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American Journal of Physics

Zero range potential approximation in quantum scattering problems

Eliton Popovicz Seidel, Felipe Arretche

Published Sep 18, 2019• 1 citation
Publisher

Abstract

This article presents the Zero Range Potential (ZRP) approximation as an alternative to treat quantum scattering problems. The main focus is to provide an undergraduate-level derivation of the ZRP boundary condition that describes the interaction between an electron and a potential, using the spherical-potential-well analytical solutions. The validity of the approximation is discussed qualitatively and quantitatively. Finally, the ZRP approximation is applied to the scattering of an electron by an atom, comparing the results obtained with modern experimental measurements.

DOI
10.1119/1.5125110

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This article presents the Zero Range Potential (ZRP) approximation as an alternative to treat quantum scattering problems. The main focus is to provide an undergraduate-level derivation of the ZRP boundary condition that describes the interaction between an electron and a potential, using the spherical-potential-well analytical solutions. The validity of the approximation is discussed qualitatively and quantitatively. Finally, the ZRP approximation is applied to the scattering of an electron by an atom, comparing the results obtained with modern experimental measurements.
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 describe a method for evaluating analytical long-range contributions to scattering lengths for some potentials used in atomic physics. We assume that an interaction potential between colliding particles consists of two parts. The form of a short-range component, vanishing beyond some distance from the origin (a core radius), need not be given. Instead, we assume that a set of short-range scattering lengths due to that part of the interaction is known. A long-range tail of the potential is chosen to be an inverse power potential, a superposition of two inverse power potentials with suitably chosen exponents or the Lent potential. For these three classes of long-range interactions a radial Schrodinger equation at zero energy may be solved analytically with solutions expressed in terms of the Bessel, Whittaker and Legendre functions, respectively. We utilize this fact and derive exact analytical formulae for the scattering lengths. The expressions depend on the short-range scattering lengths, the core radius and parameters characterizing the long-range part of the interaction. Cases when the long-range potential (or its part) may be treated as a perturbation are also discussed and formulae for scattering lengths linear in strengths of the perturbing potentials are given. It is shown that for some combination of the orbital angular momentum quantum number and an exponent of the leading term of the potential the derived formulae, exact or approximate, take very simple forms and contain only polynomial and trigonometric functions. The expressions obtained in this paper are applicable to scattering of charged particles by neutral targets and to collisions between neutrals. The results are illustrated by accelerating convergence of scattering lengths computed for e - -Xe and Cs-Cs systems.
Electron scattering by rare-gas dimers is studied for very low incident energies using the zero range potential (ZRP) method. Beyond the traditional ZRP, we consider an alternative formulation accounting for the atomic polarization, inspired in the modified effective range theory. The scattering calculations are reported in fixed nuclei, rigid rotor, and rovibrational approximations, the first two being analytical. An expression for the electron-molecule scattering length is obtained. We find that short-range interactions are the dominant mechanism for rotational transitions in electron scattering by rare-gas dimers, while the long-range interactions may be neglected. Our results show how the elastic, rotational, and rovibrational cross sections depend on the electron-atom scattering length, on the molecular parameters, and also on the inclusion of polarization effects. The principle of detailed balance is discussed in the context of the ZRP method. Finally, we show that the rovibrational coupling considerably affects the rotational cross sections when the rotational constant becomes comparable to the vibrational constant.
We study the problem of a nonrelativistic quantum particle moving on a real line with an idealized and localized singular interaction with zero range at x = 0 (i.e., a point interaction there). This kind of system can be described in two ways: (i) by considering an alternative free system (i.e., without the singular potential) but excluding the point x = 0 (In this case, the point interaction is exclusively encoded in the boundary conditions.) and (ii) by explicitly considering the singular interaction by means of a local singular potential. In this paper we relate, compare, and discuss, in a simple and pedagogical way these two equivalent approaches. Our main goal in this paper is to introduce the essential ideas about point interactions in a very accesible form to advanced undergraduates.