Energy Bands and Wave Functions in Periodic Potentials

Abstract

In preparation for treatment of the one-dimensional wave equation with perturbed periodic potentials, this paper discusses the case of perfectly periodic potentials. Solutions for the entire range of $x$ are constructed by joining together "cell solutions"—solutions defined for a single period of the potential. Several types of cell solution are discussed. The concept of "self-matching cell solutions" leads to a simple demonstration of the band structure of permitted energy levels. The effective momentum of the particle is defined for energies in the forbidden bands, and a parameter $\sigma \mathrm{}\left(E\right)\mathrm{}$ of interest in the treatment of perturbed periodic systems is introduced. Expansions are given for solutions of the wave equation near the band edges, in both the permitted and forbidden bands.