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Somewhere I read that spin quantum number is a particularly interesting theory of quantum mechanics as what it really implies is that particles like electrons do not come back to the initial state of observation after one rotation, but if they rotate 2 times then the same state is obtained. Is it really so?

Even if it/or it is not why do the electrons need two types of spin quantum number? Are two electrons in same orbital supposed to rotate in opposite direction?

But then I also hear that electrons are particles and since particles cannot have an axis of rotation they do not actually rotate hence do not "spin". Another proof of this was that the observed magnetic moment of electron is much more than what was expected theoretically from the spinning of electrons hence definitely some phenomenon other than just spinning/rotating must take place.

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    $\begingroup$ A particle also cannot produce a diffraction pattern when sent through a thin slit because particles have no extension, but one grows tired of telling particles what they can and cannot do. $\endgroup$ – David H Oct 23 '13 at 16:55
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It is true, you have to "rotate twice" (or by $720^\circ$) to recover the original state. You can prove this in the following way. Let

$$|a\rangle=|+\rangle\langle+|a\rangle+|-\rangle\langle -|a\rangle$$

be a general ket. Consider now a rotation by a finite angle $\theta$ around the $z$ axis. I remind here that if a ket of a spin $1/2$ system is $|a\rangle$, the ket after rotation is $|a\rangle_{R}=\mathcal{D}_{z}(\theta)|a\rangle$. Where the rotation operator is defined as

$$\mathcal{D}_{z}(\theta)=exp\left(-\frac{iS_{z}\theta}{\hbar}\right)$$

Also, you can rotate about the $x$ or $y$ axis. For the sake of completeness $S_z$ is $S_{z}=\frac{\hbar}{2}[(|+\rangle\langle+|)-(|-\rangle\langle-|)]$. With of all this in place, we what to see the effect of the rotation operator on our general ket.

$$exp\left(-\frac{iS_{z}\theta}{\hbar}\right)|a\rangle=\mathrm{e}^{-i\theta/2}|+\rangle\langle+|a\rangle+\mathrm{e}^{i\theta/2}|-\rangle\langle -|a\rangle$$

Now it is easy to see that if we let a rotation of $2\pi$ we get

$$|a\rangle_{R_{z}(2\pi)}\rightarrow -|a\rangle$$

And if you ask if the minus sign is an observable, the answer is yes. This was observed with the help of neutron interferometry experiments. But for $\theta=4\pi$ we recover the initial state.

With regard to your second question, the Pauli exclusion principle is your answer. You cannot have to electrons in the same orbital with the same quantum numbers. For more details on spin you can read the wiki page here. Also you can look up the Stern-Gerlach experiment. Another link for the S-G experiment here.

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I think the best way to understand spin is to look at it from a complete abstract point of view: do not try to find classical analogs to it.

What you find when you perform experiments with electrons is that they interact with a magnetic fields as if they had an intrinsic magnetic moment with two possible values: a positive one and a negative one (loosely speaking, spin up and spin down, respectively). This is what you find. This is what you get from nature, period.

Having found this, the only think that you can say is that the electron has another degree of freedom, i.e., it has another way to store information about its state - for instance, you can say that an electron has a specific momentum and a specific spin. In other words, imagine that the electron is a person and you want to describe that person. So, you say that the person has brown eyes, brown hair, big nose and so one. This is what you get from the first look. But then, you put the person in a specific social situation (the magnetic field, in the case of the electron...) and you find more characteristics of the person - he/she is friendly or selfish or something else. The spin is basically it: it is another characteristic of the electron that becomes explicit when you put it in specific situations.

About the rotation: pay close attention to the rotation operator that you are using when you apply it to your state. That operator doesn't "live" in "our" 3D world - it is a rotation operator in the space of spin states. Putting it in a more tangly way, classical objects (living in our 3D world) rotate using a representation of the group SO(3), whereas spin states rotate with a representation of the group SU(2).

Hope this helps.

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