The Wigner equation

[1] is a differential equation describing the dynamical evolution of the

Wigner distribution function where the subscripts

denote the dependence on

. Expanding the potential

in a Taylor series

we can write the dynamical equation for the Wigner distribution function as follows

One can notice that the first three terms correspond to the classical

Vlasov equation. The additional terms are formally proportional to

and can be interpreted as quantum corrections. The Wigner equation is equivalent to the Liouville equation for the density operator or the

Schrödinger equation for the wavefunction.

References:- [1]^ E.P. Wigner: On the quantum correction for thermodynamic equilibrium. In: Phys. Rev. 40, 1932, pp. 749–759.