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

American Journal of Physics

Analytical and numerical solution of the Schrödinger equation for a surface potential barrier including spin-orbit interaction

M. M. Moya, J. S. Helman

Published May 1, 1979• 1 citation
Publisher

Abstract

We consider the problem of an electron interacting with a surface potential barrier which includes the spin-orbit interaction. An exact solution of the Schrödinger equation is given for the case of a step potential. For smooth barrier profiles, the time- dependent Schrödinger equation is numerically solved, and the time evolution of the spin-up and spin-down components of a wave packet is displayed in a series of pictures.

DOI
10.1119/1.11815

Related papers

Transit Time of Electron Tunneling Through a Potential Barrier

T. Figielski

The average transit time of electron tunneling through a potential barrier is newly defined and examined without using the concept of a wave packet and without solving explicitly the time dependent Schrödinger equation.

Acta Physica Polonica A•Oct 1, 1992•0 citations

A Finite Element Method Solver for Time-Dependent and Stationary Schrödinger Equations with a Generic Potential

A. Soba

A general finite element solution of the Schrödinger equation for a one-dimensional problem is presented. The solver is applicable to both stationary and time-dependent cases with a general user-selected potential term. Furthermore, it is possible to include external magnetic or electric fields, as well as spin-orbital and spin-magnetic interactions. We use analytically soluble problems to validate the solver. The predicted numerical auto-states are compared with the analytical ones, and selected mean values are used to validate the auto-functions. In order to analyze the performance of the time-dependent Schrödinger equation, a traveling wave package benchmark was reproduced. In addition, a problem involving the scattering of a wave packet over a double potential barrier shows the performance of the solver in cases of transmission and reflection of packages. Other general problems, related to periodic potentials, are treated with the same general solver and a Lagrange multiplier method to introduce periodic boundary conditions. Some simple cases of known periodic potential solutions are reported.

Communications in Computational Physics•May 1, 2009•1 citation

Semiclassical computations of time-dependent tunneling

M. Kira, Ilkka Juhani Tittonen, W. K. Lai +1 more

In this paper we consider the time evolution of wave-packet tunneling through a potential barrier. Using a path-summation approach, we can describe the quantum evolution in terms of particle trajectories both in the allowed and forbidden regions. Near the barrier edges, where the potential changes rapidly, the wave aspects dominate. These make the trajectories branch into infinite families of paths, which have to be included in the summation. Combining these features, we treat correctly the complementary aspects of the full quantum process, and this allows us to obtain a computational method that gives accurate numerical results with considerably less expenditure of computer time than a direct integration of the Schr\"odinger equation. We test the method on a simple rectangular potential by comparing the calculations with both conventional wave scattering results and computations from the Schr\"odinger equation. A general potential can be approximated by a staircase, which allows the application of our method. A simple adiabatic extension is shown to work excellently for the case of a periodically modulated barrier.

Physical Review A•Apr 1, 1995•23 citations

Exact solution of Schrödinger equation for time-dependent ultra-short barrier

Swati Mudra, Aniruddha Chakraborty

Abstract In this paper, the time-dependent Schrödinger equation for an ultra-short barrier has been solved. Here, the height of the barrier has been taken as time-dependent with different possible cases. A new mathematical technique using Green’s function method is presented. These mathematical calculations are purely analytical and give exact results.

Physica Scripta•Jul 9, 2019•1 citation

The Time-Independent Schrödinger Equation

Gary E. Bowman

Abstract After revisiting the time-dependent Schrödinger equation and introducing the Hamiltonian operator, the time-independent Schrödinger equation (TISE) is obtained. Conditions on acceptable wavefunctions are then stated. The method for solution of the TISE is laid out, and the infinite potential well is solved as an example. A discussion of the physical interpretation of the solutions — as energy eigenstates in position space — follows. This provides an opportunity to introduce overall and relative phases, and to point out that the latter are responsible for time evolution. The TISE is then applied to the important problems of scattering from a step potential and tunneling through a rectangular potential barrier. Use of the TISE for such problems greatly simplifies calculations, but it can also instill misconceptions; both aspects are discussed.

Oxford University Press eBooks•Nov 1, 2007•0 citations

Quantum solutions for a symmetric double square well

Edward A. Johnson, H. T. Williams

The quantum behavior of a wave packet in a one-dimensional infinite square well with a finite barrier in the center is considered. Computer generated plots are presented that lead to useful analytic approximations for finding eigenvalues of the Schrödinger equation and for explaining the time dependence of wave packets.

American Journal of Physics•Mar 1, 1982•7 citations
We consider the problem of an electron interacting with a surface potential barrier which includes the spin-orbit interaction. An exact solution of the Schrödinger equation is given for the case of a step potential. For smooth barrier profiles, the time- dependent Schrödinger equation is numerically solved, and the time evolution of the spin-up and spin-down components of a wave packet is displayed in a series of pictures.
A general finite element solution of the Schrödinger equation for a one-dimensional problem is presented. The solver is applicable to both stationary and time-dependent cases with a general user-selected potential term. Furthermore, it is possible to include external magnetic or electric fields, as well as spin-orbital and spin-magnetic interactions. We use analytically soluble problems to validate the solver. The predicted numerical auto-states are compared with the analytical ones, and selected mean values are used to validate the auto-functions. In order to analyze the performance of the time-dependent Schrödinger equation, a traveling wave package benchmark was reproduced. In addition, a problem involving the scattering of a wave packet over a double potential barrier shows the performance of the solver in cases of transmission and reflection of packages. Other general problems, related to periodic potentials, are treated with the same general solver and a Lagrange multiplier method to introduce periodic boundary conditions. Some simple cases of known periodic potential solutions are reported.
In this paper we consider the time evolution of wave-packet tunneling through a potential barrier. Using a path-summation approach, we can describe the quantum evolution in terms of particle trajectories both in the allowed and forbidden regions. Near the barrier edges, where the potential changes rapidly, the wave aspects dominate. These make the trajectories branch into infinite families of paths, which have to be included in the summation. Combining these features, we treat correctly the complementary aspects of the full quantum process, and this allows us to obtain a computational method that gives accurate numerical results with considerably less expenditure of computer time than a direct integration of the Schr\"odinger equation. We test the method on a simple rectangular potential by comparing the calculations with both conventional wave scattering results and computations from the Schr\"odinger equation. A general potential can be approximated by a staircase, which allows the application of our method. A simple adiabatic extension is shown to work excellently for the case of a periodically modulated barrier.
Abstract After revisiting the time-dependent Schrödinger equation and introducing the Hamiltonian operator, the time-independent Schrödinger equation (TISE) is obtained. Conditions on acceptable wavefunctions are then stated. The method for solution of the TISE is laid out, and the infinite potential well is solved as an example. A discussion of the physical interpretation of the solutions — as energy eigenstates in position space — follows. This provides an opportunity to introduce overall and relative phases, and to point out that the latter are responsible for time evolution. The TISE is then applied to the important problems of scattering from a step potential and tunneling through a rectangular potential barrier. Use of the TISE for such problems greatly simplifies calculations, but it can also instill misconceptions; both aspects are discussed.