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As well known, spin could not be thought of as a rotational motion in classical mechanics, i.e. it's an intrinsic property.

But how to prove it? i.e. how to mathematically/experimentally show that the statement, spin could not have any classical (rotational) correspondence, was true?

Edit: "It" here meant:

  1. Is necessarily for spin to be some intrinsic quantity? or rather could be explained, even if it's non classical like Andrei has mentioned.

  2. In experimental setup, could we do a experiment and then claim that it's true for all spin? Because, as Cinaed Simson has mentioned, Stern Gerlach experiment was one experiment to prove this for single electron, i.e. a particular spin $1/2$ system. However, it did not experimentally exclude other spin system jut by itself.

  3. Also, just because the system collapse(in Stern Gerlach), doesn't mean there could not be a orbital motion, right? Quote: Ben Crowell:"...even-even nuclei usually have a first excited state that is spin 2, and this spin is explained correctly in most cases (for open-shell nuclei) as arising completely from the orbital motion of the nucleons (not their spin-1/2's)..." But I didn't quite understand what he meant by even(boson) and odd(fermion). What does even/odd has to do with orbital motion?

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  • $\begingroup$ Have you tried determining the speed at which the electron would be rotating to match its intrinsic magnetic moment (assuming a really small radius of the electron)? $\endgroup$ – Aaron Stevens Nov 18 '19 at 21:32
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    $\begingroup$ As stated, this is not quite true. For example, even-even nuclei usually have a first excited state that is spin 2, and this spin is explained correctly in most cases (for open-shell nuclei) as arising completely from the orbital motion of the nucleons (not their spin-1/2's). I think what you want to assert is that half-integer spins can't be explained as purely orbital motion. Even a spin 1/2 like that of the proton is actually partly the intrinsic spin of the quarks and partly their orbital angular momentum. If this sounds OK, please edit the question appropriately. $\endgroup$ – Ben Crowell Nov 18 '19 at 21:43
  • $\begingroup$ @AaronStevens: I don't see how that really helps. Classical models of the electron fail for a lot of reasons, but that doesn't prove that no half-integer spin can ever be produced by orbital motion. $\endgroup$ – Ben Crowell Nov 18 '19 at 21:47
  • $\begingroup$ @BenCrowell Yeah for some reason I was seeing "electron" here. Not sure why :) $\endgroup$ – Aaron Stevens Nov 18 '19 at 21:58
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    $\begingroup$ Maybe the OP could clarify what they're asking. Cinaed Simson's comment seems to be interpreting the question as asking (A) how we know that angular momentum is quantized. As you can see from my comment, I took it as asking (B) how we know that half-integer spins can't be generated by orbital motion. If A, then I think the part about "could not have any classical (rotational) correspondence" is wrong. If B, then I think the failure to specify half-integer spins is wrong. OP, please edit to clarify. $\endgroup$ – Ben Crowell Nov 18 '19 at 23:49
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ShoutOutAndCalculate,

Please take a look on this paper:

Proposed physical explanation for the electron spin and related antisymmetry

Cetto, A.M., de la Peña, L. & Valdés-Hernández, A. Quantum Stud.: Math. Found. (2019) 6: 45. https://doi.org/10.1007/s40509-017-0152-8

You can read it free here:

https://arxiv.org/pdf/1707.08674.pdf

The paper presents a counterexample to the claim that spin cannot be originating in a classical context:

"The calculations presented here confirm the physical picture of the spin of the electron as a helicoidal motion (a zitterbewegung) around the local mean trajectory, adding an effective structure to the originally pointlike particle."

The theory that allowed these calculations is called "Stochastic Electrodynamics". It is in fact classical electromagnetism with a certain assumption regarding the initial state.

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