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Annals of Physics

Semiclassical Trajectory-Coherent Approximation in Quantum Mechanics I. High-Order Corrections to Multidimensional Time-Dependent Equations of Schrödinger Type

Vladislav G. Bagrov, Владимир Владимирович Белов, Andrey Yu. Trifonov

Published Mar 1, 1996• 79 citations
Publisher
DOI
10.1006/aphy.1996.0027

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In this paper we present a method for obtaining the WKB approximation for non-separable multidimensional potentials. At its leading order one has to solve the classical Hamilton-Jacobi equation with zero energy for which a very efficient method is proposed. The essence of this method lies in the recognition that in the vicinity to the potential minimum the solution of Schrödinger equation has to approach that of coupled harmonic oscillators. Quantum corrections to the semiclassical result are then obtained very easily and to an arbitrarily high order. The method is applied to the calculation of the tunneling splitting and tunneling lifetime. We show that classical turning points are part of the dynamical problem; they could not be determined simply from looking at the form of the potential, but are obtained from the solution of the pertinent equations of motion.
The quantum trajectories in the de Broglie-Bohm formulation of quantum mechanics depend on an additional quantum potential derived from the full wave solution of Schrödinger's equation. The task of supplying collectively all the correct quantum results strongly alters the characteristics of the corresponding classical trajectories, which underlie semiclassical approximations to the evolving wave function. Both classical and quantum trajectories are here considered to be conservative with no influence of an external environment, even though this is the source of eventual classicality in quantum systems, that is, decoherence. The concept of integrability, closely correspondent in classical and quantum mechanics, is not preserved by the quantum trajectories. General systems, in which classical chaotic motion participates, are much harder to treat semiclassically, but quantum trajectories can be chaotic even for integrable systems. This discrepancy between the character of classical and quantum trajectories in the de Broglie-Bohm interpretation does not clarify the singular classical-quantum transition.
It is by now well recognized how the quantum time evolution for time-dependent Hamiltonians that are quadratic both in position and momentum variables generates ‘‘squeezing.’’ On the other hand, ‘‘squeezed states’’ and their ‘‘high-n generalizations’’ are known to provide a simple classical mechanical description of general quantum time evolution. In this paper, these ideas are combined to show the relevance of ‘‘generalized squeezed states’’ in the semiclassical limit of the evolution of time-dependent wave packets for general time-dependent Hamiltonians, including the multidimensional case.