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In a proof of the equivalence of the canonical and grand canonical ensembles (in the thermodynamic limit) in Jochen Rau's Statistical Physics and Thermodynamics (highly recommended!), the author evaluates the grand canonical partition function as $$Z_G = \textrm{Tr}(\exp(-\beta \hat{H} + \alpha \hat{N})).$$ In evaluating this trace, it seems to me that the author uses a joint eigenbasis of $\hat{H}$ and $\hat{N}$ and, thus, the fact that they commute.

My question is, what sorts of restrictions exist on $\hat{H}$ (i.e. on the type of system to which this implies) for $[\hat{H},\hat{N}] = \hat{0}$?

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    $\begingroup$ Glad you like the book ;) You can also try out his QM book... Anyway, this was asked before, let me search the relevant PSE posts. But what kind of restrictions do you mean? The only answer I can imagine is: The Hamiltonian must preserve the particle number (and thus commute with the number operator). $\endgroup$
    – Tobias Fünke
    Commented Apr 20, 2023 at 15:15
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    $\begingroup$ See e.g. this and this. $\endgroup$
    – Tobias Fünke
    Commented Apr 20, 2023 at 15:16
  • $\begingroup$ It is superb, thank you again for the suggestion! Is his QM book worthwhile/at the level of Ballentine too? Also, I'm not sure I completely follow the answers in your links (none were accepted by their respective OPs); is the conclusion I am to derive that Rau is simply assuming a system for which $[\hat{H},\hat{N}] = \hat{0}$ is true? @TobiasFünke $\endgroup$
    – EE18
    Commented Apr 20, 2023 at 15:56
  • $\begingroup$ I really like the operational approach of the author, also in the context of QM. But IMHO it is, for a physicist at least, supplementary, because he does not discuss many things usual QM (intro) courses discuss (but instead discusses many things not found in "standard" books). Anyway: Yes, I mean if you have a system where the Hamiltonian does not conserve the particle number (say through creation or annihilation of particles), then its does not commute with the respective number operator. So yes, you basically have to assume this... $\endgroup$
    – Tobias Fünke
    Commented Apr 20, 2023 at 16:02
  • $\begingroup$ ...as the most upvoted answer in the second link states: If you want to describe equilibrium conditions, such that the density operator is constant in time, you are forced to conclude that $H$ preserves the particle number (in the context of the grand canonical ensemble) as long as $\mu \neq 0$. $\endgroup$
    – Tobias Fünke
    Commented Apr 20, 2023 at 16:03

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