# Spin, Demystified

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?

Diagram showing the possible spin angular momentum values for 1/2 spin particles (for example, electrons) (Photo credit: Wikipedia)

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 $\hbar$, Planck’s constant divided by $2\pi$.

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.

English: Standard model of elementary particles: the 12 fundamental fermions and 4 fundamental bosons. (Photo credit: Wikipedia)

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:

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## 19 thoughts on “Spin, Demystified”

1. john zande

David, may I suggest (request) an article regarding the shape of the Universe.

NASA: “We now know that the universe is flat with only a 0.4% margin of error.” This is the infinite flat model, but what does this mean, exactly? I’m having trouble wrapping my head around it, and no one I know explains these things better than you.

Take this question: If, for arguments sake, I ride my spaceship “up” (as in a 3D box) how then does it level out to follow a flat universe?

I would imagine the universe ‘should’ be well shaped, like a receding funnel. If mass bends space then my brain tells me the combined mass of the universe should bend itself…?

Help me obi wan kenobi you’re our only hope.

1. David Yerle Post author

Request accepted! I’ll write about this tomorrow. I will not give you a summary in this comment because right now I cannot think of any simple, quick way to explain this. But I’ll try.
The last sentence made me laugh out loud.

2. elkement

Very well explained… In relation to the comments on my blog today I have a question:
You are a great science writer – your blog is on par with articles in Scientific American and the like, and in contrast to journalists who had specialized in science writing you really know what you are talking about.
Probably you have answered this already in one of your previous posts – did you every consider to work as a professional science writer? Or would you dread writing if deadlines would be attached to it and editors … who would define what kind of topics are appropriate?

1. David Yerle Post author

Actually, I never thought seriously about it! I always saw myself as more of a writer, period. It crossed my mind a couple of times, but I always dismissed the idea, assuming the people who actually do science writing have PhDs (I only have half a PhD) and connections. Also, I’m Spanish, so probably not very well qualified for science writing in English…
That said, if someone sees my blog and offers me a job, I’d love that. I read tens of papers everyday anyway, might as well make some money from it!
By the way, thanks for the praise!

1. elkement

I am not an expert on the science writing community – but Margaret Wertheim springs to my mind as I enjoyed her recent book ‘Physics on the Fringe’. She has a BSc in physics and a BA in math – not a PhD.
But probably it is all about proper networking and knowing the right people though…

3. bloggingisaresponsibility

So “forces” are just particles, or are the particles those that transmit forces?

I’m also trying to wrap my mind around the idea that volume-less points actually exist in real life. I understand them as necessary abstractions in geometry…

1. David Yerle Post author

Well, there are not such things as forces. Two particles exchange bosons and, each time this happens, they move. When we see this happen, we interpret it as a force, but all there is is particle exchange.
One way to wrap your head around the volume-less points is to think that space cannot really be continuous. In fact, it is most likely an emergent concept which arises from particle interactions. This way, it makes sense for particles to have zero volume, in the sense that at some stage the very notion of volume breaks down!

4. livelysceptic

Thank you for posting this, David. It makes more sense than anything I have read on ‘spin’ so far, even though imagining volume-less points spinning is a bit dizzying for the uninitiated. 🙂

5. Johannes Nelson

This is awesome. The title is misleading to me, though, because there was nothing about ‘spin’ as I knew of it that needed demystification. To me hula hooping was about as complex as it got! Now, however, spin has been taken to a whole new level!

I knew that particles were responsible for forces, but I didn’t know that these were all called bosons. Does this mean that the elusive graviton would be a boson? Also, are there particles responsible for the other forces (weak and strong nuclear)? For some reason, I think I always attributed these forces to positrons and electrons, but reading this I now know that that is ridiculous. I guess what I am asking is, how extensively can the category of ‘Boson’ be broken down, if it all? I am interested because I have always tried to better understand the Higgs-Boson. I get the general idea, but the specifics confuse me.

I am with BR on the volume-less points idea. To me, it seems totally logical that these things would actually spin, but I think that that is because I cannot really intuitively grasp the idea that these points have no volume.

1. David Yerle Post author

Yes, the graviton is a boson and it has spin 2. Here are the particles responsible for each force:
Electromagnetic force: photon (spin 1)
Weak force: Z0 and W+- bosons (spin 1). These guys have mass, so they can only interact with things that are very close (hence the “weak” name)
Strong force: gluons (spin 1) These guys can interact with each other, making the force really sticky. Quarks interact with this force and make up protons and neutrons. This force actually gets stronger with distance, that’s why we’re unable to split protons!
Gravity: gravitons (spin 2)
And the Higgs boson, which has spin 0 and interacts with everything.
Electrons can interact with photons and the messengers for the weak force. Quarks can interact with everything.
I will write an article on the Higgs boson one of these days!

1. Johannes Nelson

Thank you for this. I had a framework in mind, but no specifics. What do these interactions look like? I mean, what actually happens when a boson interacts something to apply force? Is it absorbed?

I would absolutely love a post on Higgs-Boson! I look forward to it, really.

1. David Yerle Post author

Yup, bosons are absorbed by the particles they’re interacting with and then some other boson is re-emitted. Bosons are actually indistinguishable so probable it makes no sense to say some “other” boson…

6. tk

Well, hope that You don’t mind pointing that out but I think that spin has nothing to do with special relativity.

My conclusion comes from the fact, that one can construct equations that have spin, but are Galilei (non-relatively, non Lorentz) invariant.

Greetings,
T.K.

1. David Yerle Post author

Hi T.K,
I think I mentioned this in my article. If I’m not mistaken (I wrote this a while ago), I said that spin had to be included in classical QM equations by hand (which it had to be) but that it arises naturally when trying to find a Lorentz-invariant QM equation. In other words, even though spin is not incompatible with classical physics, it does not “naturally” arise in the way that it does with the Dirac equation. Of course, one could argue that the Klein-Gordon equation is relativistic and has no spin, but at least spin does arise in some relativistic QM equations. It does not do so in classical ones: you have to artificially duplicate the equations in order to add spin.
Cheers,
David

2. David Yerle Post author

Hi TK,
It looks like I replied prematurely: I read your blog post and saw what you were getting at. Actually, I was completely unaware of the Lévy-Leblond equation, but on hindsight it make sense to also “square root” the Schroedinger equation. Thanks for enlightening me!
I will update the post accordingly.
Cheers,
David