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.

Video 18. Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013) Source.

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