Today I’m going to explain why you are not falling through your chair right now using
one simple fact, and one object. The fact is that all electrons are the same as each
other, and the object is a structurally critical item of my clothing. There’s a chance this
episode could get a little weird.
By now we’ve established that quantum spin is very weird. We talked all about that recently
- how electrons have spin but aren’t really rotating. And about how you need to turn an
electron around twice - 720 degrees - to get it back to its starting position. They are,
we say, spin-½ - because one normal rotation only gets you halfway around. That particular
weirdness is not just another cute case of quantum mechanics being a bit silly. The fact
that some particles have this property is the entire reason that stuff in our universe
has structure, and that matter doesn’t immediately collapse. It’s the source of the Pauli exclusion
principle, and today I’m going to show you exactly why this simple property makes it
possible for us to have nice things in our universe.
Particles with spin-½ - or more generally any half-integer spin - 3/2, 5/2, etc. are
called fermions, and include all the particles that we think of as matter - from electrons
to quarks to the neutrinos. The other spin behavior is to have integer spin - spin-1
for example is the much more sensible case of a single 360 degrees rotation back to the
starting point. Integer-spin particles are bosons, and they are the force carrying particles
like photons with spin 1 or the Higgs particle with spin 0.
Now it’s possible to stack as many bosons on top of each other as you like. For example
in a laser beam, there’s no limit to the number of photons you can add - all of them
in the same quantum state. But not fermions - no two fermions can share the same quantum
state, which is why electrons can’t occupy the same energy states in atoms. Without this,
electrons in multi-electron atoms would all fall into the same lowest energy state - all
atoms would be the minimum possible size, and there would be no such thing as chemistry.
No nice things like solids or molecules.
The non-overlap-ability of fermions is called the Pauli exclusion principle. I’m going
to show you why this is the inevitable behavior of groups of particles that have two properties:
1) this weird rotational symmetry, and 2) indistinguishability - which is just the fact
that all electrons, for example, are exactly the same - there is no observable change when you
swap two electrons. Combining spin behavior and indistinguishability gives us something
called the spin statistics theorem, which sounds complicated but I’m going to derive
it with just some basic arithmetic and my belt. What did you think I was going to say?
Before we get to my belt, let me remind you about spinors. We talked all about it in the
recent episode - but for now just know that it’s just the type of wavefunction that
fermions have, and has this property that it returns to its starting state with a 720
degree rotation, not 360.
In that previous episode we saw this amazing animation of an object rotating while being
fixed by bands to it's surrounding environment. Crazily the bands disentangle every 720 degrees. So a spinor’s rotational
weirdness is not necessarily all that “quantum” - it’s a natural function of how it’s
connected to the universe. So allow me to introduce you to the belt trick, first
conceived by Paul Dirac himself.
It goes like this: hold the ends of the belt in each hand so the belt is flat. Let’s
think of the belt buckle as a particle - say an electron - and the belt is its connection
to whatever - the universe, or to maybe another electron. Now rotate the electron a full
720 degrees so you have a double twist. Now I’m going to untwist it without rotating
either end of the belt. Watch. Now at this point I am letting go of the buckle - but
the important thing is that the orientation of the belt ends don’t change with respect
to each other. And there you go, the twists are gone. So the system under these sorts
of rotations is a spinor because a 720 twist is topologically equivalent to no twist - simple
translation of the ends transforms between the two states.
On the other hand, if we rotate one end 360 we can’t untangle the belt if we keep the
ends fixed.
We can think both ends of the belt as spinor particles like electrons, and in that case we can do another
experiment. What happens if the ends exchange positions? If we’re careful again not to
rotate either end with respect to each other then the belt ends up with a single twist
in it - equivalent to a 360 degree rotation of one end. So it seems that for spinors,
a 360 degree rotation is equivalent to the particles switching places. This is slightly
worrying - if we’re using our hands as analogies for electrons then we just demonstrated that
it makes a difference if we switch their locations - and that's a problem if electrons are supposed to be
indistinguishable which is one of the critical ingredients of the spin statistics theorem.
We’ll come back to how electrons can have this property and yet still be indistinguishable.
Okay, let’s summarize what we’ve learned. There are objects that require a 720 degree rotation
to return to their original configuration in terms of their relationship to other objects.
We call these things spinors. For these same objects - spinors - a 360 degree rotation
is the same thing as swapping places with a second spinor, at least in terms of the
relationship between those spinors.
Okay. How do we connect all of this to actual electrons? Well electrons don’t really rotate in the classical sense. They’re quantum objects
described by a quantum wavefunction. A wavefunction is this thing that holds information about
the probability of a given property being observed. For example, the wavefunction representing
the position of a particle can look like a sine wave moving through space. If you have
two such wavefunctions overlapping - like two photons in a laser beam, a shift in one
of them by half a wave cycle puts the two out of phase with each other. In that case they
actually cancel each other out and you effectively have no photons. Mathematically, a half-cycle
phase shift corresponds to putting a negative sign in front of the wavefunction. That makes
sense right - because adding the two wavefunctions causes them to sum to zero.
