Quantum Statistical Mechanics

Abstract

In general, the process that we use to prepare a system of atoms, molecules or whatever we want for a measurement produces many copies, but not all in the same quantum state. A pure state in which all the molecules, say, are in the same state, is a limiting case. In general, the system will be in a mixed state. One reason for that is that thermal excitations are unavoidable. Let the possible states be vectors of a Hilbert space with a basis $\{\langle \psi _{n}|\}.$ The expectation value of an operator $\hat{A}$ is

$$ \langle \hat{A}\rangle =\sum _{n}P_{n}\langle \psi _{n}|\hat{A}|\psi _{n}\rangle ,$$

where $P_{n}$ is the (classical) probability of finding the system in $\langle \psi _{n}|$.

The classical approach by Gibbs (Sect. 5.20) in the quantum context fully reveals its power.