The system in grand canonical emsemble together with the surrounding reservoir is isolated, thus have conserved particle number $N$. However, the system itself only has fixed average $\langle N\rangle$, and the quantum mechanical operator $\hat{N}$ necessarily does not commute with the Hamiltonian. Why can we assign both energy and particle number to each state?
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1$\begingroup$ You don't assign energy and particle number to each state, right? You just evaluate $\mathrm{Tr}[\exp [-\beta(H - \mu N)]]$ in a basis which diagonalizes at most one of them. $\endgroup$– Connor BehanCommented Nov 8, 2021 at 2:54
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$\begingroup$ @ConnorBehan Yes, but when evaluating it, it seems that we always simply write out $\sum_n \exp[-\beta(En-\mu N_n)]$ as if we are doing it in a basis which diagonalizes both of them? $\endgroup$– RicknJerryCommented Nov 8, 2021 at 4:49
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$\begingroup$ I've only seen that done in classical stat-mech systems. $\endgroup$– Connor BehanCommented Nov 8, 2021 at 5:25
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1$\begingroup$ Possibly related: How can μ be nonzero if grand canonical density operator commutes with $\hat H$ ? and Does non-conservation of number of particles imply zero chemical potential?. $\endgroup$– Tobias FünkeCommented Nov 8, 2021 at 9:09
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