= Robertson uncertainty principle
{c}
{title2=$\Delta A\,\Delta B\geq\tfrac12|\langle[A,B]\rangle|$}
= Robertson uncertainty relation
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{synonym}
The <Robertson uncertainty principle> bounds the product of the <standard deviations> of two <observables> in the same <quantum state>. For a normalized <pure state> $|\psi\rangle$ and <self-adjoint operators> $A,B$, write their <expectation values> and <variances> as
$$
\langle A\rangle=\langle\psi|A|\psi\rangle,\qquad (\Delta A)^2=\|(A-\langle A\rangle I)\psi\|^2=\langle A^2\rangle-\langle A\rangle^2,
$$
and similarly for $B$, where $I$ is the <identity operator>. The <quantum uncertainties> $\Delta A,\Delta B$ then obey
$$
\Delta A\,\Delta B\geq\frac12|\langle[A,B]\rangle|,\qquad [A,B]=AB-BA.
$$
Here $[A,B]$ is the <commutator>, and every <expectation value> is taken in the same $|\psi\rangle$. These <standard deviations> describe the outcome spreads when $A$ and $B$ are measured on separate ensembles prepared in that same <quantum state>. They are not instrument errors or the change caused by a preceding <measurement in quantum mechanics>. The definition and bound are developed in https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/005979fa741c3ea2e0430456b70caf93_MIT8_05F13_Chap_05.pdf[Barton Zwiebach, Uncertainty Principle and Compatible Observables, 2013, §§1–2].
For <unbounded self-adjoint operators>, the <operator domains> matter. Assume $\psi\in D(A)\cap D(B)$, so that both second moments are finite; $\langle A^2\rangle$ here means the second moment of the measured <probability distribution>, equal to $\|A\psi\|^2$, even if $A^2\psi$ is undefined. To use the displayed operator <commutator>, also assume $\psi\in D(AB)\cap D(BA)$. Without these product-domain assumptions, the always meaningful version on $D(A)\cap D(B)$ uses $|\operatorname{Im}\langle A\psi|B\psi\rangle|$ as its right-hand side. These qualifications prevent a formal <commutator> identity from being applied to an inadmissible <quantum state>.
The brief proof is <Cauchy-Schwarz inequality> applied to $u=(A-\langle A\rangle I)\psi$ and $v=(B-\langle B\rangle I)\psi$:
$$
\Delta A\,\Delta B=\|u\|\|v\|\geq|\langle u|v\rangle|\geq|\operatorname{Im}\langle u|v\rangle|=\frac12|\langle[A,B]\rangle|.
$$
The final equality uses <self-adjointness> and the product-domain assumptions. Thus a geometric statement about <inner products> becomes a constraint on experimental spreads. For nonzero $\Delta A,\Delta B$, equality requires $u,v$ to be proportional with a purely imaginary proportionality factor; the <equality case of the Heisenberg uncertainty relation> turns this condition into a <Gaussian wave packet>.
The bound is useful because computing a single <commutator> and its <expectation value> can constrain two entire outcome distributions. Its right-hand side depends on the <quantum state>: the <spin uncertainty state dependence> example gives both a positive saturated bound and a zero bound for the same noncommuting <observables>. Robertson's paper, titled "The Uncertainty Principle", appeared on 1 July 1929 in Physical Review, volume 34, starting at page 163. https://doi.org/10.1103/PhysRev.34.163[H. P. Robertson, The Uncertainty Principle, 1929]{ref}.
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