The situation is somewhat similar to a classical spinning charged sphare, although this similarity easily breaks down.
Correct.
> But why do we call it spin?
Because it is a physical quantity whose units are those of angular momentum, and we have to call it something.
> What are the possible values?
+/- h/4pi where h is Planck's constant. (It is usually written has h-bar/2 where h-bar is h/2pi.)
> Is it a magnitude or a vector?
It's a vector that always points in a direction corresponding to the orientation of the apparatus you use to measure it.
> Is there a reason we call it "spin" instead of "taste" or some other arbitrary name?
Yes. See above.
> How do you change it?
You can change an electron spin by measuring it along a different axis than the last time you measured it. The result you get will be one of two possible values. You can't control which one you get.
> What happens to it when particles interact?
Their spins become entangled.
Do you know an example of a process that moves angular momentum from one kind of spin to the other?
It is techinically a two-component spinor, which is why the direction of the spin 'moves' if you measure it along different x,y,z axes. It is also quantized unlike a normal vector: All fermions have quantized half-integer spin magnitudes and all bosons have integer magnitudes.
Magnetic fields can be used to change the spin.
When particles interact, opposing spins tend to pair up in each electron orbital which cancels the magnetic field. This is why permanent magnets must have unpaired electron orbitals.
It's more akin to the direction or axis of the spin being changed, and simply measuring the spin along a certain access will change it: https://en.wikipedia.org/wiki/Spin_(physics)#Measurement_of_...
Spin being an intrinsically quantum mechanical concept, I'm afraid the microscopic mechanism by which that transfer occurs will only be explainable in a quantum mechanical context. Here it will appear as a term in the Hamiltonian coupling the spin of an electron to its motion in a potential.
https://en.m.wikipedia.org/wiki/Einstein%E2%80%93de_Haas_eff...
Spin is not an object really 'spinning', but in fact, neither is angular momentum, and momentum isn't really an object 'moving'. Let's be clear about what momentum is in quantum mechanics: there is, say, an electron field, and along a particular path it oscillates as `e^i(Et-px)/hbar`. The momentum determines how frequently the wave oscillates as you move in space; the energy determines how much it oscillates in time (the E and p operators pull E and p scalars out of this, and the Schrodinger equation says that E^2 - p^2 = m^2 (sorta; we take the positive solution of a low-momentum expansion...).
Anyway, the point is, momentum means "the wave function oscillates as you change the position'. Angular momentum means "the wave function oscillates as you change the angle'. The base orbital angular momentum state looks like `e^i m φ`, and the operator that extracts `m` is `- i d_φ`. Etc.
The other thing about wave functions is that they are continuous everywhere. The presence of a particle is something like "having a knot" in the field -- there is a region where there must be some net object, because if you integrate around the region you see non-zero net flux. That kind of thing. So to have "intrinsic angular momentum of 1/2" means that, if you integrate around a region where a fermion is, you'll see a net rotation of the wave function by half a phase.
Now, that seems nonsensical, because if you integrate around a point, you should get back to the value you started at. And in fact, you do, but the two are distinguishable: the reconciliation for this is related to the fact that SO(3) is not simply-connected; if you produce a path of rotations that takes every vector back to where it started (such as XYZXYZ, where X rotates around the X axis), any path that performs one loop does not deform the identity path, but one that does two does -- which makes these states physically different. So if you gave me a wave function, I could bucketize all of the points in, say, the 'z' direction into ones that are in the 'identity' element (relative to a point of my choosing) vs the 'anti-identity' element. These have opposite spins.
I am still working on tightening up the model, and I haven't quite figured out how this causes the magnetic field to send such a particle in a different direction. But the rest feels close, to me, and doesn't rely on any hand-waving statements like "because it seems to work this way".
So it can be written as a vector? No?
Spinors are difficult to describe in an HN post since they require a good amount of linear algebra, but my favorite explanation is probably here: http://www.weylmann.com/spinor.pdf
That's right. It's not a vector because it doesn't "transform" like a vector.
If you take a vector and rotate about an axis by 360 degrees, you get the same vector.
If you take a spinor and rotate it by 360 degrees you get a spinor which is "flipped". You have to rotate the spinor by 720 degrees to get back to the same spinor.
This is intrinsically weird, but that's QM.
(I don't exactly understand if this 'is' a knot, in a sense. I guess it is.)