Weyl's problem: A computational approach

Abstract

The distribution of eigenvalues of the wave equation in a bounded domain is known as Weyl's problem. We describe several computational projects related to the cumulative state number, defined as the number of states having wavenumber up to a maximum value. This quantity and its derivative, the density of states, have important applications in nuclear physics, degenerate Fermi gases, blackbody radiation, Bose–Einstein condensation, and the Casimir effect. Weyl's theorem states that, in the limit of large wavenumbers, the cumulative state number depends only on the volume of the bounding domain and not on its shape. Corrections to this behavior are well known and depend on the surface area of the bounding domain, its curvature, and other features. We describe several projects that allow readers to investigate this dependence for three bounding domains—a rectangular box, a sphere, and a circular cylinder. Quasi-one- and two-dimensional systems can be analyzed by considering various limits. The projects have applications in statistical mechanics, but can also be integrated into quantum mechanics, nuclear physics, or computational physics courses.