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A massive thank you to Dr. Andrew Mitchell for all his help and expertise. If you want to delve further into this topic then we highly recommend his lecture on Classical physics derived from quantum mechanics: https://www.youtube.com/watch?v=2iuLwWDYC4U&t=1s
We’re incredibly grateful to Prof. David Kaiser, Prof. Steven Strogatz, Prof. Geraint F. Lewis, Elba Alonso-Monsalve, Prof. Christopher S. Baird, Prof. Anthony Bloch, and Prof. Stephen Bartlett for their invaluable contributions to this video.
A special thanks to Mahesh Shenoy from FloatHeadPhysics for his help with this video. Check out his excellent intuitive video on the UV Catastrophe here (Sep 2024): https://www.youtube.com/watch?v=ALEDjjAVZSY
▀▀▀
0:00 What path does light travel?
2:40 Black Body Radiation
6:47 How did Planck solve the ultraviolet catastrophe?
9:42 The Quantum of Action
13:25 De Broglie’s Hypothesis
15:16 The Double Slit Experiment
20:00 How Feynman Did Quantum Mechanics
25:01 Proof That Light Takes Every Path
31:16 The Theory of Everything
▀▀▀
Sign up to our Patreon to submit your Q&A questions: https://ve42.co/PatreonLA2
Try Snatoms! A molecular modelling kit I invented where the atoms snap together.
https://ve42.co/SnatomsV
▀▀▀
References:
Armin Hermann (Nov 1974). The Genesis of Quantum Theory (1899-1913). - https://ve42.co/genquanttheor
Richard P. Feynman (1985). QED The Strange Theory of Light and Matter. - https://ve42.co/qed
A fantastic book on the history of the Principle of Least Action: Rojo, A. and Bloch, A. The
Principle of Least Action: History and Physics. - https://ve42.co/Bloch2018
Coopersmith, J. (2017). The lazy universe: an introduction to the principle of least action. Oxford University Press. - https://ve42.co/LazyU
PBS Space Time. (Nov 2021) - https://www.youtube.com/watch?v=Q_CQDSlmboA
Sabine Hossenfelder. (May 2022) - https://www.youtube.com/watch?v=A0da8TEeaeE
Max Planck (1901). On the Law of Distribution of Energy in the Normal Spectrum. Annalen der Physik - https://ve42.co/lawdistenergy
Paul Ehrenfest (1911). Welche Züge der Lichtquantenhypothese spielen in der Theorie der Wärmestrahlung eine wesentliche Rolle?. Wiley Online Library - https://ve42.co/quantthermrad
Nils-Erik Bomark and Reidun Renstrøm (Aug 2023). The Ultraviolet myth. Proceedings of Science - https://ve42.co/uvmyth
Planck's law. In Wikipedia - https://ve42.co/planckslaw
PBS Space Time. (Jul 2017). - https://www.youtube.com/watch?v=vSFRN-ymfgE&t=10s
Einstein and The Photoelectric Effect. (Jan 2005) via APS News - https://ve42.co/einsteinphotoelec
Louis de Broglie (1924). On the Theory of Quanta. - https://ve42.co/theoryquanta
Larry Sorensen (2012). Path Integrals 1. University of Washington - https://ve42.co/pathintegrals
Feynman’s Method of “A Particle Exploring All Po
A massive thank you to Dr. Andrew Mitchell for all his help and expertise. If you want to delve further into this topic then we highly recommend his lecture on Classical physics derived from quantum mechanics: https://www.youtube.com/watch?v=2iuLwWDYC4U&t=1s
We’re incredibly grateful to Prof. David Kaiser, Prof. Steven Strogatz, Prof. Geraint F. Lewis, Elba Alonso-Monsalve, Prof. Christopher S. Baird, Prof. Anthony Bloch, and Prof. Stephen Bartlett for their invaluable contributions to this video.
A special thanks to Mahesh Shenoy from FloatHeadPhysics for his help with this video. Check out his excellent intuitive video on the UV Catastrophe here (Sep 2024): https://www.youtube.com/watch?v=ALEDjjAVZSY
▀▀▀
0:00 What path does light travel?
2:40 Black Body Radiation
6:47 How did Planck solve the ultraviolet catastrophe?
