that the coherent states $|0 \lambda'\rangle$ with $\lambda = 0$ (but with arbitrary $\lambda' = \sqrt{M \omega/2} (\langle R_x \rangle + i \langle R_y \rangle)$ form an overcomplete set of degenerate ground states with the energy $\omega/2$. These states represent Gaussian wave packets centered at the points $(\langle R_x \rangle,\langle R_y \rangle)$ determined by $\lambda'$. Unlike for a free particle, these Gaussian wave packets do not spread. Semi-classically speaking, the charged particle is in a “circular orbit” of quantized sharp radius $\sqrt{2/M \omega} |\lambda| = 0$ with an uncertain position $(\langle R_x \rangle,\langle R_y \rangle)$ of the center. Since the ground state is infinite degenerate, the charged particle can spontaneously select any average position $(\langle R_x \rangle,\langle R_y \rangle)$ at which it can stay with average velocity zero in a state of minimal uncertainty. Again, this is reminiscent of the spontaneous breakdown of translation invariance.

Particle on a Torus
===================

In this section we put the problem in a finite periodic volume. This explicitly breaks rotation invariance, but leaves translation invariance intact (at least at the classical level), and leads to an energy spectrum with finite degeneracy. In order to clarify some subtle symmetry properties, we also discuss issues of 