 Solution

ID: 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 gives

\frac{U}{N} = \frac{\int _{0}^{ϵ_{F}} ϵ ^{3/2} d ϵ}{\int _{0}^{ϵ_{F}} ϵ ^{1/2} d ϵ} = \frac{3}{5} ϵ_{F} .

Degeneracy 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.

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