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Physica Scripta

Quantum scattering by a spherical barrier with an arbitrary coupling strength

Pedro C. Azado, A. C. Maioli, Alexandre G. M. Schmidt

Published May 4, 2021• 9 citations
Publisher

Abstract

Abstract A solution to the Lippmann-Schwinger equation for the scattering of a plane wave by a spherical barrier is presented. The spherical barrier is described by a boundary-wall with coupling constant γ = γ ( θ , ϕ ). A general method is presented, along with several possible examples for the coupling constant such as sin θ and cos θ dependencies, Janus particle and a ring-shaped surface. The exact scattering amplitude for the general case is also presented.

DOI
10.1088/1402-4896/abfde4

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Abstract A solution to the Lippmann-Schwinger equation for the scattering of a plane wave by a spherical barrier is presented. The spherical barrier is described by a boundary-wall with coupling constant γ = γ ( θ , ϕ ). A general method is presented, along with several possible examples for the coupling constant such as sin θ and cos θ dependencies, Janus particle and a ring-shaped surface. The exact scattering amplitude for the general case is also presented.
An exact quantum formalism for atom scattering from a sinusoidal hard-wall surface is presented. The Lippmann-Schwinger equation is solved for a scattering kernel consistent with the hard-wall boundary conditions on the Schr\"odinger equation. It results in an infinite-dimensional matrix equation for the Fourier coefficients of the scattering kernel which can be solved in a finite-dimensional limit to convergence. The results show either rainbow or specular patterns depending on the surface roughness and incident kkk vector, as predicted by semiclassical and coupled-channel calculations. Bragg-like structure is present with the periodicity of the amplitude of the sinusoidal hard wall and the effects of multiple scattering are evidenced at large surface amplitudes.
Abstract In quantum mechanics textbooks, a single-particle scattering theory is introduced. In the present work, a generalized scattering theory is presented, which can be in principle applied to the scattering problems of arbitrary number of particle. In laboratory frame, a generalized Lippmann-Schwinger scattering equation is derived. We emphasized that the derivation is rigorous, even for treating infinitesimals. No manual operation such as analytical continuation is allowed. In the case that before scattering N particles are plane waves and after the scattering they are new plane waves, the transition amplitude and transition probability are given and the generalized S matrix is presented. It is proved that the transition probability from a set of plane waves to a new set of plane waves of the N particles equal to that of the reciprocal process. The generalized theory is applied to the cases of one- and two-particle scattering as two examples. When applied to single-particle scattering problems, our generalized formalism degrades to that usually seen in the literature. When our generalized theory is applied to two-particle scattering problems, the formula of the transition probability of two-particle collision is given. It is shown that the transition probability of the scattering of two free particles is identical to that of the reciprocal process. This transition probability and the identity are needed in deriving Boltzmann transport equation in statistical mechanics. The case of identical particles is also discussed.
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 investigate the non-relativistic scattering of a plane wave by a vertical segment formulating the problem in terms of the Lippmann–Schwinger equation in two spatial dimensions. Adjusting the coupling strength function we show how to implement the scattering by a system of multiple slits and by a Cantor set. We present detailed calculations of the scattered wave function for the line segment, as well as for the single, double, and multiple slits. We define reflection and transmission functions that are position-dependent in a defined region. From these results, we obtain the probability densities and differential and total cross-sections for these problems.
Abstract We study the non-relativistic quantum mechanical scattering of a plane wave by a shield barrier and an elliptical aperture modeled as Dirac delta functions running along a coordinate surface of the sphero-conal coordinate system. The scattering problem is formulated via Lippmann-Schwinger (LS) equation in the position representation. In order to solve the LS equation, we first calculate the free Green's function of the problem and obtain its bilinear expansion in terms of the eigenfunctions of the scalar Helmholtz equation—which are products of spherical Bessel (or first kind Hankel) functions and Lamé polynomials. Such bilinear expansion allows us to obtain an integral equation with a separable kernel and solve the scattering problem. Then, we calculate the wavefunctions in the internal and external domains and the scattering amplitudes.