The single particle energy eigenstates of a hypothetical system are nondegenerate and have energies given by a = o: where & = 6.25 x 10-21 eV. The index a runs over the positive integers. The system contains N = 1020 fermions and is at temperature T = 300 K. [20 points] a. Show that the difference between energy levels is small enough so that the total energy can be calculated by integrating the density of energy states over the range of populated energies, and write an expression for D (E). b. Show that the Fermi energy &F = Neo and calculate the Fermi temperature. How does T compare to the Fermi temperature? c. Provide good estimates of the total energy U and the heat capacity, expressing your answers in terms of N, o, k, and T.