Phase-Space Dynamics

Phase-space dynamics rewrites density-operator time evolution as an equation for the Wigner function $W(x,p,t)$. The result looks close to classical Liouville flow, but with quantum corrections controlled by the star product:

$$\frac{\partial W}{\partial t} = \{H_W,W\}_M.$$

Here $H_W$ is the Weyl symbol of the Hamiltonian and $\{\cdot,\cdot\}_M$ is the Moyal bracket. This is not a semiclassical approximation. It is an exact rewriting of closed-system unitary dynamics, provided the Wigner-Weyl transforms are defined with the same conventions.

The payoff is conceptual and practical: phase-space dynamics shows precisely when quantum evolution resembles classical Hamiltonian flow and where noncommutative corrections enter.

From Density Operators to Wigner Functions

For a closed system with Hamiltonian $H$, the density operator satisfies the Liouville–von Neumann equation

$$\frac{\partial\rho}{\partial t} = \frac{1}{i\hbar}[H,\rho].$$

Taking the Weyl transform gives

$$\left( \frac{\partial\rho}{\partial t} \right)_W = \frac{1}{i\hbar} \left( H_W\star\rho_W-\rho_W\star H_W \right).$$

With the convention

$$W(x,p,t)=\frac{1}{2\pi\hbar}\rho_W(x,p,t),$$

the same equation becomes

$$\frac{\partial W}{\partial t} = \frac{1}{i\hbar} \left( H_W\star W-W\star H_W \right).$$

By definition, this is

$$\frac{\partial W}{\partial t} = \{H_W,W\}_M.$$

The equation preserves normalization because it is the Wigner transform of unitary density-operator evolution:

$$\frac{d}{dt} \int dx\,dp\,W(x,p,t)=0,$$

assuming boundary terms vanish or periodic boundary conditions are used consistently.

Moyal Evolution Equation

For one degree of freedom, the star product is

$$A\star B = A \exp\left[ \frac{i\hbar}{2} \left( \overleftarrow{\partial_x}\overrightarrow{\partial_p} - \overleftarrow{\partial_p}\overrightarrow{\partial_x} \right) \right] B.$$

The Moyal bracket is its antisymmetric part:

$$\{A,B\}_M = \frac{1}{i\hbar} \left( A\star B-B\star A \right).$$

For smooth functions, the formal expansion is

$$\{A,B\}_M = \{A,B\}_{\rm PB} - \frac{\hbar^2}{24}A\Lambda^3B + O(\hbar^4),$$

where

$$A\Lambda B = \frac{\partial A}{\partial x} \frac{\partial B}{\partial p} - \frac{\partial A}{\partial p} \frac{\partial B}{\partial x}.$$

Thus phase-space quantum dynamics begins with classical Hamiltonian flow and then adds even powers of $\hbar$ through higher derivatives.

Hamiltonians of Kinetic Plus Potential Form

For

$$H=\frac{\hat p^2}{2m}+V(\hat x),$$

the Weyl symbol is

$$H_W(x,p)=\frac{p^2}{2m}+V(x).$$

The Wigner–Moyal equation is

$$\begin{aligned} \frac{\partial W}{\partial t} &= - \frac{p}{m} \frac{\partial W}{\partial x} + \sum_{\ell=0}^{\infty} \frac{(-1)^\ell}{(2\ell+1)!} \left( \frac{\hbar}{2} \right)^{2\ell} \\ &\quad\times \frac{d^{2\ell+1}V}{dx^{2\ell+1}} \frac{\partial^{2\ell+1}W}{\partial p^{2\ell+1}}. \end{aligned}$$

The $\ell=0$ term gives

$$V'(x)\frac{\partial W}{\partial p}.$$

Together with the kinetic term, this is the classical Liouville equation:

$$\frac{\partial W}{\partial t} = - \frac{p}{m} \frac{\partial W}{\partial x} + V'(x) \frac{\partial W}{\partial p} + \text{quantum corrections}.$$

The first correction is

$$- \frac{\hbar^2}{24} V'''(x) \frac{\partial^3W}{\partial p^3}.$$

Higher corrections involve higher odd derivatives of $V$ and higher odd momentum derivatives of $W$.

