Angular momentum operator

{wiki}

= Quantum angular momentum
{synonym}
{title2}

Basically the operators are just analogous to the classical ones e.g. the classical:
$$
L_z = x p_y - y p_x
$$
becomes:
$$
\hat{L}_z = -i \hbar \left( x\pdv{}{y} - y\pdv{}{x} \right)
$$

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 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[https://youtube.com/watch?v=NwbvTa2xV9k]
{title=Quantum Mechanics 7a - Angular Momentum I by <ViaScience> (2013)}
