Wigner Function

The Wigner function is defined for a pure state as

(1)\begin{align} W(x,p)~\stackrel{\mathrm{def}}{=}~\frac{1}{\pi\hbar}\int_{-\infty}^\infty \psi^*(x+y)\psi(x-y)e^{2ipy/\hbar}\,dy\, \end{align}

For a mixed state, one may define it from the density matrix

(2)\begin{align} W(x,p)=\frac{1}{\pi\hbar}\int_{-\infty}^\infty \langle x+y| \hat{\rho} |x-y \rangle e^{-2ipy/\hbar}\,dy \end{align}

In optics, instead of psi, on has the electric field. This is computed via

(3)\begin{align} \vec E(k,\vec X) = iek \int_{-\infty}^{\infty} [\vec \beta - \frac{\vec n}{R}[1+\frac{i}{kR}]]e^{ik(c\tau + R)}d\tau \end{align}

Here, $\vec \beta(\tau)$ is the velocity vector. $\vec n(\tau)$ points from the electron to the observation point. R is the distance between electron and observation point.

k is the wavenumber $k=\frac{2\pi}{\lambda} = \frac{\omega}{c}$

Here, we assume that kR is large.