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Journal of Physics A Mathematical and General

Equiprobability of large-energy classical paths for one-dimensional motion in potential wells of infinite depth

M. Fusco Girard

Published Feb 21, 1995• 1 citation
Publisher

Abstract

The semiclassical propagator in configuration space for a particle with one-dimensional motion in a potential well of infinite depth is investigated. It is shown that, if the motion's period and its derivative are asymptotically decreasing functions of the energy, the large-energy classical paths give the same contribution (in modulus) to the propagator, and are therefore asymptotically equiprobable. This extends the results previously obtained for the quartic anharmonic oscillator.

DOI
10.1088/0305-4470/28/4/033

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The semiclassical propagator in configuration space for a particle with one-dimensional motion in a potential well of infinite depth is investigated. It is shown that, if the motion's period and its derivative are asymptotically decreasing functions of the energy, the large-energy classical paths give the same contribution (in modulus) to the propagator, and are therefore asymptotically equiprobable. This extends the results previously obtained for the quartic anharmonic oscillator.
The semiclassical propagator for the one-dimensional anharmonic oscillator is investigated, in the configuration space, by means of the so-called Van Vleck formula, which expresses it as a sum over all the denumerably infinite classical paths connecting given points in the same time. Analytical formulae for the paths' contributions are given, together with some numerical results. It is shown that in the general case the amplitudes of the contributions asymptotically approach the same value, while the phases oscillate; the Van Vleck series therefore does not converge, but in the generic case it can be resumed. Finally, the conditions under which one of the paths gives a dominant contribution to the semiclassical propagator are discussed.
This paper shows how to calculate the terms of a semiclassical (WKB) expansion of the quantum-mechanical propagator corresponding to the quartic anharmonic-oscillator potential, V=mω2q2/2 +λq4/4. This nonperturbative treatment expresses each term in the series as a path integral, which is then evaluated in the framework of a formalism, introduced by C. DeWitt-Morette, which does not entail the usual time-slicing operation followed by a limiting procedure. The Gaussian measure used absorbs all the quadratic terms in the expansion of the action functional about a classical path. The covariance of this Gaussian measure is Feynman’s Green function for the small-disturbance operator of the system. This function can be obtained by varying the constants of integration in the classical solution, and therefore the coefficients of the expansion depend only on this classical solution. If the latter is chosen to be the one which tends to its harmonic counterpart when λ→0, then it is seen that the propagator also tends to its harmonic counterpart when λ→0.
We introduce a very general approximation to the quantum propagator that is based on the assumption that the most important contributions to the complex semiclassical propagator evolve from real classical trajectories that almost satisfy the desired boundary conditions. Our results for two systems --- the autocorrelation function for the quartic anharmonic oscillator and the photodissociation spectrum of CO2{\mathrm{CO}}_{2}CO2​ --- show that these nearly real contributions yield an excellent approximation to the quantum propagator for quite long times. The approach taken here is applicable to problems with many (e.g., several hundred) degrees of freedom, and hence promises to provide an accurate and useful representation of the quantum dynamics for a wide variety of physically interesting systems.