| context | $\varphi$ … classical confined phase volume |
| definiendum | $D(E):=\varphi'(E) $ |
In bounded space, the computations involve a counting energy levels, or energy Eigenvalues $\varepsilon_r$ in the quantum case, which we index by $r$. When energy eigenvalue degeneracy of the Schrödinger equation have to be counted, the volume $V$ is enters the theory. A standard approximation is
| $\sum_r\dots\ \ \longrightarrow\ \ (2S+1)\frac{V}{(2\pi)^3}\int\mathrm d^3k \dots $ |
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where $S$ is the particle spin.
We now state the density of states of ideal quantum gases in a finite volume $V$ for energies $E\ge 0$.
For fermions with $\varepsilon({\bf k})=\frac{\hbar^2}{2m}{\bf k}^2$:
| $ D(E) = 2\pi\ (S+1)\frac{V}{(2\pi)^3}\left(\frac{\hbar^2}{2m}\right)^{-3/2}\cdot\sqrt{E} $ |
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For bosons with $\varepsilon({\bf k})=\hbar c \|{\bf k}\|$ and spin $S=1$:
| $ D(E) = 2\pi\ 2\frac{V}{(2\pi)^3}(\hbar c)^{-3}\cdot E^2 $ |
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More generally, for an dispersion relation $E=E_0+c_k\,k^p$ in an $n$-dimensional space (volume of the space being $c_n\,k^n$), the density is
$D(E) = \dfrac{c_n}{c_k^r}\dfrac{\mathrm d}{{\mathrm d}E}(E-E_0)^r=r\,\dfrac{c_n}{c_k^r}(E-E_0)^{r-1}$, where $r:=\tfrac{n}{p}$.
Wikipedia: Density of states