In solids one deals with a large number of interacting particles and hence the problem of determining the electron wave functions and the energy levels is extremely complicated. A simple quantum mechanical picture of an electron in a crystal can, however, be obtained by assuming that the atomic nuclei are at rest in the crystalline state and that the electron is in a periodic potential which has periodicity of the lattice. The periodic potential may be considered to be caused by a fixed nuclei plus some average potential due to all other electrons. The solution of Schrodinger equation for this potential gives a set of state or levels, which may be occupied by the electrons according to Pauli’s exclusive principle.
The potential energy of an electron at a distance r from an atomic nucleus of charge
Z e is given by