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Physical Review A

Eigenchannel method for nonrelativistic scattering from zero-range potentials

Radosław Szmytkowski, Marek Gruchowski

Published Dec 28, 2004• 11 citations
Publisher

Abstract

An eigenchannel method for nonrelativistic quantum scattering from an arbitrary system of zero-range potentials is presented. Eigenchannel vectors are defined as characteristic vectors of an auxiliary Hermitian matrix spectral problem with a positive semidefinite weight; real eigenvalues to this problem are the negative cotangents of eigenphase shifts. The eigenchannel vectors and the eigenphase shifts are used to construct particular solutions, called eigenchannels, of the pertinent Schr\"odinger equation. Analysis of the asymptotics of the eigenchannels in the far zone leads to a definition of eigenchannel harmonics, which for the problem at hand play the role analogous to that played by spherical harmonics for scattering in a central field. Representations of a far-field amplitude (for scattering of particles emitted from a point source) as well as a scattering amplitude, a scattering kernel, total, and total averaged cross sections (for scattering of a parallel beam of projectiles) in terms of the eigenphase shifts and the eigenchannel harmonics are derived. As an illustrative example, scattering of a plane wave from a system of four identical zero-range potentials, located in vertices of a regular tetrahedron, is worked out in detail.

DOI
10.1103/physreva.70.062719

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An eigenchannel method for nonrelativistic quantum scattering from an arbitrary system of zero-range potentials is presented. Eigenchannel vectors are defined as characteristic vectors of an auxiliary Hermitian matrix spectral problem with a positive semidefinite weight; real eigenvalues to this problem are the negative cotangents of eigenphase shifts. The eigenchannel vectors and the eigenphase shifts are used to construct particular solutions, called eigenchannels, of the pertinent Schr"odinger equation. Analysis of the asymptotics of the eigenchannels in the far zone leads to a definition of eigenchannel harmonics, which for the problem at hand play the role analogous to that played by spherical harmonics for scattering in a central field. Representations of a far-field amplitude (for scattering of particles emitted from a point source) as well as a scattering amplitude, a scattering kernel, total, and total averaged cross sections (for scattering of a parallel beam of projectiles) in terms of the eigenphase shifts and the eigenchannel harmonics are derived. As an illustrative example, scattering of a plane wave from a system of four identical zero-range potentials, located in vertices of a regular tetrahedron, is worked out in detail.
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.
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.
Most of our knowledge about the structure of matter and interaction between particles is derived from scattering experiments. From a theoretical point of view, scattering problems are concerned with the continuous (and positive) energy eigenvalues and unbound eigenfunctions of the Schro¨dinger equation. We have already encountered one-dimensional examples of scattering in Chapter 4, in the discussion of reflection and transmission of an incident particle with definite momentum from a potential step or barrier. In this chapter we consider scattering from a more formal point of view. We will confine our discussion to elastic scattering (scattering without loss or gain of energy by the projectile) from central potentials although many of the concepts and results are applicable to inelastic scattering.
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.
In this chapter we discuss the quantum theory of scattering of incident particles from the target where an incident beam has point particles, scattering is elastic, the scattering potential is time independent, there is no multiple scattering, and the interaction between an incident particle and a scattering center is described by the potential that depends only on relative positions of particles. Scattering cross-section and frame of reference, asymptotic expansion of wave function and formal expression for scattering amplitude, partial wave analysis, integral equation, and the Born approximation are covered in the chapter, in a broader sense. Free particle and asymptotic solutions, scattering amplitude and phase shift, optical theorem, scattering length, phase shift and its interpretation, scattering by a square well potential and by hard sphere potential, and interpretation of the phase shift and its expression are explained with the use of partial wave analysis. Scattering from bare and screened potentials within the Born approximation, validity of the Born approximation, and the transformation from the center of a mass coordinate system to a laboratory coordinate system are written towards the end of the chapter.
The problem of determining the Schr\"odinger wave function of a nonrelativistic particle that is either scattered by a potential of a finite range or that is bound to it is reformulated in a novel way. It is shown that in either case the wave function must satisfy a certain boundary condition on the surface that delimits the effective range of the potential. For scattering states the boundary condition is analogous to the mathematical formulation of the Ewald-Oseen extinction theorem of classical electromagnetic theory. The new formulation is illustrated by determining the scattering states and the bound states for a central potential. It is also shown that a boundary condition that is used in band-structure calculations in solids is an immediate consequence of our quantum-mechanical extinction theorem for bound states.