= Solution
For the usual nonrelativistic dispersion, the three-dimensional <density of states> is $g(\epsilon)=C\epsilon^{1/2}$. Filling all states through the <Fermi energy> gives
$$
\boxed{\frac UN=\frac{\int_0^{\epsilon_F}\epsilon^{3/2}d\epsilon}{\int_0^{\epsilon_F}\epsilon^{1/2}d\epsilon}=\frac35\epsilon_F.}
$$
Degeneracy factors cancel. More generally $\epsilon\propto p^s$ gives $U/N=3\epsilon_F/(3+s)$; an ultrarelativistic gas would give $3\epsilon_F/4$. The nonrelativistic result is the standard massive ideal-gas interpretation.
Back to article page
