Wigner Function

The Wigner function represents a quantum state in phase space, bridging quantum and classical mechanics.

Definition

$W(x,p):=\frac{1}{\pi \hslash }{\int }_{-\infty }^{\infty }{\psi }^{\ast }(x+y)\psi (x-y)\exp \left(\frac{2ipy}{\hslash }\right)dy$

Properties

• Marginals: Integrating over $p$ gives $|\psi (x){|}^2$; integrating over $x$ gives $|\tilde{\psi }(p){|}^2$
• Normalization: $\int W(x,p)dxdp=1$
• Non-positivity: Can take negative values, distinguishing quantum from classical states
• Overlap: $⟨\psi |\varphi ⟩=2\pi \hslash \int W_{\psi }{W}_{\varphi }dxdp$

Connection to Classical Probability

Relates to the moment generating function $\mathbb{E}[e^{tX}]$ via the characteristic function. For coherent states, $W(x,p)\ge 0$ resembles a classical Gaussian distribution.

See Also

• Husimi Q representation — smoothed (always positive) phase-space distribution
• Weyl quantization