system
Given the following post, respond as if you're commenting on Reddit:
human
Do electrons physically spin the way wheels and tops spin? Or does the word spin mean something different when it comes to electrons?

They say electrons can have a spin of +1/2 or -1/2.  Does this number represent how the electron rotates about an axis, or does it mean something else?  What do the numbers +1/2 and -1/2 mean; why couldn't they have used +1 and -1 instead?
gpt
Different people will give different answers, because it has been taught differently over the years in introductory quantum mechanics classes. Current sophomore-level quantum mechanics dogma is that spin is an intrinsic quantity unrelated to anything you've ever encountered in the macroscopic world, and indeed that is a good way to learn it at that level -- because quantized angular momentum works a bit differently from the classical case that everyone is familiar with by their teen years, and breaking the intuitions built up from macroscopic experience is a major part of the quantum mechanics curriculum.  

Upper-division and graduate-level quantum mechanics classes cover spherical harmonic interference in more detail (spherical harmonics are functions that are to spherical coordinates what the sinusoids are to cartesian coordinates), and it becomes more obvious that the electron wave functions rotate around the origin just as do wave functions of more complicated objects (such as wheels) -- it's just that they are restricted by the single-value rule from any amount of spin other than +1/2 or -1/2 quantum of angular momentum around any axis.

The relationship between the electron's angular momentum and magnetic moment is "anomalous" -- it does not correspond to the relationship you would find for a massy sphere filled with charge, nor for a massy sphere covered with charge on its surface.  It corresponds closely to the relationship you would find for a tiny cylinder covered on the curved surface with charge.

A very counterintuitive element of electron spin is that electrons always have to spin.  Angular momentum comes in quanta, so you can only gain or lose angular momentum in integer amounts of the universal quantum of angular momentum.  But the electron happens to intrinsically contain half of that quantum -- so it can never achieve zero angular momentum.

The +1/2 and -1/2 numbers refer to how many quanta of angular momentum the electron is carrying, along a particular axis (which is usually obvious from context -- e.g. along or against a magnetic field, or along or against the bulk angular momentum of an atom that contains the electron).  The whole point is that the electron can never be *not* spinning along *any* axis -- at best it can be in a superposed state where it's turning both clockwise and anticlockwise around that axis, with probability 0.5 for each case.  If you measure the spin (say) along the Z axis, you'll find that it is either CW or CCW about the Z axis -- and if you measure it again (say) along the Y axis, you'll find that it is either CW or CCW about that axis, too.  See, counterintuitive.