## Wigner equation

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The Wigner equation [1] is a differential equation describing the dynamical evolution of the Wigner distribution function$P_W(x,p,t)$
 (1)
where the subscripts $\pm$ denote the dependence on $x_\pm := x \pm \frac {y}{2}$. Expanding the potential $V$ in a Taylor series
 (2)
we can write the dynamical equation for the Wigner distribution function as follows
 (3)
One can notice that the first three terms correspond to the classical Vlasov equation. The additional terms are formally proportional to $\hbar$ 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.