Consider a system of N particles in a cubic box of volume . The
particles are assumed not to interact with each other. Thus, the
Hamiltonian in Cartesian coordinates may be taken to be where we
are assuming that all particles are of the same type. The
microcanonical partition function is Since the Hamiltonian is
independent of the coordinates, the 3N coordinate integrations
can be done straightforwardly. The range of each one is 0 to L.
Thus, these integrations give an overall factor : To do the
momentum integrals, we first change variables to Substitution
into the partition function gives The 3N dimensional integral
can now be seen to an integration over the surface of a sphere
defined by the equation Therefore, it proves useful to transform
to 3N dimensional spherical coordinates, , where where is the
3N-1 solid angle integral over the 3N-1 angles. The partition
function now becomes: At this point, we use an identity of
-functions: We will also make use of the fact that N is large,
so that we may take . Usi