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Theoretical and Mathematical Physics

The method of amplitude functions in two-dimensional scattering theory

Vasily Veniaminovitch Pupyshev

Published Apr 1, 2017• 6 citations
Publisher

Abstract

We present a formulation and mathematical justification of the method of amplitude functions. This method allows solving the radial problem of the two-dimensional scattering of a quantum particle by the sum of a Coulomb potential and a certain short-range or long-range central potential.
DOI
10.1134/s0040577917040043

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The concept of a generalized scattering amplitude is introduced and applied to quantum potential scatterings. After an outgoing spherical wave is factored out from a wave function, the remaining wave envelope is defined as the generalized scattering amplitude. The transformed Schr\"odinger equation, in terms of the generalized scattering amplitude, can be solved numerically by using a finite-difference method over the entire scattering domain. Example problems of scatterings by a spherical potential well and the (${\mathit{e}}^{\mathrm{\ensuremath{-}}}$,H) static field interaction potential are solved to validate the theoretical formulation and numerical method. The far-field solutions for the ordinary scattering amplitude and differential cross section agree very well with those obtained from the partial-wave analysis. The radial profiles of the generalized scattering amplitude and particle density function over the entire scattering region are also presented, and their properties are discussed. These results demonstrate that the method can be used to yield a complete and accurate solution to scattering problems involving arbitrary interaction potentials.
We prove that radial wave functions of a charged quantum particle moving in a two-dimensional plane of the three-dimensional coordinate space and scattering by a Coulomb center at rest in the same plane are governed by the Coulomb equation with a half-integer index. We investigate the structure of these functions and consider three physically interesting limits: the non-Coulomb limit and high- and low-energy limits. We explicate the basic differences between two- and three-dimensional Coulomb scattering.
The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.