Energy Bands and Wave Functions in Periodic Potentials

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 (E) 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.