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 ( 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.
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An angular momentum operator generates rotations of a quantum state. Its three components satisfyand cyclic permutations. Since distinct components do not commute, a state generally cannot have definite values of all three at once. The total commutes with every component, allowing simultaneous eigenstates of and one chosen projection.
Their eigenvalues are and , where . The quantization axis fixes which projection is labeled. Orbital angular momentum has integer for a scalar spatial wavefunction, while spin can have half-integer . Combining electronic and nuclear angular momenta explains hyperfine structure; a Stern-Gerlach experiment makes discrete magnetic projections visible by spatially separating different responses to a field gradient.
In quantum mechanics, the angular momentum operator is an important operator that describes the angular momentum of a quantum system, similar to how the linear momentum operator describes the linear momentum. Angular momentum is a key concept in both classical and quantum physics, and it plays a crucial role in the behavior of atomic and subatomic particles. There are several forms of angular momentum in quantum mechanics, including orbital angular momentum and spin angular momentum.