Angular Momentum and Spin

Abstract

Determine the uncertainty relations between the orbital angular momentum $\hat L = \left( {{{\hat L}_x},{{\hat L}_y},{{\hat L}_z}} \right)$ and the components of the position and of the momentum operators $\hat r = \left( {\hat x,\hat y,\hat z} \right),\hat p = \left( {{{\hat p}_x},{{\hat p}_y},{{\hat p}_z}} \right)$. Then, find the operator ${\hat L_z}$ in spherical polar coordinates and explain why the operators $\hat \phi$ (azimuthal angle) and ${\hat L_z}$ can be measured simultaneously. What are the functions of $\hat \phi$ whose commutator with ${\hat L_z}$ has a physical sense?