9:42 The Quantum of Action
13:25 De Broglie’s Hypothesis
15:16 The Double Slit Experiment
20:00 How Feynman Did Quantum Mechanics
25:01 Proof That Light Takes Every Path
31:16 The Theory of Everything
▀▀▀
Sign up to our Patreon to submit your Q&A questions: https://ve42.co/PatreonLA2
Try Snatoms! A molecular modelling kit I invented where the atoms snap together.
https://ve42.co/SnatomsV
▀▀▀
References:
Armin Hermann (Nov 1974). The Genesis of Quantum Theory (1899-1913). - https://ve42.co/genquanttheor
Richard P. Feynman (1985). QED The Strange Theory of Light and Matter. - https://ve42.co/qed
A fantastic book on the history of the Principle of Least Action: Rojo, A. and Bloch, A. The
Principle of Least Action: History and Physics. - https://ve42.co/Bloch2018
Coopersmith, J. (2017). The lazy universe: an introduction to the principle of least action. Oxford University Press. - https://ve42.co/LazyU
PBS Space Time. (Nov 2021) - https://www.youtube.com/watch?v=Q_CQDSlmboA
Sabine Hossenfelder. (May 2022) - https://www.youtube.com/watch?v=A0da8TEeaeE
Max Planck (1901). On the Law of Distribution of Energy in the Normal Spectrum. Annalen der Physik - https://ve42.co/lawdistenergy
Paul Ehrenfest (1911). Welche Züge der Lichtquantenhypothese spielen in der Theorie der Wärmestrahlung eine wesentliche Rolle?. Wiley Online Library - https://ve42.co/quantthermrad
Nils-Erik Bomark and Reidun Renstrøm (Aug 2023). The Ultraviolet myth. Proceedings of Science - https://ve42.co/uvmyth
Planck's law. In Wikipedia - https://ve42.co/planckslaw
PBS Space Time. (Jul 2017). - https://www.youtube.com/watch?v=vSFRN-ymfgE&t=10s
Einstein and The Photoelectric Effect. (Jan 2005) via APS News - https://ve42.co/einsteinphotoelec
Louis de Broglie (1924). On the Theory of Quanta. - https://ve42.co/theoryquanta
Larry Sorensen (2012). Path Integrals 1. University of Washington - https://ve42.co/pathintegrals
Feynman’s Method of “A Particle Exploring All Po
Category
🗞
NewsTranscript
00:00As a 42 year old who's spent most of my life studying physics,
00:03I must admit that I had a big misconception.
00:06I believed that every object has one single trajectory through space.
00:11One single path.
00:12But in this video, I will prove to you that this is not the case.
00:17Everything is actually exploring all possible paths all at once.
00:22So let's start with a simple thought experiment.
00:25Say you're at a beach when all of a sudden you see your friend struggling out in the water.
00:31You want to go help him as quickly as possible, so which path should you take to get there?
00:38The shortest path is a straight line, so you could head directly towards him.
00:43But you can run faster than you can swim, and this path requires more swimming.
00:48So alternatively, you could run down the beach to minimize the distance through the water.
00:54But now the total distance is longer than it needs to be.
00:58So the optimal path, it turns out, is somewhere in between.
01:03To be precise, it depends on the speeds at which you can run and swim.
01:08Now you might recognize this mathematical relationship
01:11because it is the exact same law that governs light passing from one medium into another.
01:17So light also takes the fastest path from point A to point B.
01:21What's weird about this is that as humans,
01:24we can see where we want to go and then figure out the fastest route.
01:28But light? I mean, how does light know how to travel to minimize its journey time?
01:34Now here is where my misconception comes in.
01:37I shine a laser beam, the light just goes in one direction.
01:39I throw a ball, the ball just goes in one direction, you know.
01:42I would have answered there is nothing strange about this.
01:45Light sets off from point A in some direction,
01:48and then a little while later it encounters a new medium.
01:52And due to local interactions with that medium, it changes direction ending up at point B.
01:58If you later find that of all the possible paths,
02:01light took the shortest time to get from A to B,
02:04I wouldn't think it was optimizing for anything.
02:07I would just think that's what happens when light obeys local rules.
02:11But now I will prove to you that light doesn't set out in only one direction.
02:16Instead, it really does explore all possible paths.
