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Wigner's function: Observables

Let a (pure or mixed) quantum state of a one-dimensional particle be represented by a Wigner function \(W(x,p)\). Here, \(x\) and \(p\) are the particle's position and momentum, respectively. Density operator The density operator \(\rho\) representing the state can be expressed in terms of the Wigner function: \[\rho = \int_{-\infty}^{+\infty} dx \int_{-\infty}^{+\infty} dx' \int_{-\infty}^{+\infty} dp \, | x' \rangle \, e^{i p (x' - x) / \hbar} W \left( \frac{x + x'}{2}, p \right) \langle x | \,.\] Proof: According to the definition of the Wigner function, \[W(x,p) = \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} dx' \, e^{-i p x' / \hbar} \langle x + \tfrac{1}{2} x' | \rho | x - \tfrac{1}{2} x' \rangle \,.\] Hence, \[\begin{align} \int_{-\infty}^{+\infty} dp \, e^{i p \xi' / \hbar} W(\xi,p) &= \frac{1}{2 \pi \hbar} \int_{-\infty}^{+\infty} d\xi'' \, \langle \xi + \tfrac{1}{2} \xi'' |\rho | \xi - \tfrac{1}{2}

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