Today is electricity and magnetism midterm day, so I'm just going to jot down a skeleton of a thought process about the quantum mechanical phase operator research I've been reading for the last few days, and then I have to run.
In matrix rperesentation, the derivative of a polynomial can be represented as:
3. Nieto's article
4. Nieto RMP article
CARRUTHERS P. & NIETO M. (1968). Phase and Angle Variables in Quantum Mechanics, Reviews of Modern Physics, 40 (2) 411-440. DOI: 10.1103/RevModPhys.40.411
5. Discrete Fourier Transform