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What is Spin? | Quantum Mechanics

September 16, 2019


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!

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