Numerical matrix method for quantum periodic potentials

TL;DR

This paper introduces a numerical matrix approach to solve quantum periodic potential problems by approximating the potential with an infinite well and expressing wave functions as superpositions, successfully applied to various Kronig-Penney models.

Contribution

The paper presents a novel numerical matrix method for solving quantum periodic potentials, extending its application to complex Kronig-Penney models with additional features.

Findings

Accurately computes energy levels of periodic potentials.

Effectively handles complex Kronig-Penney models.

Provides a versatile approach for quantum periodic systems.

Abstract

A numerical matrix methodology is applied to quantum problems with periodic potentials. The procedure consists essentially in replacing the true potential by an alternative one, restricted by an infinite square well, and in expressing the wave functions as finite superpositions of eigenfunctions of the infinite well. A matrix eigenvalue equation then yields the energy levels of the periodic potential within an acceptable accuracy. The methodology has been successfully used to deal with problems based on the well-known Kronig-Penney (KP) model. Besides the original model, these problems are a dimerized KP solid, a KP solid containing a surface, and a KP solid under an external field. A short list of additional problems that can be solved with this procedure is presented.