### Wigner function: States

Consider a quantum particle moving along the $$x$$ axis. Let the particle's state be represented by a density operator $$\rho$$. If the particle is in a pure state $$| \psi \rangle$$, then $$\rho = | \psi \rangle \langle \psi |$$.

The particle's state can also be described in phase space, with position and momentum variables $$x$$ and $$p$$, by means of an appropriate quasi-probability distribution. The Wigner function $$W(x,p)$$ is one of the most prominent choices.

### Definition

The Wigner function is defined as $W(x,p) = \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dx' \, e^{-i p x' / \hbar} \langle x + \tfrac{1}{2} x' |\rho | x - \tfrac{1}{2} x' \rangle \,.$ Equivalently, it can be written as an integral over momentum: $W(x,p) = \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dp' \, e^{i p' x / \hbar} \langle p + \tfrac{1}{2} p' |\rho | p - \tfrac{1}{2} p' \rangle \,.$

Proof: Making use of the completeness relation $$\int_{-\infty}^{+\infty} dp \, | p \rangle \langle p | = 1$$, we write $W(x,p) = \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dp' \int_{-\infty}^{+\infty} dp'' \int_{-\infty}^{+\infty} dx' \, e^{-i p x' / \hbar} \langle x + \tfrac{1}{2} x' | p' \rangle \langle p' |\rho | p'' \rangle \langle p'' | x - \tfrac{1}{2} x' \rangle \,.$ Taking into account that $$\langle x | p \rangle = (2 \pi \hbar)^{-1/2} e^{i p x / \hbar}$$, we obtain $W(x,p) = \frac{1}{(2 \pi \hbar)^2} \int_{-\infty}^{+\infty} dp' \int_{-\infty}^{+\infty} dp'' \int_{-\infty}^{+\infty} dx' \, e^{-i p x' / \hbar} e^{i p' (x + x'/2) / \hbar} \langle p' |\rho | p'' \rangle e^{-i p'' (x - x'/2) / \hbar} \,.$ The right-hand side can be rearranged as $\frac{1}{(2 \pi \hbar)^2} \int_{-\infty}^{+\infty} dp' \int_{-\infty}^{+\infty} dp'' \, e^{i (p' - p'') x / \hbar} \langle p' |\rho | p'' \rangle \underbrace{ \int_{-\infty}^{+\infty} dx' \, e^{i \big( (p' + p'')/2 - p \big) x' / \hbar} }_{2 \pi \hbar \delta \big( (p' + p'')/2 - p \big)}$ and so $W(x,p) = \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dp' \int_{-\infty}^{+\infty} dp'' \, e^{i (p' - p'') x / \hbar} \langle p' |\rho | p'' \rangle \delta \big( (p' + p'')/2 - p \big) \,.$ Changing integration variables to $$\tau = (p' + p'') / 2$$ and $$\eta = p' - p''$$, we obtain $W(x,p) = \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} d\eta \, e^{i \eta x / \hbar} \int_{-\infty}^{+\infty} d\tau \, \langle \tau + \tfrac{1}{2} \eta |\rho | \tau - \tfrac{1}{2} \eta \rangle \delta(\tau - p) \,,$ and, finally, $W(x,p) = \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} d\eta \, e^{i \eta x / \hbar} \langle p + \tfrac{1}{2} \eta |\rho | p - \tfrac{1}{2} \eta \rangle \,.$

### Key properties

The following properties of the Wigner function make it resemble a phase-space density in classical mechanics.

• The Wigner function is real-valued.

Proof: Complex conjugating, $$W^* = \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dx' \, e^{i p x' / \hbar} \langle x - \tfrac{1}{2} x' |\rho | x + \tfrac{1}{2} x' \rangle$$, and then changing the integration variable from $$x'$$ to $$-x'$$, one gets $$W^* = W$$.

• Integration of the Wigner function over position (respectively momentum) yields the probability distribution for momentum (respectively position): $\int_{-\infty}^{+\infty} dx \, W(x,p) = \langle p | \rho | p \rangle \,.$ $\int_{-\infty}^{+\infty} dp \, W(x,p) = \langle x | \rho | x \rangle \,.$ Proof: \begin{align} \int_{-\infty}^{+\infty} dx \, W(x,p) &= \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dp' \, \langle p + \tfrac{1}{2} p' |\rho | p - \tfrac{1}{2} p' \rangle \underbrace{ \int_{-\infty}^{+\infty} dx \, e^{i p' x / \hbar} }_{2 \pi \hbar \delta(p')} \\ &= \langle p | \rho | p \rangle \,, \end{align} \begin{align} \int_{-\infty}^{+\infty} dp \, W(x,p) &= \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dx' \, \langle x + \tfrac{1}{2} x' |\rho | x - \tfrac{1}{2} x' \rangle \underbrace{ \int_{-\infty}^{+\infty} dp \, e^{-i p x' / \hbar} }_{2 \pi \hbar \delta(x')} \\ &= \langle x | \rho | x \rangle \,. \end{align}

• The overlap between any two pure quantum sates $$| \psi_1 \rangle$$ and $$| \psi_2 \rangle$$ is given by $|\langle \psi_1 | \psi_2 \rangle |^2 = 2 \pi \hbar \int_{-\infty}^{+\infty} dx \int_{-\infty}^{+\infty} dp \, W_1(x,p) W_2(x,p) \,,$ where $$W_1$$ (respectively $$W_2$$) is the Wigner function corresponding to $$| \psi_1 \rangle$$ (respectively $$| \psi_2 \rangle$$).

