ABSTRACT
We now look at the situation in which the wave equation is solved in the presence of a periodic potential. The result is a qualitative change in the form of solutions. In this chapter we focus exclusively on the electron properties as defined by the allowable solutions of the wave equation under these conditions. The lattice is present only to the extent that it provides a background for finding the allowed energy levels for the electrons. An important development of the model that arises as a result of the periodic potential is that the electrons inside the material are separated into two types: low energy ‘bound’ electrons which are spatially constrained to occupy the localized energy wells, and ‘free’ electrons which have higher energy and can migrate throughout the material. The result is allowed energy ‘bands’ separated by unallowed energy ‘gaps’. The higher energy ‘free’ electrons are, in general terms, the same as the electrons described in the Sommerfeld model. The ‘bound’ electrons here represent a new addition to the quantum-mechanical description of the electrons in the material.
