In analogy to the classical canonical ensemble, the quantum canonical ensemble is defined by
Thus, the quantum canonical partition function is given by
and the thermodynamics derived from it are the same as in the classical case:
etc. Note that the expectation value of an observable A is
Evaluating the trace in the basis of eigenvectors of H (and of 
), we
obtain
The quantum canonical ensemble will be particularly useful to us in many things to come.