Possible values are half integer numbers: 0, 1/2, 1, 3/2, and so on.
The approach shown in this section: Section "Spin comes naturally when adding relativity to quantum mechanics" shows what the spin number actually means in general. As shown there, the spin number it is a direct consequence of having the laws of nature be Lorentz invariant. Different spin numbers are just different ways in which this can be achieved as per different Representation of the Lorentz group.
Video 1. "Quantum Mechanics 9a - Photon Spin and Schrodinger's Cat I by ViaScience (2013)" explains nicely how:
- incorporated into the Dirac equation as a natural consequence of special relativity corrections, but not naturally present in the Schrödinger equation, see also: the Dirac equation predicts spin
- photon spin can be either linear or circular
- the linear one can be made from a superposition of circular ones
- straight antennas produce linearly polarized photos, and Helical antennas circularly polarized ones
- a jump between 2s and 2p in an atom changes angular momentum. Therefore, the photon must carry angular momentum as well as energy.
- cannot be classically explained, because even for a very large estimate of the electron size, its surface would have to spin faster than light to achieve that magnetic momentum with the known electron charge
- as shown at Video "Quantum Mechanics 12b - Dirac Equation II by ViaScience (2015)", observers in different frames of reference see different spin states
Originally done with silver in 1921, but even clearer theoretically was the hydrogen reproduction in 1927 by T.E. Phipps and J.B. Taylor.
The hydrogen experiment was apparently harder to do and the result is less visible, TODO why: physics.stackexchange.com/questions/33021/why-silver-atoms-were-used-in-stern-gerlach-experiment
Needs an inhomogenous magnetic field to move the atoms up or down: magnetic dipole in an inhomogenous magnetic field. TODO how it is generated?
Best mathematical explanation: Section "Spin comes naturally when adding relativity to quantum mechanics".
Physics from Symmetry by Jakob Schwichtenberg (2015) chapter 3.9 "Elementary particles" has an amazing summary of the preceding chapters the spin value has a relation to the representations of the Lorentz group, which encodes the spacetime symmetry that each particle observes. These symmetries can be characterized by small integer numbers:As usual, we don't know why there aren't elementary particles with other spins, as we could construct them.
Leads to the Klein-Gordon equation.
Leads to the Dirac equation.
Leads to the Proca equation.
Theorized for the graviton.
More interestingly, how is that implied by the Stern-Gerlach experiment?
physics.stackexchange.com/questions/266359/when-we-say-electron-spin-is-1-2-what-exactly-does-it-mean-1-2-of-what/266371#266371 suggests that half could either mean:
- at limit of large
lfor the Schrödinger equation solution for the hydrogen atom the difference between each angular momentum is twice that of the eletron's spin. Not very satisfactory.
- it comes directly out of the Dirac equation. This is satisfactory. :-)
Initially a phenomenological guess to explain the periodic table. Later it was apparently proven properly with the spin-statistics theorem, physics.stackexchange.com/questions/360140/theoretical-proof-of-paulis-exclusion-principle.
And it was understood more and more that basically this is what prevents solids from collapsing into a single nucleus, not electrical repulsion: electron degeneracy pressure!
Video "The Biggest Ideas in the Universe | 17. Matter by Sean Carroll (2020)" at youtu.be/dQWn9NzvX4s?t=3707 says that no one has ever been able to come up with an intuitive reason for the proof.