Robertson-Schrödinger uncertainty relation ($(\Delta A{)}^2(\Delta B{)}^2-C_{AB}^2\ge |⟨[A,B]⟩{|}^2/4$)

The Robertson-Schrödinger uncertainty relation keeps the real part discarded in the proof of the Robertson uncertainty principle. Write $A^{\prime }=A-⟨A⟩I$ and $B^{\prime }=B-⟨B⟩I$. Their quantum covariance is $C_{AB}=\mathrm{Re}⟨A^{\prime }\psi |B^{\prime }\psi ⟩$. On the same finite-moment and product operator domains used above, the Cauchy-Schwarz inequality givesFor position and momentum, form the real, symmetrized covariance matrixA larger coordinatewise product can therefore encode a tilted covariance ellipse, rather than a larger determinant. This explains why the equality case of the Heisenberg uncertainty relation is more restrictive than equality in the Robertson-Schrödinger uncertainty relation.

For self-adjoint operators and an admissible pure state, the quantum covariance is the real, symmetrized second moment of two centered observables. The displayed expression uses their anticommutator; equivalently $C_{AB}=\mathrm{Re}⟨A^{\prime }\psi |B^{\prime }\psi ⟩$, which requires only the individual finite second moments. It reduces to ordinary covariance for commuting observables. For noncommuting observables, its definition does not assert that both have simultaneous sharp values drawn from a joint classical probability distribution.