Problem: Nearly Free Electrons in One Dimension

Electrons of mass m are confined to one dimension. A weak periodic potential, described by the Fourier series , is applied.

(a) Under what conditions will the nearly free-electron approximation work? Assuming that the condition is satisfied, sketch the three lowest energy bands in the first Brillouin zone. Number the energy bands (starting from one at the lowest band).

(b) Calculate (to first-order) the energy gap at (between the first and second band) and k=0 (between the second and third band).