To obtain the values of α and β, first take the norm of each operator, recognizing that J + and J − are a Hermitian conjugate pair ( J ± ± = J ∓ ∓ † † ): ⟨ ⟨ j m | J + † † J + | j m ⟩ ⟩ = ⟨ ⟨ j m | J − − J + | j m ⟩ ⟩ = ⟨ ⟨ j ( m + 1 ) | α α ∗ ∗ α α | j ( m + 1 ) ⟩ ⟩ = | α α | 2 , ⟨ ⟨ j m | J − − † † J − − | j m ⟩ ⟩ = ⟨ ⟨ j m | J + J − − | j m ⟩ ⟩ = ⟨ ⟨ j ( m − − 1 ) | β β ∗ ∗ β β | j ( m − − 1 ) ⟩ ⟩ = | β β | 2 .