Density of States

$N=\int \mathrm{d}N=\int \frac{\mathrm{d}N}{dx}\mathrm{d}x=\int D(x)\mathrm{d}x$

• $D(x)$ is the density of states and can be in terms of energy, frequency, or wavenumber

$D(\epsilon )=\frac{\mathrm{d}N}{\mathrm{d}\epsilon }=\frac{\pi }{2}{\left(\frac{2m}{{\hslash }^2}{\left(\frac{L}{\pi }\right)}^2\right)}^{\frac{3}{2}}{\epsilon }^{\frac{1}{2}}$

• Density of states for non-relativistic fermions in 3D

DOS	1D	2D	3D
E ~ K (relativistic)	const	E	$E^2$
E ~ K${}^2$ (non-relatavistic)	$\frac{1}{\sqrt{E}}$	const	$\sqrt{E}$