Liouville's theorem (Hamiltonian)

In Hamiltonian mechanics, Liouville's theorem, named after the French mathematician Joseph Liouville, predicts how probability distributions evolve over time. Typically, ρ is the probability that the system will be found in an infinitesimal volume <math>d\tau<math> of phase space, τ standing for both position and momentum coordinates. In a system of N particles, τ is a convenient shorthand for the set of variables

<math>\{\,x_1,x_2,\ldots,x_N,y_1,y_2,\ldots,y_N,z_1,z_2,\ldots,z_N;p_{x1},p_{x2},\ldots,p_{xN},\ldots,p_{zN}\,\}.

<math>

In a system with Hamiltonian H and distribution function ρ, the theorem states that

<math>\frac{\partial}{\partial t}\rho=-\{\,\rho ,H\,\}<math>

where the curly braces denote a Poisson bracket.

An interesting corollary of this theorem is that time evolution preserves volumes in phase space. If a system is known to begin within a particular volume of phase space, then after an interval of time passes, the system will reside in a subspace of equal volume.

Canonical quantization yields a quantum-mechanical version of this theorem. This procedure, often used to devise quantum analogues of classical systems, involves describing a classical system using Hamiltonian mechanics. Classical variables are then re-interpreted as quantum operators, while Poisson brackets are replaced by commutators. In this case, the resulting equation is

<math>\frac{\partial}{\partial t}\rho=-\frac{i}{\hbar}[\rho,H]<math>

where ρ is the density matrix.