structural:reformatting, structural:summarization
Uncertainty relation with three angular momentum components

For a particle of spin- j {\displaystyle j} the following uncertainty relation holds σ J x 2 + σ J y 2 + σ J z 2 ≥ j, {\displaystyle \sigma _{J_{x}}^{2}+\sigma _{J_{y}}^{2}+\sigma _{J_{z}}^{2}\geq j,} where J l {\displaystyle J_{l}} are angular momentum components. The relation can be derived from ⟨ J x 2 + J y 2 + J z 2 ⟩ = j (j + 1), {\displaystyle \langle J_{x}^{2}+J_{y}^{2}+J_{z}^{2}\rangle =j(j+1),} and ⟨ J x ⟩ 2 + ⟨ J y ⟩ 2 + ⟨ J z ⟩ 2 ≤ j. {\displaystyle \langle J_{x}\rangle ^{2}+\langle J_{y}\rangle ^{2}+\langle J_{z}\rangle ^{2}\leq j.} The relation can be strengthened as σ J x 2 + σ J y 2 + F Q / 4 ≥ j, {\displaystyle \sigma _{J_{x}}^{2}+\sigma _{J_{y}}^{2}+F_{Q}/4\geq j,} where F Q {\displaystyle F_{Q}} is the quantum Fisher information.