What is spin? It’s something that apparently some particles have that you can measure...

but what is it?

In the video I’ll explain how we know spin exists, but also why we don’t understand

it.

Spin was a latecomer to the quantum mechanics party. Even after Schrodinger wrote his famous

equation, and everything seemed to be working people didn’t realise it existed. Then people

realised that some particles seem to have odd magnetic properties the original quantum

mechanics didn’t predict, and the source of that was labeled spin.

To understand this we’re going to need one piece of classical electromagnetism theory:

Whenever a charged particle moves, it creates a magnetic field. This is pretty amazing,

and you can see it for yourself, by putting a compass next to a charge carrying wire.

In fact, if you have charged particles going in a small loop, this not only creates a magnetic

field, that field pretty much looks like it’s from a bar magnet. So then, we expect charged

particles moving around, in circles, to act as little magnets. Since that’s something

we know from classical physics it was built into quantum mechanics too from the start.

But eventually some experiments showed that there must be another source of magnetism.

I’m going to explain one of those, a simplified version of the stern-gerlach experiment.

Let’s say we have a bunch of electrons by themselves. They are charged, but they’re

not doing little loops, so they shouldn’t be magnets. We test this by putting them in

a Stern-Gerlach apparatus.

I’m not going to explain how the machine works, but what is does is this. If a small

magnet is placed inside it measures it’s orientation. If the magnets North pole is

up, it exerts an upward force on the magnet. If the magnet is the other way around, it

exerts a downward force. Of course there are other orientations. If the magnet is like

this, the magnet is still somewhat pointing up, so the force is still up, but it’s smaller.

It is proportional to how much this magnet is pointing in the up direction. Similarly

for this magnet pointing down. Say a magnet is on its side, then it has no force at all.

But these aren’t the only ways the magnet can be oriented. It could be forward or backward

or left or right, or any combination, so what happens there? Well, this machine only measures

how much the magnet is up or down. If the magnet is like this, it’s pointing a bit

upward. On the other hand, if a magnet was just pointing left or right or forward or

backward, it wouldn’t feel any force and so it would go straight. So what expect is,

if we shot through a bunch of magnets in random orientations, they all land between here and

here and in a pretty smooth distribution.

Of course, if we threw in a non magnet, it wouldn’t feel anything, and just go through.

So that’s what we should expect with electrons.

But we do the experiment and find something really odd. The electrons do experience forces

as if they’re magnets, but they don’t spread out smoothly either. half of them go

up to the same height, and the other half go down, again, to the same height.

Let’s look at this second mystery first. If electrons really are magnets, we expect

they’re randomly oriented, but these results would be like saying that for some reason,

half of them where pointing exactly up and half exactly down. Well, maybe we did something

very wrong when we prepared them that meant this happened. So to test that, let’s do

something clever. Let’s turn the stern gerlach machine on its side. What that does is, now

the machine measures how much the magnet is pointing to the left or the right, instead

of up and down. The more to the left a magnet is pointing, the more to the left it goes.

If a magnet isn’t pointing either left or right, it just goes through.

Now let’s use the same source of electrons we had before. We think that they’re all

pointing up or down, so if they go through this machine, they shouldn’t be deflected

at all. What happens? Half of them go right, half

of them go left. So then were they actually oriented left and right all along? But that

can’t be right either.

Let me explain how to think of this in the framework we built up in the last couple of

videos.

I can measure the up or downness of a particle, so that is an observable. What are it’s

eigenstates? Well, this experiment showed that the particle can only be fully up or

fully down, and nothing in between, so there are only two eigenstates. We’ll call them

spin up and spin down. Now we can apply the quantum mechanical principle that, to fully

describe the wavefunction of this particle, we only need to describe it in one basis,

so I can fully write the state of this particle in terms of up and down. This is weird, because

in classical physics this wouldn’t be enough to tell you the state of a magnet. I can’t

just tell you how much it’s pointing up or down, I need to also tell you how left

or right it is, and also how forward or backward. But in quantum mechanics, this is enough.

Let’s see what happens when we do some arbitrary combination. Here, since the square of this

number is bigger, it has a bigger chance of being spin down, but still can be spin up.

It doesn’t go ¾ of the way down ever though, it’s only ever up or down.

But what about if I flipped the machine on it’s side to measure left rightness?

In Quantum mechanics, if you tell me the wavefunction in terms of up and down, I should be able

to figure out what the wavefunction is in terms of left and right. To be able to do

that, we just need to know how to convert up and down to left and right, but how? The

coefficents should be equal in size, if we use the classical analogy. A particle that’s

fully up can’t be pointing either left or right at all. A quantum particle that’s

up, but is being measured for left and rightness must go one way or the other, but at least,

it shouldn’t be biased towards one side.

Now we just need to decide the sign of each of these. If we agree to call this way right

and that left, then this turns out to be the right way to do it.

And this is something you’ll recognise from the last two videos.

Ok, so now I’ve shown you how quantum mechanics deals with spin, let’s return to the must

more difficult question, ‘what is spin?’ The electrons aren’t doing little loops,

so why are they magnetic? This is what was originally proposed: the electrons are spinning

on their own axis. The idea is, if you have a charged ball that’s spinning, that means

bits of charge are moving in a circle, and so it creates a magnetic field like a bar

magnet. But we know this is wrong. If this really was true, we can calculate how large

the electron should be- it’s bigger than the whole atom. It also makes some other incorrect

conclusions. These days we think the electron isn’t a ball, it’s an infinitely small

point- and that can’t spin.

I think this image of the electron spinning, is really more harmful than helpful- not just

because it makes all kinds of incorrect predictions, but because feeling like we understand something

stops us from asking about what it is. But, making this video I realised there’s lots

of other physics terms I don’t understand. What is energy? What’s charge? It seems

like we define those things by how we measure them, and then same is true for spin, spin

is that thing that makes some particles act a bit like magnets. I really hope that, in

the future as we understand more physics, more of these terms can be understood in terms

of deeper physics. There is some hope for spin. I told you that spin had to be added

to quantum mechanics in an ad hoc way. This is true, but when Dirac tried to merge standard

quantum mechanics and relativity, something in his new equation acted a lot like spin.

While this was awesome, I don’t feel like it solves the mystery of spin, because while

it comes out of the maths, it still doesn’t explain what it is, or why relativity demands

it exists. Maybe there are some properties of physics we can never understand, and maybe

spin is one of them.

There was so much more I could have said about spin but I ran way over time. So, I thought

it would be fun if you guys taught each other instead, so I’ll set this as homework.

In the description, I’ve put a whole lot of questions I didn’t answer in the video.

I would love if you picked one of more of these, or found your own and did some research.

You can present your work in any format you want: as a Youtube comment, on any of these

social media accounts, on your own website etc. It would be kind of amazing if you want

to make it into a video, but I know that’s daunting.

As for the content, you should know that some of the questions I asked in the description

are ones I don’t know the answer to despite trying to find out, so it’s absolutely fine

to just make an attempt- this doesn’t have to be a huge thing. Also, a lot of sources

claim to answer questions but leave big gaps, or make potentially unjustified assumptions.

And some of these questions are still unanswered, even though there are some interesting leads.

So then your task isn’t to necessarily definitively answer your question. In fact, I’d way rather

you point out why you don’t understand something, or discuss several possible solutions. Don’t

worry about being dumb. Not knowing things isn’t dumb, pretending you know things is

dumb.

I’d love if you read each other’s responses. If you want to give any constructive feedback

and start a discussion, then awesome I’m encourage that so much- but remember be kind.

Anyway, good luck, I’m really looking forward to learning things from you!