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Wigner's 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, \[|W(x,p)| \le \sum_n p_n |W_n(x,p)| \le \frac{1}{\pi \hbar} \sum_n p_n = \frac{1}{\pi \hbar} \,.\]