The Wigner distribution function is a quasi-probability distribution function used to study quantum mechanics in phase space.

1Definition

The Wigner Transform maps a quantum operator $\hat A$ to a classical function in phase space. Eugene Wigner used this to associate to a quantum system a real phase space function, now called Wigner distribution function (WDF) [1]. It is defined to be the symbol associated to the density operator$\hat \rho$ describing the quantum system

$$P_W(x,p) := \int \!\!d^3y \; e^{\frac {i}{\hbar }p y} \; \langle x-\frac {y}{2} | \hat \rho | x+\frac {y}{2} \rangle .$$

(1)

In the case of a mixed state, the density operator can be written as convex combination of the pure state wavefunctions$\psi _n$

$$\hat \rho = \sum _n \lambda _n |\psi _n\rangle \langle \psi _n|, \quad \lambda _n \geq 0, \quad \sum _n \lambda _n = 1.$$

(2)

For mixed states the WDF thus reads

$$P_W(x ,p) = \int \!\!d^3y \; e^{\frac {i}{\hbar }p y} \, \sum _n \lambda _n \psi _n^\ast ( x + \frac {y}{2}) \psi _n(x - \frac {y}{2}),$$

(3)

while for a pure state it reduces to

$$P_W(x ,p) = \int \!\!d^3y \; e^{\frac {i}{\hbar }p y} \, \psi ^\ast ( x + \frac {y}{2}) \psi (x - \frac {y}{2}).$$

(4)

2Properties

The WDF has many similarities to the classical distribution function:
$P_W(x,p)$ is a real function, as follows by taking the complex conjugate and changing variables $y \mapsto -y$. It is normalized to $1$ in the following sense

$$\int \!\!d^3x \! \int \!\!\frac {d^3p}{(2\pi \hbar )^3}\; P_W(x,p) = 1.$$

(5)

When integrated over all momenta, it gives the probability density:

$$\int \!\!\frac {d^3p}{(2\pi \hbar )^3} P_W = \int \!\!d^3y \; \delta ^3(y) \sum _n \lambda _n \psi _n^\ast {}(x+\frac {y}{2}) \psi _n{}(x-\frac {y}{2}) = \sum _n \lambda _n |\psi _n( x)|^2,$$

(6)

whereas when integrated over all positions, it yields the momentum distribution

$$\int \!\!d^3x\; P_W = \sum _n \lambda _n \int \!\!d^3x_- \!\! \int \!\!d^3x_+ \; e^{\frac {i}{\hbar }px_+} e^{-\frac {i}{\hbar }px_-} \psi _n^\ast {}_+ \psi _n{}_- = \sum _n \lambda _n \left |\tilde {\psi }_n\!\left ( \frac {p}{\hbar }\right )\right |^2,$$

(7)

where the subscripts $\pm$ denote the dependence on $x_\pm := x \pm \frac {y}{2}$. The WDF thus has the attractive property that the marginal distributions, obtained by integrating over either the position or momentum variables, do reproduce the correct non-negative position and momentum probability distributions respectively, as specified by quantum mechanics.
Compared to the classical distribution function, the Wigner distribution function has the peculiar property that it may assume negative values. For this reason it is usually called a quasi-probability distribution and cannot be interpreted as a phase space probability density in the sense of classical mechanics.
The dynamical evolution of the Wigner distribution function is described by the Wigner equation, reducing to the classical Vlasov equation in the formal $\hbar \to 0$ limit.