Robertson uncertainty relation for the angular momentum components

[Yan Campo]

TL;DR Summary

I would like any explanation about Robertson the uncertainty relation for the angular momentum components and compatibility between the components

I'm studying orbital angular momentum in the quantum domain, and I've come up with the Robertson uncertainty relation for the components of orbital angular momentum. Therefore, I read that it is necessary to pay attention to the triviality problem, because in the case where the commutator is zero, the product of the standard deviations is zero, so the variance is also zero. This means that we don't have information about one of the observables and, therefore, we don't know the incompatibility between the two, I think. But, I can't see any kind of problem in using the Robertson uncertainty relation in the orbital angular momentum components. Can anyone explain to me, or give me an example about this? I really want to understand.

 

[anuttarasammyak]

I am afraid there is no angle operator such that
$$[\hat{\theta},\hat{L}]=i\hbar$$
to which we apply Roberson uncertainty relation.

 

[physical_chemist]

Yan, the Robertson uncertainty principle is regarding two operator have a common complete set of eigenfunctions, i.e., in such basis both operators are diagonal. This is usually expressed, for example, as

$$\Delta A\Delta B \geq \frac{1}{2}\left | \int \psi^{*}[A,B]\psi d\tau\right |$$

But, in the case of angular momentum components, it does not mean that some of the eigenfunctions of $L_{z}$ cannot also be simultaneous eigenfunctions of $L_{x}$ and $L_{y}$. See the case of $Y_{0}^{0}(\theta,\phi)$ spherical harmonic. In such case, it is allowed to have $\Delta L_{x} = 0$, $\Delta L_{y} = 0$ and $\Delta L_{z} = 0$.