02:20And the same is true for electrons and protons, all quantum particles.
02:25So the fact that we see things on single well-defined trajectories
02:29is in a way the most convincing illusion nature has ever devised.
02:34And the way it works all comes down to a quantity known as the action.
02:40In a previous video, we showed how an obscure scientist, Muppertwee,
02:44made an ad hoc proposal that there should be a quantity called action,
02:48which he defined as mass times velocity times distance.
02:53And he claimed that everything always follows the path that minimizes the action.
02:59Hamilton later showed that this action is equivalent to the integral over time
03:04of kinetic energy minus potential energy.
03:07Action was useful and an alternative way of solving physics problems,
03:12especially when Newton's laws get too cumbersome.
03:15But then around the turn of the 20th century,
03:18action showed up at the heart of a scientific revolution, the birth of quantum mechanics.
03:25It all started with electric lighting in Germany.
03:28Think about what it's like in the 1890s, right?
03:30Electricity being more widely available, at least in urban centers,
03:34and things like, you know, light bulbs.
03:36They were new. They were literally the hot new thing.
03:39Germany wanted to replace all their gas street lights with electric light bulbs.
03:44So an important question was,
03:46how do you maximize the visible light given off by a hot filament?
03:50Scientists at a German research institute, the PTR,
03:54studied how much light different materials emitted as a function of temperature.
03:59At low temperatures, each material gave off its own characteristic spectrum,
04:03mostly in the infrared.
04:05But above about 500 degrees Celsius, all materials started to glow in the same way,
04:12with an almost identical distribution of light.
04:16The hotter the object, the more energy was emitted at every wavelength.
04:20And the peak of the distribution shifted to the left.
04:25But they still didn't understand how it worked theoretically.
04:28So that was sort of the next step, right?
04:30If you can understand how it works theoretically,
04:33then you can use that theory to potentially design new products.
04:36They started by imagining the simplest object possible,
04:40one that would absorb all light that falls onto it,
04:43and perfectly emit radiation based on its temperature.
04:47They came up with a hole in a metal cube.
04:52This hole is a perfect blackbody,
04:54because any light that shines onto it will go straight in,
04:57bounce around inside, and eventually be absorbed.
05:00But this also makes it a perfect emitter.
05:02Any radiation inside the cube can escape through the hole unimpeded.
05:07Theorists reasoned that electrons in the walls of the cube
05:10would wiggle around emitting electromagnetic waves.
05:14These waves would then bounce off the other walls.
05:18When you have two waves of the same frequency,
05:21where one travels to the right and the other to the left,
05:24they can interfere in such a way that they create places
05:27where there's no wave amplitude, those are nodes,
05:31and places where there's maximum wave amplitude, the antinodes.
05:35Waves like this are called standing waves,
05:37because they don't really move left or right.
05:40And inside a cavity, given enough time and reflections,
05:44it is only these standing waves that survive,
05:47all the other ones just cancel out.
05:50So a sort of order emerges from the chaos.
05:56In two dimensions, standing waves look something like this.
06:00For shorter wavelengths or higher frequencies,
06:03you can fit more and more different vibrational modes inside this cube.
06:08So that in three dimensions, the total number of modes
06:11is proportional to frequency cubed, or one over lambda cubed.
06:16The expectation was there would be more and more waves inside the cube
06:20the shorter the wavelength.
06:22This led directly to the Rayleigh-Jeans law.
06:26At longer wavelengths, it matched the experimental data pretty well.
06:30But at shorter wavelengths, the theory diverged from experiment.
06:35In fact, it predicted that at the shortest wavelengths,
06:38an infinite amount of energy would be emitted.
06:42This, for obvious reasons, became known as the ultraviolet catastrophe.
06:49The person to solve this problem was Max Planck.
06:53But Planck almost didn't make it into studying physics,
06:56because when he was 16 years old, he went up to his professor and asked him,
07:01well, maybe I could do a career in physics.
07:04To which his professor responded that he'd better find another field to do research in,
07:08because physics was essentially a complete science.
07:11You know, there was just a few tiny little problems that they had to clean up.
07:15But besides that, it was over.
07:19But Planck didn't listen.
07:21By 1897, he was a professor himself,
07:24and for the next three years,
07:25he struggled to find a theoretical explanation for blackbody radiation.
