Apply BornVon Karman cyclic boundary conditions to Bloch wav

Apply Born-Von Karman cyclic boundary conditions to Bloch wave-functions. (a) show that for a given value of the band-index all the distinct states have wave vectors in the first Brillouin zone of the reciprocal lattice. (b) how many distinct states (that is states with different Bloch wave-functions) fit into the first Brillouin zone? (c) comment on the relation between the number of states in the first Brillouin zone and degrees of freedom of electrons described by Bloch wave-functions. (d) compare with the case of the Brillouin zone of phonon modes and comment on the similarities and differences of the numbers of states in the two cases (phonon modes and Bloch states with a given band-index)

Solution

Check this link :

arxiv.org/pdf/0806.3198