Past exam of the advanced Quantum Condensed Matter Physics course of the University of Cambridge / 2015 / Paper 1 / 1 / b / Solution

For the unpolarized gas write $x=n/2$. The positive direct Hartree energy before background cancellation and the kinetic energy areTheir ratio is thereforeThis compares the separate direct and kinetic terms. The net Hartree-plus-background contribution after the stipulated cancellation is zero, so it would be incorrect to interpret $r_s$ as that residual contribution divided by $T$. The exchange-energy magnitude in the unpolarized state is $E_H/2$, not $E_H$.