Wigner Function

The Wigner function represents a quantum state by a real function on classical phase space. For one degree of freedom, it assigns to a density operator $\rho$ a function

$$W_\rho(x,p).$$

It is normalized and has the correct position and momentum marginals, but it is not an ordinary probability density because it can be negative. The Wigner function is therefore a quasiprobability distribution: probability-like enough to connect quantum mechanics with classical phase space, but quantum enough to retain interference, uncertainty, and noncommutativity.

Throughout this page, the position-momentum Fourier convention is

$$\langle x\vert p\rangle = \frac{1}{\sqrt{2\pi\hbar}}e^{ipx/\hbar}.$$

Different sign conventions shift signs in the Fourier factors, so always check the convention before comparing formulas.

Definition for Density Operators

For a density operator $\rho$, define

$$W_\rho(x,p) = \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty}dy\, e^{-ipy/\hbar} \left\langle x+\frac{y}{2} \middle| \rho \middle| x-\frac{y}{2} \right\rangle.$$

The variables $x$ and $p$ are phase-space coordinates. The integration variable $y$ measures an off-diagonal separation in the position-space density matrix. Thus the Wigner function packages both diagonal probability information and off-diagonal coherence.

It is often useful to write the position-space density matrix as

$$\rho(x_1,x_2) = \langle x_1\rvert\rho\lvert x_2\rangle.$$

Then

$$W_\rho(x,p) = \frac{1}{2\pi\hbar} \int dy\, e^{-ipy/\hbar} \rho\left( x+\frac{y}{2}, x-\frac{y}{2} \right).$$

The center coordinate $x$ and separation coordinate $y$ are the natural variables for the Wigner transform.

Pure States

For a pure state $\rho=\lvert\psi\rangle\langle\psi\rvert$,

$$\rho(x_1,x_2)=\psi(x_1)\psi^*(x_2),$$

so

$$W_\psi(x,p) = \frac{1}{2\pi\hbar} \int dy\, e^{-ipy/\hbar} \psi\left(x+\frac{y}{2}\right) \psi^*\left(x-\frac{y}{2}\right).$$

This formula shows why $W_\psi$ contains phase information. It is not built only from $\lvert\psi(x)\rvert^2$; it uses products of the wavefunction at two nearby points.

For a statistical mixture

$$\rho=\sum_a w_a\lvert\psi_a\rangle\langle\psi_a\rvert, \qquad w_a\geq0, \qquad \sum_a w_a=1,$$

linearity gives

$$W_\rho(x,p)=\sum_a w_a W_{\psi_a}(x,p).$$

The mixture weights are ordinary probabilities, but the individual Wigner functions may themselves have negative regions.

Normalization and Marginals

The Wigner function is normalized as

$$\int dx\,dp\,W_\rho(x,p)=\operatorname{Tr}\rho.$$

For a normalized state, $\operatorname{Tr}\rho=1$.

The position marginal is an ordinary probability density:

$$\int dp\,W_\rho(x,p) = \langle x\rvert\rho\lvert x\rangle.$$

The momentum marginal is also an ordinary probability density:

$$\int dx\,W_\rho(x,p) = \langle p\rvert\rho\lvert p\rangle.$$

These two marginals are one reason the Wigner function is so useful. It puts the $x$ and $p$ descriptions into one phase-space object without claiming that $x$ and $p$ have a simultaneous classical joint probability distribution.

Reality

If $\rho$ is Hermitian, then $W_\rho(x,p)$ is real. To see this, take the complex conjugate:

$$\begin{aligned} W_\rho(x,p)^* &= \frac{1}{2\pi\hbar} \int dy\, e^{ipy/\hbar} \left\langle x-\frac{y}{2} \middle| \rho \middle| x+\frac{y}{2} \right\rangle. \end{aligned}$$

Changing variables $y\to -y$ returns the original expression. A real Wigner function should not be mistaken for a positive probability density; real functions can be negative.

Expectation Values and Weyl Symbols

The Wigner function becomes a phase-space calculus when paired with the Weyl symbol of an operator. Schematically, if $A_W(x,p)$ is the Weyl symbol corresponding to an operator $A$, then

$$\operatorname{Tr}(\rho A) = \int dx\,dp\, W_\rho(x,p)A_W(x,p),$$

with the same convention used for both transforms.

