Uncertainty relations of angular momentum and the angular position

Recently I happened to read an article on the bbs ( bulletin board system ) about the uncertainty principle of the angular momentum and the angular position. The author was confused by some linkage between the quantum and classical limits, which is not so interesting to me. However, his confusion, at some point, brought me to reconsider the uncertainty relations between the angular momentum ( more precisely, Lz) and the angular position ( more precisely, the azimuthal angle ). First, there is an upper limit for the angle. This makes angle very different from the position in translational sense, which can span all over the space to infinity. Second, the angular momentum operator is no more Hermitian on the state \hat{phi}|n> where |n> is the eigenstate of Lz and \hat{phi} is the "physical" angle operator. There are two ways to get around it. One is to keep it as usual but carefully trace the boundary terms: Don't miss it! The other is to create an periodic angle operator which keep Lz always Hermitian. The latter approach seems more rigirous. However, the former way is very physically meaningful. I trid to find some research papers that discussed this issue. The most original seems to be written by Dr. Judge. He adopt the first approach. The report is only one page long. Excellent! I like a short article because I am not that patient. This is very interesting paper.
**** D. Judge, Phys. Lett. 5, 189 (1963)***

I read it and took notes below.
http://www.phys.ufl.edu/~chungwei/paper/uncertainty_angular.pdf