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Given equilibrium, why does one only consider a mixed state where the pure states are eigenfunctions of the hamiltonian, i.e. states with an energy eigenvalue?

And for the second question, I quote David Tong's ``Lectures on Statistical Physics'' ( freely and legally accessible on http://www.damtp.cam.ac.uk/user/tong/statphys.html more concretely part 1 page 5):

Why would we conceptually expect the delta E? Say my system is isolated and initially in a certain energy eigenstate, then it will never hop to another energy eigenstate, no matter how close it is to the previous one, right? So what's the physics of this claim?[itex]\textrm{In quantum systems, the energy levels will be discrete. However, with many

particles}[/itex]

[itex]\textrm{the energy levels will be finely spaced and can be effectively treated as

a continuum.}[/itex]

[itex]\textrm{When we say that $\Omega(E)$ counts the number of states with energy

$E$ we implicitly}[/itex]

[itex]\textrm{mean that it counts the number of states with energy between $E$ and $E + \delta E$ where $\delta E$}[/itex]

[itex]\textrm{is small compared to the accuracy of our measuring

apparatus but large compared to the spacing of the levels.}[/itex]

Thank you! They're simple questions, but every introducing book on this subject seems to ignore these points completely, making me wonder if my questions are somehow unjust. PF to the rescue.