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Chinese Physics B

The analytical transfer matrix method for quantum reflection

许田, Cao Zhuang-Qi, Fang Jing-Huai

Published Apr 1, 2010• 8 citations
Publisher

Abstract

In this paper, the analytical transfer matrix method (ATMM) is applied to study the properties of quantum reflection in three systems: a sech2 barrier, a ramp potential and an inverse harmonic oscillator. Our results agree with those obtained by Landau and Lifshitz [Landau L D and Lifshitz E M 1977 Quantum Mechanics (Non-relativistic Theory) (New York: Pergamon)], which proves that ATMM is a simple and effective method for quantum reflection.

DOI
10.1088/1674-1056/19/4/040307

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In this paper, the analytical transfer matrix method (ATMM) is applied to study the properties of quantum reflection in three systems: a sech2 barrier, a ramp potential and an inverse harmonic oscillator. Our results agree with those obtained by Landau and Lifshitz [Landau L D and Lifshitz E M 1977 Quantum Mechanics (Non-relativistic Theory) (New York: Pergamon)], which proves that ATMM is a simple and effective method for quantum reflection.
We calculate transition and reflection coefficients of arbitrary barrier by analytical transfer matrix method(ATMM),and then apply the method to the quantum reflection above the barrier.The results in step-exponential barrier and parabolic barrier by ATMM are more accurate than those by WKB approximation.Furthermore,when the energy of the particle is greater than the barrier maximum,only ATMM rather than WKB can get precise results.The results show that reflection coefficients gradually change to zero,where quantum reflection happens.These problems can be solved by ATMM exactly.
Based on the similarity between time-independent Schrodinger equation and Helmholtz equation, the analytical transfer matrix method was utilized to analyze macroscopic quantum tunneling effect, and the potential filed and eigenvalue were corresponded to the refractive index and propagation constant, respectively.Without solving the Schrodinger equation and any approximation,the exact analytic expressions of reflection coefficient and reflection probability of the particle were obtained.Initial results show the equations obtained here are exact and have clear physical meanings.It is expected to be applied in the researches of many basic quantum phenomena,such as quantum tunneling effect,quantum reflection, quantum particles and tunneling resonance.
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, 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.