A simple method to obtain the equilibrium solution of Wigner-Boltzmann equation with all higher order quantum corrections

A simple method has been introduced to render the equilibrium solution of the Wigner equation for all orders of the quantum correction. This process generates a recursion relation involving the coefficients of the different powers of the momentum. The technique relies on an appropriate guess for the trial solution and differs from the Wigner’s original work. The solution is yielded in a compact exponential form with a polynomial of momentum in the argument and regains Wigner’s form when expansion of the exponential factor is carried out. The study is also important in the investigation of various closed as well as open quantum mechanical systems. In addition, this solution may be employed to obtain the nonequilibrium one particle Wigner distribution in the relaxation-time approximation and under near-equilibrium condition.