Proof: Let $$\rho_1 = | \psi_1 \rangle \langle \psi_1 |$$ and $$\rho_2 = | \psi_2 \rangle \langle \psi_2 |$$. Then \begin{align} \int_{-\infty}^{+\infty} &dx \int_{-\infty}^{+\infty} dp \, W_1(x,p) W_2(x,p) \\ &= \frac{1}{(2 \pi \hbar)^2} \int_{-\infty}^{+\infty} dx \int_{-\infty}^{+\infty} dp \int_{-\infty}^{+\infty} dx' \, e^{-i p x' / \hbar} \langle x + \tfrac{1}{2} x' |\rho_1 | x - \tfrac{1}{2} x' \rangle \int_{-\infty}^{+\infty} dx'' \, e^{-i p x'' / \hbar} \langle x + \tfrac{1}{2} x'' |\rho_2 | x - \tfrac{1}{2} x'' \rangle \\ &= \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dx \int_{-\infty}^{+\infty} dx' \int_{-\infty}^{+\infty} dx'' \, \langle x + \tfrac{1}{2} x' |\rho_1 | x - \tfrac{1}{2} x' \rangle \langle x + \tfrac{1}{2} x'' |\rho_2 | x - \tfrac{1}{2} x'' \rangle \delta(x' + x'') \\ &= \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dx \int_{-\infty}^{+\infty} dx' \, \langle x + \tfrac{1}{2} x' |\rho_1 | x - \tfrac{1}{2} x' \rangle \langle x - \tfrac{1}{2} x' |\rho_2 | x + \tfrac{1}{2} x' \rangle \,. \end{align} Changing integration variables to $$\xi = x + \tfrac{1}{2} x'$$ and $$\xi' = x - \tfrac{1}{2} x'$$, one gets \begin{align} 2 \pi \hbar \int_{-\infty}^{+\infty} dx \int_{-\infty}^{+\infty} dp \, W_1(x,p) W_2(x,p) &= \int_{-\infty}^{+\infty} d\xi \int_{-\infty}^{+\infty} d\xi' \, \langle \xi |\rho_1 | \xi' \rangle \langle \xi' |\rho_2 | \xi \rangle \\ &= \operatorname{Tr} \rho_1 \rho_2 \\ &= |\langle \psi_1 | \psi_2 \rangle |^2 \,. \end{align}

The following properties of the Wigner function show that it is not a probability density in the classical sense.

• The Wigner function can take negative values.

Example: Consider a pure state given by a superposition of two Gaussian wave packets: $\langle x | \psi \rangle = N \left( e^{-\lambda (x-a)^2} + e^{-\lambda (x+a)^2} \right)$ with parameters $$a, \lambda > 0$$ and normalization constant $$N > 0$$. The Wigner function at $$x = 0$$ reads \begin{align} W(0,p) &= \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dx' \, e^{-i p x' / \hbar} \langle \tfrac{1}{2} x' | \psi \rangle \langle \psi | -\tfrac{1}{2} x' \rangle \\ &= \frac{N^2}{2 \pi \hbar} \int_{-\infty}^{+\infty} dx' \, e^{-i p x' / \hbar} \left( e^{-\lambda (x'/2 - a)^2} + e^{-\lambda (x'/2 + a)^2} \right)^2 \,. \end{align} Integration yields $W(0,p) = 2 \sqrt{\frac{2 \pi}{\lambda}} e^{-p^2 / 2 \lambda \hbar^2} \left( \cos \frac{2 p a}{\hbar} + e^{-2 \lambda a^2} \right) \,.$ Therefore, $$W(0,p) < 0$$ if $$p$$ satisfies $\cos \frac{2 p a}{\hbar} < -e^{-2 \lambda a^2} \,.$

• The Wigner function is bounded: $-\frac{1}{\pi \hbar} \le W(x,p) \le \frac{1}{\pi \hbar} \,.$ Proof: Let $$\{ | n \rangle \}$$ be an orthonormal basis diagonalizing the density operator, i.e., $\rho = \sum_n p_n | n \rangle \langle n |$ with $$0 \le p_n \le 1$$ and $\sum_n p_n = 1 \,.$ For simplicity, we assume the spectrum of $$\rho$$ to be discrete. Then, the Wigner function is given by $W(x,p) = \sum_n p_n W_n(x,p) \,,$ where $W_n(x,p) = \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dx' \, e^{-i p x' / \hbar} \langle x + \tfrac{1}{2} x' | n \rangle \langle n | x - \tfrac{1}{2} x' \rangle \,.$ For $$x$$ and $$p$$ fixed, we define states $$| u_n \rangle$$ and $$| v_n \rangle$$ as $\langle x' | u_n \rangle = \frac{e^{i p x' / 2 \hbar}}{\sqrt{2}} \langle x - \tfrac{1}{2} x' | n \rangle \,,$ $\langle x' | v_n \rangle = \frac{e^{-i p x' / 2 \hbar}}{\sqrt{2}} \langle x + \tfrac{1}{2} x' | n \rangle \,.$ Then, $W_n(x,p) = \frac{1}{\pi \hbar} \langle u_n | v_n \rangle \,.$ In view of the Cauchy-Schwarz inequality, as well as the fact that $$\langle u_n | u_n \rangle = \langle v_n | v_n \rangle = 1$$, we have $| \langle u_n | v_n \rangle | \le \sqrt{\langle u_n | u_n \rangle \langle v_n | v_n \rangle} = 1 \,.$ This implies $| W_n(x,p) | \le \frac{1}{\pi \hbar}$ and, finally, $|W(x,p)| \le \sum_n p_n |W_n(x,p)| \le \frac{1}{\pi \hbar} \sum_n p_n = \frac{1}{\pi \hbar} \,.$