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arXiv:2204.05153 · Physical Review A

Propagating-wave approximation in two-dimensional potential scattering

Farhang Loran, Alí Mostafazadeh

Published Sep 9, 2022• 2 citations
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Abstract

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.

DOI
10.1103/physreva.106.032207

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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 vvv with an associated energy-dependent nonlocal potential Vk{\mathcal{V}}_{k}Vk​ that does not couple to the evanescent waves. The scattering solutions \ensuremathψ(r)\ensuremath{\psi}(\mathbf{r})\ensuremathψ(r) of the Schr"odinger equation, (\ensuremath−\ensuremath∇2+Vk)\ensuremathψ(r)=k2\ensuremathψ(r)(\ensuremath{-}{\ensuremath{\nabla}}^{2}+{\mathcal{V}}_{k})\ensuremath{\psi}(\mathbf{r})={k}^{2}\ensuremath{\psi}(\mathbf{r})(\ensuremath−\ensuremath∇2+Vk​)\ensuremathψ(r)=k2\ensuremathψ(r), have the remarkable property that their Fourier transform \ensuremathψ\ifmmode~\else˜\fi(p)\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(\mathbf{p})\ensuremathψ\ifmmode~\else˜\fi​(p) vanishes unless p\mathbf{p}p corresponds to the momentum of a classical particle whose magnitude equals kkk. 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.
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A development of the scattering amplitude in potential scattering which is based upon replacing a smooth potential by a sequence of potential steps is given. The Schrödinger equation is solved exactly in each interval over which the potential is constant and by matching boundary conditions an expression relating the coefficients of two linearly independent solutions at small radius to those at large radius is found. From this the scattering amplitude is calculated. The limits of validity are found to be dependent upon the value of the expression (r/2)ddrln(2mℏ2[E−V]). Limiting cases of the amplitude are shown to be a modified WKB approximation and the Born approximation. A particular application to the Yukawa potential is discussed. The discussion is restricted to attractive potentials.
We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multidelta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and the Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schrodinger equation for an effective nonunitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.
We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multi-delta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schr\"odinger equation for an effective non-unitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.
The scattering properties of any complex scattering potential, , can be obtained from the dynamics of a particular non-unitary two-level quantum system . The application of the adiabatic approximation to yields a semiclassical treatment of the scattering problem. We examine the adiabatic series expansion for the evolution operator of and use it to obtain corrections of arbitrary order to the semiclassical formula for the transfer matrix of v. This results in a high-energy approximation scheme that unlike the semiclassical approximation can be applied for potentials with large derivatives.