Is there an angular momentum-angle uncertainty relation in Quantum Mechanics?

Question

In classical mechanics, for each generalized coordinate and conjugate momentum to each coordinate, there is the Poisson bracket identity $\{q_i,p_j\}=\delta_{ij}$. Let's choose polar coordinates with each point as $(r,\theta)$, and the conjugate momentum to $r$ is $p_r$ and the conjugate momentum to $\theta$ is $L$. Now, in the transition to quantum mechanics we replace the dynamical variables by Hermitian operators, and the values of the commutators are equal to the values of the Poisson brackets multiplied by $i \hbar$. My question is: it is possible to build a well-behaved angle operator $\hat{\theta}$ such that $[\hat{\theta},\hat{L}]= i\hbar$ and therfore has uncertainty relation $\sigma_{\theta}\sigma_L\geq\frac{\hbar}{2}$? Does an angular momentum-angle uncertainty relation exist?

Answer 1

Short answer is, no there is no well-behaved angle operator $\hat \theta$ such that $[\hat{\theta},\hat{L}]= i\hbar$.

Since there are eigenstates of the angular momentum $L$ the uncertainty relation $\sigma_{\theta}\sigma_L\geq\frac{\hbar}{2}$ that would ensue implies that $\sigma_{\theta}$ should become infinite for such an eigenstate (just like $\sigma_{p}$ is infinite for an eigenstate of $x$ and $\sigma_{x}$ is infinite for an eigenstate of $p$). Which, for a well-behaved angle is impossible.

If a small uncertainty $\sigma_{\theta}$ can indeed be obtained with a large $\sigma_{L}$, because one can describe a "wave packet" well localised in angle by using many harmonics, the converse is not true.

In fact trying to reduce the number of harmonics to get $\sigma_{L}$ small would spread the "wave packet" not just over $2\pi$, but round and round, so that $\sigma_{\theta}$ grows indefinitely. But this, of course, prevents the hoped for $\hat \theta$ from being well-behaved, it has to be multivalued, the same position being represented by $\theta+2k\pi$ for many values of the integer $k$.

See essentially the same discussion here in particular, and more places as per Qmechanic's comment above.