Quantum mechanics of the two-dimensional circular billiard plus baffle system and half-integral angular momentum

Abstract

We examine the quantum mechanical eigensolutions of the two-dimensional infinite well or quantum billiard system consisting of a circular boundary with an infinite barrier or baffle along a radius. Because of the change in boundary conditions, this system includes quantized angular momentum values corresponding to half-integral multiples of $\hbar/2$. We discuss the resulting energy eigenvalue spectrum and visualize some of the novel energy eigenstates found in this system. We also discuss the density of energy eigenvalues, $N(E)$, comparing this system to the standard circular well. These two billiard geometries have the same area (A=$\pi R^2$), but different perimeters ($P=2\pi R$ versus $(2\pi + 2) R$), and we compare both cases to fits of $N(E)$ which make use of purely geometric arguments involving only $A$ and $P$. We also point out connections between the angular solutions of this system and the familiar pedagogical example of the one-dimensional infinite well plus $\delta$-function potential.