Kind: captions
Language: en
As a 42-year-old who spent most of my
life studying physics, I must admit that
I had a big misconception. I believed
that every object has one single
trajectory through space, one single
path. But in this video, I will prove to
you that this is not the case.
Everything is actually exploring all
possible paths all at once. So, let's
start with a simple thought experiment.
Say you're at a beach when all of a
sudden you see your friend struggling
out in the water. You want to go help
him as quickly as possible. So which
path should you take to get
there? The shortest path is a straight
line. So you could head directly towards
him. But you can run faster than you can
swim. And this path requires more
swimming. So alternatively, you could
run down the beach to minimize the
distance through the water. But now the
total distance is longer than it needs
to be. So the optimal path, it turns
out, is somewhere in between. To be
precise, it depends on the speeds at
which you can run and swim. Now, you
might recognize this mathematical
relationship because it is the exact
same law that governs light passing from
one medium into another. So light also
takes the fastest path from point A to
point B. What's weird about this is that
as humans, we can see where we want to
go and then figure out the fastest
route. But light, I mean, how does light
know how to travel to minimize its
journey time? Now, here is where my
misconception comes in. I shine a laser
beam, the light just goes in one
direction. I throw a ball, the ball just
goes in one direction. You know, I would
have answered, there is nothing strange
about this. Light sets off from point A
in some direction and then a little
while later it encounters a new medium
and due to local interactions with that
medium, it changes direction ending up
at point B. If you later find that of
all the possible paths, light took the
shortest time to get from A to B, I
wouldn't think it was optimizing for
anything. I would just think that's what
happens when light obeys local rules.
But now I will prove to you that light
doesn't set out in only one direction.
Instead, it really does explore all
possible paths. And the same is true for
electrons and protons, all quantum
particles. So the fact that we see
things on single well-defined
trajectories is in a way the most
convincing illusion nature has ever
devised. And the way it works all comes
down to a quantity known as the action.
In a previous video, we showed how an
obscure scientist Mertui made an ad hoc
proposal that there should be a quantity
called action, which he defined as mass
time velocity time distance. And he
claimed that everything always follows
the path that minimizes the
action. Hamilton later showed that this
action is equivalent to the integral
over time of kinetic energy minus
potential energy.
Action was useful and an alternative way
of solving physics problems, especially
when Newton's laws get too cumbersome.
But then around the turn of the 20th
century, action showed up at the heart
of a scientific revolution, the birth of
quantum
mechanics. It all started with electric
lighting in Germany. Think about what
it's like in the 1890s, right?
Electricity being more widely available,
at least in urban centers, and things
like, you know, light bulbs. They were
new. They were literally the hot new
thing. Germany wanted to replace all
their gas street lights with electric
light bulbs. So, an important question
was, how do you maximize the visible
light given off by a hot filament?
Scientists at a German research
institute, the PTR, studied how much
light different materials emitted as a
function of temperature.
At low temperatures, each material gave
off its own characteristic spectrum,
mostly in the infrared. But above about
500° C, all materials started to glow in
the same way with an almost identical
distribution of light. The hotter the
object, the more energy was emitted at
every wavelength, and the peak of the
distribution shifted to the
left. But they still didn't understand
how it worked theoretically. So that was
sort of the next step, right? Is if you
can understand how it works
theoretically, then you can use that
theory to potentially design your
products. They started by imagining the
simplest object possible, one that would
absorb all light that falls onto it and
perfectly emit radiation based on its
temperature. They came up with a hole in
a metal cube. This hole is a perfect
black body because any light that shines
onto it will go straight in. bounce
around inside and eventually be
absorbed. But this also makes it a
perfect emitter. Any radiation inside
the cube can escape through the hole
unimpeded. Theorists reasoned that
electrons in the walls of the cube would
wiggle around emitting electromagnetic
waves. These waves would then bounce off
the other walls.
When you have two waves of the same
frequency where one travels to the right
and the other to the left, they can
interfere in such a way that they create
places where there's no wave amplitude,
those are nodes, and places where
there's maximum wave amplitude, the
anti-nodes. Waves like this are called
standing waves because they don't really
move left or right. And inside a cavity,
given enough time and reflections, it is
only these standing waves that survive.
