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 Mathematical Physics

Transfer matrices for one-dimensional potentials

Asher Peres

Published May 1, 1983• 22 citations
Publisher

Abstract

The one-dimensional Schrödinger equation can be written as a first-order multicomponent equation by considering ψ and dψ/dx, or combinations thereof, as independent variables. A potential barrier is then represented by a matrix belonging to one of the homomorphic groups SU(1,1), SO(2,1), Sp(2,R), or SL(2,R). The relationship between these groups is clarified. In various applications, one of them may turn out more convenient than others. In particular, SO(2,1), which is obtained by using as a basis some bilinear combinations of ψ and dψ/dx, leads to remarkable results: The Schrödinger wavefunction is represented by a trajectory on a unit hyperboloid; a periodic potential corresponds to a pseudorotation around a fixed axis; a random potential gives a random walk on the hyperboloid. This method can also be used to calculate bound states (in potential wells) and may have many other interesting applications.

DOI
10.1063/1.525837

Related papers

Solving the Schrödinger Equation by the Transfer-Matrix Method

Juliana Almeida, Tatiana Miguel Rodrigues de Souza, Alexys Bruno‐Alfonso

The transfer-matrix method was proposed in the 1940s to solve linear ordinary differential equations.It is broadly studied due to its efficiency and simplicity for solving one-dimensional physical problems.In this work, the method is applied to solve the timeindependent Schrödinger equation, in order to improve both the formalism and the computational treatment.Closed-form solutions are derived for quantum wells and periodic potentials of arbitrary profiles.Numerical results are given for symmetric finite-barrier quantum wells and sinusoidal potentials.

Matemática Contemporânea•Jan 1, 2024•0 citations

Phase-space Propagation of the Time-independent Schrödinger Equation for One-dimensional Rectangular Potential Systems

Ivan E. Solarte, Santiago Barreiro-Medina, Carlos A. Arango

Abstract A thorough analysis of one-dimensional rectangular potential systems, encompassing both barriers and wells, is provided within the framework of the phase-space propagation method for the Time-Independent Schrödinger Equation (TISE). An initial value representation approach is adopted, involving an ensemble of initial conditions invariant under phase-space flow on the left side of the barrier or well. The system’s periodicity in the free-motion regions, coupled with the simplicity of the potential, facilitates the derivation of analytical expressions for the evolution of the phase-space state across the barrier or well and onto the right side. The transfer matrix method is employed to obtain explicit analytical expressions for the phase-space state of the one-dimensional barrier and its transmission coefficient. A similar approach is extended to the potential energy well, where transcendental equations for both bound and virtual (anti-bound) states are derived through a geometrical analysis of the phase-space flow of the TISE and the stable and unstable manifolds. Furthermore, a detailed analysis of the transmission coefficient is conducted. It is shown that the energy curves for the nth bound state and the $$(n+2)$$ th virtual state tend toward the same asymptotic limit. This limit corresponds to the energy curve (or line) of the $$(n+1)$$ th resonance, offering deeper insights into the system’s energy spectrum and its behavior in the limit of infinite potential wells.

International Journal of Theoretical Physics•Jul 22, 2025•0 citations

Exact solution of the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential barrier

Wayne W. Lui, Masao Fukuma

A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.

Journal of Applied Physics•Sep 1, 1986•208 citations

Wave Mechanics and Its Simple Applications

Ajay Sharma

The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.

CRC Press eBooks•Jun 18, 2021•0 citations

Reflection and transmission coefficients from the superposition of various potentials

Petarpa Boonserm, Tritos Ngampitipan, Kunlapat Sansuk

Abstract The reflection and transmission coefficients describe the behavior of the matter wave incident on a potential barrier. They can be expressed in terms of the probability with which the matter wave can be reflected or transmitted. The central equation accounting for the behavior of the matter wave is the Schrödinger equation. The Schrödinger equation is the second order partial differential equation. However, in a stationary state, the Schrödinger equation is reduced to the time independent Schrödinger equation. This time independent Schrödinger equation is the second order linear ordinary differential equation. Since the time independent Schrödinger equation is linear, superposition of any of the two solutions to the time independent Schrödinger equation is also a solution. In this paper, we focus on the superposition of various potentials. The reflection and transmission coefficients from the superposition of various potentials are obtained. A comparison between the exact coefficients and those obtained by the 2 × 2 transfer matrix is made. The relationship between the transmission coefficient of the superposed potential and that of each individual potential is found. The results show that the transmission coefficient obtained from the 2 × 2 transfer matrix is of a lower bound on the exact transmission coefficient.

Journal of Physics Conference Series•Nov 1, 2019•2 citations

A New Approach to Green's Function of a Particle in One Dimension

Hirotsugu Matsuda

A new method for solving the Schrödinger equation of a particle moving in one dimension in a given static potential is presented in this paper. We start from the observation that the density matrix corresponds to a Laplace transform of Green's function, and satisfies a diffusion-type equation which may be regarded as representing a certain limit of a discrete Markoffian random walk process. Our method of solving the Schrödinger equation is to divide the space occupied by the system into non-overlapping regions, in each of which the Schrödinger equation can be solved. We then introduce a path concept so as to synthesize the solutions for the various regions and thereby formulate the one-particle Green's function. As an application of this formulation, the energy bands of a one-dimensional infinite system are treated. The condition for the existence of impurity band is derived, which qualitatively supports Anderson's theory on the absence of diffusion in certain random lattices.

