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 Schrödinger 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 Cramér{Rao inequality and Fisher information. © 2023 World Scientific Publishing Co. Pte Ltd. All rights reserved.