This substituting in the time independent Schrödinger equation with a simple effective Hamiltonian H ^ ^ = − − ℏ ℏ 2 2 m ∇ ∇ 2 + V ( r ) induces a periodicity with the wave function: ψ ψ ( r + ∑ ∑ i N i a i ) = ψ ψ ( r )