All the other ones just cancel
out. So a sort of order emerges from the
chaos. In two dimensions, standing waves
look something like this. For shorter
wavelengths or higher frequencies, you
can fit more and more different
vibrational modes inside this
cube. So that in three dimensions the
total number of modes is proportional to
frequency cubed or one over lambda
cubed. The expectation was there would
be more and more waves inside the cube
the shorter the wavelength. This led
directly to the genes law. At longer
wavelengths it matched the experimental
data pretty well. But at shorter
wavelengths the theory diverged from
experiment. In fact, it predicted that
at the shortest wavelengths, an infinite
amount of energy would be
emitted. This, for obvious reasons,
became known as the ultraviolet
catastrophe. The person to solve this
problem was Max Plunk. But Plunk almost
didn't make it into studying physics
because when he was 16 years old, he
went up to his professor and asked him,
"Well, maybe I could do a career in
physics." to which his professor
responded that he'd better find another
field to do research in because physics
was essentially a complete science. You
know, there was just a few tiny little
problems that they had to clean up. But
besides that, it was
over. But Plunk didn't listen. By 1897,
he was a professor himself, and for the
next 3 years, he struggled to find a
theoretical explanation for blackbody
radiation. He tried approach after
approach, but no matter what he tried,
nothing worked. He said, "I was ready to
sacrifice every one of my previous
convictions about physical
laws." Then in a quote act of
desperation, he did something no one had
thought to try. According to classical
physics, the energy of an
electromagnetic wave depends only on its
amplitude, not its wavelength or
frequency. and it could take any
arbitrary value. So any atom could emit
any wavelength of light with an
arbitrarily small amount of
energy. But Plunk tried restricting the
energy so that it could only come in
multiples of a smallest amount, a
quantum. And he made the energy of one
quantum directly proportional to its
frequency, E= HF, where H is just a
constant.
Think about what this does to the
radiation coming from the black body. At
a given temperature, the atoms in the
cavity have a range of energies. Some
have a little bit, a few have a lot, and
most have their energy somewhere in
between. For long wavelength, low
frequency radiation, the energy HF of
one quantum is small. So all of the
atoms have enough energy to emit this
wavelength and the spectrum matches the
gene's prediction very well. But at
shorter wavelengths, higher frequencies,
the energy of a quantum increases and
now not all of the atoms have enough
energy to emit that wavelength. This is
why experiment diverges from the
classical prediction. The spectrum peaks
and then starts to fall because fewer
and fewer atoms have enough energy to
emit one quantum of that radiation. And
there comes a point when none of the
atoms have enough energy to emit one
quantum. So here the spectrum must drop
to
zero. With this approach, Plonc got a
new formula for the radiation spectrum.
Now all that was left for him to do was
to tune the parameter H. And when he did
this just right, he got his formula to
match up perfectly with
experiment. But he was sort of troubled
by his own formula because to him it was
just a mathematical trick. He had no
clue why it worked. It was purely
formal. And most importantly, he had no
clue what this H represented. I mean, he
had introduced a new physical constant
without any
reason. He wrote a theoretical
interpretation had to be found at any
cost, no matter how high. So from that
moment on, he dedicated himself to
finding one. He later reflected that
after some weeks of the most strenuous
work of my life, light came into the
darkness and a new undreamed of
perspective opened up before
me. He introduces what we now call
plank's constant and it has the units of
action. Planck's constant h is a quantum
of action.
Plunk later proposed that anytime any
change happened in nature, it would be
some whole multiple of this quantum of
action. So, it's kind of spooky. This
breakthrough that starts the ball
rolling toward quantum theory brings
action in, not energy and not force.
Action gives you a
hint. At first, the quantum of action
got little attention. That is until a
26-year-old patent clerk came on the
scene. In 1905, Albert Einstein claimed
that Plonc's theory wasn't just a
mathematical trick. It was telling us
that light actually comes in discrete
packets or photons, each with an energy
HF. Einstein used this insight to
explain the photoelectric effect. How
light can eject electrons from metal,
but only when the frequency is high
enough. If the frequency is too low, no
electrons will be emitted regardless of
the
intensity. The idea of quantization
spread. 8 years later, Neil's Boore was
trying to understand how an atom is
stable. If it has a positive charge in
the center and negative electrons
whizzing around it, why don't they just
spiral into the nucleus, radiating their
energy as they go? And what he wants to
do is he says there's something fishy
about something being discreet. That
seems to be the new ambiguous weirdo
lesson of the new chrono of action. Bore
realizes that as the electron goes
around the nucleus, it has an angular
momentum. Mass time velocity time
radius. So angular momentum has the same
units as action. And so what he decides
to do is discretise the orbital angular
momentum for no good reason. He says,
"Let me slap that on and say angry
electron can only be in one unit, two
units, three units of the same quantity
h." And because it's talking about
motion or circle, the factors 2 pi come
in. So it's really nh over 2 pi, what we
now call nh
bar. This comes out of nowhere. There
seems like absolutely no good reason why
angular momentum should be quantized.
