If B A ψ and A B ψ are defined, then by subtracting the means and re-inserting in the above formula, we deduce Δ Δ ψ ψ A ^ ^ Δ Δ ψ ψ B ^ ^ ≥ ≥ 1 2 | ⟨ [ A ^ ^ , B ^ ^ ] ⟩ ψ ψ | where ⟨ X ^ ^ ⟩ ψ ψ = ⟨ ψ ψ | X ^ ^ | ψ ψ ⟩ is the operator mean of observable X in the system state ψ and Δ Δ ψ ψ X ^ ^ = ⟨ ⟨ X ^ ^ 2 ⟩ ⟩ ψ ψ − − ⟨ ⟨ X ^ ^ ⟩ ⟩ ψ ψ 2 .