Consider the orthogonality relation in (9mlj) (j|qn) = eign-qm)Rj Ns j N-1 eitan-ambia j=0 (qm|j) (jan) = j thereby proving orthogonality. N Na where qn = n7, q = n =n are the discrete wavevectors, N = L/a is the number of sites in the chain and a is the lattice spacing. By substituting R;= ja and treating this expression as a geometric series, show that = = 8nm, (2.145) (1)(N-1)(n-m) sin[(n m)] sin[ (n m)] N - = 8nm,