For simple symmetrically ordered observables, this looks like a classical phase-space average. For noncommuting products, the operator ordering matters. Ordinary multiplication of phase-space functions does not reproduce arbitrary operator products; the full phase-space formulation uses a noncommutative product and the Moyal bracket.

Gaussian Wave Packet

Consider the normalized minimum-uncertainty Gaussian

$$\psi(x) = \frac{1}{(2\pi\sigma_x^2)^{1/4}} \exp\left[ - \frac{(x-x_0)^2}{4\sigma_x^2} + \frac{ip_0(x-x_0)}{\hbar} \right].$$

Its Wigner function is

$$W_\psi(x,p) = \frac{1}{2\pi\sigma_x\sigma_p} \exp\left[ - \frac{(x-x_0)^2}{2\sigma_x^2} - \frac{(p-p_0)^2}{2\sigma_p^2} \right],$$

where

$$\sigma_p=\frac{\hbar}{2\sigma_x}.$$

Since $\sigma_x\sigma_p=\hbar/2$, the prefactor may also be written as $1/(\pi\hbar)$. This Wigner function is positive everywhere and centered at $(x_0,p_0)$. It is the cleanest example of a quantum state that looks like a localized blob in classical phase space, while still respecting the uncertainty principle.

Harmonic-Oscillator Examples

For the harmonic oscillator ground state,

$$H=\frac{p^2}{2m}+\frac12m\omega^2x^2,$$

the Wigner function is

$$W_0(x,p) = \frac{1}{\pi\hbar} \exp\left[ - \frac{m\omega x^2}{\hbar} - \frac{p^2}{m\hbar\omega} \right].$$

This is a positive Gaussian. Its widths reproduce

$$(\Delta x)^2=\frac{\hbar}{2m\omega}, \qquad (\Delta p)^2=\frac{m\hbar\omega}{2}.$$

Coherent states displace this Gaussian in phase space without changing its shape. This is the phase-space version of why coherent states are classical-like oscillator states.

Number states beyond the ground state are different. If

$$r^2= \frac{m\omega x^2}{\hbar} + \frac{p^2}{m\hbar\omega},$$

then the first excited state has

$$W_1(x,p) = \frac{1}{\pi\hbar} \left( 2r^2-1 \right) e^{-r^2}.$$

At the origin, $W_1(0,0)=-1/(\pi\hbar)$. This is a simple explicit example of Wigner negativity.

Negativity

The Wigner function can be negative because it is a representation of a noncommutative quantum state, not a joint probability distribution for simultaneous sharp values of $x$ and $p$.

Negative regions often signal interference in phase space. Superpositions of well-separated wave packets, for example, have Wigner functions with oscillatory interference fringes between the positive lobes. Those fringes can be essential even when the position probability density only shows two separated peaks.

Negativity is a useful marker of nonclassicality, but it should not be overinterpreted. A positive Wigner function does not mean that the state is classical in every sense, and negativity depends on the chosen quasiprobability representation. The precise operational meaning depends on the measurement and resource-theory context.

Measurements and Tomography Preview

The Wigner function is not directly measured by observing a single particle at a phase-space point. Instead, experiments reconstruct it from families of compatible measurements, such as quadrature distributions in optical homodyne tomography or related phase-space reconstruction methods.

The mathematical reason is the marginal property: rotated quadratures give line projections of phase-space distributions. In favorable settings, enough marginals reconstruct the full Wigner function, much as computed tomography reconstructs an image from projections. The experimental details belong to quantum optics and continuous-variable measurement theory, but the conceptual point is already visible here: the Wigner function organizes many measurable distributions into one phase-space object.

Classical Limit

The Wigner function is one of the cleanest bridges to the classical limit. For states whose Wigner functions are broad compared with $\hbar$-scale interference features and for Hamiltonians whose higher quantum corrections are negligible, phase-space evolution approaches classical Liouville evolution.

For quadratic Hamiltonians, such as the harmonic oscillator, Wigner functions move exactly according to the classical phase-space flow. For anharmonic potentials, quantum corrections enter through higher derivatives and powers of $\hbar$. The Moyal bracket is the canonical object that makes this statement precise.

This does not mean that the Wigner function turns quantum mechanics into classical statistical mechanics. It explains how classical-looking phase-space dynamics can emerge from a quantum representation while preserving the possibility of interference and negativity.

