In classical mechanics, we have the Liouville theorem stating that the Hamiltonian dynamics is volume-preserving.

What is the quantum analogue of this theorem?

  • 3
    $\begingroup$ Did you look at the Wikipedia article? It pretty clearly states the quantum analogon - it's the von Neumann equation. Do you have a more specific question about that? $\endgroup$
    – ACuriousMind
    Mar 27, 2017 at 12:13

2 Answers 2


It's subtle. The theorem is not there: quantum flows are compressible (Moyal, 1949).

I'll follow Ch. 0.12 of our book, Concise Treatise of Quantum Mechanics in Phase Space, 2014.

The analog of the Liouville density of classical mechanics is the Wigner function in phase space quantum mechanics. Its evolution equation (generalizing Liouville's) is $$ {\partial f \over \partial t} =\{\!\!\{H ,f\}\!\!\}~, $$ where the double (Moyal) brackets indicate a celebrated quantum modification of Poisson brackets by terms of $O(\hbar^2)$, and serve to prove Ehrenfest's theorem for the evolution of expectation values.

For any phase-space function $k(x,p)$ with no explicit time-dependence, $$ \begin{eqnarray} \frac{d\langle k \rangle }{dt} &=& \int\! dx dp~ {\partial f \over \partial t} k \nonumber \\ &=& {1\over i\hbar} \int\! dx dp~ ( H\star f- f\star H)\star k \nonumber\\ &=& \int\! dx dp~ f \{\!\!\{k,H\}\!\!\} = \left \langle \{\!\!\{ k,H\}\!\!\} \right \rangle , \end{eqnarray} $$ where the star product and its manipulations are detailed in said text.

Moyal stressed (discovered?) that his eponymous quantum evolution equation above contrasts to Liouville's theorem (collisionless Boltzmann equation) for classical phase-space densities, $$ {d f_{cl}\over dt}= {\partial f_{cl} \over \partial t} + \dot{x} ~\partial_x f_{cl} + \dot{p}~ \partial_p f_{cl} =0~. $$

Specifically, unlike its classical counterpart, in general, $f$ does not flow like an incompressible fluid in phase space, thus depriving physical phase-space trajectories of meaning, in this context. (Only the harmonic oscillator evolution is trajectoral, exceptionally.)

For an arbitrary region $\Omega$ about some representative point in phase space, the efflux fails to vanish, $$ \begin{eqnarray} {d \over dt}\! \int_{\Omega}\! \! dx dp ~f&=& \int_{\Omega}\!\! dx dp \left ({\partial f \over \partial t}+ \partial_x (\dot{x} f) + \partial_p (\dot{p} f ) \right)\\ &= & \int_{\Omega}\! \!\! dx dp~ (\{\!\!\{ H,f\}\!\!\}-\{H,f\})\neq 0 ~.\nonumber \end{eqnarray} $$

That is, the phase-space region does not conserve in time the number of points swarming about the representative point: points diffuse away, in general, at a rate of O($\hbar^2$), without maintaining the density of the quantum quasi-probability fluid; and, conversely, they are not prevented from coming together, in contrast to deterministic (incompressible flow) behavior.

Still, for infinite $\Omega$ encompassing the entire phase space, both surface terms above vanish to yield a time-invariant normalization for the WF.

The $O(\hbar^2)$ higher momentum derivatives of the WF present in the MB (but absent in the PB---higher space derivatives probing nonlinearity in the potential) modify the Liouville flow into characteristic quantum configurations. So, negative probability regions moving to the left amount to probability flows to the right, etc... Wigner flows are a recondite field, cf. Steuernagel et al, 2013.