07:30He tried approach after approach, but no matter what he tried, nothing worked.
07:36He said, I was ready to sacrifice every one of my previous convictions about physical laws.
07:44Then, in a quote, active desperation, he did something no one had thought to try.
07:50According to classical physics,
07:51the energy of an electromagnetic wave depends only on its amplitude,
07:56not its wavelength or frequency.
07:58And it could take any arbitrary value.
08:01So any atom could emit any wavelength of light with an arbitrarily small amount of energy.
08:08But Planck tried restricting the energy
08:10so that it could only come in multiples of a smallest amount, a quantum.
08:16And he made the energy of one quantum directly proportional to its frequency,
08:21E equals hf, where h is just a constant.
08:27Think about what this does to the radiation coming from the blackbody.
08:30At a given temperature, the atoms in the cavity have a range of energies.
08:35Some have a little bit, a few have a lot, and most have their energy somewhere in between.
08:40For long wavelength, low frequency radiation, the energy hf of one quantum is small.
08:47So all of the atoms have enough energy to emit this wavelength,
08:51and the spectrum matches the Rayleigh gene's prediction very well.
08:55But at shorter wavelengths, higher frequencies, the energy of a quantum increases.
09:01And now, not all of the atoms have enough energy to emit that wavelength.
09:06This is why experiment diverges from the classical prediction.
09:10The spectrum peaks and then starts to fall,
09:13because fewer and fewer atoms have enough energy to emit one quantum of that radiation.
09:19And there comes a point when none of the atoms have enough energy to emit one quantum.
09:24So here, the spectrum must drop to zero.
09:29With this approach, Planck got a new formula for the radiation spectrum.
09:33Now, all that was left for him to do was to tune the parameter h.
09:36And when he did this just right, he got his formula to match up perfectly with experiment.
09:44But he was sort of troubled by his own formula,
09:47because to him it was just a mathematical trick.
09:50He had no clue why it worked.
09:51It was purely formal.
09:53And most importantly, he had no clue what this h represented.
09:58I mean, he had introduced a new physical constant without any reason.
10:03He wrote, a theoretical interpretation had to be found at any cost, no matter how high.
10:09So from that moment on, he dedicated himself to finding one.
10:14He later reflected that after some weeks of the most strenuous work of my life,
10:18light came into the darkness, and a new, undreamed of perspective opened up before me.
10:27He introduces what we now call Planck's constant, and it has the units of action.
10:33Planck's constant h is a quantum of action.
10:38Planck later proposed that anytime any change happened in nature,
10:42it would be some whole multiple of this quantum of action.
10:46So it's kind of spooky.
10:48This breakthrough that starts the ball rolling toward quantum theory
10:53brings action in, not energy and not force.
10:57Action gives you a hint.
11:01At first, the quantum of action got little attention.
11:05That is, until a 26-year-old patent clerk came on the scene.
11:09In 1905, Albert Einstein claimed that Planck's theory wasn't just a mathematical trick.
11:15It was telling us that light actually comes in discrete packets, or photons, each with an energy hf.
11:22Einstein used this insight to explain the photoelectric effect,
11:26how light can eject electrons from metal, but only when the frequency is high enough.
11:31If the frequency is too low, no electrons will be emitted, regardless of the intensity.
11:37The idea of quantization spread.
11:41Eight years later, Niels Bohr was trying to understand how an atom is stable
11:45if it has a positive charge in the center and negative electrons whizzing around it.
11:50Why don't they just spiral into the nucleus, radiating their energy as they go?
11:54And what he wants to do is, he says, there's something fishy about something being discrete.
11:59That seems to be the new, ambiguous, weirdo lesson of the new quantum of action.
12:04Bohr realizes that as the electron goes around the nucleus,
12:08it has an angular momentum, mass times velocity times radius.
12:13So angular momentum has the same units as action.
12:18And so what he decides to do is discretize the orbital angular momentum, for no good reason.
12:23He says, let me slap that on and say,
12:25angular momentum of the electron can only be in one unit, two units, three units of the same quantity,
12:30can only be in one unit, two units, three units of the same quantity, h.
12:34And because it's talking about motion in a circle, the factors 2 pi come in,
12:38so it's really n h over 2 pi, what we now call n h bar.
