Fractional Angular Momentum and Quasi-Probability Densities for Angular Degrees of Freedom

In the present update work we consider properly defined two-parameter quasi-probability densities that, e.g., can be used as witness for quantum behaviour for a class of pure states as expressed in terms of a self-adjoint observable angular position and the corresponding angular momentum operator L. It is shown that negative values of the corresponding quasi-probability densities may reveal the quantum nature of superpositions of angular momentum eigenstates with a fractional mean value of L but in an ambiguous manner. For a suitable choice of parameters these quasi-probability densities are positive and are in accordance with Borns rule in quantum mechanics. It is also shown that experimental data of the uncertainties for the angular position and L observables can be sufficient to reveal some unique quantum-mechanical features of such states without necessarily making use of quasi-probability densities.