A suitable factorization is given by C l = p r + i ℏ ℏ ( l + 1 ) r − − i μ μ ω ω r with F l = − − ( 2 l + 3 ) μ μ ω ω ℏ ℏ and G l = − − ( 2 l + 1 ) μ μ ω ω ℏ ℏ . Then E l + 1 n ′ = E l n + F l − − G l 2 μ μ = E l n − − ω ω ℏ ℏ , and continuing this, E l + 2 n ′ = E l n − − 2 ω ω ℏ ℏ E l + 3 n ′ = E l n − − 3 ω ω ℏ ℏ ⋮ ⋮ Now the Hamiltonian only has positive energy levels as can be seen from ⟨ ⟨ ψ ψ | 2 μ μ H l | ψ ψ ⟩ ⟩ = ⟨ ⟨ ψ ψ | C l ∗ ∗ C l | ψ ψ ⟩ ⟩ + ⟨ ⟨ ψ ψ | ( 2 l + 3 ) μ μ ω ω ℏ ℏ | ψ ψ ⟩ ⟩ = ⟨ ⟨ C l ψ ψ | C l ψ ψ ⟩ ⟩ + ( 2 l + 3 ) μ μ ω ω ℏ ℏ ⟨ ⟨ ψ ψ | ψ ψ ⟩ ⟩ ≥ ≥ 0. This means that for some value of l the series must terminate with C l max | n l max ⟩ ⟩ = 0 , and then E l max n = − − F l max 2 μ μ = ( l max + 3 2 ) ω ω ℏ ℏ . This is decreasing in energy by ω ω ℏ ℏ unless C l | n , l ⟩ ⟩ = 0 for some value of l . Identifying this value as n gives E l n = − − F l = ( n + 3 2 ) ω ω ℏ ℏ .