12:42This comes out of nowhere.
12:44There seems like absolutely no good reason why angular momentum should be quantized.
12:50But by doing it, Bohr finds the correct energy levels of the hydrogen atom.
12:54When an electron jumps from a higher orbit to a lower one,
12:58the energy difference is given off as a photon of a particular color of light,
13:03exactly reproducing the hydrogen spectrum.
13:06And that was a pretty startling thing to have fall out.
13:09I think that really was compelling.
13:11And take some quantity with units of action
13:14and apply some, again, kind of ad hoc discretization or quantization to it.
13:20Now, although it worked spectacularly well, no one could make sense of it.
13:25That is, until 11 years later.
13:29For his PhD, Louis de Broglie was contemplating the recent discoveries in physics.
13:34And his big insight was that if light could be both a wave and a particle,
13:39then maybe matter particles could also be waves.
13:44He proposed that everything, electrons, basketballs, people,
13:48absolutely everything has a wavelength.
13:51And he defined this wavelength analogously to light,
13:54as Planck's constant divided by the particle's momentum, or mass times velocity.
14:02Now, if an electron is a wave, the only way it could stay bound to a nucleus in an atom
14:09is if it exists as a standing wave.
14:11That requires that a whole number of wavelengths fit around the circumference of the orbit.
14:17You could have one wavelength, or two wavelengths, or three, and so on.
14:21So, the circumference 2 pi r must be equal to some multiple n times the wavelength.
14:28We can sub in de Broglie's expression for the wavelength to get that 2 pi r equals nh over mv.
14:34But we can rearrange this to get that mvr, the angular momentum, is equal to nh over 2 pi.
14:43That is precisely Bohr's quantized angular momentum condition.
14:51But now we have a good physical reason why it's quantized.
14:55Because electrons are waves, and they must exist as standing waves to be bound in atoms.
15:00Because they want to have constructive interference of a stable orbit back.
15:04That's pretty good. You get a dissertation out of that. That's pretty good.
15:07It is this wave nature of quantum objects that means they no longer have
15:11a single path through space. Instead, they must explore all possible paths.
15:19Now, I have thought about and taught the double slit experiment hundreds of times
15:23without fully realizing this implication.
15:26In the double slit experiment, I feel like the mental thing that I'm doing in my head is like,
15:30okay, well, the beam is not perfectly straight,
15:32and of course it's going to intersect both of those slits,
15:35because they're really close together, you know?
15:37But then I heard this story about a professor teaching the double slit experiment,
15:41and it makes everything so clear.
15:45So the professor starts by explaining the setup.
15:48Electrons are fired one at a time through two slits to be detected at a screen.
15:54Now, because you can't say for certain which slit the particle went through,
15:57quantum mechanics tells us it must go through both at the same time.
16:01So to get the probability of finding a particle somewhere on the screen,
16:05you simply add up the amplitude of the wave going through one slit
16:09with the amplitude of the wave going through the other slit, and square it.
16:15But that's when a student raised his hand.
16:18What if you add a third slit?
16:21Well, you just add up the amplitudes of the waves going through each of the three slits,
16:25and you can work out the probability.
16:28The professor wanted to continue, but then the student interjected again.
16:32What if you add a fourth slit and a fifth?
16:36The professor, who is now clearly losing his patience, replies,
16:40I think it's clear to the whole class that you just
16:42add up the amplitudes from all the slits.
16:44It's the same for six, seven, etc.
16:49But now the bold student pressed his advantage.
16:53What if I make it infinite slits?
16:57So that the screen disappears.
16:59And then I add a second screen with infinite slits, and a third, and a fourth.
17:06The student's point was clear.
17:10Even when we're not doing a double slit experiment,
17:12when it's just light or particles traveling through empty space,
17:16they must be exploring all possible paths.
17:20Because this is exactly how the math would work
17:22if you had infinite screens, each with infinite slits.
17:25You have to add up the amplitude from each slit.
17:28That's just the way it works.
17:31According to the story, the student was Richard Feynman.
17:35And while the story is made up, the logic is flawless.
17:39Because if you believe in a double slit experiment,
17:41that you can tell which of the two slits the particle went through,
17:45then you have to consider the possibility that it goes through both.
