The angular momentum operator is L = r × × p and obeys the canonical quantization relations [ L i , L j ] = i ℏ ℏ ϵ ϵ i j k L k defining the Lie algebra for so(3), where ϵ ϵ i j k is the Levi-Civita symbol. Under gauge transformations, the angular momentum transforms as ⟨ ⟨ ψ ψ | L | ψ ψ ⟩ ⟩ → → ⟨ ⟨ ψ ψ ′ ′ | L ′ ′ | ψ ψ ′ ′ ⟩ ⟩ = ⟨ ⟨ ψ ψ | L | ψ ψ ⟩ ⟩ + q ℏ ℏ c ⟨ ⟨ ψ ψ | r × × ∇ ∇ Λ Λ | ψ ψ ⟩ ⟩ .