Physic Labs
Step 1 of 4: Uncertainty = norm of the fluctuation
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(ΔA)2=⟨ψ∣A^2∣ψ⟩−⟨ψ∣A^∣ψ⟩2=∥ (A^−⟨A^⟩)ψ ∥2(\Delta A)^2 = \langle\psi|\hat A^2|\psi\rangle - \langle\psi|\hat A|\psi\rangle^2 = \|\,(\hat A - \langle\hat A\rangle)\psi\,\|^2
Analysis

For every observable AA, the uncertainty ΔA\Delta A is the standard deviation: its square is the squared norm of the state vector after applying the 'deviation from mean' operator A^−⟨A^⟩\hat A - \langle\hat A\rangle. Notation: set ∣f⟩=(A^−⟨A^⟩)∣ψ⟩|f\rangle = (\hat A - \langle\hat A\rangle)|\psi\rangle and ∣g⟩=(B^−⟨B^⟩)∣ψ⟩|g\rangle = (\hat B - \langle\hat B\rangle)|\psi\rangle; then (ΔA)2=⟨f∣f⟩(\Delta A)^2 = \langle f|f\rangle and (ΔB)2=⟨g∣g⟩(\Delta B)^2 = \langle g|g\rangle.