Plugging this into the above inequalities, we get σ σ x 2 σ σ p 2 ≥ ≥ | ⟨ ⟨ f ∣ ∣ g ⟩ ⟩ | 2 ≥ ≥ ( ⟨ ⟨ f ∣ ∣ g ⟩ ⟩ − − ⟨ ⟨ g ∣ ∣ f ⟩ ⟩ 2 i ) 2 = ( i ℏ ℏ 2 i ) 2 = ℏ ℏ 2 4 or taking the square root σ σ x σ σ p ≥ ≥ ℏ ℏ 2 .