Microstates of the canonical ensemble In the micro canonical ensemble the microstates of a system in an arbitrary macrostate, are also eigenstates of the Hamiltonian. Does the same apply to the microstates of the canonical ensemble? Are they eigenstates of the the Hamiltonian? I would expect them not to be, since here the energy is not constant. But I am not sure
 A: At the core of the statistical mechanics using ensembles, there is the possibility of assigning a probability to the set of all possible mechanical states of the system (microstates).
Therefore, the starting point is the identification of such microstates. In principle, any complete set of commuting observables could be used. However, for equilibrium macrostates, one knows (von Neumann's equation) that it is possible and convenient to use eigenstates of the energy for all possible energies of the mechanical system.
Such a statement does not depend on which energy eigenstates contribute to a macrostate. Therefore, they are used to label the microstates, in the case of a microcanonical ensemble, where only one particular value of the energy is picked up, but also for canonical and garn-canonical ensembles, where some finite probability is assigned to the microstates of any possible value of the energy.
A: When talking about the canonical ensemble, one has to distinguish

*

*the Hamiltonian of the system of interest, $H_S$

*the Hamiltonian of the system of interest + bath/thermostat/reservoir, $H_{tot} = H_S + H_B + V_{SB}$
$H_{tot}$ is treated in a microcanonical ensemble framework, and hence we are discussing its eigenstates.  Generally it will not commute with $H_S$, since there is some interaction between the system and the bath. The derivation of the canonical ensemble is however based on solid reasoning that the interaction energy is smaller than the energy of the system, and can be neglected in tehrmodynamic limit (roughly speaking, the energy of the system is proportional to its volume, whereas the interaction energy is proportional to its surface, i.e., scales as volume to power $2/3$.)
Hence, once this logic is accepted and we talk about microcanonical ensemble, the microstates of the system are its eigenstates.