But by doing it, bore finds the correct
energy levels of the hydrogen atom. When
an electron jumps from a higher orbit to
a lower one, the energy difference is
given off as a photon of a particular
color of light exactly reproducing the
hydrogen spectrum. And that was a pretty
startling thing to have fall out. So I
think that really was compelling and
take some quantity with units of action
and apply some again kind of ad hoc um
discretization or quantization to it.
Now although it worked spectacularly
well, no one could make sense of
it. That is until 11 years
later. For his PhD, Louis de Bruy was
contemplating the recent discoveries in
physics. And his big insight was that if
light could be both a wave and a
particle, then maybe matter particles
could also be waves.
He proposed that everything, electrons,
basketballs, people, absolutely
everything has a wavelength. And he
defined this wavelength analogously to
light as plank's constant divided by the
particles momentum or mass time
velocity. Now, if an electron is a wave,
the only way it could stay bound to a
nucleus in an atom is if it exists as a
standing wave. That requires that a
whole number of wavelengths fit around
the circumference of the orbit. You
could have one wavelength or two
wavelengths or three and so on. So the
circumference 2 pi r must be equal to
some multiple n* the wavelength. We can
sub in de bru's expression for the
wavelength to get that 2 pi r= nh / mv.
But we can rearrange this to get that
mvr the angular momentum is equal to nh
/ 2
pi. That is precisely bor's quantized
angular momentum condition. But now we
have a good physical reason why it's
quantized because electrons are waves
and they must exist as standing waves to
be bound in atoms because they want to
have constructive interference of a
stable orbit. Bang. That's pretty good.
Good. You get a dissertation out of
that. That's pretty good. It is this
wave nature of quantum objects that
means they no longer have a single path
through space. Instead, they must
explore all possible
paths. Now, I have thought about and
taught the double slit experiment
hundreds of times without fully
realizing this implication. In the
double slit experiment, I feel like the
mental thing that I'm doing in my head
is like, okay, well, the beam is not
perfectly straight and of course it's
going to intersect both of those slits
because they're really close together,
you
know? But then I heard this story about
a professor teaching the double slit
experiment and it makes everything so
clear. So the professor starts by
explaining the setup. Electrons are
fired one at a time through two slits to
be detected at a screen. Now, because
you can't say for certain which slit the
particle went through, quantum mechanics
tells us it must go through both at the
same time. So, to get the probability of
finding a particle somewhere on the
screen, you simply add up the amplitude
of the wave going through one slit with
the amplitude of the wave going through
the other slit and square
it. But that's when a student raised his
hand. What if you add a third slit?
Well, you just add up the amplitudes of
the waves going through each of the
three slits and you can work out the
probability. The professor wanted to
continue, but then the student
interjected again. What if you add a
fourth slit and a fifth? The professor,
who is now clearly losing his patience,
replies, I think it's clear to the whole
class that you just add up the
amplitudes from all the slits. It's the
same for 6, 7, etc.
But now the bold student pressed his
advantage. What if I make it infinite
slits so that the screen
disappears and then I add a second
screen with infinite slits and a third
and a
fourth. The students point was clear.
Even when we're not doing a double slit
experiment, when it's just light or
particles traveling through empty space,
they must be exploring all possible
paths. Because this is exactly how the
math would work if you had infinite
screens, each with infinite slits. You
have to add up the amplitude from each
slit. That's just the way it
works. According to the story, the
student was Richard Feman. And while the
story is made up, the logic is
flawless. Because if you believe in a
double slit experiment that you can't
tell which of the two slits the particle
went through, then you have to consider
the possibility that it goes through
both. By that same logic, anytime any
particle goes from place one to place
two, you have to consider all the
possible paths it could take to get
there, including ones that go faster
than the speed of light, including ones
that go back in time, and including ones
that go to the sun and back. Feel like
it can't go to the sun and back. You
have to restrict it to be local, right?
So, the math doesn't do that. I mean,
you could see that just in the double
slit experiment, right? And we'll do
light because then there's no funky
business with the speed. If you're going
to say like this path interferes with
this path, then these distances are
different, right? And so clearly they
can't have the same speed. So you need
to consider paths that have different
speeds.
Fineman's way of doing quantum mechanics
suggests that anything going from one
place to another is connected in every
possible way. The internet is kind of
like that, too. Connecting us to
anything, anywhere at any time. At least
in theory. There are still artificial
barriers like geob blocks and country
restrictions that block off parts of the
internet. But fortunately, there's
today's sponsor, NordVPN, which can help
knock down those barriers. Just connect
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sponsoring this part of the video. And
now, let's get back to Fineman's crazy
way of doing quantum mechanics.
