Quantum mechanics has a lot of weird stuff - but there’s one thing that everyone agrees
that no one understands. I’m talking about quantum spin. Let’s find out how chasing
this elusive little behavior of the electron led us to some of the deepest insights into
the nature of the quantum world.
There’s a classic demonstration done in undergraduate physics courses - the physics
professor sits on a swivel stool and holds a spinning bicycle wheel. They flip the wheel
over and suddenly begin to rotate on the chair. It’s a demonstration of the conservation
of angular momentum. The angular momentum of the wheel is changed in one direction,
so the angular momentum of the professor has to increase in the other direction to leave
the total angular momentum the same.
Believe it or not, this is basically the same experiment - suspend a cylinder of iron from
a thread and switch on a vertical magnetic field. The cylinder immediately starts rotating
with a constant speed. At first glance this appears to violate conservation of angular
momentum because there was nothing spinning to start with. Except there was - or at least
there sort of was. The external magnetic field magnetized the iron, causing the electrons
in the iron’s outer shells to align their spins. Those electrons are acting like tiny
bicycle wheels, and their shifted angular momenta is compensated by the rotation of
the cylinder.
That explanation makes sense if we imagine electrons like spinning bicycle wheels - or
spinning anything. Which might sound fine because electrons do have this property that
we call spin. But there’s a huge problem: electrons are definitely NOT spinning like
bicycle wheels. And yet they do seem to possess a very strange type of angular momentum that
somehow exists without classical rotation. In fact the spin of an electron is far more
fundamental than simple rotation - it’s a quantum property of particles, like mass
or the various charges. But it doesn’t just cause magnets to move in funny ways - it turns
out that quantum spin is a manifestation of a much deeper property of particles - a property
that is responsible for the structure of all matter. We’ll unravel all of that over a
couple of episodes - but today we’re going to
Today we’re going to talk about what spin really is and get a little closer to
understanding what this weird property of nature.
The experiment with the iron cylinder is called the Einstein de-Haas effect, first performed
by, well, Einstein and de-Haas in 1915. It wasn’t the first indication of the spin-like
properties of electrons. That came from looking at the specific wavelengths of photons emitted
when electrons jump between energy levels in atoms. Peiter Zeeman, working under the
great Hendrik Lorenz in the Netherlands, found that these energy levels tend to split when
atoms are put in an external magnetic field. This Zeeman effect was explained by Lorentz
himself with the ideas of classical physics. If you think of an electron as a ball of charge
moving in circles around the atom, that motion leads to a magnetic moment - a dipole magnetic
field like a tiny bar magnet. The different alignments of that orbital magnetic field
relative to the external field turns one energy level into three.
Sounds reasonable. But then came the anomalous Zeeman effect. In some cases, the magnetic
field causes energy levels to split even further - for reasons that were, at the time, a complete
mystery. One explanation that sort of works is to say that each electron has its own magnetic
moment - by itself it acts like a tiny bar magnet. So you have the alignment of both
the orbital magnetic moment and the electron’s internal moment contributing new energy levels.
But for that to make sense, we really need to think of electrons as balls of spinning
charge - but that has huge problems. For example, in order to produce the observed
magnetic moment they’d need to be spinning faster than the speed of light. This was first
pointed out by the Austrian physicist Wolfgang Pauli. He showed that, if you assume electrons
have a maximum possible size given by the best measurements of the day, then their surfaces
would have to be moving faster than light to give the required angular momentum. And
that’s assuming that electrons even have a size - as far as we know they are point-like
- they have zero size, which would make the idea of classical angular momentum even more
nonsensical. Pauli rejected the idea of associating such
a classical property like rotation to
the electron, instead insisting on calling it a “classically non-describable two-valuedness”.
OK, so electrons aren’t spinning, but somehow they act like they have angular momentum. And this
is how we think about quantum spin now. It’s an intrinsic angular momentum that plays into
the conservation of angular momentum like in the Einstein de-Haas effect, and it also
gives electrons a magnetic field. An electron’s spin is an entirely quantum mechanical property,
and has all the weirdness you’d expect from the weirdest of theories. But before we dive
into that weirdness, let me give you one more experiment that reveals the magnetic properties
that result from spin.
This is the Stern-Gerlach experiment - proposed by Otto Stern in 1921 and performed by Walther
Gerlach a year later. In it silver atoms are fired through a magnetic field with a gradient
- in this example stronger towards the north pole above and getting weaker going down.
A lone electron in the outer shell of the silver atoms grants the atom a magnetic moment.
