Editing Canonical commutation relation (section)

== Uncertainty relation and commutators ==
All such nontrivial commutation relations for pairs of operators lead to corresponding [[uncertainty principle|uncertainty relations]],<ref name="robertson">{{cite journal |first=H. P. |last=Robertson |title=The Uncertainty Principle |journal=[[Physical Review]] |volume=34 |issue=1 |year=1929 |pages=163–164 |doi=10.1103/PhysRev.34.163 |bibcode = 1929PhRv...34..163R }}</ref> involving positive semi-definite expectation contributions by their respective commutators and anticommutators.  In general, for two [[Self-adjoint operator|Hermitian operators]] {{mvar|A}} and {{mvar|B}}, consider expectation values in a system in the state {{mvar|ψ}}, the variances around the corresponding expectation values being {{math|1=(Δ''A'')<sup>2</sup> &equiv; {{langle}}(''A'' − {{langle}}''A''{{rangle}})<sup>2</sup>{{rangle}}}}, etc.

Then
<math display="block"> \Delta A \, \Delta B \geq \frac{1}{2} \sqrt{ \left|\left\langle\left[{A},{B}\right]\right\rangle \right|^2 + \left|\left\langle\left\{ A-\langle A\rangle ,B-\langle B\rangle \right\} \right\rangle \right|^2} ,</math>
where {{math|1=[''A'', ''B''] &equiv; ''A B'' &minus; ''B A''}} is the [[Commutator#Ring theory|commutator]] of {{mvar|A}} and {{mvar|B}}, and {{math|1={''A'', ''B''} &equiv; ''A B'' + ''B A''}} is the [[anticommutator]].

This follows through use of the [[Cauchy–Schwarz inequality]], since
{{math|{{!}}{{langle}}''A''<sup>2</sup>{{rangle}}{{!}} {{!}}{{langle}}''B''<sup>2</sup>{{rangle}}{{!}} &ge; {{!}}{{langle}}''A B''{{rangle}}{{!}}<sup>2</sup>}}, and {{math|1=''A B'' = ([''A'', ''B''] + {''A'', ''B''})/2 }}; and similarly for the shifted operators {{math|''A'' − {{langle}}''A''{{rangle}}}} and {{math|''B'' − {{langle}}''B''{{rangle}}}}. (Cf. [[uncertainty principle derivations]].)

Substituting for {{mvar|A}} and {{mvar|B}} (and taking care with the analysis) yield Heisenberg's familiar uncertainty relation for {{mvar|x}} and {{mvar|p}}, as usual.