17:49By that same logic, any time any particle goes from one slit to the other,
17:54any time any particle goes from place one to place two,
17:57you have to consider all the possible paths it could take to get there.
18:01Including ones that go faster than the speed of light,
18:04including ones that go back in time,
18:05and including ones that go to the sun and back.
18:08Really, it can't go to the sun and back.
18:09You have to restrict it to be local, right?
18:11So the math doesn't do that.
18:13I mean, you could see that just in the double slit experiment, right?
18:16And we'll do light, because then there's no funky business with the speed.
18:20If you're going to say, like, this path interferes with this path,
18:23then these distances are different.
18:25Right.
18:25And so clearly, they can't have the same speed.
18:28So you need to consider paths that have different speeds.
18:34Feynman's way of doing quantum mechanics suggests that
18:37anything going from one place to another is connected in every possible way.
18:42And the internet is kind of like that too,
18:44connecting us to anything, anywhere, at any time.
18:47At least in theory.
18:49There are still artificial barriers like geoblocks and country restrictions
18:53that block off parts of the internet.
18:55But fortunately, there's today's sponsor, NordVPN,
18:58which can help knock down those barriers.
19:00Just connect to one of their thousands of servers,
19:02for example, this one in the US,
19:04and it looks as if you're accessing the internet from there.
19:08The team and I travel a lot to make these videos,
19:10and using a VPN is a game changer.
19:13It allows us to access the news sites and articles we need,
19:16no matter where in the world we are.
19:18Personally, I also love that NordVPN allows me to stay up to date
19:21with how the Canucks are doing back in Canada.
19:24Not very well at the moment.
19:27Canucks have a real shot at the cup this year.
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19:56I want to thank NordVPN for sponsoring this part of the video.
19:59And now let's get back to Feynman's crazy way of doing quantum mechanics.
20:04So according to Feynman,
20:06anytime a particle, a photon, or even a macroscopic object moves from 0.1 to 0.2,
20:11it has some chance to take any path.
20:14And as preposterous as it might sound,
20:17he found that we need to include all these paths in our calculation,
20:21where each path is weighted the same.
20:25So why then do we not see all those crazy paths?
20:28Well, that's because we still need to add up their amplitudes.
20:33For simplicity, imagine we only have three paths.
20:36Then here's what we're going to do.
20:38First, let's take this one.
20:40When the particle wave starts following it, we start a stopwatch.
20:44It goes around and around very fast.
20:46And when it gets to the endpoint, we hit stop.
20:49We'll do the same for the other two paths.
20:56And then we add up the arrows, square the result,
21:00and that is then proportional to the probability the particle took those paths to get there.
21:05In this case, the arrow and square are pretty small.
21:08So the probability of the particle going from 1 to 2 using these paths is small.
21:13Compare that with these three paths, for example.
21:17Well, now the arrow is much larger.
21:21And this is important.
21:22The larger the resulting arrow, the higher the probability of that event happening.
21:27Now, in these examples, the stopwatch is not actually measuring time.
21:31Instead, it measures something called the phase.
21:35Just as in the double slit experiment,
21:36when a wave takes a different path from point one to point two,
21:40it will end up there with a different phase.
21:44And this phase is what determines the amplitude of the wave at that point.
21:50Mathematically, we can write the amplitude, our stopwatch,
21:53as e to the i phi, where phi is the phase.
21:59As the particle wave follows a path, its phase increases, winding the vector around.
22:05So now the big question is, how do we measure time?
22:08Now the big question is, how much does the phase change for each path?
22:13Well, to answer that, imagine we split up the path into many tiny sections,
22:17each one so small that it's effectively straight.
22:21Then, in each section, the particle wave goes a distance delta x in a time delta t.
22:27And the increase in phase is easy to compute.
22:30It just depends on the wavelength and frequency of the wave.
22:34To find the total increase in phase for the whole path,
22:37we just add up all the little phase increases of all the individual sections.
22:42But we can sub in lambda equals h over mv from de Broglie.
22:46And using e equals hf, we can sub in for frequency.
22:50We can also simplify by writing h over 2 pi as h bar to get this expression.
22:57Then we can take delta t to the right.
23:00And if we make delta t infinitesimally small, then we can replace this sum with an integral.
23:06But now dx by dt is just velocity, so we can write this as mv squared.
