Physical Interpretation of Phase Space Volume

Perhaps one of the most important results of the whole of Classical Mechanics is that the volume occupied by an ensemble in the phase space remains constant in time. Another very interesting result is that it is also invariant under canonical transformations (as the Jacobian of the canonical transformation is generically unity).

Both these results compel me to think that there should be some extremely simple and deep physical meaning to the volume of the phase space. Intuitively, it seems to me that since the number of points occupied in the phase space is just the number of microstates of the ensemble, the volume of the phase space also denotes just the number of microstates of the ensemble up to some (not-so-clear-to-me) scaling factor. My question, thus, is that what is this scaling factor? Is it just arbitrary (i.e., to be decided by some choice of units etc.) or is there a definite scaling factor?

• The scaling factor between phase space volume and microstates is $h$, which can be seen from the WKB approximation $\int p \, dx = n h$ or more generally by Weyl's law. I asked a somewhat related question here. Aug 28 '18 at 8:09
• You get a lot of great stuff for free here. For example, the classical adiabatic invariant is the phase space volume, so this must correspond in QM to the number of microstates. But that determines the entropy, so a thermodynamic adiabatic process leaves the entropy invariant. Aug 28 '18 at 8:10
• @knzhou Thanks for your comments your question is quite related and the discussion there seems helpful. And yes, I agree that this thing relates to a lot of results in statistical mechanics and quantum mechanics. Aug 28 '18 at 8:16

At least for quantum systems, this factor is not arbitrary, but usually taken to be $\sim\hbar^d$, where $d$ is the number of phase space dimensions. In this context, phase space volume may be linked to "conservation of information".