussian. For this model we can compute the characteristic function of the action of the channel on a Gaussian state explicitly and we can as well derive a representation of the propagator in an integral form. This allows us to compare the exact results with semiclassical approximations and perturbation theory and evaluate their accuracy. We finally apply these results to the study of the evolution of the von Neumann entropy of a state.