To simplify things, the solution V ( z ) is considered as an infinite well. Thus the solutions for the z direction are the energies ε ε z = n z 2 π π 2 ℏ ℏ 2 2 m ∗ ∗ L 2 , n z = 1 , 2 , 3... and the wavefunctions are sinusoidal. For the x and y directions, the solution of the Schrödinger equation can be chosen to be the product of a plane wave in y -direction with some unknown function of x , i.e., ψ ψ x y = u ( x ) e i k y y . This is because the vector potential does not depend on y and the momentum operator p ^ ^ y therefore commutes with the Hamiltonian. By substituting this Ansatz into the Schrödinger equation one gets the one-dimensional harmonic oscillator equation centered at x k y = ℏ ℏ k y e B .