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

TL;DR

This paper explores how classical symmetries influence quantum angular momentum quantization in two-dimensional systems with magnetic flux, revealing that gauge fields shift spectra without altering fundamental quantization rules.

Contribution

It demonstrates that classical rotational symmetry determines non-integer angular momentum quantization and shows gauge fields shift spectra without changing quantization rules, generalizing the anyon model.

Findings

Classical symmetry dictates non-integer angular momentum quantization.

Magnetic flux shifts the spectrum of angular momentum without changing quantization rules.

The anyon model is a special case of arbitrary quantization in this framework.

Abstract

We study the quantum-classical correspondence in terms of coherent wave functions of a charged particle in two-dimensional central-scalar-potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level-space of angular momentum being greater or less than $\hbar$ is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily $2\pi$-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The quantum mechanical model of anyon proposed by Wilczek (Phys. Rev. Lette. 48, 1144) becomes a special case of the arbitrary-quantization.