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In our atomic physics class, we saw that the spin-orbit coupling term arises from the scalar product of the magnetic moment of the electron (proportional to its spin), and the magnetic field created by the nucleus rotating around the electron. But this implies that in the electron's reference frame, it still sees it's own spin? How is this possible?

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That is... an exceptionally Zen sort of question... since you are talking about viewing spin from the internal perspective of a point object. Pauli pretty much threw up his hands on such issues and said "just use the math!" – Terry Bollinger Jan 25 '13 at 18:26
    
It is not exactly analogous, but the question that lets me sleep soundly after thinking about this is "Can a billiard ball still have angular momentum in it's own (inertial) reference frame?" – dmckee Jan 25 '13 at 18:37
    
What exactly is confusing you? Rotating reference frames are privileged frames in special relativity, so I don't see a problem with an electron seeing its own spin. – Mark Mitchison Jan 25 '13 at 18:38
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The spin is the angular momentum of the electron in its own reference frame, the amount of rotation around its "axis". So it's surely nonzero. If the angular momentum were zero in the electron's frame, then the spin would be zero - it wouldn't exist! But it surely does exist. motls.blogspot.cz/2012/12/… – Luboš Motl Jan 25 '13 at 18:41
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@ChristopherYeates: I don't expect you to obtain a meaningful answer from mainstream physics. There is no logic here you could follow without contradictions. Lubos Motl gave you a positive answer, but notice the quotation marks used with the word axis. Likewise, all speculations on this subject from the QM - being purely mathematical - should be treated this way ... – bright magus Sep 12 '14 at 12:59

Spin is an intrinsic property of quantum objects that, unlike a particle's orbital momentum, does not depend on the frame of reference you are considering. Another intrinsic quantity of that kind would be charge, which is also just a fundamental number you assign to a particle, no matter its state of motion.

One possible source of confusion when talking about spin is the fact that it is similar to classical or orbital angular momentum in many ways, e.g. it follows the same mathematical structures (for example addition of angular momenta). It is, however, still different. Electrons, and elementary particles in general, are, to the best of our knowledge point particles. Our classical intuition of something "spinning around its own axis" requires extended objects, a notion which definitely breaks down at the fundamental level of quantum mechanics.

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