Wannier function the Wannier functions of a hand are defined In terms of the Bloch functions of the same baud by where rn is a lattice point.

(a) Prove that Wannier functions about different lattice points n, m are orthogonal:

f dV w*(r ? r) w (r - r_m) = 0, n ? m. (43). This orthogonality property makes the functions often of greater use than atomic orbital?s centered on different lattice sites, because the Tatter are not generally orthogonal.?

(b) The Wannier functions are peaked around the lattice sites. Show that for ?k = N^-1/2 e^lkxu₀(x) the Wannier function is for N atoms on a line of lattice constant a.

Transcribed Image Text:

w(r — r„) = N− ¹/² Σ exp( — ik · r₂) 4k(r) . sin 7(x-x₂)/a π(x-x₁)/a w(x-x₁) = u(x)