Is the only atom that has a closed form solution, which allows for very good predictions, and gives awesome intuition about the orbitals in general.
It is arguably the most important solution of the Schrodinger equation.
In particular, it predicts:
The explicit solution can be written in terms of spherical harmonics.
Video 1. A Better Way To Picture Atoms by minutephysics (2021) Source. Renderings based on the exact Schrödinger equation solution for the hydrogen atom that depict wave function concentration by concentration of small balls, and angular momentum by how fast the balls rotate at each point. Mentions that the approach is inspired by de Broglie-Bohm theory.
In the case of the Schrödinger equation solution for the hydrogen atom, each orbital is one eigenvector of the solution.
Remember from time-independent Schrödinger equation that the final solution is just the weighted sum of the eigenvector decomposition of the initial state, analogously to solving partial differential equations with the Fourier series.
This is the table that you should have in mind to visualize them:
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?
Video 1. Quantum Numbers, Atomic Orbitals, and Electron configurations by Professor Dave Explains (2015) Source. He does not say the key words "Eigenvalues of the Schrödinger equation" (Which solve it), but the summary of results is good enough.
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.
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 .
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 , and so the only value for is 0
  • the electrons in p-orbitals such as 2p, 3p and 4p are , and so the possible values for are -1, 0 and 1
  • the electrons in d-orbitals such as 2d are , and so the possible values for are -2, -1, 0 and 1 and 2
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.
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?

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