[UPSC CSE 2008] Statistical Physics (Q308, 20M)

Show that at T = 0, the Fermi distribution function has a value 1 for energies less than the Fermi energy $ \epsilon_F $ and is zero above it. For a system of non-interacting electrons at T = 0, show that the ground state energy of the system of N particles is $ \frac{3}{5} N \epsilon_F $.