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American Journal of Physics

Two-dimensional time-dependent quantum-mechanical scattering event

Ian Galbraith, Yin Sing Ching, E. Abraham

Published Jan 1, 1984• 50 citations
Publisher

Abstract

Nonrelativistic quantum-mechanical scattering in two dimensions is studied numerically by integrating the time-dependent Schrödinger equation. A partial-wave analysis is used to discuss the numerical results. A potential barrier, a square well and a single slit are considered as the scattering potentials.
DOI
10.1119/1.13811

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A self-contained discussion of nonrelativistic quantum mechanical potential scattering in two dimensions is presented. The discussion includes, among other topics, partial wave decomposition in coordinate and momentum space, Lippmann–Schwinger integral equations of scattering for the scattering wavefunction and the transition operator, optical theorem, and the unitarity relation for the transition operator. The present definition of the scattering amplitude in terms of the asymptotic wavefunction differs from the usual definition. The present definition has certain advantages, for example, in writing the optical theorem and in studying the analytical properties of the scattering amplitude.
Quantum mechanics describes phenomena that are key to understanding processes that cannot be explained by classical physics. This paper analyzes the behavior of particles in the presence of a potential barrier through numerical solutions to both time-dependent and time-independent Schrödinger equations. These results enabled a detailed understanding of the energy structure and temporal dynamics of wave functions, including the phenomenon of quantum tunneling. The obtained data have a wide range of applications in technology: from semiconductor devices to quantum computing.