Quantum and classical solutions for a 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 2 π− 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 θ w , p=1,2,… for the sector and μ q= qπ 2π−θ w , q=1,2… for the pacman. We derive a `duality' relation between both fractional indices given by λ p= pμ q 2μ q−q and μ q= qλ p 2λ p−p . We find that the average of the angular momentum L ̂ z is zero but the average of 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.