At the scale of atoms and particles, the universe stops behaving the way our intuition says it should. Things can be in two states at once, a particle can interfere with itself, and two objects can stay mysteriously linked across any distance. Quantum mechanics is our astonishingly accurate — and astonishingly strange — theory of this hidden layer of reality.This course builds the weirdness up gently. You'll get an intuitive feel for what quantum mechanics actually claims, meet the famous double-slit experiment that sits at the heart of the mystery, explore superposition through Schrödinger's poor cat, and confront entanglement — the 'spooky action at a distance' that troubled even Einstein. Honest note: the maths works flawlessly, but what it all means is still genuinely debated by physicists, so expect wonder rather than tidy answers.
Quantum physics has a mystique of being complicated and hard to understand, in fact Richard Feynmann who won the Nobel prize for his work on quantum electrodynamics said: “If you think you understand quantum physics, you don’t understand quantum physics”. Which is kind of disheartening for us because if he didn’t understand it, what chance do the rest of us have? Fortunately this quote is a little misleading. We do in fact understand quantum physics really well, in fact it is arguably the most successful scientific theory out there, and has let us invent technologies like computers, digital cameras, LED screens, lasers and nuclear power plants. And you know, you don’t really want to build a nuclear power plant if you don’t really understand how it works. So quantum physics is the part of physics that describes the smallest things in our Universe: molecules, atoms, subatomic particles thing like that. Things down there don’t quite work the same way that we are used to up here. This is fascinating because you and everything around you is made from quantum physics, and so this is really how the whole universe is actually working. I’ve drawn these protons, neutrons and electrons as particles, but in quantum mechanics we really describe everything as waves. By the way I'm using quantum physics and quantum mechanics interchangeably, they are the same thing. So instead of an electron looking like this, it should look something like this. This is called a wave-function. But this wave-function isn’t a real physical wave like wave on water or a sounds wave. A quantum wave is an abstract mathematical description. To get the real world properties like position or momentum of an electron we have to do mathematical operations on this wave-function, so for the position we take the amplitude and square it, which for this wave would look something like this. This gives us a thing called a probability distribution which tells us that you are more likely to find the electron here than here, and when we actually measure where the electron is, an electron particle pops up somewhere within this area. So with quantum physics we don’t know anything with infinite detail, we can only predict probabilities that things will happen, and it looks like this is a fundamental feature of the Universe which was quite a departure from the clockwork, deterministic universe in classical physics, the kind of thing Newton derived. This wave-function model predicts what subatomic particles will do incredibly well, but weirdly we've got no idea if this wave-function is literally real or not. No one has ever seen a quantum wave because whenever we measure an electron all we ever see is a point like electron particle. So there is like this hidden quantum realm where the waves exist, and then the world we can see, which is where all the waves have turned into particles. And the barrier between these is a measurement. We say a measurement ‘collapses’ the wave function, but we don’t actually have any physics to describe how the wave collapses. This is a gap in our knowledge that we have dubbed the measurement problem, and this is one of the things that Feynmann was referring to with his quote. Another confusing thing is how exactly to picture an electron. It seems to be a wave until you measure it, and then it is a particle, so what actually is it? This is known as particle-wave duality, and here is an example of it in action: the famous double slit experiment. Imagine spraying a paintball gun at a wall with two openings in it, you’d expect to see two columns of paint go through and hit the wall behind. But if you shrink this all down to the size of electrons you see something quite different. You can fire one electron at a time at the slits and they appear on the back wall, but as they build up over time you get a whole pattern of stripes, instead of just two bands, this pattern of stripes is called an interference pattern, something you only see with waves. The idea is that it is the electron-wave that goes through both slits at the same time, and then the waves from each slit overlap with each other, and where the waves add together you have a high probability of the electron popping up at the wall, but where the waves cancel out the probability is very low. So actually on the back wall the highest probability of finding the electron is in the middle of the slits, and then it goes down and up again, and down and up again and this is the interference pattern. So when you fire one electron after another they follow this probability distribution and this interference pattern starts building up, and that's exactly what we see in experiments. So this shows that electrons behave like waves in this experiment. A question is what actually happens to this spread-out electron-wave when you do a measurement? It seems like it goes from this spread out wave to this localised particle, but like I said, there's nothing in quantum mechanics that tells us how the wave-function collapses. And this is not only true