Consider a degenerate gas of N fermions in a 2-dimensional box
of area A = L 2 and with T = 0. For a particle
in a 2D box, the allowed energies are given by:
E = (h2/ 8mL2)
(nx2 + ny2 ).
(Note: A 2-dimensional electron gas can actually be physically
constructed in a device called a 2D quantum well. The electrons are
free to move on the interface between two different materials, but
cannot move away from the interface.)
a) Derive a formula for the Fermi energy εf of
this system in terms of the area A of the box and the number of
particles N.
b) Calculate the average energy per particle in the gas at T =
0, U/N. (If you were not able to do a), you may use the standard 3D
formula for εf to complete this part.)