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

Siberian Journal of Physics

One-Dimensional Periodic Potential in Quantum Mechanics

Vladimir K. Ignatovich

Published Jul 1, 2008• 0 citations
Publisher

Abstract

An elegant mathematical method is demonstrated with the help of simplest one-dimensional problems of quantum mechanics. This method is then applied to calculation of scattering on one-dimensional periodic systems. Generalization of the method for calculation of scattering in three dimensional periodic media and for spinor particles is pointed out.

DOI
10.54362/1818-7919-2008-3-2-99-107

Related papers

Green functions for generalized point interactions in one dimension: A scattering approach

Alexandre G. M. Schmidt, Bin Kang Cheng, M. G. E. da Luz

Recently, general point interactions in one dimension has been used to model a large number of different phenomena in quantum mechanics. Such potentials, however, require some sort of regularization to lead to meaningful results. The usual ways to do so rely on technicalities that may hide important physical aspects of the problem. In this work we present a method to calculate the exact Green functions for general point interactions in one dimension. Our approach differs from previous ones because it is based only on physical quantities, namely, the scattering coefficients R and T to construct G. Renormalization or particular mathematical prescriptions are not invoked. The simple formulation of the method makes it easy to extend to more general contexts, such as for lattices of N general point interactions, on a line, on a half-line, under periodic boundary conditions, and confined in a box.

arXiv:quant-ph/0211193•Physical Review A•Dec 31, 2002•32 citations

One-dimensional multiple scattering in quantum mechanics

Douglas Lessie, Joseph Spadaro

The formal theory of multiple scattering in quantum mechanics is adapted to one dimension and applied to the basic problem of scattering by a double delta-function potential. The multiple scattering method is shown to provide a much more powerful analysis than is obtained by either the one-dimensional partial wave approach or by the standard differential solution to the Schrödinger equation.

American Journal of Physics•Oct 1, 1986•16 citations

The remarkable capabilities of recursive relations

V. K. Ignatovich

The recursive relations method is presented. To illustrate its capability a new way is presented to solve the one-dimensional Schrödinger equation with a stepwise or any periodic potential (semi-infinite or finite). The method is shown to be also applicable to other physical problems, for instance, to molecular gas diffusion in linear and branched guides and to the scattering of a quantum mechanical particle on a spherically symmetric nut-shell potential.

American Journal of Physics•Oct 1, 1989•14 citations

One-dimensional scattering by a locally periodic potential

Hai-Woong Lee, Adam Zysnarski, Phillip L. Kerr

One-dimensional scattering of a particle or a wave packet by a finite number of periodic potential barriers is studied. The fundamental physical mechanism that determines the transmission coefficient is shown to be quantum interference between waves with different numbers of inner reflections. The tendency of the transmission coefficient to saturate with respect to an increase in the number of barriers is due to the free-particle nature of a particle subject to a completely periodic potential.

American Journal of Physics•Aug 1, 1989•27 citations

Solving the one-dimensional quantum mechanics problems using the matter wave multireflection

W. Polak

This article presents a simple, yet general and exact analytic-numerical method of solving one-dimensional quantum mechanics problems. It is based on the concept of dividing an arbitrary potential into convenient segments and analyzing the multiple reflection of a plane matter wave between neighboring potential walls. Amplitude reflection and transmission coefficients of each segment are the basis for the analysis. A simple iterative technique for calculating them by a square barrier approximation is given. This method is applicable to various potential barriers, wells, and periodic structures including continuous variations of potential energy and particle effective mass.

Journal of Applied Physics•Jul 15, 1993•1 citation

Reflections in One-Dimensional Wave Mechanics

Rolf W. Landauer

A method is presented for solving the one-dimensional Schroedinger equation. The method gives the wave function in the form of an infinite series. The first term of the series is the WBK approximation. Each of the higher terms in turn represents the reflections generated by the preceding term. The expansion is obtained by approximating the potential by a "staircase" potential, which is constant over short regions and changes its value discontinuously between these regions. The method is applied to periodic potentials. By taking the WBK approximation and the first reflection term into account, the zones of the periodic potential can be found in terms of a coefficient which gives the probability that a particle will be reflected in passing through one cell of the periodic potential.

Physical Review•Apr 1, 1951•34 citations
An elegant mathematical method is demonstrated with the help of simplest one-dimensional problems of quantum mechanics. This method is then applied to calculation of scattering on one-dimensional periodic systems. Generalization of the method for calculation of scattering in three dimensional periodic media and for spinor particles is pointed out.
Recently, general point interactions in one dimension has been used to model a large number of different phenomena in quantum mechanics. Such potentials, however, require some sort of regularization to lead to meaningful results. The usual ways to do so rely on technicalities that may hide important physical aspects of the problem. In this work we present a method to calculate the exact Green functions for general point interactions in one dimension. Our approach differs from previous ones because it is based only on physical quantities, namely, the scattering coefficients R and T to construct G. Renormalization or particular mathematical prescriptions are not invoked. The simple formulation of the method makes it easy to extend to more general contexts, such as for lattices of N general point interactions, on a line, on a half-line, under periodic boundary conditions, and confined in a box.
This article presents a simple, yet general and exact analytic-numerical method of solving one-dimensional quantum mechanics problems. It is based on the concept of dividing an arbitrary potential into convenient segments and analyzing the multiple reflection of a plane matter wave between neighboring potential walls. Amplitude reflection and transmission coefficients of each segment are the basis for the analysis. A simple iterative technique for calculating them by a square barrier approximation is given. This method is applicable to various potential barriers, wells, and periodic structures including continuous variations of potential energy and particle effective mass.
A method is presented for solving the one-dimensional Schroedinger equation. The method gives the wave function in the form of an infinite series. The first term of the series is the WBK approximation. Each of the higher terms in turn represents the reflections generated by the preceding term. The expansion is obtained by approximating the potential by a "staircase" potential, which is constant over short regions and changes its value discontinuously between these regions. The method is applied to periodic potentials. By taking the WBK approximation and the first reflection term into account, the zones of the periodic potential can be found in terms of a coefficient which gives the probability that a particle will be reflected in passing through one cell of the periodic potential.