The problem of quantum statistical mechanics is the quantum mechanical
treatment of an *N*-particle system. Suppose the corresponding
*N*-particle classical system has Cartesian coordinates

and momenta

and Hamiltonian

Then, as we have seen, the quantum mechanical problem consists of determining the state vector from the Schrödinger equation

Denoting the corresponding operators, and , we note that these operators satisfy the commutation relations:

and the many-particle coordinate eigenstate is a tensor product of the individual eigenstate :

The Schrödinger equation can be cast as a partial differential equation by multiplying both sides by :

where the many-particle wave function is . Similarly, the expectation value of an operator is given by

- The density matrix and density operator
- Time evolution of the density operator
- The quantum equilibrium ensembles
- A simple example - the quantum harmonic oscillator

Tue May 9 19:40:24 EDT 2000