Quantum and classical solutions for free particle in wedge billiards

Abstract

We have studied the quantum and classical solutions of a particle constrained to move inside a sector circular billiard with angle $θ_w$ and its pacman complement with angle $2π-θ_w$. In these billiards rotational invariance is broken and angular momentum is no longer a conserved quantum number. The "fractional" angular momentum quantum solutions are given in terms of Bessel functions of fractional order, with indices $λ_p={pπ\over {θ_w}}$, $p=1,2,...$ for the sector and $μ_q={qπ\over {2π- θ_w}}$, $q=1,2...$ for the pacman. We derive a ``duality'' relation between both fractional indices given by $λ_p={{pμ_q} \over {2μ_q - q}}$ and $μ_q = {{qλ_p} \over {2λ_p - p}}$. We find that the average of the angular momentum $\hat L_z$ is zero but the average of $\hat L^2_z$ has as eigenvalues $λ_p^2$ and $μ_q^2$. We also make a connection of some classical solutions to their quantum wave eigenfunction counterparts.