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Journal of Physics A Mathematical and General

Propagator for narrow potential barriers

Paolo Moretti

Published May 10, 2005• 4 citations
Publisher

Abstract

The Schrödinger equation in integral form is applied to the one-dimensional scattering problem in the case of a narrow potential barrier. Since the kernel can be considered, in a first approximation, separable, an explicit expression for the propagator is found by means of the complementary error function. The problem of a particle confined in a half-space and interacting with a narrow potential barrier is also considered and solved in an approximate way.

DOI
10.1088/0305-4470/38/21/013

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The Schrödinger equation in integral form is applied to the one-dimensional scattering problem in the case of a narrow potential barrier. Since the kernel can be considered, in a first approximation, separable, an explicit expression for the propagator is found by means of the complementary error function. The problem of a particle confined in a half-space and interacting with a narrow potential barrier is also considered and solved in an approximate way.
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The Schrödinger equation for the scattered particle is converted into an integral equation, in which the kernel is obtained by the use of the condition that the events described by the solutions of the equation must be causally related, i.e., that causes must always precede the effects which they produce. The integral equation is iterated once, and it is found that its solution corresponds to the usual description of a scattering process in terms of an undeflected wave traversing the system plus a radially scattered wave moving outwards. In the second part, it is shown that a simple relation exists between solutions of Schrödinger's equation of the sort just described and the solution of the initial-value problem associated with the equation of motion for a free particle. By this means a solution of the scattering problem is set up which describes how the wave function develops as the wave which it describes propagates through the system. The series which results was first given by Feynman. As an example, the first Born approximation for the scattering problem is developed by each approach.
Time dependence for barrier penetration is considered in the phase space. An asymptotic phase-space propagator for nonrelativistic scattering on a one-dimensional barrier is constructed. The propagator has a form universal for various initial state preparations and local potential barriers. It is manifestly causal and includes time-lag effects and quantum spreading. Specific features of quantum dynamics which disappear in the standard semiclassical approximation are revealed. The propagator may be applied to calculation of the final momentum and coordinate distributions, for particles transmitted through or reflected from the potential barrier, as well as for elucidating the tunneling time problem. \textcopyright{} 1996 The American Physical Society.