Extensions to the Kronig-Penney model

The Kronig-Penney model uses a periodic array of rectangular wells to deduce the existence of semiconductor energy bands from the solution of the corresponding Schroedinger equation. As part of this procedure, it is necessary to then solve a messy transcendental equation; this difficulty was surmounted by replacing the rectangular array with delta function potentials. The author describes two methods for presenting this topic at the undergraduate level. The first involves solving the one-dimensional Schroedinger equation with a rather complicated potential proposed by W. Prokovjev (1929). The second method uses a quadratic potential in the radial Schroedinger equation