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 ${\theta }_w$ and its pacman complement with angle $2\pi -{\theta }_w$. 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 ${\lambda }_p=\frac{p\pi }{{\theta }_w}$, $p=1,2,...$ for the sector and ${\mu }_q=\frac{q\pi }{2\pi -{\theta }_w}$, $q=1,\mathrm{2...}$ for the pacman. We derive a ``duality'' relation between both fractional indices given by ${\lambda }_p=\frac{p{\mu }_q}{2{\mu }_q-q}$ and ${\mu }_q=\frac{q{\lambda }_p}{2{\lambda }_p-p}$. We find that the average of the angular momentum ${\hat{L}}_z$ is zero but the average of ${\hat{L}}_z^2$ has as eigenvalues ${\lambda }_p^2$ and ${\mu }_q^2$. We also make a connection of some classical solutions to their quantum wave eigenfunction counterparts.

10 pages and two PostScript figures

References in corpus (1)

• Classical and quantum chaos in a circular billiard with a straight cut

Cited by in corpus (1)

• Rotational Symmetry of Classical Orbits, Arbitrary Quantization of Angular Momentum and the Role of Gauge Field in Two-Dimensional Space