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:

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 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?

Determines energy. This comes out directly from the resolution of the Schrödinger equation solution for the hydrogen atom where we have to set some arbitrary values of energy by separation of variables just like we have to set some arbitrary numbers when solving partial differential equations with the Fourier series. We then just happen to see that only certain integer values are possible to satisfy the equations.

Fixed total angular momentum.

The direction however is not specified by this number.

To determine the quantum angular momentum, we need the magnetic quantum number, which then selects which orbital exactly we are talking about.

Fixed quantum angular momentum in a given direction.

Can range between $±l$.

E.g. consider gallium which is 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p1:

- the electrons in s-orbitals such as 1s, 2d, and 3d are $l=0$, and so the only value for $m_{l}$ is 0
- the electrons in p-orbitals such as 2p, 3p and 4p are $l=1$, and so the possible values for $m_{l}$ are -1, 0 and 1
- the electrons in d-orbitals such as 2d are $l=2$, and so the possible values for $m_{l}$ are -2, -1, 0 and 1 and 2

The z component of the quantum angular momentum is simply:
so e.g. again for gallium:

$L_{z}=m_{l}ℏ$

- s-orbitals: necessarily have 0 z angular momentum
- p-orbitals: have either 0, $−ℏ$ or $+ℏ$ 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. $z$ is normally used by convention.

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 $±1/2$ but rather mixes up both the azimuthal quantum number and spin. What is the reason?