Progress of Theoretical Physics•Apr 1, 1962•12 citations
The one-dimensional Schrödinger equation can be written as a first-order multicomponent equation by considering ψ and dψ/dx, or combinations thereof, as independent variables. A potential barrier is then represented by a matrix belonging to one of the homomorphic groups SU(1,1), SO(2,1), Sp(2,R), or SL(2,R). The relationship between these groups is clarified. In various applications, one of them may turn out more convenient than others. In particular, SO(2,1), which is obtained by using as a basis some bilinear combinations of ψ and dψ/dx, leads to remarkable results: The Schrödinger wavefunction is represented by a trajectory on a unit hyperboloid; a periodic potential corresponds to a pseudorotation around a fixed axis; a random potential gives a random walk on the hyperboloid. This method can also be used to calculate bound states (in potential wells) and may have many other interesting applications.
The transfer-matrix method was proposed in the 1940s to solve linear ordinary differential equations.It is broadly studied due to its efficiency and simplicity for solving one-dimensional physical problems.In this work, the method is applied to solve the timeindependent Schrödinger equation, in order to improve both the formalism and the computational treatment.Closed-form solutions are derived for quantum wells and periodic potentials of arbitrary profiles.Numerical results are given for symmetric finite-barrier quantum wells and sinusoidal potentials.
Abstract A thorough analysis of one-dimensional rectangular potential systems, encompassing both barriers and wells, is provided within the framework of the phase-space propagation method for the Time-Independent Schrödinger Equation (TISE). An initial value representation approach is adopted, involving an ensemble of initial conditions invariant under phase-space flow on the left side of the barrier or well. The system’s periodicity in the free-motion regions, coupled with the simplicity of the potential, facilitates the derivation of analytical expressions for the evolution of the phase-space state across the barrier or well and onto the right side. The transfer matrix method is employed to obtain explicit analytical expressions for the phase-space state of the one-dimensional barrier and its transmission coefficient. A similar approach is extended to the potential energy well, where transcendental equations for both bound and virtual (anti-bound) states are derived through a geometrical analysis of the phase-space flow of the TISE and the stable and unstable manifolds. Furthermore, a detailed analysis of the transmission coefficient is conducted. It is shown that the energy curves for the nth bound state and the (n+2)(n+2)(n+2) th virtual state tend toward the same asymptotic limit. This limit corresponds to the energy curve (or line) of the (n+1)(n+1)(n+1) th resonance, offering deeper insights into the system’s energy spectrum and its behavior in the limit of infinite potential wells.
A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.
The Schrödinger equation assumes that a particle behaves as a wave and yields a solution in terms of wave function and the energy of the particle under consideration. Once the wave function is known, then everything about the particle can be deduced from the wave function. This chapter covers the development of the Schrödinger wave equation and its applications to one-dimensional systems. Time-dependent and time-independent Schrödinger equations, bound and scattering states, probability density, probability current and expectation values, the solution of the time-independent 1D-Schrödinger equation for a free particle, infinite as well as finite potential wells, potential steps, and the finite potential barrier are discussed in detail in the chapter. The phenomenon of tunneling, relevance and the importance of the free particle, potential wells and potential barriers to contemporary fields of physics and chemistry, periodic solids and their band structures, the Kronig-Penney model, confined states in quantum wells, and wires and dots are other topics presented in innovative manners in the chapter. The importance of the topics presented here is demonstrated through carefully chosen solved examples and unsolved exercises from contemporary fields of sciences.
Abstract The reflection and transmission coefficients describe the behavior of the matter wave incident on a potential barrier. They can be expressed in terms of the probability with which the matter wave can be reflected or transmitted. The central equation accounting for the behavior of the matter wave is the Schrödinger equation. The Schrödinger equation is the second order partial differential equation. However, in a stationary state, the Schrödinger equation is reduced to the time independent Schrödinger equation. This time independent Schrödinger equation is the second order linear ordinary differential equation. Since the time independent Schrödinger equation is linear, superposition of any of the two solutions to the time independent Schrödinger equation is also a solution. In this paper, we focus on the superposition of various potentials. The reflection and transmission coefficients from the superposition of various potentials are obtained. A comparison between the exact coefficients and those obtained by the 2 × 2 transfer matrix is made. The relationship between the transmission coefficient of the superposed potential and that of each individual potential is found. The results show that the transmission coefficient obtained from the 2 × 2 transfer matrix is of a lower bound on the exact transmission coefficient.
A new method for solving the Schrödinger equation of a particle moving in one dimension in a given static potential is presented in this paper. We start from the observation that the density matrix corresponds to a Laplace transform of Green's function, and satisfies a diffusion-type equation which may be regarded as representing a certain limit of a discrete Markoffian random walk process. Our method of solving the Schrödinger equation is to divide the space occupied by the system into non-overlapping regions, in each of which the Schrödinger equation can be solved. We then introduce a path concept so as to synthesize the solutions for the various regions and thereby formulate the one-particle Green's function. As an application of this formulation, the energy bands of a one-dimensional infinite system are treated. The condition for the existence of impurity band is derived, which qualitatively supports Anderson's theory on the absence of diffusion in certain random lattices.