Density of states

A system at some energy at most $E$ has some number $N(E)$ of quantum states available to it. The density of states is how this number changes with energy

The density of states is related to the multiplicity, since over some energy $[E, E + \delta E]$ the number of states is $\Gamma = \frac{\d N}{\d E} \delta E$.

Classically for most systems $N(E) = \infty$. We can approximate the quantum result while still doing mostly classical calculations by assuming that each state has some volume $h^3$ in phase space ($p$-$v$ space), and that states cannot “overlap”.

When energy is quadratic in the parameters of the system (quantum number, momentum, etc), the number of accessible states looks like a ball in some high dimension. The density of states is then its surface area.

The volume of an $n$-dimensional ball is

Where $\Gamma(x + 1) = x!$ for integer $x$.