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The classical concept of electron spin is wrong. From Stern Gerlach experiment we have seen that two separate beams are produced and that is why it is considered that electron has some property in magnetic field which is named as "SPIN" although it has no similarity with classical spin. So my question is from where angular momentum arises? For a moment I consider well there is no similarity also with classical angular momentum. But then why it plays an important role to generate total angular momentum?

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    $\begingroup$ Spin is an intrinsic property of an electron. It consist in a term of angular momentum that can be summed in different ways to the orbital angular moment (for instance, take a look at spin-orbit coupling, LS coupling, JJ coupling,...). Experiments have shown that this term is necessary (just as an example, it can explain the phenomenon of ferromagnetism in crystals where you have a quenching of the orbital angular momentsum) but, as you said, it can't be interpreted in classical terms. I think that "why" and "from where it arises" is not a completely proper question: experiments show that... $\endgroup$ – JackI Apr 23 '17 at 5:48
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    $\begingroup$ Nature is like that, and that's it. On the other hand, question about its importance is wonderful but really vast (and dependent on your background in quantum mechanics): I invite you to take a look at spin - orbit coupling, methods for summing the angular momentum and classical and quantum theories about magnetism. $\endgroup$ – JackI Apr 23 '17 at 5:52
  • $\begingroup$ You may want to take a look at the paper What is Spin? by Ohanian. $\endgroup$ – Mo_ Apr 23 '17 at 12:01
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In fact there are similarities between spin and classical angular momentum, which is why historically the result of the Stern-Gerlach experiment was understood in terms of some sort of angular momentum.

The energy levels of an atom placed in a weak magnetic field will split according to their $m$ values. For a state of angular momentum $\ell$, there will be $2\ell+1$ magnetic substates, and their energy difference will increase linearly with the applied external magnetic field (provided the field remains sufficiently weak). Basically, this is the Zeeman effect.

The same holds for spin. The energy level of the last electron in a silver atom - this electron is in an $\ell=0$ state - is split into $2$ sublevels with spacing proportional to the applied external magnetic field. This indicates that $s=1/2$ (so that $2s+1=2$). Since half-integer values of orbital angular momentum are not possible because the magnetic quantum number $m$ must be integer, spin cannot be orbital angular momentum but otherwise "behaves as if it were angular momentum".

Just to be sure this was not some complicated multi-electron effect, Phipps and Taylor [ T. E. Phipps, and J.B. Taylor "The Magnetic Moment of the Hydrogen Atom". Physical Review. 29 (2) (1927): 309–320.] redid the Stern-Gerlach experiment with hydrogen in its ground state and got the same type of splitting.

Finally, in levels where $\ell\ne 0$, the splitting is precisely as if spin and orbital angular momentum could be combined, so it makes perfect sense to think of spin as angular momentum, albeit not "classical" or orbital angular momentum.

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The spin was deduced from theoretical ideas and from the structure of the atoms:

The existence of spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which particles are observed to possess angular momentum that cannot be accounted for by orbital angular momentum alone.

The Stern-Gerlach-experiment was carried out in 1922 and at this time it was known that subatomic particles have intrinsic (permanent existing) magnetic dipole moments. The alignment of the particles magnetic dipole moments in a changing external magnetic field induces electromagnetic radiation, moves the particles sideways and disalignes the particles again. This is constantly repeated and is called the Lorentz force.

Since the macroscopic phenomenon of gyroscope precession is well know since a long, long time and the reason for this behavior could be explained only since the discover of the magnetic dipole moment and the photon emission during deflection the cause and effect are interchanged.

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First claiming a rotation of the particles (the intrinsic spin) it was stated that by this the particles receive a magnetic dipole moment.

In our days it would be better to show that the magnetic dipole moment runs the phenomenons of EM induction (and the gyroscopic effect).

And the theoretical deduction about the two possible states of the magnetic dipole moments for charged particles is indeed right. It is observed that for an electron the direction of the magnetic dipole moment and the directions of the photon emission and the direction of the resulting deflection is unique AND is opposite to the emission and deflection of the positron (and the proton).

So your question

Why electron has spin angular momentum?

should be replayed that charges have magnetic dipole moments and in an varying external magnetic field they get deflected and this is historical connected with a spin angular momentum.

And honored scholars please tell me which of the mentioned phenomenons and deduction steps is not well established knowledge.

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  • $\begingroup$ What do the "Output axis" and "Input axis" correspond to here in terms of spin? $\endgroup$ – ZeroTheHero Apr 23 '17 at 16:48
  • $\begingroup$ @ZeroTheHero According to the linked Wikipedia page about Gyroscope: "The axle of the spinning wheel defines the spin axis. The rotor is constrained to spin about an axis, which is always perpendicular to the axis of the inner gimbal. So the rotor possesses three degrees of rotational freedom and its axis possesses two. The wheel responds to a force applied to the input axis by a reaction force to the output axis."The sketch is from the same page $\endgroup$ – HolgerFiedler Apr 23 '17 at 16:56
  • $\begingroup$ I know what's a gyroscope but your link talks about - for instance - a gimbal and I can't see how this fits into the picture of the electron spin. $\endgroup$ – ZeroTheHero Apr 23 '17 at 16:57
  • $\begingroup$ @ZeroTheHero "First claiming a rotation of the particles (the intrinsic spin) it was stated that by this the particles receive a magnetic dipole moment. In our days it would be better to show that the magnetic dipole moment runs the phenomenons of EM induction (and the gyroscopic effect)." $\endgroup$ – HolgerFiedler Apr 23 '17 at 16:59
  • $\begingroup$ What does a gimbal have to do with the spin of an electron? From Wikipedia: A gimbal is a pivoted support that allows the rotation of an object about a single axis. $\endgroup$ – ZeroTheHero Apr 23 '17 at 19:01

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