Free Particle Shear

For a free particle,

$$H_W=\frac{p^2}{2m}.$$

All quantum corrections vanish, and the Wigner equation is

$$\frac{\partial W}{\partial t} = - \frac{p}{m} \frac{\partial W}{\partial x}.$$

The solution is a phase-space shear:

$$W(x,p,t) = W(x-pt/m,p,0).$$

This looks exactly like the classical ensemble solution. It does not mean the state has become classical. A Wigner function with negative interference fringes will shear just as exactly as a positive Gaussian.

Quadratic Hamiltonians

For the harmonic oscillator,

$$H_W(x,p) = \frac{p^2}{2m} + \frac12m\omega^2x^2.$$

Because $H_W$ is quadratic, all third and higher derivatives vanish. The Moyal bracket equals the Poisson bracket exactly, so

$$\frac{\partial W}{\partial t} = - \frac{p}{m}\frac{\partial W}{\partial x} + m\omega^2x\frac{\partial W}{\partial p}.$$

The solution is a rotation in scaled phase space:

$$W(x,p,t) = W(x_{-t},p_{-t},0),$$

where

$$x_{-t} = x\cos\omega t - \frac{p}{m\omega}\sin\omega t,$$

and

$$p_{-t} = p\cos\omega t + m\omega x\sin\omega t.$$

This exact classical-looking flow is a special property of quadratic Hamiltonians. It explains why Gaussian oscillator states, coherent states, and squeezed states have especially simple phase-space motion.

Anharmonic Potentials

For anharmonic potentials, the higher terms in the Moyal expansion matter. Consider

$$V(x)=\frac{\lambda}{4}x^4.$$

Then

$$V'(x)=\lambda x^3, \qquad V'''(x)=6\lambda x,$$

and all higher odd derivatives vanish. The Wigner equation becomes

$$\frac{\partial W}{\partial t} = - \frac{p}{m}\frac{\partial W}{\partial x} + \lambda x^3\frac{\partial W}{\partial p} - \frac{\hbar^2\lambda x}{4} \frac{\partial^3W}{\partial p^3}.$$

The first two terms are the classical Liouville flow for the quartic oscillator. The last term is the leading and, for this polynomial potential, only Moyal correction. It is responsible for phase-space interference effects that no classical ensemble can reproduce.

Expectation Values During Evolution

Expectation values are computed using Weyl symbols:

$$\langle A\rangle(t) = \operatorname{Tr}(\rho(t)A) = \int dx\,dp\,W(x,p,t)A_W(x,p).$$

One may evolve either the Wigner function or the symbol, mirroring Schrödinger and Heisenberg pictures. In symbol language, the Heisenberg equation becomes

$$\frac{dA_W}{dt} = \{A_W,H_W\}_M + \left( \frac{\partial A}{\partial t} \right)_W.$$

For $A_W=x$ and $A_W=p$, the Moyal bracket gives the usual Ehrenfest-level equations

$$\frac{d}{dt}\langle x\rangle = \frac{\langle p\rangle}{m}, \qquad \frac{d}{dt}\langle p\rangle = -\langle V'(x)\rangle,$$

where the second equation still contains the quantum expectation of $V'(x)$, not generally $V'(\langle x\rangle)$.

Numerical Phase-Space Evolution Preview

Numerical Wigner evolution is attractive because it displays transport, interference, and negativity directly in phase space. Common approaches include:

• pseudo-spectral grids in $x$ and $p$ for derivative terms;
• split evolution between kinetic shear and potential terms;
• truncating the Moyal series for smooth potentials and controlled regimes;
• solving the equivalent density-operator or wavefunction problem as a benchmark;
• using exact free-particle and harmonic-oscillator flows as validation tests.

There are also real hazards. High-order momentum derivatives amplify grid noise, boundary reflections can corrupt normalization, negative regions require enough resolution to avoid artificial smoothing, and truncated Moyal equations may violate positivity properties of the underlying density operator. A reliable numerical calculation should monitor

$$\int dx\,dp\,W,$$

reality of $W$, comparison against Hilbert-space evolution, and convergence with grid size.

Relation to Classical Limit

The formal classical limit is obtained by dropping the higher Moyal corrections:

$$\frac{\partial W}{\partial t} \approx \{H,W\}_{\rm PB}.$$

This approximation is justified when the state and Hamiltonian are smooth on phase-space scales set by $\hbar$ and when the time evolution does not generate fine interference structure. It can fail dramatically for superpositions, tunneling, chaotic stretching, and long-time evolution even when $\hbar$ is small.

