Editing Density matrix (section)

== Wigner functions and classical analogies ==
{{main|Phase-space formulation}}
The density matrix operator may also be realized in [[phase space]]. Under the [[Wigner quasi-probability distribution#The Wigner–Weyl transformation|Wigner map]], the density matrix transforms into the equivalent [[Wigner quasi-probability distribution|Wigner function]],
: <math> W(x,p) \,\ \stackrel{\mathrm{def}}{=}\ \, \frac{1}{\pi\hbar} \int_{-\infty}^\infty \psi^*(x + y) \psi(x - y) e^{2ipy/\hbar} \,dy.</math>

The equation for the time evolution of the Wigner function, known as [[phase space formulation#Time evolution|Moyal equation]], is then the Wigner-transform of the above von Neumann equation,
: <math>\frac{\partial W(x, p, t)}{\partial t} = -\{\{W(x, p, t), H(x, p)\}\},</math>
where <math>H(x,p)</math> is the Hamiltonian, and <math>\{\{\cdot,\cdot\}\}</math> is the [[Moyal bracket]], the transform of the quantum [[commutator]].

The evolution equation for the Wigner function is then analogous to that of its classical limit, the [[Liouville's theorem (Hamiltonian)#Liouville equations|Liouville equation]] of [[classical physics]]. In the limit of a vanishing Planck constant <math>\hbar</math>, <math>W(x,p,t)</math> reduces to the classical Liouville probability density function in [[phase space]].