If N is the number of ions in the lattice, then we have the relation: aN = L. Replacing in the boundary condition and applying Bloch's theorem will result in a quantization for k : ψ ψ ( 0 ) = e i k ⋅ ⋅ 0 u ( 0 ) = e i k L u ( L ) = ψ ψ ( L ) u ( 0 ) = e i k L u ( L ) = e i k L u ( N a ) → → e i k L = 1 ⇒ ⇒ k L = 2 π π n → → k = 2 π π L n ( n = 0 , ± ± 1 , … … , ± ± N 2 ) .