Basically the operators are just analogous to the classical ones e.g. the classical:
becomes:

$L_{z}=xp_{y}−yp_{x}$

$L^_{z}=−iℏ(x∂y∂ −y∂x∂ )$

Besides the angular momentum in each direction, we also have the total angular momentum:

$L^_{2}=L^_{x}+L^_{y}+L^_{z}$

Then you have to understand what each one of those does to the each atomic orbital:

- total angular momentum: determined by the azimuthal quantum number
- angular momentum in one direction ($z$ by convention): determined by the magnetic quantum number

There is an uncertainty principle between the x, y and z angular momentums, we can only measure one of them with certainty at a time. Video 1. "Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013)" justifies this intuitively by mentioning that this is analogous to precession: if you try to measure electrons e.g. with the Zeeman effect the precess on the other directions which you end up modifing.

TODO experiment. Likely Zeeman effect.