For electrons with an energy >> mc2, where m is the rest mass of the election, the energy is given by pc,

For electrons with an energy ε >> mc2, where m is the rest mass of the election, the energy is given by ε ≈ pc, where p is the momentum. For electrons in a cube of volume V = L3 the momentum is of the form (πh/L), multiplied by (ax2 + ny2 + nz2), exactly as for the non-relativistic limit.
(a) Show that in this extreme relativistic limit the Fermi energy of a gas of N electrons is given by

εF = hπc(3n/π)1/3,

where n = N/V.
(b) Show that the total energy of the ground stale of the gas is

U0 = ¾ NεF.