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American Journal of Physics

Scattering by a Coulomb field in two dimensions

Qiong-Gui Lin

Published Oct 1, 1997• 18 citations
Publisher

Abstract

The scattering cross section in a Coulomb field in two dimensions is obtained by solving the Schrödinger equation in parabolic coordinates. In contrast with the three-dimensional case, the result is different from the corresponding classical one and does not coincide with the Born approximation in general. The quantum-mechanical cross section depends on ℏ and reduces to the classical one in the limit ℏ→0.

DOI
10.1119/1.18694

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The scattering cross section in a Coulomb field in two dimensions is obtained by solving the Schrödinger equation in parabolic coordinates. In contrast with the three-dimensional case, the result is different from the corresponding classical one and does not coincide with the Born approximation in general. The quantum-mechanical cross section depends on ℏ and reduces to the classical one in the limit ℏ→0.
Parabolic coordinates allow Schrödinger’s equation to be solved analytically for a particle confined to the xy plane and incident with velocity v on a potential C/r. In sharp contrast to the three-dimensional case, the cross section assumes its classical value only in the rationally foreseeable classical regime ‖C‖/ℏv≫1, but not otherwise; and in this low-velocity regime the enhancement factor (probability density at the origin) for an attractive potential tends to 2 (independently of the value of ‖C‖), instead of diverging as in three dimensions like some inverse power of v.
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.
Synopsis: Students are introduced to the ideas of particle scattering in quantum mechanics in a one-dimensional context. ƒe integral Schrödinger equation is deduced using a Wronskian technique, and reflection and transmission amplitudes are defined. Several simple scattering problems are solved, and the Born series for the energy-wave function is explained. A three-dimensional spherical basis is introduced, which allows the integral Schrödinger equation in this context to be formulated. ƒe relation between plane waves and spherical waves is found from a solution to the unit-source Helmholtz equation, again using the Wronskian technique. Partial waves, phase shifts, and cross sections are defined, and finite range scattering is considered. ƒe chapter concludes with a consideration of identical particle scattering and an application to the protonproton case, where a dramatic manifestation of the underlying substructure emerges at high energies.
Synopsis: Students are introduced to the ideas of particle scattering in quantum mechanics in a one-dimensional context. The integral Schrödinger equation is deduced using a Wronskian technique, and reflection and transmission amplitudes are defined. Several simple scattering problems are solved, and the Born series for the energy-wave function is explained. A three-dimensional spherical basis is introduced, which allows the integral Schrödinger equation in this context to be formulated. The relation between plane waves and spherical waves is found from a solution to the unit-source Helmholtz equation, again using the Wronskian technique. Partial waves, phase shifts, and cross sections are defined, and finite range scattering is considered. The chapter concludes with a consideration of identical particle scattering and an application to the proton–proton case, where a dramatic manifestation of the underlying substructure emerges at high energies.