I have written a bunch of articles on quantum mechanics and the notion of spin keeps popping up. Unfortunately, in most of them it was arrived at tangentially, so I didn’t have the time to tackle it properly. I hope to remedy this today.
Spin is nothing but angular momentum. Just like momentum is a measure of how fast an object is moving, angular momentum is a measure of how fast it is spinning. Anything that spins has a certain angular momentum that can be calculated quite straightforwardly with a mathematical formula I won’t bother you with (or maybe I will: it’s mwr, where w is the angular velocity and r is the distance to the center. For more than one particle or for a continuous solid you have to add (integrate) over all the particles).
Spinning doesn’t sound very mysterious, so why all the fuss? Well, the fuss comes from the fact that physicists realized, sometime during the first half of the 20th century, that some particles seemed to have an intrinsic angular momentum. That is, they seem to have an angular momentum that was always the same, regardless of their situation. As if the particles had no other choice but to spin all the time at exactly the same speed. Another baffling aspect was that those particles were considered to be point-like. So how can something with no volume spin around itself?
Believe it or not, it took the unification of special relativity and quantum mechanics to solve the mystery. Until then, physicists just added spin to their equations by hand, taking into account something they had observed but which they had not idea why it happened. *
It was Dirac who combined for the first time the equations of quantum mechanics and relativity in a way that made sense. What he found out was that particles who obeyed his equations had to have a spin which was exactly one half of , Planck’s constant divided by .
As relativistic quantum mechanics advanced, physicists discovered particles only came in two types: some of them had to have half-integer spin (1/2) whereas some of them had to have an integer spin (0, 1, 2). The first ones were called Fermions; the latter ones, bosons.
Before we go into the difference between Fermions and Bosons, I’d like to delve into the meaning of spin. Does it make sense to say that electrons “spin” if they are point-like particles? Or is spin just a mathematical quantity that behaves just like an angular momentum, without the actual spinning?
Different physicists will give you different answers. I think it all boils down to your definition of spinning: if you consider spinning in the classical, everyday sense, then electrons may no really be spinning. If you decide that things that spin do it in the quantum sense, even when they’re big (after all, the whole world is quantum, not only at the microscopic scale) then electrons spin as much as you do. Also, even if you decide that only everyday spinning qualifies as spinning, electrons may still spin: if superstring theory is true, for example, particles are not point-like but extended. Something like that is perfectly capable of spinning.
Physicists soon discovered that fermions and bosons behaved in quite different ways. Fermions obey something called the “Fermi exclusion principle” (hence the word “Fermion”), which states that no two fermions can be in the same quantum state. That’s why electrons don’t instantly fall to the lowest energy level of an atom, but occupy progressively higher ones: once a state is taken, no more electrons may go in. One may say that Fermi’s exclusion principle is responsible for Chemistry.
Bosons, on the other hand, obey something called “Bose-Einstein statistics” (hence the word “Boson”). There is no limit to the number of bosons that can occupy one quantum state at one time. Have you ever read about something called a “Bose-Einstein condensate?” It is based exactly on this. It consists of getting a bunch of bosons and cooling them down so much that they all fall to the lowest energy level. Since they all occupy the same state, the whole bunch (which could be made of billions of particles) behaves like a single particle. Which is pretty cool.
Fermions make up all of our matter. Protons and neutrons (which are made of quarks) are Fermions, as well as electrons and positrons (which are like electrons with a positive charge). Bosons, on the other hand, are force-carriers. They correspond to the usual forces plus some new ones: gravity, the electromagnetic field, the weak force and the strong force. Bosons can be absorbed by fermions and can thus cause them to move, giving the appearance of a force. In fact, forces can be seen as an exchange of bosons between two particles (which can be bosons or fermions).
I hope you now understand the notion of spin, at least a little better than you did before this article. Here are the takeaway points:
1. Spin is rotation.
2. Spin is a consequence of mixing relativity and quantum mechanics.*
3. There are two types of particles with half-integer and integer spin, which have very different properties.
*Reader TK points out that you can get spin also from the equations of non-relativistic quantum mechanics, in a way analogous to the way it is done relativistically. For more information, visit his blog post on the matter: