Is there a categorical difference between "discretized" statistical mechanics and quantum mechanics from a theory perspective?
For example, we can count possible configurations of a system using the binomial formula when position and time are discretized. I showed in this post that this state counting leads us to the familiar result for an $NVE$ ensemble in statistical mechanics when Stirling's approximation is applied.
Or, stated another way, is discretizing the methods in statistical mechanics satisfactory in reproducing the main results of QM?
Edit a year later
I don't like how I worded this question, nor how I created this post, which didn't even ask a question. I was mainly wondering if there was anything interesting about how I derived $\Omega_0 = \Omega_A \Omega_B$, for the case of discretized space.