Solving the Schrodinger Equation of an Electron in a Periodic Crystal Potential through Elliptic Functions

TL;DR

This paper introduces a novel method using elliptic functions to solve the Schrödinger equation for an electron in a periodic crystal potential, providing insights into eigenfunctions and band structures.

Contribution

It develops an elliptic function-based approach to analytically solve the Schrödinger equation in periodic potentials, enhancing understanding of electronic band structures.

Findings

Eigenfunctions are shown to be real under the method.

Valence and conduction band structures are characterized.

The approach effectively models double periodic lattice planes.

Abstract

In this study, the Schrodinger equation of a valence electron in a periodic crystal potential is formulated and solved using the elliptic function formalism. The method allows double periodic lattice planes to be represented in the Gauss plane. The reality of the obtained eigenfunctions and the structure of the valence and conduction bands are also investigated.