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In grand canonical ensemble, the state of the system $|\psi(t)\rangle$(in Schrodinger picture) evolves by original Hamitonian $\hat H$ or grand canonical Hamiltonian $\hat H -\mu \hat N$, that is

$$i \partial_t |\psi(t)\rangle= \hat H |\psi(t)\rangle$$

or

$$i \partial_t |\psi(t)\rangle= (\hat H -\mu \hat N)|\psi(t)\rangle$$

And why ?

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    $\begingroup$ Schrodinger's equation applies in general to an isolated quantum system, which is quite the opposite of a system described by the grand canonical ensemble! $\endgroup$ – pppqqq Nov 14 '16 at 19:37
  • $\begingroup$ @pppqqq So this question is ill-defined? But how to describe the evolution of a microscopic state in grand canonical ensemble? $\endgroup$ – 346699 Nov 14 '16 at 20:31
  • $\begingroup$ Yes, I think that your question underlies some misconceptions. I'll try to give a concise answer. $\endgroup$ – pppqqq Nov 14 '16 at 20:37
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It looks like OP's question underlies a couple of misconceptions. Let me explain:

Schrodinger's equation. Schrodinger's equation describes the evolution of the state of an isolated quantum system, which is known to be in the state $\vert \psi (0)\rangle$ at $t=0$. It applies only to isolated quantum systems. While, for a microscopic system, it is sometimes possible to neglect its entanglement with the rest of the universe, this is certainly false for the macroscopic systems described by statistical mechanics (indeed, under normal conditions, it is even false for a dust particle of the size of some $\mu \text m$).

The evolution of the microstate. In the general case, where the system can't be described by a ket, the state (and its evolution) is given by a density matrix $\rho (t)$. However, equilibrium statistical mechanics deals only with the situation in which the system has attained equilibrium and it can be described by a constant canonical density matrix, at least for the purpose of calculating thermodynamic quantities. In particular the concept of "evolution of a microstate" is out of question in statistical mechanics. It would ofcourse be meaningful to ask "how does the system reach the equilibrium", which is the subject of nonequilibrium statistical mechanics, but once you assume that the system is in the grand-canonical ensemble there's no room for evolution.

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  • $\begingroup$ Finally, let me say that the concept of "microstate", which is clear-cut in classical stat-mech, is a bit vague and confusing in quantum stat-mech, since in the latter we would consider a basis such as the energy eigenstates as a "complete set of microstates". $\endgroup$ – pppqqq Nov 14 '16 at 21:13
  • $\begingroup$ If I want to write $\rho(t)=\exp(-\beta (H-\mu N))$ in Heisenberg picture, I need to use $\hat H$ or $\hat H -\mu \hat N$ $\endgroup$ – 346699 Nov 15 '16 at 1:32
  • $\begingroup$ If you want to model the evolution of the density matrix you will have to write something more complicated that $\rho (t+t') = U(t') \rho (t)U(t')^{\dagger}$. The evolution for an open quantum system is not unitary and is given in general by a master equation. A good place to start reading something about open quantum systems is, in my opinion, Schlossauer: springer.com/us/book/9783540357735 $\endgroup$ – pppqqq Nov 15 '16 at 6:23

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