If you took undergraduate quantum mechanics, at some point you were introduced to the concept of "spin." If you're like me, you left that class feeling you were shown a magic trick but not how it worked. Don’t worry; you are not alone. The reason you weren't told more is not that a better explanation was left for graduate quantum mechanics. You weren't told more because there isn't a lot more to know.

Of course, there is the magnetic dipole moment. Shoot a beam of electrons through a magnetic field with a gradient, and the beam will bend thanks to the influence of the field on a moving charge. Furthermore, it will also split into two sub-beams -- one containing "spin-up" electrons and the other containing "spin-down" electrons. An oriented magnetic dipole moment in the macro world corresponds to circulating current. Hence, the image of a spinning electron, presumably with a non-uniformly distributed charge that gives rise to the magnetic moment.

But if you do the math with some sort of estimate of electron size, the charge has to be spinning at many times the speed of light, which is not possible. Actually, it's not clear the electron actually has a size, so what is spinning anyway? Not that the spin isn't real -- you can induce electron spin transitions in an atom with polarized light, and photon spin (circular polarization) is real. But the intuitive explanation for the dipole moment is completely wrong, and we still don't have any concept of just what is spinning.

When we dig deeper, we learn that electrons -- like most familiar particles -- have a spin of ½. I'm sorry, ½ of what? Physicists, when confused, look at symmetries to understand behavior. A common example is rotational symmetry. A simple square rotated through 90 degrees looks identical to the un-rotated square. The same applies to 180 degree, 270 degree, and 360 degree rotations. Virtually everything looks the same if you rotate it through 360 degrees -- it's like no rotation at all.

But not spins of ½. Those you have to rotate through 720 degrees to get back to where you started, which is where the ½ comes from -- a spin of 2 is rotationally symmetric at 180 degrees, a spin of 1 at 360 degrees, and a spin of ½ at 720 degrees.

A neat trick to demonstrate this type of rotational symmetry (and a way to celebrate your inner geek at parties) is something called the plate-trick. Lay a plate flat on your hand, and then rotate it under your arm, over your head, and back down to the original position, all the time keeping the plate flat. If you watch carefully, you will notice that you rotated the plate through 720 degrees -- the same kind of rotational symmetry as spin ½. So we've discovered a real-world, albeit contrived, example.

But what does this mean for electrons? If you rotate polarized electrons through 360 degrees, why isn't that the same as no rotation? In fact, it is almost the same, but the sign of the wave function reverses -- you have to rotate through 720 degrees to get back to exactly the same wave function. The sign reversal has no observable effect if you're just looking at the intensity of the beam, but it can be seen in the circular polarization of light emitted from atoms excited by the electrons. Here again, spin behavior is, at best, on the fringes of intuition.

What about another well known property of spin ½ particles -- Fermi-Dirac statistics -- where no two indistinguishable particles can occupy the same state? And what does this have to do with spin ½? The answer is subtly related to the rotation example, but you have to go to relativistic quantum mechanics to see why.

First, you are now dealing with a wave function for two particles, and that wave function must be invariant under Lorentz transformations to comply with relativity. Swapping the particles corresponds to a Lorentz rotation in spacetime, which means that the sign of the wave function should change on a swap, which sounds familiar. But if they are indistinguishable particles (here electrons with spins pointing in the same direction), the wave function must be equal to its negative, which is only possible if it is zero. In other words, two electrons cannot occupy the same state.

Unfortunately, this proof only demonstrates the consistency of relativistic quantum theory with Fermi-Dirac behavior -- it doesn't explain the behavior in terms of more fundamental principles. And it doesn't eliminate other possible explanations. Once again, we are left grasping at smoke.

Someone once said that spin is sort of the geometric equivalent of the square root in arithmetic. It took us a long time to come to terms with that concept, and it can be argued that we grew into acceptance more than true understanding. Perhaps the same can be said for spin. Certainly it is ground-zero in quantum mechanics -- in fact, in relativistic quantum mechanics -- further removed from intuitive macro behavior than other behaviors we consider quantum.

As of now, there is no underlying theory to spin -- you just have to accept it -- like Zen.

Why should we care? Well, spin-effects are important in nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI), and the giant magneto-resistive (GMR) effect used in disk drive heads. They are also growing in importance in topological insulators with application in spintronics, where the phase reversal described above is thought to be important. More exotically, spin is one of the primary mechanisms to create entangled states, and is therefore relevant to quantum computing. Pretty useful for something we really don't understand!

And, finally, entangled states give rise to a paradox, since measurement on one component must have impact on the other component in negligible time -- thereby violating the speed-of-light limit (this has been confirmed experimentally). It seems as if relativistic quantum mechanics isn't perhaps quite as buttoned-up as we thought it was.

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