and C l C l ∗ ∗ = 2 μ μ H l + 1 + G l for scalars F l and G l . The vector C l C l ∗ ∗ C l | n l ⟩ ⟩ may be evaluated in two different ways as C l C l ∗ ∗ C l | n l ⟩ ⟩ = ( 2 μ μ E l n + F l ) C l | n l ⟩ ⟩ = ( 2 μ μ H l + 1 + G l ) C l | n l ⟩ ⟩ , which can be re-arranged as H l + 1 ( C l | n l ⟩ ⟩ ) = [ E l n + ( F l − − G l ) / ( 2 μ μ ) ] ( C l | n l ⟩ ⟩ ) , showing that C l | n l ⟩ ⟩ is an eigenstate of H l + 1 with eigenvalue E l + 1 n ′ = E l n + ( F l − − G l ) / ( 2 μ μ ) . If F l = G l , then n ′ = n , and the states | n l ⟩ ⟩ and C l | n l ⟩ ⟩ have the same energy.