Quantum theory of free electrons

exercise 3_4927

Consider a one-dimension (1D) metal of length " a " containing N electrons. Treat the metal as having a uniform potential within the box. You are to compare and contrast two cases:

( i ) The allowed values of the electron wave vector, k x , are determined by requiring "periodic boundary conditions."

( ii ) The allowed values of k x are determined by having the potential outside the box infinitely high (and repulsive) so that the electron wave function must vanish at the boundaries ( x = 0 and x = a ).

(a) Find the allowed values of k x for the two cases.

(b) If one plots the allowed values of k x as points on a line in k -space, what is the spacing between points on the line for the two cases?

(c) If all the states are filled by the N electrons (i.e. at absolute zero, T = 0), what is the energy of the highest energy electron in the two cases? Give your answer in terms of N, ħ , m , and a . Please note that the answers should be identical!

(d) Derive the density of states in energy g ( E ) for the two cases, and show that they are identical.