What Is the Mathematical Definition of the Microcanonical Partition Function?

[Pacopag]

Homework Statement

Does anyone know the mathematical definition of the microcanonical partition function?
I've seen
$$\Omega = {E_0\over{N!h^{3n}}}\int d^{3N}q d^{3N}p \delta(H - E)$$
where H=H(p,q) is the Hamiltonian. This looks like a useful definition.
Only thing is I don't know what $$E_0$$ is.

Homework Equations

The Attempt at a Solution

 

[genneth]

The microcanonical partition function is just a count of the number of states that satisfy extensive constraints on volume, energy, etc. The probability of each state is then trivially one over the partition function.

 

[Pacopag]

But in the case of a classical system the number of states is uncountable because the position and momenta are continuous.

 

[genneth]

In which case you can still calculate the phase space volume and the probability distribution is uniform over that volume --- it's the obvious generalisation.

 

[Pacopag]

Ok. Good. Now I see why my "hint" was to bring the constant energy surface in phase space into a sphere (because I know how to find the volume of a sphere).

Thank you very much genneth.