  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.
