Hydrodynamical formulation of Wigner-Dunkl quantum mechanics

The hydrodynamical form of Wigner-Dunkl quantum mechanics has been formulated. The main step to get such formulation was to derive a pair of Madelung-like equations (called Madelung-Wigner-Dunkl equations) from the Schrodinger equation appearing in Wigner-Dunkl quantum mechanics. Some aspects of both equations contained in the pair, i.e. the inhomogeneous continuity equation and the quantum Hamilton-Jacobi equation, have been investigated and discussed. With certain assumption, the relation between the expectation value of the quantum potential appearing in the quantum Hamilton-Jacobi equation and the so-called Fisher information has been derived. Then an uncertainty relation between momentum and position which may be tighter than the standard uncertainty relation has been derived by using Cramer-Rao inequality and Fisher information.