American Journal of Physics

Exactly solvable three–dimensional scattering problem

Cornel Eftimiu

Published Aug 1, 1983• 1 citation

Abstract

Schrödinger’s equation is solved exactly for a potential vanishing outside a prolate spheroid. The potential is then reconstructed from the first Born approximation for the scattering amplitude.

DOI: 10.1119/1.13151

Related papers

Schrödinger Scattering Amplitude. III

Alex Grossmann, Tai Tsun Wu

The methods of an earlier paper are used to obtain a domain of analyticity for the Schrödinger scattering amplitude minus the first Born term. The connection between the scattering integral equation and the Schrödinger equation is also studied.

Journal of Mathematical PhysicsJul 1, 196223 citations

Quantum scattering of an exactly solvable anisotropic potential in paraboloidal coordinate systes

Philip Jordan D. Blancas, Eric A. Galapon

The differential scattering cross section of an anisotropic potential is computed by solving the time independent Schrodinger equation in paraboloidal coordinate system.

Proceedings of the Samahang Pisika ng PilipinasOct 22, 20120 citations

An exactly solvable one-dimensional three-body problem with hard cores

Robert Nyden Hill

Three identical particles in one dimension interact via a potential which is infinite whenever one or more of the interparticle separations is less than a or greater than b, and zero when all interparticle separations lie between a and b. Their Schrödinger equation is solved by reducing it to the exactly solvable problem of the two dimensional Helmholtz equation inside an equilateral triangle.

Journal of Mathematical PhysicsMay 1, 19804 citations

Method of obtaining a closed-form exact 3D scattering solution of the Schrödinger equation

N. Grama, Cornelia Grama, Ioan Zamfirescu

The European Physical Journal PlusOct 29, 20220 citations

Prolate Spheroidal Wave Function as Exact Solution of the Schrödinger Equation

J. Akande, Kolawolé Kêgnidé Damien Adjaï, L. H. Koudahoun +2 more

Jan 1, 20172 citations

Propagating-wave approximation in two-dimensional potential scattering

Farhang Loran, Alí Mostafazadeh

arXiv:2204.05153Physical Review ASep 9, 20222 citations

American Journal of Physics

Exactly solvable three–dimensional scattering problem

Cornel Eftimiu

Published Aug 1, 1983• 1 citation

Publisher

Abstract

Schrödinger’s equation is solved exactly for a potential vanishing outside a prolate spheroid. The potential is then reconstructed from the first Born approximation for the scattering amplitude.

DOI: 10.1119/1.13151

Related papers

Schrödinger Scattering Amplitude. III

Alex Grossmann, Tai Tsun Wu

Journal of Mathematical PhysicsJul 1, 196223 citations

Quantum scattering of an exactly solvable anisotropic potential in paraboloidal coordinate systes

Philip Jordan D. Blancas, Eric A. Galapon

The differential scattering cross section of an anisotropic potential is computed by solving the time independent Schrodinger equation in paraboloidal coordinate system.

Proceedings of the Samahang Pisika ng PilipinasOct 22, 20120 citations

An exactly solvable one-dimensional three-body problem with hard cores

Robert Nyden Hill

Journal of Mathematical PhysicsMay 1, 19804 citations

Method of obtaining a closed-form exact 3D scattering solution of the Schrödinger equation

N. Grama, Cornelia Grama, Ioan Zamfirescu

The European Physical Journal PlusOct 29, 20220 citations

Prolate Spheroidal Wave Function as Exact Solution of the Schrödinger Equation

J. Akande, Kolawolé Kêgnidé Damien Adjaï, L. H. Koudahoun +2 more

In quantum mechanics, the wave function and energy are required for the complete characterization of fundamental properties of a physical system subject to a potential energy. It is proved in this work, the existence of a Schrodinger equation with position-dependent mass having the prolate spheroidal wave function as exact solution, resulting from a classical quadratic Lienard-type oscillator equation. This fact may allow the extension of the current one-dimensional model to three dimensions and increase the understanding of analytical features of quantum systems.In quantum mechanics, the wave function and energy are required for the complete characterization of fundamental properties of a physical system subject to a potential energy. It is proved in this work, the existence of a Schrodinger equation with position-dependent mass having the prolate spheroidal wave function as exact solution, resulting from a classical quadratic Lienard-type oscillator equation. This fact may allow the extension of the current one-dimensional model to three dimensions and increase the understanding of analytical features of quantum systems.

