On the momentum eigenfunctions for the periodic problem

Abstract

The problem of calculating the eleotronic energy levels in solids has attracted great interest, because of its considerable practical importance for a good understanding of the properties of metals and semiconductors.Among the methods derived for this purpose , many are based on a plane wave expansion of the electronic wavefunction. With such methods energy and wavefunctions are usually calculated only for a few values of the crystal momentum of the electron. It was pointed out that, from the computation required to find the energy -that is the solution of a determinantal equation - it is possible to derive with little labour the rate of change with crystal momentum, of the energy and wave-functions, at least for a certain direction of the momentum.The calculation of these derivatives has not been carried out so far, and the purpose of the present work is to perform it for a simple one-dimensional case, and try to assess the value of the method for a band structure calculation of an actual substance.