For the wave functions, recall that p ^ ^ y commutes with the Hamiltonian. Then the wave function factors into a product of momentum eigenstates in the y direction and harmonic oscillator eigenstates | ϕ ϕ n ⟩ ⟩ shifted by an amount x 0 in the x direction: Ψ Ψ ( x , y , z ) = e i ( k y y + k z z ) ϕ ϕ n ( x − − x 0 ) where k z = p z / ℏ ℏ . In sum, the state of the electron is characterized by the quantum numbers, n, k y and k z.