10.5.2: The canonical ensemble

In analogy to the classical canonical ensemble, the quantum canonical ensemble is defined by

\[\begin{align*} \rho &= e^{-\beta H} \\[4pt] f(E_i) &= e^{-\beta E_i} \end{align*} \]

Thus, the quantum canonical partition function is given by

\[\begin{align*} Q(N,V,T) &= {\rm Tr}(e^{-\beta H}) \\[4pt] &= \sum_i e^{-\beta E_i} \end{align*} \]

and the thermodynamics derived from it are the same as in the classical case:

\[\begin{align*} A (N, V, T ) &= -{1 \over \beta}\ln Q(N,V,T) \\[4pt] E (N, V, T )  &=-{\partial \over \partial \beta}\ln Q(N,V,T) \\[4pt]  P (N, V, T) &= {1 \over \beta}{\partial \over \partial V}\ln Q(N,V,T) \end{align*}\]

etc. Note that the expectation value of an observable \(A\) is

\[\langle A \rangle = {1 \over Q}{\rm Tr}(Ae^{-\beta H}) \nonumber \]

Evaluating the trace in the basis of eigenvectors of \(H\) (and of \( {\rho } \) ), we obtain

\[\begin{align*}  \langle A \rangle &= {1 \over Q}\sum_i \langle E_i\vert Ae^{-\beta H} \vert E_i \rangle \\[4pt] &= {1 \over Q}\sum_i e^{-\beta E_i} \langle E_i\vert A\vert E_i\rangle \end{align*}\]

The quantum canonical ensemble will be particularly useful to us in many things to come.