system
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human
What do the values of spin mean? (i.e 1, 1/2, etc)

When a particle rotates (even though it doesn't make sense to say that such a small object can rotate, but the effect is the same) it creates a tiny magnetic field around it. That's why you can have spin up or down. Two poles. But what do the values mean, exactly, and how are they calculated?
gpt
The spin actually has an existence and meaning independent of magnetic fields.

In classical physics, we use *angular momentum* to describe how much rotational motion there is -- something rotating around a bigger radius has a greater angular momentum, something rotating at more revolutions per second has a greater angular momentum.  Angular mometum is conserved, adn this arises because the equations of physics are unchanged by rotations of the coordinate system.

In quantum mechanics, this same invariance under rotations leads to the quantum analogue of angular momentum.  We often call this *orbital angular momentum*, even though things aren't really orbiting, borrowing from the classical picture.

But now you add relativity to quantum mechanics, and something remarkable happens.  Yes, things still have orbital angular momentum (the thing that generalizes classical rotational motion), but it turns out that they can have an additional piece of angular momentum that doesn't come from motion in any sense at all, but that they just get to have by existing.  We call this angular momentum things just have because they exist *spin*.

This is not unprecedented.  Relativity teaches us that in addition to the energy things get to have from their motion (kinetic energy) and position (potential energy), they can also have energy just by existing (this is what mass represents!).  It turns out something analogous happens with angular momentum.

Remember, angular momentum is conserved, even with relativity and quantum mechanics, because the laws of physics are unchanged by rotations of the coordinate system.  In 3 spatial dimensions, the mathematical properties of rotations restrict the possible values of *spin* to 0, 1/2, 1, 3/2, etc. (More properly, we start with the Poincaré group in 3+1 dimensions and work towards this conclusion from there.)

When something has spin *s*, it means the total intrinsic angular momentum it has just for existing has magnitude (hbar)sqrt{*s*(*s*+1)}, where hbar=Planck's constant/2(pi).  If you then pay attention to the direction of angular momentum and ask how much of it lies along any particular direction, if it has spin *s*, the only amounts it can have along that direction are (hbar)*s*, (hbar)(*s*-1),(hbar)(*s*-2),...,-(hbar)*s*.  These results all follow from the connection between angular momentum and rotations.  

As an example, when *s*=1/2, along any particular direction, the only values you can get are (hbar)(1/2) and -(hbar)(1/2), and those are what we mean by spin up and spin down.  The Delta baryons, which have spin 3/2, can have along any particular direction (hbar)(3/2), (hbar)(1/2), -(hbar)(1/2), or -(hbar)(3/2).