23:13Now we know that in the simplest case, the total energy E is just kinetic plus potential energy.
23:19And subbing that in, we're left with the integral over time
23:22of kinetic energy minus potential energy.
23:26But wait a second, that is just the classical action.
23:31So it's action that determines how fast the stopwatch turns.
23:36As the particle moves along a trajectory,
23:39its action increases, and that is what increases the phase.
23:43And what's important to note is that h bar is tiny.
23:46It's about 10 to the minus 34 joule seconds,
23:49which is way smaller than the action of any everyday object.
23:54That means the phase of ordinary objects on ordinary paths
23:58spins around zillions of times, eventually pointing in some random direction.
24:04If you consider a slightly different path,
24:06the action may be slightly different, say 0.01 joule seconds different.
24:13That doesn't seem like much, but divide it by h bar,
24:16and the arrow will spin around 10 to the 32 more times.
24:20So again, it will just point in some random direction.
24:24This is what happens to almost all of the possible paths.
24:28So when you add up the phases, they just cancel out.
24:32They destructively interfere.
24:35The only exception is for the paths closest to the path of least action,
24:39because these paths are at a minimum.
24:42So if you make tiny changes to the path to first order, the action doesn't change.
24:50And so for other paths that are very close to the path of least action,
24:53their arrows point in basically the same direction.
24:57They constructively interfere,
24:59and that is why those are the paths we see.
25:04This explains how light knows where to go.
25:07I mean, it doesn't.
25:08It just explores all possible paths.
25:11But the paths we end up seeing are the ones that interfere constructively.
25:16And those are the paths of least action.
25:23So really, this is how classical mechanics emerges from quantum mechanics.
25:28It's why a ball follows the trajectory it does, and how planets orbit the sun.
25:33They don't really have a precise trajectory.
25:36Instead, everything explores all possible paths.
25:40It's just that massive particles have large actions compared to h-bar,
25:44so that only paths extremely close to the true path of least action survive,
25:49which is why they're much more particle-like.
25:52If you go to much smaller particles like electrons or photons,
25:55the actions are much smaller.
25:57And so there's more of a spread in which trajectories they actually end up taking.
26:02Now, you might say, I still don't believe you.
26:05But Casper has this incredible demo that should convince you 100%
26:10that this is really how the world works.
26:14To do it, I've got a light, a mirror, and a camera.
26:18Now, there are infinitely many paths that the light could take.
26:21And according to Feynman, we have to add up the contributions of each of them,
26:25including paths that go like this.
26:28Now, you might say he's crazy.
26:31I'm not crazy. That's what happens.
26:33Another possibility is it could come here and go.
26:37Or it could come here and go.
26:38Or it could come where you'd like it to come and go.
26:42And it can go over here and go, and so on, and so on.
26:47And these are all possibilities.
26:48And every single one of these paths has their own little arrow.
26:52So what we can do is we can look at all those arrows and see where they line up.
26:56And so if I turn on this light, that's exactly where you see it reflect.
27:00So that the angle of incidence is equal to the angle of reflection.
27:04But now what I'm going to do is I'm going to cover up that spot
27:07so that we no longer see the light reflect.
27:09And then I'm going to prove that Feynman is right,
27:12that really light also goes like this.
27:15It's just that most of the time, those effects are cancelled out.
27:18Now, that sounds impossible, right?
27:21But let's zoom in to this tiny piece right here.
27:25Then we see all these different paths and all the arrows just go around and around in circles.
27:29So when you add them up, they all just cancel out.
27:33But what if I cover up about half of them like so?
27:37Well, now when I add up those arrows, you suddenly do see a large resulting arrow.
27:42And so if I can somehow cover up this mirror in many, many tiny strips,
27:46then I should be able to get the light to reflect like this.
27:50And I can do that with this piece of foil right here.
27:54On this piece of foil, there are about a thousand lines per millimeter
27:57and that should be enough to get this effect.
28:00So let me turn off the lights.
28:03So let's see.
28:04I'm going to turn it on in three, two, one.
28:09We see it!
28:10We see it!
28:12Ah!
28:13That is so cool.
28:14It actually looks a lot weirder than I was expecting it to.
28:17I was expecting more like one spot, but there's many, many spots where it's reflecting.
28:22Ooh, okay.