Common Mistakes

• Calling $W_\rho(x,p)$ a joint probability density for simultaneous sharp $x$ and $p$.
• Forgetting the factor $1/(2\pi\hbar)$ or using incompatible Fourier conventions.
• Assuming a real Wigner function must be nonnegative.
• Treating negative values as measurement probabilities.
• Ignoring operator ordering when using phase-space functions for expectation values.
• Concluding that a positive Wigner function is automatically classical in every operational sense.
• Comparing Wigner functions across different normalization conventions without translating them.

Cross-Links

• Optical Phase-Space Distributions for mode-amplitude conventions, $P$ and $Q$ ordering, optical tomography, and loss.
• Density Operators
• Fourier Transform Conventions
• Phase Space
• Phase-Space Conventions
• Marginals and Quasi-Probabilities
• Gaussian States and Wigner Functions
• Coherent States in Phase Space
• Gaussian Wave Packets
• Coherent States
• Quantum Harmonic Oscillator
• Moyal Bracket
• Classical vs Quantum Probability
• Formula Sheet
• Wigner Function Notebook

References

• E. Wigner, “On the Quantum Correction For Thermodynamic Equilibrium,” Physical Review 40, 749-759, 1932.
• M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: Fundamentals,” Physics Reports 106, 121-167, 1984.
• W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001.
• C. K. Zachos, D. B. Fairlie, and T. L. Curtright, eds., Quantum Mechanics in Phase Space, World Scientific, 2005.
• W. B. Case, “Wigner functions and Weyl transforms for pedestrians,” American Journal of Physics 76, 937-946, 2008.

Exercises

1. Prove the position marginal formula

$$\int dp\,W_\rho(x,p) = \langle x\rvert\rho\lvert x\rangle.$$

Solution

Start from the definition:

$$\int dp\,W_\rho(x,p) = \frac{1}{2\pi\hbar} \int dp\,dy\, e^{-ipy/\hbar} \left\langle x+\frac{y}{2} \middle| \rho \middle| x-\frac{y}{2} \right\rangle.$$

Use

$$\frac{1}{2\pi\hbar} \int dp\,e^{-ipy/\hbar} = \delta(y).$$

Then

$$\int dp\,W_\rho(x,p) = \int dy\,\delta(y) \left\langle x+\frac{y}{2} \middle| \rho \middle| x-\frac{y}{2} \right\rangle = \langle x\rvert\rho\lvert x\rangle.$$

2. Prove that $W_\rho(x,p)$ is real when $\rho$ is Hermitian.

Solution

Taking the complex conjugate gives

$$W_\rho(x,p)^* = \frac{1}{2\pi\hbar} \int dy\, e^{ipy/\hbar} \left\langle x-\frac{y}{2} \middle| \rho \middle| x+\frac{y}{2} \right\rangle.$$

Now substitute $y\to -y$. The integral becomes exactly the original definition of $W_\rho(x,p)$. Therefore $W_\rho(x,p)^*=W_\rho(x,p)$.

3. Check the normalization of the harmonic-oscillator ground-state Wigner function.

Solution

Use

$$W_0(x,p) = \frac{1}{\pi\hbar} \exp\left[ - \frac{m\omega x^2}{\hbar} - \frac{p^2}{m\hbar\omega} \right].$$

The Gaussian integrals are

$$\int dx\, e^{-m\omega x^2/\hbar} = \left( \frac{\pi\hbar}{m\omega} \right)^{1/2},$$

and

$$\int dp\, e^{-p^2/(m\hbar\omega)} = \left( \pi m\hbar\omega \right)^{1/2}.$$

Multiplying them and the prefactor gives

$$\frac{1}{\pi\hbar} \left( \frac{\pi\hbar}{m\omega} \right)^{1/2} \left( \pi m\hbar\omega \right)^{1/2} =1.$$

4. Use the formula for $W_1(x,p)$ to show that Wigner functions can be negative.

Solution

For the first excited oscillator state,

$$W_1(x,p) = \frac{1}{\pi\hbar} \left( 2r^2-1 \right) e^{-r^2}.$$

At the origin $r=0$, so

$$W_1(0,0) = - \frac{1}{\pi\hbar}.$$

This is negative, even though the position and momentum marginals obtained from $W_1$ are ordinary nonnegative probability densities.