Editing Canonical commutation relation (section)

== Uncertainty relation for angular momentum operators ==
For the angular momentum operators {{math|1=''L''<sub>''x''</sub> = ''y p<sub>z</sub>'' − ''z p<sub>y</sub>''}}, etc., one has that
<math display="block"> [{L_x}, {L_y}] = i \hbar \epsilon_{xyz} {L_z}, </math>
where <math>\epsilon_{xyz}</math> is the [[Levi-Civita symbol]] and simply reverses the sign of the answer under pairwise interchange of the indices. An analogous relation holds for the [[Spin (physics)|spin]] operators.

Here, for {{mvar|L<sub>x</sub>}} and {{mvar|L<sub>y</sub> }},<ref name="robertson" /> in angular momentum multiplets {{math|1=''ψ'' = {{!}}''{{ell}}'',''m''{{rangle}}}}, one has, for the transverse components of the [[Casimir invariant]] {{math|''L<sub>x</sub>''<sup>2</sup> + ''L<sub>y</sub>''<sup>2</sup>+ ''L<sub>z</sub>''<sup>2</sup>}}, the {{mvar|z}}-symmetric relations
:{{math|1={{langle}}''L<sub>x</sub>''<sup>2</sup>{{rangle}} = {{langle}}''L<sub>y</sub>''<sup>2</sup>{{rangle}} = (''{{ell}}'' (''{{ell}}'' + 1) − ''m''<sup>2</sup>) ℏ<sup>2</sup>/2 }},
as well as {{math|1={{langle}}''L<sub>x</sub>''{{rangle}} = {{langle}}''L<sub>y</sub>''{{rangle}} = 0 }}.

Consequently, the above inequality applied to this commutation relation specifies
<math display="block">\Delta L_x \, \Delta L_y \geq \frac{1}{2} \sqrt{\hbar^2|\langle L_z \rangle|^2}~, </math>
hence
<math display="block">\sqrt {|\langle L_x^2\rangle \langle L_y^2\rangle |} \geq \frac{\hbar^2}{2} \vert m\vert</math>
and therefore
<math display="block">\ell(\ell+1)-m^2\geq |m| ~,</math>
so, then, it yields useful constraints such as a lower bound on the [[Casimir invariant]]: {{math|''{{ell}}'' (''{{ell}}'' + 1) &ge; {{pipe}}''m''{{pipe}} ({{pipe}}''m''{{pipe}} + 1)}}, and hence {{math|''{{ell}}'' &ge; {{pipe}}''m''{{pipe}}}}, among others.