Asymmetric quantum wells and the solotone effect: towards an inverse method

Abstract

In previous work the author considered heat conduction in a one-dimensional composite rod. The rod consisted of three parts and two different materials. Boundary and interface conditions were imposed and the eigenvalues computed numerically. This undertaking resulted in the first demonstration, within a heat conduction context, of the presence of oscillations within the eigenvalue distribution. This so-called solotone effect arises when the underlying differential equations contain discontinuities within their coefficients. Furthermore, a novel inverse method was presented and used to accurately estimate the relative thermophysical properties of a composite rod. This was achieved by examining the periodicity of the solotone effect. In this paper we demonstrate, for the first time, the existence of a solotone effect within the energy eigenvalues of a quantum well. At first sight there appears to be little connection between classical heat conduction and quantum mechanics. Nevertheless, these phenomena are related since in both situations the underlying equations may be expressed in Sturm–Liouville form. Our analysis will focus on an asymmetric infinite square well containing a particle with position dependent mass. This arrangement is sufficiently discontinuous to give rise to a solotone effect, and we believe this effect could be used to infer information about the properties of a potential well. However, we stop short of developing a complete inverse method since we believe an investigation of this nature would make for an interesting undergraduate project.