A spinor wavefunction of the electron can “wave” through space, but it includes
another wavy part. It has its rotational degree of freedom in which a full wave cycle is a
720 degree rotation. In that case a 360 degree rotation puts a spinor perfectly out of phase
compared to its starting point. So a 360 rotation introduces a negative sign to the spinor wavefunction.
And that little negative sign is ultimately what drives the difference between fermions and bosons.
OK, so summarizing again: Electrons are spinors and so require a 720 degree rotation to be
returned to their initial state. But a 360 degree rotation shifts their phase by a half cycle
and adds a -1 to the wavefunction. But we also know from the belt trick analogy that
swapping two spinors is the same as doing a 360 degree rotation - so that should put
a negative sign in front of the combined wavefunction. And that’s the last piece of the puzzle
we need to get to the spin-statistics theorem, and the Pauli exclusion principle.
Now all we need is to do is the math.
Which you should relax about because we’re literally just doing addition and subtraction
here. Let’s think about the quantum state of an electron. We’ll call it Psi. Psi gives
the distribution of probabilities of some observable - for example, the location of
the electron around the atom. Psi is really the so-called “probability amplitude”.
The actual probability distribution is the square of that. We can never, ever observe
Psi - all we can do is map Psi^2 by making multiple measurements. The unobservability
of Psi is critical to this whole explanation, so remember that.
Now let’s put our electron in an atom - in the ground state. And add a second electron
to the first excited state. We can think of these two electrons as having a shared wavefunction - a two-particle
wavefunction we’ll call Psi(A,B) - which has two electrons A and B, in which one occupies
the ground state and one the first excited state. We’ll see what actually goes into
this wavefunction shortly.
But first let’s see how this thing should behave based on the stuff we figured out earlier.
Electrons are fermions, which means that if we swap their locations the wavefunction gets
multiplied by -1. Electron A goes into the first excited state and B goes to the ground
state. So now we have Psi(B,A) and we know that Psi(A,B) = -Psi(B,A)
Wavefunctions that change sign like this when it's particle labels are swapped are called anti-symmetric
under particle interchange, and those that don’t change sign are, unsurprisingly, called
symmetric. Fermions have antisymmetric wavefunctions, bosons, symmetric.
Remember that we wanted fermions to be indistinguishable from each other. But it seems like the two-particle
wavefunction changes if we swap the particles. Doesn’t that give us a way to “distinguish” the
swap? Actually no - no observable property is changed by the swap. Remember, that we only
“observe” the square of the wavefunction, and in that square the minus sign goes away.
The square of Psi(A,B) is equal to the square of Psi(B,A)
But although we can’t distinguish electron A from electron B through observation of these particles,
it turns out that this subtle “unobservable” property has a powerful manifestation in the
behaviour of groups of fermions.
To see that, we need to see what this two-particle wavefunction looks like in terms of the individual
wavefunctions of our two electrons. We’ll call the individual electron wavefunctions
g and f - if you like for the ground and first excited state of the atom, but this works
for any two possible wavefunctions - two possible quantum states - that our electrons could
have. If electron A is in the ground state then its wavefunction is g(A), if it’s in
the first excited state it’s f(A), and the same for electron B.
Now let’s say we don’t know which of A or B are in the ground or excited states.
The two-particle wavefunction needs to be a combination of f and g covering all the
possibilities - in fact, all possible combinations summed together in what we call a superposition.
There’s one way to do this that captures the anti-symmetric nature that we’re looking
for. It looks like this:
Which we can think of as the sum of A being in the ground state, B in the exited, then
B in the ground and A in excited. Each term in a superposition has a coefficient out front - and in this
case the coefficient for the second term is minus one. We’re choosing it because it
works. To prove it, let’s switch the particles and the wavefunction sign should flip.
We want Psi(A,B) to become Psi(B,A). And Psi(B,A)
is just this thing we saw with all the As and Bs switched.
And that is just the negative of the original. So Psi(B,A) = -Psi(A,B) - swapping electrons
flips the sign - so we’ve successfully discovered the wavefunction for a pair of fermions.
Bear with me, we’re getting VERY close now. I’m now going to show you that we can’t shove both
electrons into the same state. Let’s say we want both electrons to be in the ground state.
The two particle wavefunction would then look like this. The fs become gs.
Now both components of the superposition are the same, with the exception that the second component
has a minus sign - which causes those two components to cancel out. Essentially, two electrons shoved into
the same state end up perfectly out of phase and so destructively interfere. But you can’t
just vanish electrons - so the transition of an electron into an occupied quantum state
is impossible.
