Wigner functions in quantum mechanics with a minimum length scale arising from generalized uncertainty principle

In this paper, we generalize the concept of Wigner function in the case of quantum mechanics with a minimum length scale arising due to the application of a generalized uncertainty principle. We present the phase space formulation of such theories following GUP and show that the Weyl transform and the Wigner function satisfy most of their known properties in standard quantum mechanics. We utilize the generalized Wigner function to calculate the phase space average of the Hamiltonian of a quantum harmonic oscillator satisfying deformed Heisenberg algebra. It is also shown that averages of certain quantum mechanical operators in such theories may restrict the value of the deformation parameter specifying the degree of deformation of Heisenberg algebra. All the results presented are for pure states. The results can be generalized for mixed states.