# The canonical ensemble

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

 $\underline {\rho }$ $$e^{-\beta H}$$ $f(E_i)$ $$e^{-\beta E_i}$$

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:

 $A (N, V, T )$ $$-{1 \over \beta}\ln Q(N,V,T)$$ $E (N, V, T )$ $$-{\partial \over \partial \beta}\ln Q(N,V,T)$$ $P (N, V, T)$ $${1 \over \beta}{\partial \over \partial V}\ln Q(N,V,T)$$

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.

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