Rotational symmetry of classical orbits, arbitrary quantization of angular momentum and the role of the gauge field in two-dimensional space

We study quantum-classical correspondence in terms of the 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 well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2\pi-periodic.