We know that for a position variable $x$ and momentum $p$, the uncertainties of the two quantities are bounded by
$$\Delta x \Delta p \gtrsim \hbar$$
Now, this is usually first explained with $x$ being a simple linearly measured position and $p$ being linear momentum. But it should apply to any good coordinate and its conjugate momentum. It should, for instance, apply to angle $\phi$ about the $z$ axis, and angular momentum $L_z$:
$$\Delta \phi \Delta L_z \gtrsim \hbar$$
The thing is, $\Delta \phi$ can never be greater than $2\pi$. I mean, you have to have some value of $\phi$ and $\phi$ only runs from 0 to $2\pi$. Therefore
$$\Delta L_z \gtrsim \hbar/\Delta \phi \geq \hbar/2\pi$$
But, uh-oh! This means it is impossible for $\Delta L_z$ to be zero, and we should never be able to have angular momentum states with definite $L_z$ values.
Of course, it doesn't mean that. But I have never figured out how this is not in contradiction with the Schroedinger eqn. calculations that give us states with definite values of $L_z$. Can anyone help me out?
One answer I anticipate is that $\phi$ is sort of "abstract" in that if you chose your origin at some other point you will get completely different values of $\phi$ and $L_z$, and ipso facto, usual considerations don't apply. I don't think this will work, though. Consider a "quantum bead" sliding around on a rigid circular ring and you get the exact same problem with no ambiguity in $\phi$ or $L_z$. (Well, there will be some limited ambiguity in $\phi$, but still, there won't be in $L_z$.)