That means the external magnetic field induces a force on the atoms that depends on the direction
that these little magnetic moments are pointing relative to that field. Those that are perfectly
aligned with the field will be deflected by the most - either up or down. If these were
classical dipole fields - like actual tiny bar magnets - then the ones that were only
partially aligned with the external field should be deflected by less. So a stream of
silver atoms with randomly aligned magnetic moments is sent through the magnetic field.
You might expect a blur of points where the silver atoms hit the detector screen - some
deflected up or down by the maximum, but most deflected somewhere in between due to all
the random orientations. But that's not what’s observed. Instead, the atoms hit the screen
in only two spots corresponding to the most extreme deflections.
Let’s keep going. What if we remove the screen and bring the beam of atoms back together.
Now we know that the electrons have to be aligned up or down only. Let’s send them
through a second set of Stern-Gerlach magnets, but now they’re oriented horizontally. Classical
dipoles that are at 90 degrees to the field would experience no force whatsoever. But
if we put our detector screen we see that the atoms again land in two spots - now also
oriented horizontally.
So not only do electrons have this magnetic moment without rotation, but the direction
of the underlying magnetic momentum is fundamentally quantum.
The direction of this "spin" property
is quantized - it can only take on specific values. And that direction depends on the
direction in which you choose to measure it. Here we see an example of Pauli's two-valuedness
manifesting as something like the direction of a rotation axis, or the north-south pole
of the magnetic dipole.
But actually this two-valuedness is far deeper than that. To understand why we need to see
how spin is described in quantum mechanics. It was again Pauli who had the first big success
here. By the mid 1920s physicists were very excited about a brand new tool they’d been
given - the Schrodinger equation. This equation describes how quantum objects behave as evolving
distributions of probability - as wavefunctions.It was proving amazingly successful at describing
some aspects of the subatomic world. But the equation as Schrodinger first conceived it
did not include spin. Pauli managed to fix this by forcing the wavefunction to have two
components - motivated by this ambiguous two-valuedness of electrons.
The wavefunction became a very
strange mathematical object called a spinor, which had been invented just a decade prior.
And just one year after Pauli’s discovery, Paul Dirac found his own even more complete fix
of the Schrodinger equation - in this case to make it work with Einstein’s special
theory of relativity - something we’ve discussed before. Dirac wasn’t even trying to incorporate
spin, but the only way the equation could be derived was by using spinors.
Now spinors are exceptionally weird and cool, and really deserve their own episode. But
let me say a couple of things to give you a taste. They describe particles that have
very strange rotation properties. For familiar objects, a rotation of 360 degrees gets it
back to its starting point. That’s also true of vectors - which are just arrows pointing
in some space. But for a spinor you need to rotate it twice - or 720 degrees - to get
back to its starting state.
Here’s an example of spinor-like behavior. If I rotate this mug without letting go my
arm gets a twist. A second rotation untwists me.
We can also visualize this with a cube attached to nearby walls with ribbons. If we rotate
the cube by 360 degrees, the cube itself is back to the starting point, but the ribbons
have a twist compared to how they started. Amazingly, if we rotate another 360 - not
backwards but in the same direction - we get the whole system back to the original state.
Another thing to notice is that the cube can rotate any number of times, with any number
of ribbons attached, and it never gets tangled.
So think of electrons as being connected to all other points in the universe by invisible
strands. One rotation causes a twist, two brings it back to normal. To get a little
more technical - the spinor wavefunction has a phase that changes with orientation angle
- and a 360 rotation pulls it out of phase compared to its starting point.
To get some insight into what spin really is,
think not about angular momentum, but regular or linear momentum.
A particle's momentum is fundamentally connected to its position.
By Noter's theorem, the invariance of the laws of motion to changes in
coordinate location gives us the law of the conservation of momentum. For related reasons
in quantum mechanics position and momentum are conjugate variables.
Meaning you can represent a particle wavefunction in terms of either of these properties.
And by Heisenberg's uncertainty principle
increasing your knowledge of one, means increasing the unknowability of the other.
If position is the companion variable of momentum, what's the companion of angular momentum?
Well it's angular position. In other words the orientation of the particle.
So one way to think about the angular momentum of an electron
is not from classical rotation,
but rather from the fact that they have a rotational degree of freedom
which leads to a conserved quantity associated with that.
They have undefined orientation, but perfectly defined angular momentum.
Some physicists think that spin is more physical than this. Han Ohanian,
author of one of the most used quantum textbooks.
shows that you can derive the right values of the electron spin angular momentum and magnetic moment
by looking at the energy and charge currents in the so called Dirac field.
That's the quantum field surrounding the Dirac spinor aka the electron,
imply that even if the electron
is point like, it's angular momentum can arise from an extended though still tiny region.
