For any two self-adjoint operators A^=A^∗, B^=B^∗ on H and any unit vector ψ in the domain of A^B^ and B^A^, the standard deviations σA=⟨ψ∣(A^−⟨A^⟩)2∣ψ⟩ and σB (expectations taken in state ψ) satisfy σAσB≥21⟨[A^,B^]⟩, where [A^,B^]=A^B^−B^A^.
Why is it true?
Two observables can only be measured with simultaneous perfect precision if their operators commute; the size of the commutator is a direct measure of how incompatible they are. Cauchy–Schwarz turns "these two vectors cannot both be short" into a hard numerical bound, and separating the inner product into its real and imaginary parts is exactly what isolates the commutator (the genuinely quantum part) from the anticommutator (a classical-looking correlation term).
Proof sketch
Step 1 (centered operators). Let ΔA^=A^−⟨A^⟩I and ΔB^=B^−⟨B^⟩I, both still self-adjoint since ⟨A^⟩,⟨B^⟩ are real scalars. By definition σA2=⟨ψ∣ΔA^2∣ψ⟩=∥ΔA^ψ∥2 and likewise σB2=∥ΔB^ψ∥2.
Step 2 (Cauchy–Schwarz). Apply the Cauchy–Schwarz inequality ∣⟨f∣g⟩∣2≤⟨f∣f⟩⟨g∣g⟩ to f=ΔA^ψ and g=ΔB^ψ: ∣⟨ΔA^ψ∣ΔB^ψ⟩∣2≤σA2σB2.
Step 3 (split into real and imaginary parts). Write ⟨ΔA^ψ∣ΔB^ψ⟩=⟨ψ∣ΔA^ΔB^∣ψ⟩. Since ΔA^,ΔB^ are self-adjoint, complex-conjugating flips the operator order: ⟨ΔA^ΔB^⟩∗=⟨ΔB^ΔA^⟩. Hence the anticommutator expectation ⟨{ΔA^,ΔB^}⟩=⟨ΔA^ΔB^⟩+⟨ΔB^ΔA^⟩=2Re⟨ΔA^ΔB^⟩ is real, while the commutator expectation ⟨[ΔA^,ΔB^]⟩=⟨ΔA^ΔB^⟩−⟨ΔB^ΔA^⟩=2iIm⟨ΔA^ΔB^⟩ is purely imaginary. Also [ΔA^,ΔB^]=[A^,B^], since the constant shifts cancel in the commutator.
Step 4 (recombine). By Pythagoras applied to real and imaginary parts, ∣⟨ΔA^ΔB^⟩∣2=(Re⟨ΔA^ΔB^⟩)2+(Im⟨ΔA^ΔB^⟩)2=41∣⟨{ΔA^,ΔB^}⟩∣2+41∣⟨[A^,B^]⟩∣2≥41∣⟨[A^,B^]⟩∣2, dropping the manifestly nonnegative anticommutator term. Combining with Step 2, σA2σB2≥41∣⟨[A^,B^]⟩∣2, and taking square roots (both sides nonnegative) gives σAσB≥21⟨[A^,B^]⟩.