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Chemistry LibreTexts

The canonical ensemble

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

\[ \underline {\rho } \] $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \(e^{-\beta H}\)  
\[f(E_i) \] $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \(e^{-\beta E_i}\)  


Thus, the quantum canonical partition function is given by

\[Q(N,V,T) = {\rm Tr}(e^{-\beta H}) = \sum_i e^{-\beta E_i}\]


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

 

\[A (N, V, T ) \] $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \(-{1 \over \beta}\ln Q(N,V,T)\)  
\[E (N, V, T ) \] $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$ \(-{\partial \over \partial \beta}\ln Q(N,V,T)\)  
\[ P (N, V, T) \] $\textstyle =$ data-cke-saved-style =$ data-cke-saved-style =$ data-cke-saved-style =$  \({1 \over \beta}{\partial \over \partial V}\ln Q(N,V,T)\)  


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

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


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

\[\langle A \rangle = {1 \over Q}\sum_i \langle E_i\vert Ae^{-\beta H} \vert E_i \rangle = {1 \over Q}\sum_i e^{-\beta E_i} \langle E_i\vert A\vert E_i\rangle\]


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