For a Hamiltonian $H=p^2/(2m)+V(x)$, the above evolution equation amounts to an Eulerian probability transport continuity equation, $$\frac{\partial f}{\partial t} +\partial_x J_x + \partial_p J_p=0~,$$ where, for $\mathrm{sinc}(z)\equiv \sin z/~ z$ , the phase-space flux is $$J_x=pf/m~ ,\\ J_p= -f \mathrm{sinc} \left( {\hbar \over 2} \overleftarrow {\partial _p} \overrightarrow {\partial _x} \right)~~ \partial_x V(x). $$
Note added. For a recent discussion/proof of the zeros, singularities,and negative probability density features, hence the ineluctable violations of Liouville's theorem in anharmonic quantum systems see Kakofengitis et al, 2017.

  • $\begingroup$ Forgive the naive follow-up question, but doesn't the uncertainty principle imply that one can't "compress" the probability distribution below the inequality limit? One can "squeeze" quantum states, but one can't compress them. Am I missing something? $\endgroup$
    – Tfovid
    Apr 19, 2023 at 7:18
  • $\begingroup$ Indeed, you are missing how essentially differently the uncertainty principle manifests itself in phase-space quantization... f is a quasi-probability, not a probability, and can go negative, but because of the UP, never spike! Read up on the UP chapter on the "concise" booklet linked... $\endgroup$ Apr 19, 2023 at 11:02

The quantum mechanical analogue of the Liouville theorem is given in terms of a density matrix $\rho$ (see https://en.wikipedia.org/wiki/Density_matrix) and states

$$\frac{\partial\rho}{\partial t}=\frac{i}{\hbar}\left[\rho,H\right]$$

This immediately gives us Ehrenfest's theorem, which states that for any observable $A$, the expectation value $\langle A\rangle=\text{tr}(A\rho)$ obeys the equation

$$\frac{\mathrm{d}}{\mathrm{d}t}\langle A\rangle=\frac{i}{\hbar}\langle\left[A,H\right]\rangle$$

Which, in short, says that expectation values obey the classical equations of motion.

  • $\begingroup$ The "in short" here is a bit too short for the purposes of this question, even if it's commonly stated. It is only true that Ehrenfest's theorem gives classical equations of motion insofar as Poisson brackets and commutators are compatible, which is only up to $O(\hbar^2)$ corrections to the Poisson bracket. To make this clear, if we use in your equation $A=p$ the momentum operator in 1D, and define $F(x)=−dV/dx$ the force, the equation we derive is $d \langle p \rangle/dt= \langle F(x) \rangle $ which is quite different from $F(\langle x \rangle)$ (the latter would be a classical EoM)... $\endgroup$
    – Logan M
    Mar 27, 2017 at 18:20
  • $\begingroup$ Hence the two Ehrenfest equations of motion $d \langle p \rangle / dt = \langle F(x) \rangle$ and $ d \langle x \rangle / dt = \langle p \rangle/m$ do not give a solvable set of coupled differential equations (except for a few exceptional cases), and can not be used to derive a Liouville-style theorem. If the right hand side was $F(\langle x \rangle)$, QM would actually be a rather trivial generalization of classical mechanics, but such is essentially only the case for the harmonic oscillator. $\endgroup$
    – Logan M
    Mar 27, 2017 at 18:23
  • $\begingroup$ This is true, however, the question simply asked for a quantum analogue of Liouville's theorem, and so the subtleties of Ehrenfest's theorem seem a little less than helpful in this context. $\endgroup$ Mar 27, 2017 at 18:23
  • $\begingroup$ It isn't clear to me what you mean by "Lioville's theorem" then. If you mean that the phase space distribution function is constant along the trajectories of the system, this is simply false in QM if you replace "phase space distribution function" with "Wigner function". Indeed the higher corrections in the Moyal bracket are telling you that the Lioville theorem $\frac{\partial \rho}{\partial t} = - \{ \rho, H \}$ is no longer true in QM, as Cosmas Zachos' answer describes. $\endgroup$
    – Logan M
    Mar 27, 2017 at 18:41
  • 3
    $\begingroup$ There's a crucial mistake here. The Ehrenfest theorem actually has a minus sign. $\endgroup$
    – Whyka
    Nov 4, 2017 at 23:11

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