So according to Fineman, anytime a
particle, a photon or even a macroscopic
object moves from 0.1 to 0 2, it has
some chance to take any path. And as
preposterous as it might sound, he found
that we need to include all these paths
in our calculation where each path is
weighted the
same. So why then do we not see all
those crazy paths? Well, that's because
we still need to add up their
amplitudes.
For simplicity, imagine we only have
three paths. Then here's what we're
going to do. First, let's take this one.
When the particle wave starts following
it, we start a stopwatch. It goes around
and around very fast. And when it gets
to the end point, we hit stop. We'll do
the same for the other two paths.
And then we add up the arrows, square
the result, and that is then
proportional to the probability the
particle took those paths to get there.
In this case, the arrow and square are
pretty small. So the probability of the
particle going from 1 to two using these
paths is
small. Compare that with these three
paths, for
example. Well, now the arrow is much
larger.
And this is important. The larger the
resulting arrow, the higher the
probability of that event
happening. Now, in these examples, the
stopwatch is not actually measuring
time. Instead, it measures something
called the
phase. Just as in the double slit
experiment, when a wave takes a
different path from 0.1 to two, it will
end up there with a different
phase. And this phase is what determines
the amplitude of the wave at that point.
Mathematically we can write the
amplitude our stopwatch as e to the i fi
where phi is the
phase. As the particle wave follows a
path its phase increases winding the
vector
around. So now the big question is how
much does the phase change for each
path? Well to answer that imagine we
split up the path into many tiny
sections. each one so small that it's
effectively straight. Then in each
section, the particle wave goes a
distance delta x in a time delta t. And
the increase in phase is easy to
compute. It just depends on the
wavelength and frequency of the wave. To
find the total increase in phase for the
whole path, we just add up all the
little phase increases of all the
individual sections.
But we can sub in lambda= h over mv from
de bruy. And using e= hf we can sub in
for frequency. We can also simplify by
writing h over 2 pi as h bar to get this
expression. Then we can take delta t to
the right. And if we make delta t
infinite decimally small then we can
replace this sum with an
integral. But now dx by dt is just
velocity. So we can write this as MV^ 2.
Now we know that in the simplest case
the total energy E is just kinetic plus
potential energy. And subbing that in
we're left with the integral over time
of kinetic energy minus potential
energy. But wait a second that is just
the classical action. So it's action
that determines how fast the stopwatch
turns.
As the particle moves along a
trajectory, its action increases and
that is what increases the phase. And
what's important to note is that h bar
is tiny. It's about 10 - 34 JW seconds,
which is way smaller than the action of
any everyday
object. That means the phase of ordinary
objects on ordinary paths spins around
zillions of times eventually pointing in
some random direction.
If you consider a slightly different
path, the action may be slightly
different. Say 0.01 jewel seconds
different. That doesn't seem like much,
but divide it by h bar and the arrow
will spin around 10 32 more times. So
again, it will just point in some random
direction. This is what happens to
almost all of the possible paths. So
when you add up the phases, they just
cancel out. they destructively
interfere. The only exception is for the
paths closest to the path of least
action because these paths are at a
minimum. So if you make tiny changes to
the path to first order, the action
doesn't
change. And so for other paths that are
very close to the path of least action,
their arrows point in basically the same
direction. They constructively
interfere. And that is why those are the
paths we
see. This explains how light knows where
to go. I mean it doesn't. It just
explores all possible paths. But the
paths we end up seeing are the ones that
interfere
constructively. And those are the paths
of least
action. So really this is how classical
mechanics emerges from quantum
mechanics. It's why a ball follows the
trajectory it does and how planets orbit
the sun. They don't really have a
precise trajectory. Instead, everything
explores all possible paths. It's just
that massive particles have large
actions compared to H bar, so that only
paths extremely close to the true path
of least action survive, which is why
they're much more particle-like. If you
go to much smaller particles like
electrons or photons, the actions are
much smaller. And so there's more of a
spread in which trajectories they
actually end up taking. Now, you might
say, "I still don't believe you." But
Casper has this incredible demo that
should convince you 100% that this is
really how the world
works. To do it, I've got a light, a
mirror, and a camera. Now there are
infinitely many paths that the light
could take and according to Fman we have
to add up the contributions of each of
them including paths that go like this.
Now you might say he's crazy. I'm not
crazy. That's could happen. Another
possibility is it could come here and go
or it could come here and go or it could
come where you'd like it to come and go
and it can go over here and go and so on
and so on. And these are all
possibilities. And every single one of
these paths has their own little arrow.
