Editing Quantum statistical mechanics (section)

== Gibbs canonical ensemble ==

{{main|canonical ensemble}}

Consider an ensemble of systems described by a Hamiltonian ''H'' with average energy ''E''.  If ''H'' has pure-point spectrum and the eigenvalues <math>E_n</math> of ''H'' go to +&infin; sufficiently fast, e<sup>−''r H''</sup> will be a non-negative trace-class operator for every positive ''r''.

The ''[[Gibbs canonical ensemble]]'' is described by the state 
<math display="block"> S= \frac{\mathrm{e}^{- \beta H}}{\operatorname{Tr}(\mathrm{e}^{- \beta H})}. </math>
Where &beta; is such that the ensemble average of energy satisfies  
<math display="block"> \operatorname{Tr}(S H) = E </math>

and

<math display="block">\operatorname{Tr}(\mathrm{e}^{- \beta H}) = \sum_n \mathrm{e}^{- \beta E_n} = Z(\beta) </math>

This is called the [[partition function (mathematics)|partition function]]; it is the quantum mechanical version of the [[canonical partition function]] of classical statistical mechanics. The probability that a system chosen at random from the ensemble will be in a state corresponding to energy eigenvalue <math>E_m</math> is

<math display="block">\mathcal{P}(E_m) = \frac{\mathrm{e}^{- \beta E_m}}{\sum_n \mathrm{e}^{- \beta E_n}}.</math>

The Gibbs canonical ensemble maximizes the von Neumann entropy of the state subject to the condition that the average energy is fixed.