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Progress of Theoretical Physics

Potential Scattering in Three Dimension: Backscattering and Fresnel Effects

Tatsumi Ishihara

Published Nov 1, 1976• 2 citations
Publisher

Abstract

Starting from the Lippmann-Schwinger equation, we give two kinds of new corrections in three-dimensional potential scattering. The one includes explicitly backscattering effects along the z-axis (to-and-fro motion). The other incorporates the Fresnel diffraction with the to-and-fro motion. This theory gives a clear physical image of the forward and backward propagation of a wave. Choice of the z-axis along ki + kf is used to facilitate calculation. Relation to the eikonal approximation is easily read from resultant expressions of scattering amplitude.

DOI
10.1143/ptp.56.1521

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Starting from the Lippmann-Schwinger equation, we give two kinds of new corrections in three-dimensional potential scattering. The one includes explicitly backscattering effects along the z-axis (to-and-fro motion). The other incorporates the Fresnel diffraction with the to-and-fro motion. This theory gives a clear physical image of the forward and backward propagation of a wave. Choice of the z-axis along ki + kf is used to facilitate calculation. Relation to the eikonal approximation is easily read from resultant expressions of scattering amplitude.
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.
A self-contained discussion of integral equations of scattering is presented in the case of centrally symmetric potentials in one dimension, which will facilitate the understanding of more complex scattering integral equations in two and three dimensions. The present discussion illustrates in a simple fashion the concept of partial-wave decomposition, Green’s function, Lippmann–Schwinger integral equations of scattering for wave function and transition operator, optical theorem, and unitarity relation. We illustrate the present approach with a Dirac delta potential.
We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multidelta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and the Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schrodinger equation for an effective nonunitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.