for electrons, but for everything in the Universe, so this double slit experiment has huge consequences for our model of the Universe, and it was very surprising the first time it was done. Physicists are still grappling with this question today and have come up with many interpretations of quantum mechanics to try an explain these results, and explain how reality actually works. Okay lets go back to the wave-function. Now we can use this picture to explain other features of quantum physics that you may have heard about. So this is just one possible wave-function for an electron, but there are many others. Like this one for instance. This says that the electron has a probability of being over here, and a probability of being over here, and very little probability of being in the middle. This is perfectly allowable in quantum physics and this is where the phrase ‘things can be in two places at once' comes from. This is known as superposition, which comes from the fact that this wave can be made by adding, or superimposing these two waves. The word superposition just means the adding together of waves and we already saw this in the double slit experiment, and is not really a very special phenomenon. You can even see superposition by dropping two pebbles into a pond where the ripples overlap. Now for entanglement. Let’s say two electron-waves meet. Their waves interfere with each other and become mixed up. This means that mathematically we now have one wave-function that describes everything about both electrons and they are inextricably linked, even if they move far away from each other. A measurement on one of the particles, like measuring if it is spin up or down is now correlated with a measurement on the other, even if they move billions of miles away. Einstein was very uncomfortable with this idea because if you measure one of the particles here you instantaneously know what the other will be even if it is billions of miles away, and that's got a sort of whiff of faster than light communication, which is not allowed by the theory of relativity. But it turns out you can’t actually use this to communicate information, because the measurements give you random results, but the fact that they are correlated means that somehow there is a link that stretches over that distance. This is called non-locality. Quantum tunnelling. Quantum tunnelling is where particles have a probability of moving through barriers, essentially allowing things like electrons to pass through walls. When a wave-function meets a barrier it decays exponentially in the barrier, but if the barrier is narrow enough the wave-function will exist on the other side meaning there is a probability of the particle being found there when a measurement is made. In fact the only reason you are alive is because of quantum tunnelling in the Sun which make the Sun shine. Protons normally repel each other, but they have a small probability of quantum tunnelling into each other which is what turns hydrogen into helium and releases fusion energy. All life on Earth exists because of energy from the Sun, except for life around hydrothermal vents. Now on to the Heisenberg Uncertainty principle. I said that the beginning that this wave-function contains all of the information like position and momentum of the electron, we just have to do some maths on it. The position is given by the amplitude, or height of the wave, and the momentum is given by the wavelength of the wave. But for this specific wave the position gives us a probability distribution, so we don't know exactly where the electron is. Also there is an uncertainty in the momentum because this wave is made of many different wavelengths. But we can reduce that uncertainty, let’s have a wave that only has one wavelength, so a sine wave. Now we know the momentum exactly because the wavelength has a single value, but look at the position. There is an equal probability of the electron being found anywhere in the universe. Okay let's do the opposite let’s make a wave that has only got one position. Now we know exactly where the electron is, but what is the wavelength of the wave? Now the wavelength is very uncertain. Basically only a sine wave gives you a precise momentum, and any wave that isn't a perfect sine wave, you have to build out of multiple different sine waves, and each of those multiple different sine waves has got a different wavelength, and hence you have a range of possible different values of momentum for the particle. This is Heisenberg’s Uncertainty principle, you can only know certain things precisely, but not everything. Either you have got a definite value of momentum, and don't know anything about position, or you know the position very well, but don't know anything about the momentum, or you are in some intermediate state. And this isn't a limit of our measuring apparatus, this is a fundamental property of the Universe! And finally, where does the name ‘quantum’ come from. Well a quanta is a packet of something like a chunk of something, and one of the first quantum effects people saw were atomic spectra which is where atoms give off light with specific discrete energies. It works like this. Imagine a string that is tied at both ends, like a guitar string. If you pluck it, only certain waves can exist because the ends are tied down, in this situation we say that the wavelengths are quantised to certain values. The same thing happens if you ties the ends of the string together because the waves have to match up, they can only vibrate in certain restricted ways. And this is what is happening to an electron in an atom. The electron-wave is constrained by the atom and quantised to certain wavelengths, short wavelength have high energy and long wavelengths have a lower energy. This is why the light emitted by an atom looks like a barcode because each bar of light corresponds