Question

Commutator implies division of phase space in cells of area $h$?

Answer 1

Your interpretation is not quite right. One sharp interpretation one can give to this "cutting" of phase space into cubes of size h2Nh2N (here NN is the dimension of the system's configuration space), is that it allows one to use classical phase space to count the number of energy eigenstates of the corresponding quantum hamiltonian. Instead of trying to describe what I mean, let's investigate this stuff through an example.

Consider, the one-dimensional simple harmonic oscillator. The hamiltonian is

H(q,p)=12mp2+12mω2q2

Answer 2

Explanation:

There is a detail from a very well-written answer here that interested me. Unfortunately Springer is charging 41 Euros to access the paper cited by the author.