So precise, so discrete, which makes no sense in classical mechanics!

Has been the leading motivation of the development of quantum mechanics, all the way from the:

Schrödinger equation: major lines predicted, including Zeeman effect, but not finer line splits like fine structure
Dirac equation: explains fine structure 2p spin split due to electron spin/orbit interactions, but not Lamb shift
quantum electrodynamics: explains Lamb shift
hyperfine structure: due to electron/nucleus spin interactions, offers a window into nuclear spin

NIST Atomic Spectra Database

NIST database for spectral line: physics.nist.gov/PhysRefData/ASD/lines_form.html

Let's do a sanity check.

Searching for "H" for hydrogen leads to: physics.nist.gov/cgi-bin/ASD/lines1.pl?spectra=H&limits_type=0&low_w=&upp_w=&unit=1&submit=Retrieve+Data&de=0&format=0&line_out=0&en_unit=0&output=0&bibrefs=1&page_size=15&show_obs_wl=1&show_calc_wl=1&unc_out=1&order_out=0&max_low_enrg=&show_av=2&max_upp_enrg=&tsb_value=0&min_str=&A_out=0&intens_out=on&max_str=&allowed_out=1&forbid_out=1&min_accur=&min_intens=&conf_out=on&term_out=on&enrg_out=on&J_out=on

From there we can see for example the 1 to 2 lines:

1s to 2p: 121.5673644608 nm
1s to 2: 121.56701 nm TODO what does that $2$ mean?
1s to 2s: 121.5673123130200 TODO what does that mean?

We see that the table is sorted from lower from level first and then by upper level second.

So it is good to see that we are in the same 121nm ballpark as mentioned at hydrogen spectral line.

TODO why I can't see 2s to 2p transitions there to get the fine structure?

Forbidden mechanism

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Bibliography:

phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_(Fitzpatrick)/12%3A_Time-Dependent_Perturbation_Theory/12.13%3A_Forbidden_Transitions from Introductory Quantum Mechanics by Richard Fitzpatrick (2020)

Selection rule

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phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9HE_-_Modern_Physics/06%3A_Emission_and_Absorption_of_Photons/6.2%3A_Selection_Rules_and_Transition_Times has some very good mentions:

So it appears that if a hydrogen atom emits a photon, it not only has to transition between two states whose energy difference matches the energy of the photon, but it is restricted in other ways as well, if its mode of radiation is to be dipole. For example, a hydrogen atom in its 3p state must drop to either the n=1 or n=2 energy level, to make the energy available to the photon. The n=2 energy level is 4-fold degenerate, and including the single n=1 state, the atom has five different states to which it can transition. But three of the states in the n=2 energy level have l=1 (the 2p states), so transitioning to these states does not involve a change in the angular momentum quantum number, and the dipole mode is not available.
So what's the big deal? Why doesn't the hydrogen atom just use a quadrupole or higher-order mode for this transition? It can, but the characteristic time for the dipole mode is so much shorter than that for the higher-order modes, that by the time the atom gets around to transitioning through a higher-order mode, it has usually already done so via dipole. All of this is statistical, of course, meaning that in a large collection of hydrogen atoms, many different modes of transitions will occur, but the vast majority of these will be dipole.
It turns out that examining details of these restrictions introduces a couple more. These come about from the conservation of angular momentum. It turns out that photons have an intrinsic angular momentum (spin) magnitude of $ℏ$ , which means whenever a photon (emitted or absorbed) causes a transition in a hydrogen atom, the value of l must change (up or down) by exactly 1. This in turn restricts the changes that can occur to the magnetic quantum number: $m_{l}$ can change by no more than 1 (it can stay the same). We have dubbed these transition restrictions selection rules, which we summarize as:
$Δ l = \pm 1, Δ m_{l} = 0, \pm 1$
(1)

Metastable electron

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A fundamental component of three-level lasers.

As mentioned at youtu.be/_JOchLyNO_w?t=581 from Video "How Lasers Work by Scientized (2017)", they stay in that state for a long time compared to non spontaneous emission of metastable states!

phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9HE_-_Modern_Physics/06%3A_Emission_and_Absorption_of_Photons/6.3%3A_Lasers mentions that they are kept in that excited state due to selection rules.

This is also one of mechanisms behind phosphorescence with triplet states.

Gross hydrogen emission spectrum

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One reasonable and memorable approximation excluding any fine structure is:

E_{n} = - \frac{1 3 . 6 e V}{n ^{2}}

(1)

Equation 1.

Hydrogen spectral series mnemonic

see for example example: hydrogen 1-2 spectral line.

Hydrogen 1-2 spectral line (121.6 nm, 10.2e V)

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Equation "Hydrogen spectral series mnemonic" gives for example from principal quantum number 1 to 2 a difference:

E_{n} = - 13.6 e V [\frac{1}{2 ^{2}} - \frac{1}{1 ^{2}}] = 10.2 e V

(1)

which with Planck-Einstein relation gives about 121.6 nm (

2.47 \times 1 0^{1} 5

Hz), which is a reasonable match with the value of 121.567... from the NIST Atomic Spectra Database.

Hydrogen spectral series

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Kind of a synonym for hydrogen emission spectrum not very clear if fine structure is considered by this term or not.

A line set for hydrogen spectral line.

Formula discovered in 1885, was it the first set to have an empirical formula?

Fine structure

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Split in energy levels due to interaction between electron up or down spin and the electron orbitals.

Numerically explained by the Dirac equation when solving it for the hydrogen atom, and it is one of the main triumphs of the theory.

Fine structure constant ( $α$ )

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Hyperfine structure

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Small splits present in all levels due to interaction between the electron spin and the nuclear spin if it is present, i.e. the nucleus has an even number of nucleons.

As the name suggests, this energy split is very small, since the influence of the nucleus spin on the electron spin is relatively small compared to other fine structure.

TODO confirm: does it need quantum electrodynamics or is the Dirac equation enough?

The most important examples:

hydrogen line useful in astronomy, and also the simplest possible case between 1s
caesium standard, which is used to define the second in the International System of Units since 1967.

Hydrogen line (21 centimeter line)

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21 cm is very long and very low energy, because he energy split is very small!

Compare it e.g. with the hydrogen 1-2 spectral line which is 121.6 nm!

Zeeman effect

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Split in the spectral line when a magnetic field is applied.

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?

Anomalous: evidence of spin.

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.

p splits into 3, d into 5, f into 7 and so on, i.e. one for each possible azimuthal quantum number.

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)".

Video 1.

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.