I had a question about the differences between a quantum state and a classical microstate.
Let's say we have an NVE ensemble, and we are trying to predict some property P of our ensemble. In our methods, we say that each microstate in out ensemble has some canonical coordinates $(\mathbf{r}^N, \mathbf{p}^N)$. The number of microstates in our ensembles is equal to $\Omega(N,V,E)$, and according to the a priori probability principle, the probability of each of these ensembles is $p=1/\Omega$.
Cool, that makes sense.
Now I have a quantum microcanonical ensemble, and I want to enumerate a macroscopic property P of my system. These are the notes that I was given:
My question is, isn't $\mathcal{N} = \Omega$? I thought the definition of ensemble was that it was a list of boxes which had a certain configuration (or wavefunction), and each box had a given NVE. And by our definition of density of states, the number of boxes in our ensemble was equal to the density of states $\Omega$.
From my understanding, in classical ensembles, we have phase coordinates defining our system, in quantum systems, we have wave functions defining our system. Am I right in saying that?