Angular momentum measures how much something is rotating.
For a very brief time, a few of physicists thought that Spin was a type of angular momentum
that came from the electrons spinning on their own axis.
These days we are pretty sure that isn’t the case, electrons aren’t actually spinning,
BUT spin is still thought of as a type of angular momentum.
It’s referred to as intrinsic angular momentum- angular momentum a particle just has, without
physically rotating.
The idea always struck me as odd, and the two arguments I’d heard for it seemed weak.
The first is that in Quantum mechanics, even though angular momentum and spin don’t obey
all the same equations, but they do both obey a particular one.
Then people claim that this equation defines angular momentum- hence spin is a type of
angular momentum.
I never understood why this random equation was so important.
The other argument, is what we looked at last time, the stern-gerlach experiment and similar.
There I thought the argument was, charged particles that have normal angular momentum
act like magnets.
Particles with spin also act like magnets, so let’s say that they have a type of angular
momentum too- an intrinsic type.
I thought that was unnecessary -couldn’t you just say spin is an intrinsic magnetism
instead?
Isn’t that a simpler explanation?
In the last video, I asked you to research spin for me- which you did a brilliant job
of.
I did the homework too, by reading a bunch of things and I realised I was wrong about
both those arguments!
There is a very good reason that suggests this equation defines angular momentum- I’ll
explain why in the next couple of videos.
But today I’ll talk about why I was wrong about the stern-gerlach experiment, the angular
momentum isn’t just a useless by -product, it’s actually pretty crucial.
And so, maybe spin is angular momentum!
The mysterious Stern-Gerlach apparatus is actually just a giant magnet.
But a regular magnet wouldn’t be enough, because if a little magnet was put into this
field, it would be equally pulled up and down.
Instead, the top bit is made pointy because that makes the south pole stronger, so the
magnet is pulled upward more.
If the magnet was pointing downward, the south pole strongly repels the little magnet’s
south pole, so it goes downward.
So you can start to see how this machine might measure orientation.
It works fine for these two orientations, but here’s the problem that bothered me
for weeks.
Imagine I put it in on its side.
What we want the machine to do is to not deflect this magnet either way.
That seems fine, the magnet is equally attracted and repelled from both sides, but you can
see, these forces are going to twist the magnet.
They keep twisting it... until it’s pointing in the upward direction.
Now the force on the magnet is up!
This happens to any magnet pointed at an angle, it gets twisted to point up.
So by this logic, pretty much every magnet should go straight upward...the Stern-Gerlach
machine isn’t supposed to do this... so what’s going on?
Turns out, I was wrong, this machine doesn’t work for little bar magnets for exactly this
reason- but it does work when you have a charged particle moving in a circle.
I was making a big mistake; I thought these two things were essentially the same, because
their magnetic fields are the same.
But the particles going in loops have one extra property I completely neglected: they
have angular momentum that comes from their rotation.
We can represent angular momentum with an arrow.
The length of the arrow represents how big the angular momentum is, so if we rotate the
object faster, the arrow is bigger.
The arrow also has direction, and this represents which direction the object is spinning: the
arrow always perpendicular to the plane the object is rotating in.
The key thing about angular momentum is, it doesn’t like to change.
So that means it’s hard to speed up or slow down the object, and it’s hard to tilt it,
because either of those things change the arrow.
Going back to our situation, if we have a rotating object and we put it in a magnetic
field, just like a bar magnet, the magnetic field will try to twist it.
But if it did get twisted, it’s angular momentum would change, so unlike the bar magnet,
it’s going to resist the twisting, and that has a pretty unexpected consequence.
Instead of twisting this way, its angular momentum traces out a circle.
This is called Larmor precession.
If you want to see the maths behind why that is, I’ll explain it now.
Otherwise, skip the explanation, and I’ll just tell you why any of this matters for
spin.
For this explanation we need our classical mechanics.
I’ll do my best to explain this, but it took me ages to understand this properly,
so don’t panic if this is confusing.
We need to introduce one more technical word: torque.
A torque is something that twists an object.
In our case here, it’s from the magnetic field.
Just like angular momentum, torque is a vector, and its direction is perpendicular to the
direction of the twisting.
The magnetic field is trying to twist the angular momentum this way, so the torque is
in this direction.
Our angular momentum is currently in this direction, and we want to know which way it
will be pointing in a very short time, delta t, so we use this equation from mechanics:
The new angular momentum equals the old one plus the torque times delta t.
Multiplying the torque by delta t just makes it smaller, and then we add that to our old
angular momentum.
Using vector addition, the result is this vector.
You can see the angular momentum has started to turn around this circle.
If you think about it, the magnetic field is still trying to twist it, but now in that
direction, so the new torque is this way.
If we apply the rule again, the result is a vector even further along, and if we keep
doing it, we can see that the angular momentum processes.
This was so hard for me to wrap my head around because this torque wants to make the angular
momentum point up, but it doesn’t make it point any more up than it was before instead
it makes it spin in the other direction.
So if you put this thing with angular momentum into the stern-gerlach machine, it wouldn’t
just go flip up like a bar magnet would.
In other words, the Stern-Gerlach machine only works if you have angular momentum.
When you put a particle that has spin in the Stern-Gerlach machine, it does work -so by
this analogy, they should have angular momentum- being magnetic is not enough.
So maybe it is fair to say spin is an intrinsic angular momentum... then again, all of this
is an analogy from classical physics.
Does Larmor precession even happen in quantum mechanics?
Actually it does, and is pretty awesome.
Say an electron is doing this quantum mechanical larmor precession.
If it’s gone around once, you’d expect it to be in the same state it was originally....
but it’s not.
You have to let it go around twice before it’s the same again.
Before this video ends, I want to thank you so much for doing research for me last time!
I learnt a lot, and hopefully you did too.
People told me about ideas I’ve never heard off, and recommended a whole bunch of books
too, and so since that video, I feel like my knowledge of spin and quantum mechanics
has increased a lot in the past few weeks.
I want to do these research homeworks more often- but there’s one big problem- a lot
of the sources on this type of material are really advanced and confusing.
So it could have been difficult to read for a lot of people.
How can we make sure that everyone can participate in this sort of homework if they want to?
And also, I’ve been thinking about other types of home assignments- do you guys have
any ideas of what you’d like me to try?
Now for this video’s actual homework I’m interested, did you guys find this video’s
argument convincing?
Should we call spin intrinsic angular momentum?
Is it weird to even try and explain spin with classical ideas and using classical analogies?
Is intrinsic angular momentum inherently meaningless phrase anyway?
Every time I ask these sorts of questions I get a really diverse set of opinions all
well argued so this should be fun.
Second, I mentioned that you can only get an electron back to its original state by
rotating it twice.
Last time, some people commented that this isn’t so weird- some classical things do
this.
What do you think of this?
Do you know an example?
Finally for the classical mechanics fans, I explained why a rotating thing in a stern-gerlach
machine doesn’t just flip and point up.
But now explain why the force that it experiences is proportional to how much it was pointing
up to start.
Next time I’m covering my favourite bit of all of physics, so I’m really excited.
See you soon