This statement, together with what we saw from the belt-trick previously about spinors
having anti-symmetric wavefunctions, is the pauli exclusion principle. That is, particles
with half integer spin have antisymmetric wavefunctions (the belt trick relation), and particles
with anti symmetric wavefunctions cannot have multiple particles in the same quantum state.
The full spin statistics theorem has a lot more to it. One part of it is a rather more
rigorous explanation of why spinors must have anti-symmetric wavefunctions that doesn’t
involve pant-retention technology. It boils down to the fact that you need to use spinors
in the Dirac equation - which is the quantum equation of motion for electrons and other
spin-½ particles.
That equation by itself doesn’t force you to use symmetric or anti-symmetric wavefunctions
- but if you try to use the symmetric wavefunctions of the boson then you get bad-wrong results.
Specifically, the energy level spectrum becomes unbounded from below, meaning you can continually
remove energy from the system by lowering the state of a particle forever. But if
you use the correct anti-symmetric wavefunction then everything works just works out great. So it’s
a proof by contradiction: particles described by spinors have to have an anti-symmetric
wavefunction and so must obey the Pauli exclusion principle - or you get nonsense.
So there you have it - matter has structure and you don’t fall through your chair because
electrons are indistinguishable and they obey a simple, if odd, rotational symmetry. Ironically,
to prove the structural integrity of matter I had to compromise the structural integrity
of my pants situation. Revealing mysteries of the universe sometimes comes at the risk of revealing
mysteries. But that is a risk we’ll always take for you here on space time.
Hey everyone, thanks for clicking, watching, and hopefully for liking and subscribing.
It's really, really helpful for you to do that stuff. And for those of you who go even
further and support us on Patreon - well words can't convey our gratitude. But let me try
anyway. Today I want to give a special thank you to Ethan Cohen who's supporting us at
the quasar level. Ethan, you're about as far as I can imagine from being indistinguishable
or degenerate. That makes you some exotic form of matter, presumably unbound by the
laws of known physics. So thank you for interacting with us lowly, fermionic life-forms - our
heads are spinning in gratitude for your support.
Today we’re doing comment responses from the last two episodes - the one about reverberation
mapping, where we map the stuff around black holes by watching how light bounces around.
And the episode where we took a journey into the weird, pasta-filled core of a neutron
star.
Greg Gorman says that he imagined that black holes would look more like dim stars rather than,
well, black holes because they slingshot light around from behind them. So it’s true that
black holes do this, but if you have good enough resolution you’ll always be able
to see the dark disk in the middle. That’s because the light rays will travel a straight
line after the slingshot, so at most they can appear to be coming from the edge of the event horizon.
And that’s exactly why the event horizon telescope image looks like that. Although
in that case the dark area is a little larger, because most of the light gets sling-shotted
from a bit further out - the photosphere where light can actually orbit the black hole.
Greddan6fly says that when they were in school quasars were just discovered and there was
speculation that they might be white holes. So I came along only a little after, I imagine those
were giddy exciting times! There was speculation that quasars could be swarms
of neutron stars or supernova cascades or even bizarre objects flying at crazy speeds
out of the Milky Way. Happily the winning hypothesis of enormous black holes was as awesome as
all the others.
Dave Lawrence rightly calls me out for saying that reverberation mapping can be done with
an ordinary scope. So - I don’t mean your typical toys-r-us “my first telescope”
- but even a high-end amateur telescope with an admittedly ridiculously expensive digital
detector and spectrograph could take a spectrum of a bright quasar and see the different components
of that light vary in different ways over months or years. Not advisable for a school
science project, but its still really cool that this is doable at all.
Mehul Mishra asked what would happen if you plucked one neutron out of the weird gridlocked
plasma crystal lattice of the crust of a neutron star. And then my favorite thing happened
- someone who knows more than me answered that question. Prot Eus tells us that the hole can only be
filled by an adjacent particle, which then just shifts the location of the hole. In this way
the hole sort of acts like it’s own particle, moving around the lattice. But this pseudoparticle
is less dense that its surroundings, so it eventually rises to the surface and so gets eliminated
from the lattice. And I’m very happy we sorted this one out.
Stephen Spackman also rightly calls me out for my using the expression “Up to 10%
or more” of the speed of light, regarding how fast gas can be blasted away from a quasar.
I agree that up to 10% or more can be literally any number whatsoever. Which means I'm
not wrong! But also not useful. The broad emission lines of quasars show that the gas is
typically moving away from the black hole a few perecent to 10% the speed of light, and in rare cases even faster. I will
try to speak with better error bars in the future.
And finally, Steve Bogucki tells us that Space Time made him realize that he’s more
interested in quantum physics than astrophysics. Hey, at least you didn’t devote decades
of your professional life to astrophysics before having that revelation. No, I’m
kidding. I love astrophysics like my awe-inspiring first born child. But quantum mechanics is
my weird and brilliant and deeply compelling second born and I love you all the same, I
swear.