However you explain it, we have an excellent working definition of how spin works.
We say that particles described by spinors have spin quantum numbers that are half-integers
- ½, 3/2, 5/2, etc. The electron itself has spin ½ - so does the proton and neutron.
Their intrinsic angular momenta can only be observed as plus or minus a half times
the reduced Planck constant,
projected onto whichever direction you try to measure it. We call these
particles fermions. Particles that have integer spin - 0, 1, 2, etc. are called bosons, and
include the force-carrying particles like the photon, gluons, etc. These are not described
by spinors but instead by vectors, and behave more intuitively - a 360 degree rotation brings
them back to their original state.
This difference in the rotational properties of fermions and bosons
results in profound differences
in their behavior - it defines how they interact with each other. Bosons, for example, are
able to pile up in the same quantum states, while fermions can never occupy the same state.
This anti-social behavior of fermions
manifested as the Pauli Exclusion Principle and is responsible
for us having a periodic table, for electrons living in their own energy levels and for matter
actually having structure. It’s the reason
you don’t fall through the floor right now. But why should this obscure rotational property
lead to such fundamental behavior? Well this is all part of what we call the spin statistics
theorem - which we’ll come back to in an episode very soon.
Electrons aren’t spinning - they’re doing something far more interesting. The thing
we call spin is a clue to the structure of matter - and maybe to the structure of reality
itself through these things we call spinors - strange little knots in the subatomic fabric of
spacetime.
Last time we talked about the connection between quantum entanglement and entropy - this was
a heady topic to say the least, but you guys had such incredibly insightful comments and
questions.
Joseph Paul Duffey asks whether entropy is an illusion created by our observation of
isolated components within a "larger" entangled system? Well the answer
is that entropy is sort of relative. It's high or low depending on context.
The air in a room may be perfectly mixed and
so considered “high” entropy. But if that room is warm compared to a cold environment outside,
then the total room + environment is at a relatively low entropy compared to the maximum
- if you opened the doors and let the temperature equalize.
Von Neumann entropy is different to thermodynamic entropy in that it represents the information
contained in the system and extractable in principle, versus information that’s lost
to the system by entanglement with the environment with the environment. On
the other hand, classical or thermodynamic entropy represents
information that is hidden beneath the crude properties of the system, but may in principle
be extracted. And yet von Neumann entropy has a similar contextual nature. If your system
has no entanglement with the environment then its von Neumann entropy is zero. But if you
consider a subsystem within that system then that entropy rises.
Randomaited asks the following: If entropy only increased over time, which implies it
was at its minimum at the Big Bang, does that mean there was no quantum entanglement at
the Big Bang? To answer this we’d need to know why entropy is so low at the Big Bang - and
that’s one of the central mysteries of the universe. But, I’ll give it a shot anyway.
So we can’t really talk about the t=0 beginning of time, because that moment lost in our ignorance
about quantum gravity and inflation and whatever other crazy theory we haven’t figure out
yet. But what we do know is that at some very, very small amount of time after t=0, the universe
was extremely compact - which meant hot and dense, and it was also extremely smooth. The
compact part is where the low entropy comes from. The “gravitational degrees of freedom”
were almost entirely unoccupied. On the other hand, the extreme smoothness meant that the
entropy associated with matter was extremely high. Energy was as spread out as it could
get between all of the particles and the different ways they could move. The low gravitational
entropy massively outweighed the matter entropy, so entropy was low. That smoothness seems
to suggest the particles of the early universe were already entangled - otherwise how did
they spread out their energy?
Chris Hansen makes the same point, asking if the conditions of the Big Bang meant everything
started out entangled. You’d think so - but that’s not necessarily the case. Remember
that von Neumann entropy is relative to the
system you’re talking about, and so is entanglement.
Let’s say you have a bunch of particles that are not entangled with each other but
are all entangled with another bunch of particles somewhere else. If you ignore those other
particles then it seems like there’s no entanglement in the particles of the first
system.
And yet those particles may have correlated thermodynamic properties due to their mutual
connection to the outside. In the early universe, the extreme expansion of cosmic
inflation may have permanently separated entangled regions, but left those regions with an internal
thermal equilibrium which does NOT require maximal entanglement within the regions themselves.
In other words, the universe - or our patch of it - may have started out unentangled and
at low entropy, even if it was at thermal equilibrium.
Lincoln Mwangi also dropped some knowledge, informing us that “The Cloud” - is actually
named after Dr, Shannon, the founder of the field of information theory. As with many
of these things, the word has been corrupted over time and is now routinely mispronounced.
This is very disrespectful, and I intend to write a series of op-eds to correct the matter.
Right after we upload this video to the Claude.