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Journal of Contemporary Physics (Armenian Academy of Sciences)

Connection between the immersing and transfer-matrix methods in one-dimensional scattering problems

Д. М. Седракян, А. Ж. Хачатрян, Eduard M Kazaryan, Л. Р. Седракян

Published May 9, 2009• 6 citations
Publisher

Abstract

We show that application of the immersing and transfer-matrix methods to one-dimensional problems of particles scattering leads to the system of two linear equations for the functions F and Φ expressed by means of the transmission and reflection amplitudes. The expressions of these functions are derived. The offered method is illustrated by the finding of transmission and reflection coefficients for the potential barrier with a constant height. The developed method can be applied in solving the quasi-one-dimensional and two-dimensional problems of scattering.

DOI
10.3103/s1068337209030037

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We show that application of the immersing and transfer-matrix methods to one-dimensional problems of particles scattering leads to the system of two linear equations for the functions F and Φ expressed by means of the transmission and reflection amplitudes. The expressions of these functions are derived. The offered method is illustrated by the finding of transmission and reflection coefficients for the potential barrier with a constant height. The developed method can be applied in solving the quasi-one-dimensional and two-dimensional problems of scattering.
The immersing method is used to solve the two-channel scattering problem in the case of concrete potential. In particular, we consider the particle scattering by a two-dimensional barrier which is constant in the scattering direction and is arbitrary in the transverse direction. For this case the scattering amplitudes t 1, t 2, r 1, and r 2 are determined. Expressions for the transmission and reflection amplitudes are obtained for the case of δ-potential. The behavior of the scattering amplitudes in the limit k 2 → 0 is studied. It is also shown that the ratio of products of transmission and reflection amplitudes for two channels does not depend on the coordinate of the middle of coordinate.
The immersing method is applied to solve the N-channel scattering problem. In particular, we consider the particle scattering on a multidimensional potential barrier, which is constant in the scattering direction and arbitrary in the lateral direction. For this case the scattering amplitudes t m and r m (m = 1, 2, …, N) are determined. Transition from the obtained formulas to the case of thin potential is performed. For this case analytical expressions of transmission amplitudes t m and reflection amplitudes r m are obtained. We show that the product of transmission and reflection amplitudes in the channel m does not depend on the scattering channel. It is assumed that the scattering particle falls on the potential with the longitudinal wave vector k l corresponding to the channel l.
The immersing method is applied to solve the N-channel scattering problem for concrete potential. In particular, we consider the particle scattering on a two-dimensional potential barrier, which is constant in the scattering direction and arbitrary in the cross-section direction. For this case the scattering amplitudes t m and r m (1,2, m= ...,N) are determined. A transition from the obtained formulas to the case of ™-potential is performed. For this case transmission amplitudes t m and reflection amplitudes r m are obtained. It is also shown that the product of transmission and reflection amplitudes along the channel m does not depend on the scattering channel.
The Generalized Method of Transfer Matrix (MMT) appears as an alternative to quantum phenomena research in semiconductor materials and optoelectronic components. In this paper the MMT is applied to the propagation phenomena of a particle through a well or a potential barrier to the quantum case. The relations scattering amplitudes are deduced using the coefficients of the transfer matrix (transmission and reflection). The results are compared with related literature, to determine the accuracy and scope of the MMT, namely which can be applied to study many other problems of wave propagation and electrons, even in layered media as photonic crystals.
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.