There has recently been considerable interest in one-dimensional electrical conductors. In this problem, you are asked to calculate some free-electron properties for a system of length L containing N electrons. Thus, there are n =N/L electrons per unit length.

a) Calculate the density of states per unit energy range per unit length (hint: use periodic boundary conditions and positive and negative values of k).

b) Calculate the Fermi energy e_f at T=0. Estimate e_f in electron volts for n = 10^8 /cm.

C) Calculate the energy of the gas at T =0. Give your answers in terms of n and e_f.