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Physica Scripta

Non-relativistic scattering by a shield barrier and by an elliptical aperture

Alexandre G. M. Schmidt, Anderson L. de Jesus

Published Jul 5, 2022• 3 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ac7e7e

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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.
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.
A self-contained discussion of nonrelativistic quantum mechanical potential scattering in two dimensions is presented. The discussion includes, among other topics, partial wave decomposition in coordinate and momentum space, Lippmann–Schwinger integral equations of scattering for the scattering wavefunction and the transition operator, optical theorem, and the unitarity relation for the transition operator. The present definition of the scattering amplitude in terms of the asymptotic wavefunction differs from the usual definition. The present definition has certain advantages, for example, in writing the optical theorem and in studying the analytical properties of the scattering amplitude.
Abstract In this paper, we propose a pedagogical presentation of the Lippmann–Schwinger equation as a powerful tool, so as to obtain important scattering information. In particular, we consider a one-dimensional system with a Schrödinger-type free Hamiltonian decorated with a sequence of N attractive Dirac delta interactions. We first write the Lippmann–Schwinger equation for the system and then solve it explicitly in terms of an N × N matrix. Then, we discuss the reflection and the transmission coefficients for an arbitrary number of centres and study the threshold anomaly for the N = 2 and N = 4 cases. We also study further features like the quantum metastable states and resonances, including their corresponding Gamow functions and virtual or antibound states. The use of the Lippmann–Schwinger equation simplifies our analysis enormously and gives exact results for an arbitrary number of Dirac delta potentials.
The covariant single-time equations of the quantum field theory are formulated in the relativistic configurational representation. The explicit formulae for the Green functions corresponding to the scattering states are calculated in this representation. Using the derived nonhomogeneous equations the scattering problem is solved exactly for certain potentials (combinations of zero-range potentials). The equations and their solutions are studied in the non-relativistic limit. The conditions of total reflection, available for such potentials, are investigated.
Abstract The present work aims to obtain an exact solution for the Lippmann–Schwinger equation considering a scalar particle in three-dimensional hyperspherical and hyper-pseudospherical spaces. We analytically solve the Lippmann–Schwinger equation for a potential of the boundary-wall type, written as Dirac delta distributions over a two-dimensional single barrier. Our Lippmann–Schwinger equation in space representation is a second-kind Fredholm integral equation, where a bilinear form of the Green’s function leads to a separable kernel. The probability density for the particle, considering level surfaces in both spaces, and the differential cross-section in the hyper-pseudospherical space, are also presented graphically.