Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Sichuan Teachers College

Transfer-Matrix Formalism of One-Dimensional Potential Barrier Problems

Tao Cai

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

The paper introduces a simplified and exact transfer matrix formalism.That can be used to study a series of one dimensional potential barrier problems with only two general purpose matrices one to propagate a wave function over a region of constant potential,the other to take a wave function over a discontinuity in a potential.Not only are standard results recovered for one potential barrier problem with this technique,but it is easy to extend this formalism into two,even more many potential barier cases,which facilitates wide ranging explorations of one dimensional quantum mechanics.

Related papers

Exploring one-dimensional quantum mechanics with transfer matrices

James S. Walker, J. Gathright

An exact transfer-matrix formalism is developed for analyzing and solving problems in one-dimensional quantum mechanics. We show that with only three general-purpose matrices—one to propagate a wave function over a region of constant potential, one to take a wave function over a discontinuity in a potential, and one to connect a wave function across a delta function—a rich and intriguing variety of behavior is revealed. Not only are standard results recovered with this technique, in ways suitable for presentation in the classroom, but new findings and applications are discussed as well. A primary advantage of the transfer-matrix approach is that it facilitates wide-ranging explorations of one-dimensional quantum mechanics by both students and researchers, especially when implemented with Mathematica. For those interested in pursuing independent explorations, an electronic, interactive version of this paper, complete with the figures given here and the code that generates them, is available over Internet as a Mathematica notebook.

American Journal of Physics•May 1, 1994•51 citations

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

Д. М. Седракян, А. Ж. Хачатрян, Eduard M Kazaryan +1 more

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.

Journal of Contemporary Physics (Armenian Academy of Sciences)•May 9, 2009•6 citations

Tunneling through one-dimensional piecewise-constant potential barriers

Siddhant Das

In this paper, we examine transmission through one-dimensional potential barriers that are piecewise constant. The transfer matrix approach is adopted, and a new formula is derived for multiplying long matrix sequences that not only leads to an elegant representation of the wave function but also results in much faster computation than earlier methods. The proposed method covers a broad spectrum of potentials, of which multi-barrier systems are special cases. The procedure is illustrated with a finite lattice of nonuniform rectangular barriers—non-uniformity being a novel feature, as the uniform case has been solved exactly by Griffiths and Steinke. For the nonuniform multi-barrier problem, the intervening wells strongly influence the transmission probability. Surprisingly, we find that the wells act “individually,” i.e., their influence is a function only of their width and is independent of their exact locations in a multi-barrier system. This finding leads to an observation that we have termed the “alias effect.” The exact solutions are supplemented with asymptotic formulas.

arXiv:1404.0750•American Journal of Physics•Jun 23, 2015•8 citations

SOLUTION OF SCHRÖDINGER'S EQUATION USING THE TRANSFER MATRIX METHOD FOR ONE-DIMENSIONAL POTENTIALS

Jhon H. Vargas, Raúl E. Castiblanco, Maicol Cárdenas +1 more

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.

•Jan 1, 2011•0 citations

Transfer matrix in scattering theory: a survey of basic properties and recent developments

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.

•Jan 1, 2020•40 citations

Transfer matrix in scattering theory: A survey of basic properties and recent developments

Ali Mostafazadeh (14357856)

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.

arXiv:2009.10507•Nov 30, 2023•10 citations
The paper introduces a simplified and exact transfer matrix formalism.That can be used to study a series of one dimensional potential barrier problems with only two general purpose matrices one to propagate a wave function over a region of constant potential,the other to take a wave function over a discontinuity in a potential.Not only are standard results recovered for one potential barrier problem with this technique,but it is easy to extend this formalism into two,even more many potential barier cases,which facilitates wide ranging explorations of one dimensional quantum mechanics.
An exact transfer-matrix formalism is developed for analyzing and solving problems in one-dimensional quantum mechanics. We show that with only three general-purpose matrices—one to propagate a wave function over a region of constant potential, one to take a wave function over a discontinuity in a potential, and one to connect a wave function across a delta function—a rich and intriguing variety of behavior is revealed. Not only are standard results recovered with this technique, in ways suitable for presentation in the classroom, but new findings and applications are discussed as well. A primary advantage of the transfer-matrix approach is that it facilitates wide-ranging explorations of one-dimensional quantum mechanics by both students and researchers, especially when implemented with Mathematica. For those interested in pursuing independent explorations, an electronic, interactive version of this paper, complete with the figures given here and the code that generates them, is available over Internet as a Mathematica notebook.
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.
In this paper, we examine transmission through one-dimensional potential barriers that are piecewise constant. The transfer matrix approach is adopted, and a new formula is derived for multiplying long matrix sequences that not only leads to an elegant representation of the wave function but also results in much faster computation than earlier methods. The proposed method covers a broad spectrum of potentials, of which multi-barrier systems are special cases. The procedure is illustrated with a finite lattice of nonuniform rectangular barriers—non-uniformity being a novel feature, as the uniform case has been solved exactly by Griffiths and Steinke. For the nonuniform multi-barrier problem, the intervening wells strongly influence the transmission probability. Surprisingly, we find that the wells act “individually,” i.e., their influence is a function only of their width and is independent of their exact locations in a multi-barrier system. This finding leads to an observation that we have termed the “alias effect.” The exact solutions are supplemented with asymptotic formulas.
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.
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.