PX284 - M1 - electron gas

g (k) d k = \frac{V k^{2}}{2 π^{2}} d k \times 2

where, the factor of 2 arises from the spin degeneracy ( $2 s + 1 = 2$ for $s = 1 / 2$ )

\begin{matrix} E = \frac{ℏ^{2} k^{2}}{2 m} \\ k = \sqrt{\frac{2 m E}{ℏ^{2}}} \\ ⟹ d k = \sqrt{\frac{2 m}{ℏ^{2}}} \frac{1}{2 \sqrt{E}} d E \\ ∴ g (E) d E = \frac{V}{2 π^{2}} {(\frac{2 m}{ℏ^{2}})}^{3 / 2} E^{1 / 2} d E \end{matrix}