Why can't I just think the spin as rotating?

I really don't understand why do electron have 1/2 spin number, why they are not actually spinning.

I can accept that the electrons have their own magnetic field, which is certain, but why do they have $\hbar\sqrt3/2$ of angular momentum, and I don't know what the heck is spin number.

I've read the definition of spin and spin quantum number more than a hundred times but there is no betterment. I've smashed my head in my desk more than a hundred times either.

My question is the title. Why can't I just think the spin as rotating?

What I've saw recently,

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.

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Possible duplicate: physics.stackexchange.com/q/1/2451 and links therein. – Qmechanic Sep 13 '13 at 18:58
I think this is just one of the things you have to get used to in quantum mechanics. Very briefly, while electrons (and other particles) cannot be considered as spinning because they are point particles, they have an intrinsic quantity that behaves exactly like the angular momentum and gives it a magnetic moment. We call it spin because it reminds us of the spin of a rigid body. – Javier Sep 13 '13 at 20:18
I don't think smashing your head on a desk will help matters. – user50229 Sep 14 '13 at 13:09

(1) First argument

An ordinary object that is spinning on an axis has an angular momentum which is determined by how the mass of the object is distributed about the axis, and how fast the object is spinning. For a fixed angular momentum, if the mass is distributed farther from the axis the angular velocity is lower; if the mass is distributed closer to the axis, the angular velocity is higher. Think of a spinning ice skater who turns at one, slower speed with arms extended and at another, higher speed with arms pulled in overhead (on the axis).

No size has been found for electrons; they appear to be point particles. If an electron has finite size (which happens to be too small to see), it introduces issues in any classical description, like charge self-repulsion having infinite energy or having a surface which spins faster than the speed of light. If the electron is a point particle, i.e., has no finite size, then it cannot have angular momentum due to spinning about it's own center of mass because the entire object is on its rotational axis. How to get out of this conundrum?

You cannot "see" an electron to determine whether it is spinning or not. The "spinning" of the electron is not measurable; it makes no sense to speak of it in science. However, you can measure an electron's angular momentum; it makes sense to speak of angular momentum in science. Therefore, don't think of the electron as a "spinning" object (which we can never know or observe); think of it as simply having "intrinsic" angular momentum.

(2) Second argument

The usefulness of an analogy in science is determined by whether you can make inferences or understand other features in a first system under study by way of the analogy and knowledge of a second system.

Thinking of spin as "spinning" introduces conceptual problems classically, does not generalize easily to massless objects, doesn't handle half-integer spin well (the representations of a classical spinning object do not look like a spin 1/2 object), and allows you to infer almost nothing correctly about the electron, other than feeling like you know where the angular momentum comes from. This point of view doesn't work well to start with, and is a dead end as you keep studying physics.

On the other hand, thinking of spin as intrinsic angular momentum avoids all the above-identified issues and, you will find with further study, fits nicely into the rest of physics.

(3) Regarding spin-1/2 and the value ℏ√3/2, these come from group theory and a particular choice of units for angular momentum. This can't be really explained well in a post; study group theory for physicists.

Good luck!

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Nice answer. Although you allude to it, probably the most compelling reason that spin cannot simply treated as classical rotation is that spin-1/2 objects necessarily need to be rotated through 720 degrees in order to return to initial conditions - there is absolutely no classical analog of such a behavior. – James Pattarini Feb 16 at 6:33

Regarding "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."

That comment refers to calculations related to Moments_of_inertia if the electron was considered to be a spinning ball or ring, it would have to spin faster then the speed of light. In fact the calculation proves that an electron is not a spinning ball or ring.

It is important to note that this does not preclude other types of spinning objects that could have the correct properties. Consider this_linked object (wire frame video of a spinning torus around a spinning ring) that has properties consistent with an electron. Hopefully you can get a visual picture of the 'unusual' amount of energy it takes to flip this type of object, leading to the value you see for angular momentum.

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You can't get a larger moment of inertia at a given total size then a thin ring rotating around it's symmetry axis. – dmckee Sep 17 '13 at 0:28
Concerning the link, you may want to review our guidelines on self-promotion. – Qmechanic Sep 20 '13 at 21:49
Did not intend self-promotion, not sure where else to get pictures and animations of more complex objects. Regarding the largest moment of inertia, the largest moment of inertia on the wiki page is "Thin cylindrical shell with open ends, of radius r and mass m", where I=mr². This page does not account for more complex objects as for example if a second shell was spinning around the first shell as an enclosed torus in an orthogonal direction, (objects that consist of independent spinning parts) a hard object to describe without a picture. – AnimatedPhysics Sep 20 '13 at 22:42