by Ciro Santilli (@cirosantilli, 32)
Atoms exist and last for a long time, while in classical electromagnetic theory punctual orbiting electrons should emit radiation quickly and fall into the nucleus: physics.stackexchange.com/questions/20003/why-dont-electrons-crash-into-the-nuclei-they-orbit
In other sections:
A single line in the emission spectrum.
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:
Let's do a sanity check.
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 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?
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 , 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: can change by no more than 1 (it can stay the same). We have dubbed these transition restrictions selection rules, which we summarize as:
(1)
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!
One reasonable and memorable approximation excluding any fine structure is:
(1)
Equation 1. Hydrogen spectral series mnemonic.
see for example example: hydrogen 1-2 spectral line.
Equation "Hydrogen spectral series mnemonic" gives for example from principal quantum number 1 to 2 a difference:
(1)
which with Planck-Einstein relation gives about 121.6 nm ( Hz), which is a reasonable match with the value of 121.567... from the NIST Atomic Spectra Database.
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?
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.
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:
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!
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.
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.
Video 2. Zeeman Effect - Control light with magnetic fields by Applied Science (2018) Source. Does not appear to achieve a crystal clear split unfortunately.
Amazingly confirms the wave particle duality of quantum mechanics.
The effect is even more remarkable when done with individual particles such individual photons or electrons.
Richard Feynman liked to stress how this experiment can illustrate the core ideas of quantum mechanics. Notably, he night have created the infinitely many slits thought experiment which illustrates the path integral formulation.
This experiment seems to be really hard to do, and so there aren't many super clear demonstration videos with full experimental setup description out there unfortunately.
For single-photon non-double-slit experiments see: single photon production and detection experiments. Those are basically a pre-requisite to this.
photon experiments:
Non-elementary particle:
  • 2019-10-08: 25,000 Daltons
  • interactive.quantumnano.at/letsgo/ awesome interactive demo that allows you to control many parameters on a lab. Written in Flash unfortunately, in 2015... what a lack of future proofing!
Video 1. Single Photon Interference by Veritasium (2013) Source. Claims to do exactly what we want, but does not describe the setup precisely well enough. Notably, does not justify how he knows that single photons are being produced.
Video 1. Electron Interference by the Italian National Research Council (1976) Source.
Institutional video about the 1974 single electron experiment by Merli, Missiroli, Pozzi from the University of Bologna.
Uses an electron biprism as in electron holography inside a transmission electron microscope.
Shows them manually making the biprism by drawing a fine glass wire and coating it with gold.
Then actually show the result live on a television screen, where you see the interference patterns only at higher electron currents, and then on photographic film.
This was elected "the most beautiful experiment" by readers of Physics World in 2002.
Italian title: "Interferenza di elettroni". Goddammit, those Italian cinematographers can make even physics look exciting!
It would be amazing to answer this with single particle double slit experiment measurements!
Quantum version of the Hall effect.
As you increase the magnetic field, you can see the Hall resistance increase, but it does so in discrete steps.
Figure 1. Hall resistance as a function of the applied magnetic field showing the Quantum Hall effect. Source. As we can see, the blue line of the Hall resistance TODO material, temperature, etc. It is unclear if this is just
Gotta understand this because the name sounds cool. Maybe also because it is used to define the fucking ampere in the 2019 redefinition of the SI base units.
At least the experiment description itself is easy to understand. The hard part is the physical theory behind.
The effect can be separated into two modes:
Figure 1. Source.
TODO original experiment?
Laughlin paper: 1981 Quantized Hall conductivity in two dimensions.
Shows a cool new type of matter: Abelian anyons.

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