to an electron jumping from a wave with a high energy to one with a lower energy, and at the same time emitting a quantised photon of light when it does this. So the light from an atom is quantised to discrete packets of energy. Okay so that’s all the basics of quantum physics, here are some technical notes which aren’t essential to know, but pause the screen now if you are interested in a little more mathematical detail. So to round up. In quantum physics objects are described with wave-functions, but when we measure them, what we see are particles, so this leads to particle-wave duality, and also the measurement problem. And the consequence of these wave-functions are the quantum phenomena of superposition, entanglement, quantum tunnelling, the Heisenberg uncertainty principle and energy quantisation. So if you understand these things you have got a good basic understanding of quantum physics. Despite its reputation I think quantum mechanics isn’t too difficult for most people to get the basics of what is going on. In the past I have relied upon analogies to try an explain it, but here I have just described what is actually going on which I think might be more helpful. But if you have more questions I'll be on the comments below so ask away. For me the weird thing about quantum physics is that on the one hand it is incredibly accurate and predictive but also it has got giant holes in it like the measurement problem which we just don’t understand. So we can wonder, will we ever actually understand quantum physics, or is it just too abstract for our human brains to comprehend. Well I hope this video has helped you understand a little more about how quantum physics works. And thanks to the sponsor of this video brillaint.org, who have just launched their daily problems which you can dip into if you have a spare 5 minutes each day. Each problem teaches you some interesting facts that you can then use to solve the problem. And if you enjoy that specific problem there are links to more on the same topic so you can develop your framework of knowledge around that subject. And as ever if you are confused and need more guidance, then you can join with the community discussions. So this is a simple, fun way to keep learning more. If that sounds interesting go to brilliant dot org slash dos or click on the link in the description below, and the first 200 to do so will get 20% off the annual subscription which unlocks all of their premium content. Well that's it from me, thanks for watching and I'll see you next time.
I'm going to explain [to] you. What's Known as the central mystery of quantum [when] I was a [richard] [Fineman] the American physicist said this is the central mystery of Quantum Mechanics there's lots of weird stuff [that] goes on in the quantum world Hit you with this and it basically tells you what it's all about. It's called the two-slit experiment I'll start with this imagine you have a source of light shining against the screen with two slits now for dependence in the audience this source of light has to be Monochromatic light light of a particular Wavelength well where's of course a light bulb is white light and that's made up of all the colors And [spectrum] lots of different wavelengths, but imagine this is just a single wavelength of light And you can see the like is coming out in in waves like like Ripples in a pond that's the nature of wave-like behavior as the light hits the screen it Squeezes through the two slits and each slit in turn on the other side becomes almost like a new source of light and the light spreads out it diffracts and as the waves of light overlap They will interfere with [each] other. So where a crest hits a trough. They will cancel where crest hits a crest they will amplify and so on and so [on] the back screen you end up with what's called an interference pattern a series of light and Dark Fringes where the waves have either canceled out or Worked together in Phase that's fine. That's not quantum mechanics. That's a Property of light that goes back over [200] years that we've known about since the early 19th century Imagine doing the same experiment again, but doing it not with waves, but with particles Do it with grains of sand so this is the same experiment, but I've tipped it 90 degrees Rather than waves that are spread out that wash up against the two slits and squeeze through Here you've got individual Particles of said and each particle will either go through one slit or the other and so you see there will sort of drain through And you get two bumps underneath each of the slits so the two peaks is reminiscent of Particle-like, Behavior, Whereas the the multiple Pattern of interference is wave-like behavior What if we do the same experiment with atoms? Well, so imagine we have an atom gun something can fire Atoms as a stream of atoms you can't see them because [it's] very small Let's block off one of the two slits So these two slits are you know the the dimensions and separation of the slits is chosen appropriately to show us how atoms do things and so far so [good] nothing strange here, you'll see a lot of atoms hitting the back screw, so this will now have to be some sort of photosensitive screen where Whereby when an atom hits it they'll it will give off a little flash of light to say the atom has arrived here so the atoms are arriving as these little pinpricks of light that we see of Course a lot of the atoms will be blocked by the first screen. They won't go through [that] slit But those that do get through to the other side you can [see] there's a bit of spreading of the atoms But if we didn't know anything about atoms you say well, that's fine We can understand that some [a] lot of the atoms are going clean through the [slits] some are sort of maybe bouncing off the edge of the slit and so They're sort of being deflected a bit. Which