Problem 3 - Non-interacting Gas in 1D Consider a nonvinteraeting gas of N particles of mass m conned to one dimension (e.g., on a line) of length L. As usual the allowed quantum energies are given by n37? 52 E : 2mL2 With m- : 1,13,... a) (3 points) Show that the density of states for a free electron in one dimension is 13(5) = (g) (1?)1/2 15-1/2 b.) (3 points) What is the Fermi Energy (: EF) for this system in terms of the \"density\" of t ', _ _ part1cles n L. C.) (3 points) Show that the total energy for this system at T : O is given by U 1NE i 3 F Where EF is the Fermi energy