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Journal of Physics A Mathematical and General

Exact propagators for moving hard-wall potentials

M. G. E. da Luz, Bin Kang Cheng

Published Sep 7, 1992• 20 citations
Publisher

Abstract

The semiclassical approximation has been used to evaluate the exact propagators for two one-dimensional quantum systems: (a) a free particle interacting with one hard-wall potential moving with constant velocity, and (b) a free particle inside a rigid box with one wall moving uniformly in time. The results derived from these propagators are in agreement with those of the corresponding Schrodinger equations.

DOI
10.1088/0305-4470/25/17/005

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The semiclassical approximation has been used to evaluate the exact propagators for two one-dimensional quantum systems: (a) a free particle interacting with one hard-wall potential moving with constant velocity, and (b) a free particle inside a rigid box with one wall moving uniformly in time. The results derived from these propagators are in agreement with those of the corresponding Schrodinger equations.
We devise and apply a method for the calculation of the exact propagator for the Schr\"odinger equation for the one-dimensional Hamiltonian scrH(p,x)=p2{\mathit{p}}^{2}p2/2m+V0{\mathit{V}}_{0}V0​\ensuremath{\Theta}(x). The limiting cases V0{\mathit{V}}_{0}V0​\ensuremath{\rightarrow}0 (free particle) and V0{\mathit{V}}_{0}V0​\ensuremath{\rightarrow}\ensuremath{\infty} (opaque wall) are checked. The method is also applicable to any piecewise-constant potential.
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.