Quantum Theory on a Network. I. A Solvable Model Whose Wavefunctions Are Elementary Functions

Abstract

A network model is developed for electrons in molecules and crystals. Wavefunctions and energy levels can be calculated exactly. A network which is topologically equivalent to that of the molecule or crystal of interest is constructed. One‐dimensional bonds connect various node points. The potential V(x) = −V₀ sech² γx is postulated to act along each bond so that a potential well exists at each node point (atom) of the network. For some values of V₀ the wavefunctions are elementary while generally they are hypergeometric functions. When the parameter γ is very large, the model is essentially the tight binding model and, when it is very small, one has the free‐electron network model investigated by Ruedenberg and Scherr, Griffith, Coulson, and others. The density of states, energy band structure, and other features are discussed.