Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / Section A / A7 / Solution

For the usual nonrelativistic dispersion, the three-dimensional density of states is $g(ϵ)=C{ϵ}^{1/2}$. Filling all states through the Fermi energy givesDegeneracy factors cancel. More generally $ϵ\propto p^s$ gives $U/N=3{ϵ}_F/(3+s)$; an ultrarelativistic gas would give $3{ϵ}_F/4$. The nonrelativistic result is the standard massive ideal-gas interpretation.