6.2.2 Definition of Wigner Function

Among the several transforms of the wave function or in general of the density operator, that provide a phase space view of the quantum state, the Wigner function is the one that has found more applications, mainly in statistical mechanics, but also in quantum chemistry, molecular dynamics, scattering theory, or quantum optics.

The Wigner function is (up to a constant factor) the Weyl transform of the quantum-mechanical density operator. For a particle in one dimension it takes the form

$$w(r,p) = \frac{1}{2 \pi \hbar} \int \rho\bigg(r - \frac{s}{2}, r + \frac{s}{2}\bigg) e^{\imath p s / \hbar} ds .$$ (6.40)

Here $\rho(x_1,x_2)$ denotes the von Neumann density function, i.e.,

$$\rho(x_1, x_2) = \langle x_1 \vert {\bf {\rho}} \vert x_2 \rangle$$ (6.41)

with ${\bf {\rho}}$ the density operator.

This transformation can be conveniently split into two steps. For this we define:

$$r = \frac{x_1 + x_2}{2} , \quad s = x_1 - x_2 .$$ (6.42)

We call $r$ the center of mass coordinate and $s$ the distance coordinate. These names should not be physically misleading as the von Neumann equation does not describe a two-particle problem but quantum correlations.

The inverse relation is

$$x_1 = r + \frac{s}{2} , \quad x_2 = r - \frac{s}{2} .$$ (6.43)

Calculation of the Wigner distribution involves as a first step a change to new coordinates $r$ and $s$. This gives

$$\sigma(r,s) = \rho(r + \frac{s}{2}, r - \frac{s}{2}) .$$ (6.44)

We call $\sigma(r,s)$ the center of mass (com) distribution.

In a second step this is followed by a Fourier transformation with respect to $s$.

$$w(r,p) = \int ds \sigma(r, s) e^{-\imath p s/ \hbar} .$$ (6.45)

An inverse Fourier transform gives the com distribution from the Wigner distribution:

$$\sigma(r,s) = \frac{1}{2 \pi \hbar} \int dp w(r,p) e^{\imath p s / \hbar} .$$ (6.46)

Instead of the momentum coordinate $p$ it is possible to introduce the wave vector $k = p/\hbar$ which eliminates $\hbar$ from the transformation:

$$f(r,k) = \frac{1}{2 \pi} \int ds \sigma(r, s) e^{-\imath k s} .$$ (6.47)

Note that as $\rho$ is self-adjoint,

$$\rho(x_1, x_2) = \rho^*(x_2, x_1)$$

we have

$$\sigma(r,s) = \sigma^*(r,-s)$$

and

$$w(r,p) = w^*(r,p) .$$

So the Wigner distribution is real. However, in contrast to its classical analogon it can assume negative values. As position and momentum operators do not commute and hence cannot be measured together, this does not present an inconsistency.

The Wigner distribution possesses an important $x$-$p$ duality given by the alternative definition [FJP03]

$$w(r,p) = \frac{1}{2 \pi \hbar} \int \rho \bigg( p + \frac{l}{2}, p - \frac{l}{2} \bigg) e^{\imath l r/\hbar} dl .$$ (6.48)

Here $\rho$ denotes the density function in momentum representation, that is:

$$\rho(p_1, p_2) = \langle p_1 \vert {\bf {\rho}} \vert p_2 \rangle$$

where $\vert p\rangle$ denotes the eigenvector of the momentum operator $-\imath \hbar \partial_x$ with eigenvalue $p$.