Angular momentum operator (Quantum angular momentum)

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

L_{z} = x p_{y} - y p_{x}

becomes:

\hat{L}_{z} = - i ℏ (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x})

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

\hat{L}^{2} = \hat{L}_{x} + \hat{L}_{y} + \hat{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 18. "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.

Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013)

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