The derivation shown here incorporates and builds off of those shown in Robertson, Schrödinger and standard textbooks such as Griffiths. For any Hermitian operator A ^ ^ , based upon the definition of variance, we have σ σ A 2 = ⟨ ⟨ ( A ^ ^ − − ⟨ ⟨ A ^ ^ ⟩ ⟩ ) Ψ Ψ | ( A ^ ^ − − ⟨ ⟨ A ^ ^ ⟩ ⟩ ) Ψ Ψ ⟩ ⟩ . we let | f ⟩ ⟩ = | ( A ^ ^ − − ⟨ ⟨ A ^ ^ ⟩ ⟩ ) Ψ Ψ ⟩ ⟩ and thus σ σ A 2 = ⟨ ⟨ f ∣ ∣ f ⟩ ⟩ .