is why you get a bit a bit of a spread? The first mystery of Quantum mechanics comes when we open the second slit Because now we see something that's very much like the interference pattern we got with light rather than having two bands of Spots where the atoms have gone through the two slits it's as though the atoms have gone through the slits behaving like waves and And into [and] and you get interference of the waves and you get these Bands if we know nothing about atoms or quantum mechanics You could try and rationalize and say well, you know maybe atoms behave in a very strange way and Only a certain number of them are allowed to all sit together And so you know me and my gang we all going to go on this slit, no Sorry no room for you. You go next lit above and by the way. There's this rule that no one can go in Between the two met [bands], but a few naughty atoms do so there's a bit of a scatter. Yeah, we don't there could be some forces between atoms that make them coordinate their actions in a way to give this pattern That's not mysterious. That's just we just don't know how atoms do things, but we can be clever and we can force the issue What if we were to not send the atoms all through at once, but send them through one at a time Leave enough of a gap for the atem to get through to hit the screen of course as I say some atoms will Hit the hit the first screen and not get through but those that get through will hit the back screen so let's run the experiment again slowly and Gradually you'll see as the atoms go through they'll be Look like they're just randomly arriving on the other side you keep sending atoms through one at a time and gradually That same pattern appears so each atom by itself is Somehow contributing its small part to the overall Wave-like behavior that we see in the interference pattern How does it do it how does? We know the atoms are tiny? Localized particle. We can't say it's too small to even see under a microscope. [we're] firing it at the screen with the two slits Some moment later you see a flash of light on the back screen. It's arrived in a localized point It's not spread itself out you Don't get so like a wash of a sort of a faint light across the whole screen so a little point the atom is localized arriving in a certain location and yet It somehow seems to have been aware of there being two slits not one Because it's given [rise] to this interference pattern how does one Assam do that does it split in half does it become like a cloud [that] goes through both well we can try and be Even cleverer, what if we were to spy on the atom and see where it goes? We're going to gently just observe which slit it goes through So you put a detector? Just above the upper slit that will flash or beep Whenever it sees an atom go through that top slit Sure enough you fire the atoms through one at a time 50% of the time the detector will beep The other 50% of time it doesn't the assumption being that the atoms has gone through The lower slit but of course I've been cheeky here. I haven't shown you [the] results of the experiment That's where you get 50% of the time it beeps, and you see a spot arrive Adjacent to the upper slit the other half of the time It doesn't beep but you see a spot arrive at the lowest it. So yeah it's picked out the atoms that have gone through the upper slit and Not the ones that go until each atom does go through one slit or the other But that's a different result to what we had earlier So here's the last bit of sneakiness that we can play with atoms surely now. You know we're going to get to grips with it Leave the detector there, but just very quietly go and unplug it Don't let the atoms know that you're not spying on them Make them think that you're still detecting them [someday]. Yeah, okay. We're gonna run experiment a soms, okay get ready one at a time We're going to be checking on you Alright, so run the experiment again now if you can explain this using common sense and logic Do let me know Because there's a nobel prize for you Quantum [entanglement] is the idea that particles however far apart? They [are] Still somehow their fates remain intertwined they they are still aware of each other's existence
As a 42 year old who's 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 them 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, Maupertuis, made an ad hoc proposal that there should be a quantity called action, which he defined as mass times velocity times 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 sectors. 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? 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 blackbody 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 is 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 Rayleigh-Jeans 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 Planck, but Planck 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 Planck didn't listen. By 1897, he was a professor himself, and for the next three 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 Planck 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 equals hf, where h is just a constant. Think about what this does to the radiation coming from the blackbody 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 of 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 really 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, Planck 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 Planck's constant, and it has the units of action. Planck's constant, h is a quantum of action. Planck later proposed that any time 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 Planck'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. Eight years later, Niels Bohr 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 discrete that seems to be the new ambiguous weirdo lesson of the new quantum of action. Bohr