Liouville's theorem (Hamiltonian)

Liouville's theorem (Hamiltonian)

In Hamiltonian mechanics, Liouville's theorem predicts how probability distributions evolve under time evolutions.

An interesting corollary of this theorem is that time evolutions are area/volume-preserving.

There is a quantum-mechanical version of this theorem in the Schrödinger picture.

where ρ is the density matrix.