When we study a single particle moving on a ring, we introduce the angular momentum $L$ and the angular coordinate $\phi$, which satisfy the following commutation relation $$ [L,\phi]=-i\hbar. $$ However, we also say $\phi$ itself is not a well-defined operator because $\phi$ and $\phi + 2\pi$ should be identified. One way to see this illness of $\phi$ is to consider the following expectation value of the above commutator $$ \langle n|[L,\phi]| n \rangle = -i\hbar = 0 $$ where $|n\rangle$ is an eigenstate of the angular momentum $L$.
My question is how we properly understand and use the commutation relation? (Similar things appear the study of other models involving compact variables as well)
