Explain the basic premises of the sommerfeld quantum free electron model 1. for a two-dimensional "metal", the electron energies are given by 2. 2ml2 where l2-area of the 2-d conductor, and n andny are the integer "quantum numbers", running independently from 1 to infinity. for this system: derive an expression for the density of states. find an expression for the fermi energy if the conductor contains n particles a) b) c comment on how this compares to the 3d case hint: momentum space for this system is two dimensional] 3. a. sodium metal has fermi energy ef 3.12 ev. for a cube of sodium at t-0k with side length 1.50 cm. what is the largest value of the radial vector n in momentum space? show that the radial vector n is proportional to the wavevector k for the given state. b. the magntude of the wavevector corresponding to the largest value of the radial vector n is know as the fermi wavevector, kf+. determin the value of kf for this piece of sodium. c. the fastest electrons in a metal at t-0k are moving at a velocity such that the kinetic energy is equal to the fermi energy. this is the fermi velocity. determine the fermi velocity for sodium at t-0k. d. by considering the volume in momentum space, or otherwise, calculate the number of occupied electron states in the system described in (a) above. e. 4 a. describe what the fermi-dirac distribution represents and how it can be used. b. show that f(e-ak)-1-f(ff +a) for any non-zero temperature, where δε< .ef