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Physical Review

Born Approximation and Large-Momentum-Transfer Processes in Potential Scattering

Tai Tsun Wu

Published Mar 25, 1966• 10 citations
Publisher

Abstract

Several problems of nonrelativistic scattering by analytic potentials are studied, where both the energy and the momentum transfer are large. Specifically, we find asymptotically the reflection coefficients from the one-dimensional potentials $V{e}^{x}{(1+{e}^{x})}^{\ensuremath{-}2}$, $V{e}^{\frac{x}{2}}{(1+{e}^{x})}^{\ensuremath{-}1}$, and $V\mathrm{exp}(\ensuremath{-}{x}^{2})$, and the scattering amplitudes for their three-dimensional generalizations $V{e}^{r}{(1+{e}^{r})}^{\ensuremath{-}2}$, $V{e}^{\frac{r}{2}}{(1+{e}^{r})}^{\ensuremath{-}1}$, and $V\mathrm{exp}(\ensuremath{-}{r}^{2})$. The results are compared with these obtained from Born approximation, and it is found that the first Born approximation gives the correct answer asymptotically in the cases $V{e}^{\frac{x}{2}}{(1+{e}^{x})}^{\ensuremath{-}1}$ and $V{e}^{\frac{r}{2}}{(1+{e}^{r})}^{\ensuremath{-}1}$, but not in the other cases. Conjectures about more general cases are also given.

DOI
10.1103/physrev.143.1110

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Several problems of nonrelativistic scattering by analytic potentials are studied, where both the energy and the momentum transfer are large. Specifically, we find asymptotically the reflection coefficients from the one-dimensional potentials Vex(1+ex)\ensuremath−2V{e}^{x}{(1+{e}^{x})}^{\ensuremath{-}2}Vex(1+ex)\ensuremath−2, Vex2(1+ex)\ensuremath−1V{e}^{\frac{x}{2}}{(1+{e}^{x})}^{\ensuremath{-}1}Ve2x​(1+ex)\ensuremath−1, and Vexp(\ensuremath−x2)V\mathrm{exp}(\ensuremath{-}{x}^{2})Vexp(\ensuremath−x2), and the scattering amplitudes for their three-dimensional generalizations Ver(1+er)\ensuremath−2V{e}^{r}{(1+{e}^{r})}^{\ensuremath{-}2}Ver(1+er)\ensuremath−2, Ver2(1+er)\ensuremath−1V{e}^{\frac{r}{2}}{(1+{e}^{r})}^{\ensuremath{-}1}Ve2r​(1+er)\ensuremath−1, and Vexp(\ensuremath−r2)V\mathrm{exp}(\ensuremath{-}{r}^{2})Vexp(\ensuremath−r2). The results are compared with these obtained from Born approximation, and it is found that the first Born approximation gives the correct answer asymptotically in the cases Vex2(1+ex)\ensuremath−1V{e}^{\frac{x}{2}}{(1+{e}^{x})}^{\ensuremath{-}1}Ve2x​(1+ex)\ensuremath−1 and Ver2(1+er)\ensuremath−1V{e}^{\frac{r}{2}}{(1+{e}^{r})}^{\ensuremath{-}1}Ve2r​(1+er)\ensuremath−1, but not in the other cases. Conjectures about more general cases are also given.
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.
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 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 $\ensuremath{\psi}(\mathbf{r})$ 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 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.
We develop an analytic approach to evaluating the density ρ(E,Γ)ρ({\cal E},Γ)ρ(E,Γ) of complex resonance poles with real energies E\mathcal{E}E and widths ΓΓΓ in the pure reflection problem from a one-dimensional disordered sample with white-noise random potential. We start with establishing a general link between the density of resonances and the distribution of the reflection coefficient r=∣R(E,L)∣2r=|R(E,L)|^2r=∣R(E,L)∣2, where R(E,L)R(E,L)R(E,L) is the reflection amplitude, at {\it complex} energies E=E+iηE = {\cal E} +iηE=E+iη, identifying the parameter η>0η>0η>0 with the uniform rate of absorption within the disordered medium. We show that leveraging this link allows for a detailed analysis of the resonance density in the weak disorder limit. In particular, for a (semi)infinite sample, it yields an explicit formula for ρ(E,Γ)ρ({\cal E},Γ)ρ(E,Γ), describing the crossover from narrow to broad resonances in a unified way. Similarly, our approach yields a limiting formula for ρ(E,Γ)ρ({\cal E},Γ)ρ(E,Γ) in the opposite case of a short disordered sample, with size much smaller than the localization length. This regime seems to have not been systematically addressed in the literature before, with the corresponding analysis requiring an accurate and rather non-trivial implementation of WKB-like asymptotics in the scattering problem. Finally, we study the resonance statistics numerically for the one-dimensional Anderson tight-binding model and compare the results with our analytic expressions.