  A simple method has been introduced to furnish the equilibrium solution of
the Wigner equation for all order of the quantum correction. This process
builds up a recursion relation involving the coefficients of the different
power of the velocity. The technique greatly relies upon the proper guess work
of the trial solution and is different from the Wigner s original work. The
solution is in a compact exponential form with a polynomial of velocity in the
argument and returns the Wigner s form when expansion of the exponential factor
is carried out. The study keeps its importance in studying various close as
well as open quantum mechanical system. In addition, this solution may be
employed to obtain the non equilibrium one particle wigner distribution in the
relaxation-time approximation and under near-equilibrium conditions.
