Assuming that ψ ψ n is an eigenstate of the Hamiltonian H ^ ^ ψ ψ n = E n ψ ψ n . Using these commutation relations, it follows that H ^ ^ a ψ ψ n = ( E n − − ℏ ℏ ω ω ) a ψ ψ n . H ^ ^ a † † ψ ψ n = ( E n + ℏ ℏ ω ω ) a † † ψ ψ n .