for q = 1, 0, −1, where q appears explicitly as a magnetic quantum number. This equation reveals a relationship between vector operators and the angular momentum value ℓ = 1, something we will have more to say about presently. Now the matrix elements become a product of a radial integral times an angular integral, ⟨ ⟨ n ′ ℓ ℓ ′ m ′ | r q | n ℓ ℓ m ⟩ ⟩ = ( ∫ ∫ 0 ∞ ∞ r 2 d r R n ′ ℓ ℓ ′ ∗ ∗ ( r ) r R n ℓ ℓ ( r ) ) ( 4 π π 3 ∫ ∫ sin ⁡ ⁡ ( θ θ ) d Ω Ω Y ℓ ℓ ′ m ′ ∗ ∗ ( θ θ , ϕ ϕ ) Y 1 q ( θ θ , ϕ ϕ ) Y ℓ ℓ m ( θ θ , ϕ ϕ ) )