Liouville's theorem (Hamiltonian)

The Liouville equation, by Joseph Liouville, is a description of the temporal evolution of a physical system in statistical mechanics, the Hamiltonian formalism of classical mechanics and in quantum mechanics, there also called Von- Neumann equation. The Liouville equation states clearly that the volume of any subset of the phase space is preserved under a temporal evolution, ie, that the flow is volume - and orientation-preserving even through the phase space.

Classical equation

In statistical physics, an ensemble of realizations of a physical system to be characterized by a probability density in the phase space ( " space phase density "). For the time evolution of such an ensemble

And the canonical position and momentum coordinates of the - th particle denote the phase space. Regardless of the ensemble, the total time derivative vanishes, which means that does not change along a Phasenraumtrajektorie the phase space density. Replacing and according to the Hamiltonian equations of motion, so can this situation say it shorter using the Poisson bracket:

Where H denotes the Hamiltonian and the totality of phase space coordinates.

From the Liouville equation (also called " Liouville 's theorem " ) immediately followed by the set of Liouville.

The Liouville equation can also be the introduction of the Liouvilleoperators

Can be written as follows:

Quantum mechanical equation

Here, H denotes the Hamiltonian, the density matrix, and the brackets of the commutator. This form of the equation is also called Liouville von Neumann equation.

One can formally introduce a Liouville operator as in the case of classical mechanics, defined by its action on an operator on. Then the von Neumann equation writes:

With the help of the Wigner- image can be derived in semi- classical limit is a direct relationship between the classical Poisson bracket and the Hamiltonian: