On derivation of Wigner distribution function

Wigner distribution function has much importance in quantum statistical mechanics. It finds applications in various disciplines of physics including condense matter, quantum optics, to name but a few. Wigner distribution function is introduced by E. Wigner in 1932. However, there is no analytical derivation of Wigner distribution function in the literatures, to date. In this paper, a simple analytical derivation of Wigner distribution function is presented. Our derivation is based on two assumptions, these are A) by taking the integral of Wigner distribution function, with respect to configuration space, the momentum space distribution function is obtained B) WDF is real. Similarly, and in addition to Wigner distribution function, the distribution function of Sobouti-Nasiri, which is imaginary, is also derived.