For example : $[\hat{x},\hat{p}] = i \hbar \hat{I}$ and $\Delta x \Delta p \ge \hbar/2$ but in case of number states $|n \rangle $ $$[\hat{C},\hat{n}] = i \hat{S}\\ \Delta C \Delta n \ge 0 $$ where $$\hat{C}=\frac{1}{2} [\hat{E}+\hat{E^\dagger}] \\ \hat{S}=\frac{1}{2i} [\hat{E}-\hat{E^\dagger}]\\ \hat{E}=\bigg( \frac{1}{\sqrt{\hat{a} \hat{a^\dagger}}} \bigg) \hat{a}\\ \hat{E^\dagger}=\hat{a}\bigg( \frac{1}{\sqrt{\hat{a} \hat{a^\dagger}}}\bigg) $$
and $\hat{a^\dagger} (\hat{a})$ are the creation(annhilation) operator of a Quantum Harmonic Oscillator. The C(S) are cos(sin) observables of quantum phase of a Quantised EM Field. The $\hat{E}(\hat{E^\dagger})$ are Susskind–Glogower (SG) phase operators. [Ref : GerryKnight Quantum Optics Ch. 2 Sec 7.]
When is an uncertainty product greater than zero and what does it signifiy?
