The commutators needed to proceed are [ A ± ± , L z ] = ∓ ∓ i ℏ ℏ A ∓ ∓ and [ A ± ± , L 2 ] = ∓ ∓ 2 ℏ ℏ 2 A ± ± − − 2 ℏ ℏ A ± ± L z ± ± 2 ℏ ℏ A z L ± ± . Therefore, A + | ? , ℓ ℓ , m ℓ ℓ ⟩ ⟩ → → | ? , ℓ ℓ , m ℓ ℓ + 1 ⟩ ⟩ and − − L 2 ( A + | ? , ℓ ℓ , ℓ ℓ ⟩ ⟩ ) = − − ℏ ℏ 2 ( ℓ ℓ + 1 ) ( ( ℓ ℓ + 1 ) + 1 ) ( A + | ? , ℓ ℓ , ℓ ℓ ⟩ ⟩ ) , so A + | ? , ℓ ℓ , ℓ ℓ ⟩ ⟩ → → | ? , ℓ ℓ + 1 , ℓ ℓ + 1 ⟩ ⟩ , where the "?" indicates a nascent quantum number which emerges from the discussion.