Eigenvalues of the Schrödinger equation for a periodic potential with nonperiodic boundary conditions: A uniform semiclassical analysis

Abstract

A uniform semiclassical expression for the eigenvalues of a one dimensional periodic Schrödinger equation with nonperiodic boundary conditions has been derived. The potential energy function can have any number of symmetric or asymmetric barriers and wells. The treatment is uniform in that the classical turning points can come close together, coalesce, and move into the complex plane as the energy passes through a barrier maximum. A detailed application is made to Mathieu functions of integer order; the equations themselves include the case of fractional order. Approximate semiclassical expressions are derived for the widths of the energy bands and the energy gaps of the periodic Mathieu equation when these quantities are small. The semiclassical results give a physical interpretation to formulas present in the mathematical literature and to the decrease in the splitting of a sequence of avoided crossings with increasing quantum numbers in coupled oscillator systems. Numerical calculations are reported to illustrate the high accuracy of the semiclassical formulas.