Jan 1, 20172 citations

Propagating-wave approximation in two-dimensional potential scattering

Farhang Loran, Alí Mostafazadeh

We introduce a nonperturbative approximation scheme for performing scattering calculations in two dimensions that involves neglecting the contribution of the evanescent waves to the scattering amplitude. This corresponds to replacing the interaction potential $v$ with an associated energy-dependent nonlocal potential ${\mathcal{V}}_{k}$ that does not couple to the evanescent waves. The scattering solutions $\ensuremath{\psi}(\mathbf{r})$ of the Schr\"odinger equation, $(\ensuremath{-}{\ensuremath{\nabla}}^{2}+{\mathcal{V}}_{k})\ensuremath{\psi}(\mathbf{r})={k}^{2}\ensuremath{\psi}(\mathbf{r})$, have the remarkable property that their Fourier transform $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(\mathbf{p})$ vanishes unless $\mathbf{p}$ corresponds to the momentum of a classical particle whose magnitude equals $k$ . We construct a transfer matrix for this class of nonlocal potentials and explore its representation in terms of the evolution operator for an effective nonunitary quantum system. We show that the above approximation reduces to the first Born approximation for weak potentials and, similar to the semiclassical approximation, becomes valid at high energies. Furthermore, we identify an infinite class of complex potentials for which this approximation scheme is exact. We also discuss the appealing practical and mathematical aspects of this scheme.We introduce a nonperturbative approximation scheme for performing scattering calculations in two dimensions that involves neglecting the contribution of the evanescent waves to the scattering amplitude. This corresponds to replacing the interaction potential $v$ with an associated energy-dependent nonlocal potential ${\mathcal{V}}_{k}$ that does not couple to the evanescent waves. The scattering solutions $\ensuremath{\psi}(\mathbf{r})$ of the Schr\"odinger equation, $(\ensuremath{-}{\ensuremath{\nabla}}^{2}+{\mathcal{V}}_{k})\ensuremath{\psi}(\mathbf{r})={k}^{2}\ensuremath{\psi}(\mathbf{r})$, have the remarkable property that their Fourier transform $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(\mathbf{p})$ vanishes unless $\mathbf{p}$ corresponds to the momentum of a classical particle whose magnitude equals $k$. We construct a transfer matrix for this class of nonlocal potentials and explore its representation in terms of the evolution operator for an effective nonunitary quantum system. We show that the above approximation reduces to the first Born approximation for weak potentials and, similar to the semiclassical approximation, becomes valid at high energies. Furthermore, we identify an infinite class of complex potentials for which this approximation scheme is exact. We also discuss the appealing practical and mathematical aspects of this scheme.

arXiv:2204.05153Physical Review ASep 9, 20222 citations

v

with an associated energy-dependent nonlocal potential

{\mathcal{V}}_{k}

that does not couple to the evanescent waves. The scattering solutions $\ensuremath{\psi}(\mathbf{r})$ of the Schr\"odinger equation, $(\ensuremath{-}{\ensuremath{\nabla}}^{2}+{\mathcal{V}}_{k})\ensuremath{\psi}(\mathbf{r})={k}^{2}\ensuremath{\psi}(\mathbf{r})$, have the remarkable property that their Fourier transform $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(\mathbf{p})$ vanishes unless

\mathbf{p}

corresponds to the momentum of a classical particle whose magnitude equals

k

. We construct a transfer matrix for this class of nonlocal potentials and explore its representation in terms of the evolution operator for an effective nonunitary quantum system. We show that the above approximation reduces to the first Born approximation for weak potentials and, similar to the semiclassical approximation, becomes valid at high energies. Furthermore, we identify an infinite class of complex potentials for which this approximation scheme is exact. We also discuss the appealing practical and mathematical aspects of this scheme.