The page Moyal Bracket gives the exact bracket expansion. Classical Limit of the Moyal Bracket gives the dimensionless error scales and a detailed account of when the approximation is controlled.

Common Mistakes

• Thinking that a classical-looking Wigner equation makes the Wigner function an ordinary probability density.
• Forgetting that quadratic Hamiltonians are special; their exact classical-looking flow does not generalize to arbitrary potentials.
• Dropping Moyal corrections for anharmonic systems without estimating their size.
• Confusing the Wigner equation with a stochastic Fokker-Planck equation for a classical probability distribution.
• Using coarse grids that wash out negative interference fringes and then concluding the state is classical.
• Comparing phase-space and Hilbert-space calculations without matching Fourier and Weyl-symbol conventions.

Cross-Links

• Wigner Function
• Weyl Transform
• Star Product
• Moyal Bracket
• Classical Limit of the Moyal Bracket
• Phase-Space Conventions
• Liouville–von Neumann Equation
• Ehrenfest Theorem
• Hamiltonian Mechanics Review
• Formula Sheet
• Wigner Function Notebook

References

• E. Wigner, “On the Quantum Correction For Thermodynamic Equilibrium,” Physical Review 40, 749, 1932.
• J. E. Moyal, “Quantum mechanics as a statistical theory,” Proceedings of the Cambridge Philosophical Society 45, 99, 1949.
• M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: Fundamentals,” Physics Reports 106, 121, 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, 2008.

Exercises

1. Derive the free-particle shear solution.

Solution

The free equation is

$$\frac{\partial W}{\partial t} = - \frac{p}{m} \frac{\partial W}{\partial x}.$$

Along a characteristic,

$$\frac{dp}{dt}=0, \qquad \frac{dx}{dt}=\frac{p}{m}.$$

Thus

$$p(t)=p_0, \qquad x(t)=x_0+\frac{p_0t}{m}.$$

Solving for the initial point gives

$$x_0=x-\frac{pt}{m}, \qquad p_0=p.$$

Since $W$ is constant along characteristics,

$$W(x,p,t)=W(x-pt/m,p,0).$$

2. Show why the harmonic oscillator has no Moyal corrections.

Solution

For

$$H_W(x,p)=\frac{p^2}{2m}+\frac12m\omega^2x^2,$$

all third and higher derivatives of $H_W$ vanish. The Moyal bracket expansion contains the Poisson bracket plus terms involving third and higher derivatives. Therefore every correction beyond the Poisson bracket is zero, and

$$\{H_W,W\}_M=\{H_W,W\}_{\rm PB}.$$

3. Compute the first Moyal correction for a quartic potential.

Solution

For

$$V(x)=\frac{\lambda}{4}x^4,$$

one has

$$V'''(x)=6\lambda x.$$

The first correction in the Wigner equation is

$$- \frac{\hbar^2}{24} V'''(x) \frac{\partial^3W}{\partial p^3}.$$

Substitution gives

$$- \frac{\hbar^2}{24} (6\lambda x) \frac{\partial^3W}{\partial p^3} = - \frac{\hbar^2\lambda x}{4} \frac{\partial^3W}{\partial p^3}.$$

4. Explain why exact classical-looking flow does not imply classicality.

Solution

For free and quadratic Hamiltonians, the Wigner function is transported by the same phase-space flow as a classical distribution. However, the function being transported may still be negative or contain interference fringes. Classical probability densities are nonnegative and have a different interpretation. The flow law is classical-looking, but the state represented by $W$ can remain fully quantum.

5. Derive the Wigner-picture equation for expectation values.

Solution

The expectation value is

$$\langle A\rangle(t) = \int dx\,dp\,W(x,p,t)A_W(x,p).$$

If $A$ has no explicit time dependence, then

$$\frac{d}{dt}\langle A\rangle = \int dx\,dp\, \frac{\partial W}{\partial t}A_W.$$

Using

$$\frac{\partial W}{\partial t} = \{H_W,W\}_M,$$

and cyclicity of the phase-space trace integral, this can be written equivalently as

$$\frac{d}{dt}\langle A\rangle = \int dx\,dp\, W\{A_W,H_W\}_M.$$

This is the phase-space version of the Heisenberg equation.