NEW EIGENVALUE EQUATION FOR THE KRONIG-PENNEY PROBLEM

The Kronig-Penney equation (KPE) has long been used as a pedagogic too l for explaining the formation of energy bands in a periodic potential in the form of a one-dimensional periodic array of square wells. Howe ver, the KPE does not readily reduce to the solution for an isolated s quare well in the limit of a large well-to-well separation. Moreover, the solutions at the center and the edge of the Brillouin zone are als o not readily obtainable from the KPE. Computationally, the KPE can be inconvenient as it can vary over tens of orders of magnitude as the e nergy is increased from the bottom to the top of the well. In this pap er, a new technique is developed for solving the Kronig-Penney problem and an alternative to the KPE is developed. The new eigenvalue equati on has the conceptual advantage of immediately reducing to the equatio n for an isolated square well in the limit of infinite barrier width a nd of immediately providing the equation for the top and bottom of a b and as well as the computational advantage of being on the order of un ity. (C) 1997 American Association of Physics Teachers.