So what we can do is we can look at all
those arrows and see where they line up.
And so if I turn on this light, that's
exactly where you see it reflect. So
that the angle of incidence is equal to
the angle of reflection. But now what
I'm going to do is I'm going to cover up
that spot so that we no longer see the
light reflect. And then I'm going to
prove that Fman is right. That really
light also goes like this. It's just
that most of the time those effects are
canceled out. Now, that sounds
impossible, right? But let's zoom in to
this tiny piece right here. Then we see
all these different paths and all the
arrows just go around and around in
circles. So, when you add them up, they
all just cancel out. But what if I cover
up about half of them like so? Well, now
when I add up those arrows, you suddenly
do see a large resulting arrow. And so
if I can somehow cover up this mirror in
many, many tiny strips, then I should be
able to get the light to reflect like
this. And I can do that with this piece
of foil right here. On this piece of
foil, there are about a thousand lines
per millimeter. And that should be
enough to get this effect. So let me
turn off the lights. So let's see. I'm
going to turn it on in three, two,
one. We see it. We see it. Ah, that is
so cool. It actually looks a lot weirder
than I was expecting it to. I was
expecting more like one spot, but
there's many, many spots where it's
reflecting. Oo. Okay. Okay. And just to
show I haven't been cheating you. Right
underneath is my finger. And even with
the light on, you know, we see the light
reflect. And if we remove the cover,
then what do we see?
Yeah, we see exactly the normal
reflection where it's always supposed to
go, which is right there. And then we've
got now all these extra reflections, all
these extra bits where the pattern just
lines up. So very, very cool. When I was
talking about this with a friend
actually, he said, "Yeah, but you're
using a defraction grading. That's kind
of like cheating because you get all
these other reflections right now and
this light is just going in all
directions." And so there's one other
thing I've been super super curious to
try. I also want to do this with a laser
where I shine the laser right next to it
and then if light does take every
possible path, we should also see it
come off here. It probably shouldn't
work. I actually have a laser right over
here. And we can see when I shine
it, it really does just go to one spot.
And you can see where that spot is. It's
right over there, which is about the
same place where we had our reflection.
And you can also see right now if we
look at this view, that you cannot see
the laser light at all. Right? Like I
could see the laser, but I have to bring
it down all the way over here and then
I'm able to sort of see the light. But
if I just put it up here, you can see
the reflection. Now, what I'm going to
do next is I'm going to put this foil,
this magic foil, and I'm going to put it
over here. Oh, and we can turn off this.
And now, let's see what happens when I
turn on the
[Music]
lasers. Wait, wait, wait, wait. No
way. No way. It works. It works. Wait,
what? Look where the laser is going. Oh
my god, it actually
works. What? What? This is definitely
the coolest demo I've ever done. So,
what I was doing is I was holding the
laser and I can show you right now. I
was shining it down like this way off
and you could still see it reflect. But
if I take this away, it disappears. And
if I put this back, it appears.
So that is shows really that we cannot
get rid of the area which gives zero
that it really is cancelelling out. And
if we do clever things to it, we can
demonstrate the reality of the
reflections from this part of the
mirror.
So light and by extension everything
really does explore all possible paths.
It's just that most of the time the
crazy paths destructively interfere.
That's because the actions of nearby
paths change rapidly. Now, I've studied
physics for most of my life, and I feel
like I never really appreciated how
important action and the principle of
least action are. But now, I think I
finally get it. And I finally get why if
you ask theoretical physicists what
they're working on, they'll rarely talk
about energy or forces. Most of the
time, they'll talk about action. Nobody
in particle physics approaches particle
physics from a viewpoint other than
least action. But we teach physics
historically and know least action is
almost like the new kid on the block for
understanding physics. And so yeah, we
build up to it. But in reality, I think
life's a lot easier once you realize
there's this underlying principle
because when you do, then all you have
to do is write down the correct
lrangeian so you get the right action
and out come the laws of physics. So
you've got a separate lrangeian for
classical mechanics, for special
relativity, for electronamics and so on.
It's a single mathematical framework
that once you've learned it, then you
can apply it in different places in
exactly the same way. The hunt for the
theory of everything, right? The thing
that will encompass all of physics. In
reality, what people are asking is what
what is this lrangian that can spit out
all of the laws of physics in this
universe. That's really what they're
asking. At the moment, we haven't really
found that, right? because we can we can
sticky tape things together, but we
don't know if that's the proper
mathematical structure. So, that's what
people are hunting for.