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I read about how spin and magnetism are related (mathematically) but I found that it is a concept related to classical physics which is in turn related to the idea of electrons actually spinning on their axis (which was later found impossible, at least as far as I know) but I wasn't able to find any relation between quantum spin and magnetism. Can you please help?

M.A.P.
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2 Answers2

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Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum.

Particles with spin can possess a magnetic dipole moment, just like a rotating electrically charged body in classical electrodynamics. These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous magnetic fields in a Stern–Gerlach experiment, or by measuring the magnetic fields generated by the particles themselves.

The intrinsic magnetic moment $\mu$ of a spin $1/2$ particle with charge $q$, mass $m$, and spin angular momentum $\mathbf{S}$, is $$\mu=\frac{g_sq}{2m}\mathbf{S}$$ where the dimensionless quantity $g_s$ is called the spin g-factor. For exclusively orbital rotations it would be $1$ (assuming that the mass and the charge occupy spheres of equal radius).

Young Kindaichi
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Since the electron has charge, its quantum spin creates intrinsically an Amperian current loop that gives rise to a magnetic moment. That is actually Ampère's law in Maxwell equations of Electromagnetism.

Markoul11
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  • This is still a picture that appeals a classical analogy, but, as OP points out, electrons are not literally little spinning balls or loops. OP is asking how to understand this more fundamental kind of magnetic moment. – kaylimekay Jan 02 '21 at 14:08
  • You are correct. As far as I know there is currently no formal answer to this thus quantum spin has no classical analogy and is an intrinsic (hidden mechanism) property of the electron. The Standard Model does not give an answer for this. One may look to beyond the SM theories currently to seek for an answer. – Markoul11 Jan 02 '21 at 14:16
  • In the SM, the form of the coupling of the electromagnetic gauge field to the electron is dictated by its Lorentz structure. In the low energy limit, that coupling has the same structure as a magnetic dipole interaction, see this for example. – kaylimekay Jan 02 '21 at 14:33
  • Interesting. If the electron is part of the the Lorentz group and can be described as such then this would imply that it has a none simply connected 3D topology. The best candidate topology description for the electron then would be this: link . A sphere representing its charge and a horn tube, vortex string, running throughout its two poles representing its magnetic dipole moment. This topology would also explain why g-factor is 2 and the apparent contradiction until now with classical mechanics is no more. This explains also quantum spin of e. – Markoul11 Jan 02 '21 at 19:47
  • Thanks for your answers. I learnt quite a few new things – M.A.P. Jan 15 '21 at 14:56