This is true for λ λ n = e s n ⋅ ⋅ a where s ∈ ∈ C if we use the normalization condition over a single primitive cell of volume V 1 = ∫ ∫ V | ψ ψ ( x ) | 2 d x = ∫ ∫ V | T ^ ^ n ψ ψ ( x ) | 2 d x = | λ λ n | 2 ∫ ∫ V | ψ ψ ( x ) | 2 d x and therefore 1 = | λ λ n | 2 and s = i k where k ∈ ∈ R . Finally, T ^ ^ n ψ ψ ( x ) = ψ ψ ( x + n ⋅ ⋅ a ) = e i k n ⋅ ⋅ a ψ ψ ( x ) , which is true for a Bloch wave i.e. for ψ ψ k ( x ) = e i k ⋅ ⋅ x u k ( x ) with u k ( x ) = u k ( x + A n )