Quantum mechanics, angular momentum in spherical coordinates

Question

I've seen many times the angular momentum operators $\hat L_x, \hat L_y,\hat L_z$ expressed in spherical coordinates but I've never seen the components operators $\hat L_\rho, \hat L_\theta, \hat L_\phi$. What is their expression and what Is their physical interpretation?

Answer 1

That's a hard question. The cartesian components make sense because they're used to calculate the global angular momentum (i.e. over the full spatial extent of the wavefunction), referred to a universal coordinate systems. In contrast, the spherical components $\hat L_\rho, \hat L_\theta, \hat L_\phi$ are only defined locally at each point (and change from point to point), so they would only make sense for a local angular momentum, which is very different and not necessarily well-defined.

That's not to say that it can't be done, but it's the reason you haven't seen it $-$ it's a lot of work, and you need to have a clear reason to go through the hassle. And, to be honest, there isn't a reason that I can see.