Editing Uncertainty principle (section)

===Examples===

Since the Robertson and Schrödinger relations are for general operators, the relations can be applied to any two observables to obtain specific uncertainty relations. A few of the most common relations found in the literature are given below.
* '''Position–linear momentum uncertainty relation''': for the position and linear momentum operators, the canonical commutation relation <math>[\hat{x}, \hat{p}] = i\hbar</math> implies the Kennard inequality from above: <math display="block">\sigma_x \sigma_p \geq \frac{\hbar}{2}.</math>
* '''Angular momentum uncertainty relation''': For two orthogonal components of the [[angular momentum|total angular momentum]] operator of an object: <math display="block">\sigma_{J_i} \sigma_{J_j} \geq \frac{\hbar}{2} \big|\langle J_k\rangle\big|,</math> where ''i'', ''j'', ''k'' are distinct, and ''J''<sub>''i''</sub> denotes angular momentum along the ''x''<sub>''i''</sub> axis. This relation implies that unless all three components vanish together, only a single component of a system's angular momentum can be defined with arbitrary precision, normally the component parallel to an external (magnetic or electric) field. Moreover, for <math>[J_x, J_y] = i \hbar \varepsilon_{xyz} J_z</math>, a choice <math>\hat{A} = J_x</math>, <math>\hat{B} = J_y</math>, in angular momentum multiplets, ''ψ'' = |''j'', ''m''⟩, bounds the [[Casimir invariant]] (angular momentum squared, <math>\langle J_x^2+ J_y^2 + J_z^2 \rangle</math>) from below and thus yields useful constraints such as {{nobr|''j''(''j'' + 1) ≥ ''m''(''m'' + 1)}}, and hence ''j'' ≥ ''m'', among others.

* For the number of electrons in a [[superconductor]] and the [[Phase factor|phase]] of its [[Ginzburg–Landau theory|Ginzburg–Landau order parameter]]<ref>{{Citation |last=Likharev |first=K. K. |author2=A. B. Zorin |title=Theory of Bloch-Wave Oscillations in Small Josephson Junctions |journal=J. Low Temp. Phys. |volume=59 |issue=3/4 |pages=347–382 |year=1985 |doi=10.1007/BF00683782 |bibcode=1985JLTP...59..347L|s2cid=120813342 }}</ref><ref>{{Citation |first=P. W. |last=Anderson |editor-last=Caianiello |editor-first=E. R. |contribution=Special Effects in Superconductivity |title=Lectures on the Many-Body Problem, Vol. 2 |year=1964 |place=New York |publisher=Academic Press}}</ref> <math display="block"> \Delta N \, \Delta \varphi \geq 1. </math>