One-dimension Periodic Potentials in Schr\"odinger Equation Solved by the Finite Difference Method

TL;DR

This paper applies the finite difference method to solve the one-dimensional Kronig-Penney and Dirac comb potentials in the Schrödinger equation, achieving higher accuracy and analyzing how potential parameters affect eigenvalues and wave functions.

Contribution

It introduces an improved finite difference approach for periodic potentials, providing more accurate eigenvalues and insights into parameter effects, including for the Dirac comb potential.

Findings

Eigenvalue accuracy surpasses existing methods like the filter method.

Eigenvalues vary less with wave vector as potential height increases.

Eigenvalues decrease significantly with increasing potential width for higher bands.

Abstract

The one-dimensional Kronig-Penney potential in the Schr\"{o}dinger equation, a standard periodic potential in quantum mechanics textbooks known for generating band structures, is solved by using the finite difference method with periodic boundary conditions. This method significantly improves the eigenvalue accuracy compared to existing approaches such as the filter method. The effects of the width and height of the Kronig-Penney potential on the eigenvalues and wave functions are then analyzed. As the potential height increases, the variation of eigenvalues with the wave vector slows down. Additionally, for higher-order band structures, the magnitude of the eigenvalue significantly decreases with increasing potential width. Finally, the Dirac comb potential, a periodic $\delta$ potential, is examined using the present framework. This potential corresponds to the Kronig-Penney potential's width and height approaching zero and infinity, respectively. The numerical results obtained by the finite difference method for the Dirac comb potential are also perfectly consistent with the analytical solution.