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Physical Review A

Semiclassical approximations to the coherent-state propagator for a particle in a box

Ademir Luiz Xavier Junior, Marcus A. M. de Aguiar

Published Sep 1, 1996• 20 citations
Publisher

Abstract

We compute the semiclassical coherent-state propagator for a particle moving in a one-dimensional box. In this semiclassical approach complex trajectories are stationary paths of the propagator's asymptotic expansion and play a fundamental role. A second semiclassical approximation is also introduced, which makes use of real trajectories only. An application to a seemingly simple system, the infinite well, is carried out completely for the diagonal elements, and a comparison is made among the three possible methods, those based on complex and real trajectories and the ``exact case'' that is determined by decomposing the propagator into its eigenstates. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.1808

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We compute the semiclassical coherent-state propagator for a particle moving in a one-dimensional box. In this semiclassical approach complex trajectories are stationary paths of the propagator's asymptotic expansion and play a fundamental role. A second semiclassical approximation is also introduced, which makes use of real trajectories only. An application to a seemingly simple system, the infinite well, is carried out completely for the diagonal elements, and a comparison is made among the three possible methods, those based on complex and real trajectories and the ``exact case'' that is determined by decomposing the propagator into its eigenstates. \textcopyright{} 1996 The American Physical Society.
In this work, we derived a semiclassical approximation for the matrix elements of a quantum propagator in coherent states (CS) basis that avoids complex trajectories; it only involves real ones. For that purpose, we used the symplectically invariant semiclassical Weyl propagator obtained by performing a stationary phase approximation (SPA) for the path integral in the Weyl representation. After that, for the transformation to CS representation SPA is avoided; instead a quadratic expansion of the complex exponent is used. This procedure also allows us to express the semiclassical CS propagator uniquely in terms of the classical evolution of the initial point without the need of any root search typical of van Vleck--Gutzwiller-based propagators. For the case of chaotic Hamiltonian systems, the explicit time dependence of the CS propagator has been obtained. The comparison with a realistic chaotic system that derives from a quadratic Hamiltonian, the cat map, reveals that the expression here derived is exact up to quadratic Hamiltonian systems.
A semiclassical approximation to the time evolution of coherent states may be derived from a saddle point approximation to the exact quantum propagator, and in general it involves complex classical dynamics. We generalize previous one-dimensional results to ddd dimensions, and for the case d=2d=2d=2 we present several applications. We also consider other simple approximations that depend only on real classical trajectories, but are not initial value representations. These approximations are able to reproduce interference and tunneling effects and involve propagating a few classical initial conditions compatible with the quantum uncertainties.
The semiclassical approximation to the coherent state propagator requires complex classical trajectories in order to satisfy the associated boundary conditions, but finding these trajectories in practice is a difficult task that may compromise the applicability of the approximation. In this work several approximations to the coherent state propagator are derived that make use only of real trajectories, which are easier to handle and have a more direct physical interpretation. It is verified in a particular example that these real trajectories approximations may have excellent accuracy.
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.
We study the application of the semiclassical coherent-state path-integral technique to the problem of scattering. A numerical method based on initial-value trajectories and their stability information is proposed to solve the root search for the complex trajectories that determine the semiclassical propagator. Very good agreement between the numerically exact quantum and semiclassical results for the correlation function can be achieved for scattering from one-dimensional barrier as well as well-type potentials by using just one single complex trajectory at each time.