28:23Okay, and just to show I haven't been cheating you,
28:26right underneath is my finger.
28:29And even with the light on, you know, we see the light reflect.
28:34And if we remove the cover, then what do we see?
28:40Yeah, we see exactly the normal reflection where it's always supposed to go,
28:45which is right there.
28:47And then we've got now all these extra reflections,
28:50all these extra bits where the pattern just lines up.
28:54So very, very cool.
28:56When I was talking about this with a friend, actually,
28:59he said, yeah, but you're using a diffraction grating.
29:01That's kind of like cheating.
29:02Because you get all these other reflections right now.
29:05And this light is just going in all directions.
29:08And so there's one other thing I've been super, super curious to try.
29:13I also want to do this with a laser where I shine the laser right next to it.
29:18And then if light does take every possible path, we should also see it come off here.
29:23It probably shouldn't work.
29:25I actually have a laser right over here.
29:28And we can see when I shine it, it really does just go to one spot.
29:34And you can see where that spot is.
29:35It's right over there, which is about the same place where we had our reflection.
29:40And you can also see right now, if we look at this view,
29:45that you cannot see the laser light at all, right?
29:48I could see the laser, but I have to bring it down all the way over here.
29:52And then I'm able to sort of see the light.
29:54But if I just put it up here, you can see the reflection.
29:59Now, what I'm going to do next is I'm going to put this foil, this magic foil,
30:06and I'm going to put it over here.
30:08Oh, and we can turn off this.
30:09And now let's see what happens when I turn on the laser.
30:17Wait, wait, wait, wait.
30:19No way.
30:22No way.
30:23It works!
30:24It works!
30:25Wait, what?
30:27Look where the laser is going.
30:30Oh my God.
30:31It actually works.
30:34What?
30:36What?
30:37This is definitely the coolest demo I've ever done.
30:39So what I was doing, I was holding the laser, and I can show you right now,
30:43I was shining it down like this, way off, and you could still see it reflect.
30:49But if I take this away, it disappears.
30:52And if I put this back, it appears.
30:57So that shows, really, that we cannot get rid of the area which gives zero,
31:03that it really is canceling out.
31:04And if we do clever things to it,
31:07we can demonstrate the reality of the reflections from this part of the mirror.
31:14So light, and by extension, everything, really does explore all possible paths.
31:20It's just that most of the time, the crazy paths destructively interfere.
31:24That's because the actions of nearby paths change rapidly.
31:28Now, I've studied physics for most of my life,
31:30and I feel like I never really appreciated how important action
31:34and the principle of least action are.
31:36But now, I think I finally get it.
31:39And I finally get why if you ask theoretical physicists what they're working on,
31:42they'll rarely talk about energy or forces.
31:45Most of the time, they'll talk about action.
31:48Nobody in particle physics approaches particle physics
31:51from a viewpoint other than least action.
31:54But we teach physics historically,
31:57and least action is almost like the new kid on the block for understanding physics.
32:02And so, yeah, we build up to it.
32:03But in reality, I think life's a lot easier once you realize this underlying principle.
32:09Because when you do, then all you have to do is write down the correct Lagrangian
32:13so you get the right action and outcome the laws of physics.
32:18So you've got a separate Lagrangian for classical mechanics,
32:21for special relativity, for electrodynamics, and so on.
32:25It's a single mathematical framework that once you've learned it,
32:29then you can apply it in different places in exactly the same way.
32:34The hunt for the theory of everything, right?
32:36The thing that will encompass all of physics.
32:38In reality, what people are asking is,
32:40what is this Lagrangian that can spit out all of the laws of physics in this universe?
32:45That's really what they're asking.
32:46At the moment, we haven't really found that, right?
32:48Because we can sticky tape things together,
32:52but we don't know if that's the proper mathematical structure.
32:55So that's what people are hunting for.
33:01This was our second video on the Principle of Least Action.
33:04And I know we've covered a lot, but you might still have some questions.
33:07So we are hosting a Q&A over on our Patreon.
33:11You can submit your questions there and we'll also post the Q&A there.
33:14And it's all for free.
33:16If you want to submit a question or watch the Q&A,
33:19head over to patreon.com slash veritasium and you can make a free account.
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33:33And finally, I really want to thank you for watching.