Incoming links: Azimuthal quantum number
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.
Refinement of the Bohr model that starts to take quantum angular momentum into account in order to explain missing lines that would have been otherwise observed TODO specific example of such line.
They are not observe because they would violate the conservation of angular momentum.
Introduces the azimuthal quantum number and magnetic quantum number.
Looking at the energy level of the Schrödinger equation solution for the hydrogen atom, you would guess that for multi-electron atoms that only the principal quantum number would matter, azimuthal quantum number getting filled randomly.
However, orbitals energies for large atoms don't increase in energy like those of hydrogen due to electron-electron interactions.
In particular, the following would not be naively expected:
This rule is only an approximation, there exist exceptions to the Madelung energy ordering rule.
Fixed quantum angular momentum in a given direction.
Can range between .
The z component of the quantum angular momentum is simply:so e.g. again for gallium:
- s-orbitals: necessarily have 0 z angular momentum
- p-orbitals: have either 0, or z angular momentum
Note that this direction is arbitrary, since for a fixed azimuthal quantum number (and therefore fixed total angular momentum), we can only know one direction for sure. is normally used by convention.
Quantum numbers appear directly in the Schrödinger equation solution for the hydrogen atom.
However, it very cool that they are actually discovered before the Schrödinger equation, and are present in the Bohr model (principal quantum number) and the Bohr-Sommerfeld model (azimuthal quantum number and magnetic quantum number) of the atom. This must be because they observed direct effects of those numbers in some experiments. TODO which experiments.
E.g. The Quantum Story by Jim Baggott (2011) page 34 mentions:This refers to forbidden mechanism. TODO concrete example, ideally the first one to be noticed. How can you notice this if the energy depends only on the principal quantum number?
As the various lines in the spectrum were identified with different quantum jumps between different orbits, it was soon discovered that not all the possible jumps were appearing. Some lines were missing. For some reason certain jumps were forbidden. An elaborate scheme of ‘selection rules’ was established by Bohr and Sommerfeld to account for those jumps that were allowed and those that were forbidden.
This notation is cool as it gives the spin quantum number, which is important e.g. when talking about hyperfine structure.
But it is a bit crap that the spin is not given simply as but rather mixes up both the azimuthal quantum number and spin. What is the reason?
Spectroscopic notation by Andre K (2014)
Source. Non-anomalous: number of splits matches predictions of the Schrödinger equation about the number of possible states with a given angular momentum. TODO does it make numerical predictions?
www.pas.rochester.edu/~blackman/ast104/zeeman-split.html contains the hello world that everyone should know: 2p splits into 3 energy levels, so you see 3 spectral lines from 1s to 2p rather than just one.
It also mentions that polarization effects become visible from this: each line is polarized in a different way. TODO more details as in an experiment to observe this.
Well explained at: Video "Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013)".
Experimental physics - IV: 22 - Zeeman effect by Lehrportal Uni Gottingen (2020)
Source. This one is decent. Uses a cadmium lamp and an etalon on an optical table. They see a more or less clear 3-split in a circular interference pattern,
They filter out all but the transition of interest.
- youtu.be/ZmObNFAqkBE?t=165 passes the lines through a polarizer, which shows how orbital angular momentum is carried by photon polarization
- youtu.be/ZmObNFAqkBE?t=370 says they are looking at 1D2 to 1P1 changes.