Editing Uncertainty principle (section)

===Uncertainty relation with three angular momentum components===

For a particle of [[total angular momentum]] <math>j</math> the following uncertainty relation holds
<math display="block">
\sigma_{J_x}^2+\sigma_{J_y}^2+\sigma_{J_z}^2\ge j,
</math>
where <math>J_l</math> are angular momentum components. The relation can be derived from
<math display="block">
\langle J_x^2+J_y^2+J_z^2\rangle = j(j+1),
</math>
and
<math display="block">
\langle J_x\rangle^2+\langle J_y\rangle^2+\langle J_z\rangle^2\le j.
</math>
The relation can be strengthened as<ref name="PhysRevResearch21" /><ref>{{cite journal |last1=Chiew |first1=Shao-Hen |last2=Gessner |first2=Manuel |title=Improving sum uncertainty relations with the quantum Fisher information |journal=Physical Review Research |date=31 January 2022 |volume=4 |issue=1 |pages=013076 |doi=10.1103/PhysRevResearch.4.013076|arxiv=2109.06900 |bibcode=2022PhRvR...4a3076C |s2cid=237513883 }}</ref>
<math display="block">
\sigma_{J_x}^2+\sigma_{J_y}^2+F_Q[\varrho,J_z]/4\ge j,
</math>
where <math>F_Q[\varrho,J_z]</math> is the quantum Fisher information.