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In this wikipedia article is written about Electron's spin.

it was produced by the self-rotation of the electron. When Pauli heard about the idea, he criticized it severely, noting that the electron's hypothetical surface would have to be moving faster than the speed of light in order for it to rotate quickly enough to produce the necessary angular momentum.

I am wondering how they get this statement

electron's hypothetical surface would have to be moving faster than the speed of light

I would like to get real numeric value of electron angular momentum and check this statement :)

And my question is: What is the numerical value of Electron's Angular momentum in SI $\Large\frac{kg*m^2}{s}$?

If I know this value, I can calculate rotation frequency from formula $L=I\omega$ where L - angular momentum and I - moment of inertia. and if I know frequency $\omega$, I can find a speed if I assume some electron radius like "classical electron radius" or something else.

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In SI units the electron spin is:

$$S = \frac{1}{2}\hbar \approx 5.272859\times 10^{-35} \text{J}\cdot\text{s}$$

The thing is that you can't use classical mechanics to describe spin and you can't say that the electron is actually rotating. So trying to get this value from an angular speed and moment of inertia doesn't make much sense.

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This is basically a repeat of FrodCube's answer but my point is to emphasise it is still difficult today to get rid of the spin notion for newcomers to QM, (no offence, my own knowledge on this is slight), but I have got to the stage of never thinking of spin as anything other than a math device.

So 100 years ago, it must have been extremely difficult in physical, philosophical and every other "common sense" approach to see spin as it really is, a purely mathematical​ notion. Take His Albertness's (©John R) view of every aspect of QM, Einstein never accepted it fully.

So every statement the QM pioneers made has to be taken with the realisation that these new ideas took time to become accepted, and like Bohr's semi classical orbital theory (or later the "Dirac Sea"), a lot of their ideas were plain wrong.

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    $\begingroup$ I know that it is not applicable to spin, I just want to repeat this calculation and make sure it does not make sense to use angular momentum definition to spin. $\endgroup$ – Zlelik Mar 13 '17 at 22:05
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    $\begingroup$ Sure, off topic but I doubt we will ever know what the electron actually "is". All the best $\endgroup$ – user146020 Mar 14 '17 at 0:11
  • $\begingroup$ Never say never :) $\endgroup$ – Zlelik Mar 15 '17 at 12:29
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    $\begingroup$ @Zlelik hi, my guess: ,<philosophy> If it's properties can only be described by math, and it's behaviour be described only by math, then it is math. I would love to say energy instead, but I am not sure what energy is. The definition keeps changing, at least at my naive level of knowledge. </philosophy> $\endgroup$ – user146020 Mar 15 '17 at 13:09

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