Spin quantum number visualization

Question

I'm in highschool and quite new here, so I don't know a lot of the more advanced topics of physics, although I'm trying. I'm a bit confused at exactly what goes on with a spin quantum number. I understand the basic idea of the electron having +1/2 spin direction and a -1/2 spin direction, but I'm hearing that it's not really a spin in a sense?? I'm a little confused at how this would look like. Any topic recommendations on what I should study first to properly understand this, and any videos or a certain explanation that helps any of you understand it?

Answer 1

To put it simply, spin is an internal degree of freedom found in quantum mechanical systems that arises from the $SU(2)$ symmetry of such systems. That is, quantum mechanical spin has no analog in classical dynamics other than the algebraic similarity that arises due to the fact that $SU(2)$ is a double cover of the $SO(3)$ group of rotations in ordinary space. Thus, visualizing electrons or other particles as rotating about a central axis is invariably misleading, rather one must attempt to visualize spinor rotation, a difficult mental process at best.

However, one might attempt to make a mental visualization by thinking of a vector-like object (over-simplified understanding of a spinor) which has its base attached (perpendicularly) to a Moebius strip (a topological space that is non-orientable). This configuration of objects is such that it will require two full rotations, i.e. rotation through an angle of $4\pi$ as opposed to the usual $2\pi$ in order for the spinor to return to its original orientation. You can make a Moebius strip using a piece of paper and some adhesive in order to visualize this easier.

When one considers that elementary particles like electrons are point particles it is truly difficult to see how the above visualization applies, and it becomes clear that it is best to think of spin as something purely quantum mechanical (mathematical property of the field equations that describe the particles) and not as something analogous to classical rotations.