  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.
