In the case of a discrete classical partition function defined as:

$$ Z=\sum_{q \in Q}e^{-\beta (E(q)+pV(q))} $$

It is straightforward to show that it implies the following fundamental thermodynamic relation:

$$ d\overline{E}=TdS-pd\overline{V} $$

If instead, we consider a discrete quantum mechanical partition function, defined as:

$$ Z=\text{Tr} (e^{-\beta \hat{H}}) $$

How do we write and derive the fundamental thermodynamic relation for the discrete quantum mechanical case (preferably in the matrix formulation)?

For instance, how does one write the average expectation values $\overline{E}$ and $\overline{V}$ referenced in the fundamental relations, in terms of differentials of matrices?


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