realizes that as the electron goes around the nucleus, it has an angular momentum. Mass times velocity times radius. So angular momentum has the same units as action. And so what he decides to do is discretize the orbital angular momentum. For no good reason he says, let me slap that on and say, and imagine the electron can only be in one unit, two units, three units of the same quantity H. And because it's talking about motion in a circle, the factors two pi come in. So is really nh over two pi, what we now call an h bar. This comes out of nowhere. There seems like absolutely no good reason why angular momentum should be quantized. But by doing it, Bohr 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. I think that really was compelling. Take some quantity with the unit of action and apply some, again, kind of ad hoc, discretization or quantization to it. Now, although it worked spectacularly well, no one can make sense of it. That is until 11 years later. For his PhD, Louis de Broglie 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 Planck's constant, divided by the particles momentum or mass times 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 two pi r must be equal to some multiple n times the wavelength. We can sub in de Broglies expression for the wavelength to get the two pi r equals NH over mv, but we can rearrange this to get the mvr. The angular momentum is equal to NH over two pi. That is precisely Bohr'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, have a stable orbit back. That's pretty 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 six, seven, 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 student's 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 Feynman, and while the story is made up, the logic is flawless. Because if you believe in the 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, any time 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 other side of the moon and back. I feel like I 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 and 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. Feynman's way of doing quantum mechanics suggests that anything going from one place to another is connected in every possible way. And 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 geo 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 to one of their thousands of servers, for example, this one in the US. And it looks as if you're accessing the internet from there. The team and I travel a lot to make these videos, and using a VPN is a game changer. It allows us to access the news sites and articles we need, no matter where in the world we are. And personally, I also love that NordVPN allows me to stay up to date with how the Canucks are doing back in Canada. Not very well at the moment. Canucks have a real shot at the Cup this year. But to try NordVPN for yourself, sign up at nordvpn.com/veritasium. Click that link in the description or scan this QR code. And when you do, you get a huge discount on a two year plan and an additional four bonus months for free. It's the best deal and it also comes with a 30 day money back guarantee. So head to nordvpn.com/veritasium to try it out risk free. I want to thank NordVPN for sponsoring this part of the video. And now let's get back to Feynman's crazy way of doing quantum mechanics. So according to Feynman, any time a particle, a photon, or even a macroscopic object moves from point 1 to point 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 2 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 point 1 to 2, 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 phi, 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 and 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 equals h over mv from de Broglie, and using e we can sub in for frequency. We can also simplify by writing h over two pi as h bar. To get this expression. Then we can take delta t to the right. And if we make delta t infinitesimally small, then we can replace this sum with an integral. But now Dx by Dt is just velocity. So we can write this as m b squared. 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 ten to the -34 joule 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 joule seconds different. That doesn't seem like much, but divide it by h bar and the arrow will spin around ten to the 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 past 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 hbar, 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 taken a light, a mirror and a camera. Now there are infinitely many paths that the light could take. And according to Feynman, we have to add the contributions of each them. Including paths that go like this. Now, you might say he's crazy. I'm not crazy. That's what happens. Another possibility is I 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 reflects that at 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 really Feynman is right. That really light also goes like this. It's just that most of the time, those effects are cancelled 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 321. We see it. 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. Oh, 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 diffraction grating. That's kind of like cheating. 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 out 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 and we can turn off this. And now let's see what happens when I turn on the laser. 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 it shows really that we cannot get rid of the area which gives zero that it really is canceling 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 no 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 this underlying principle, because when you do, then all you have to do is write down the correct Lagrangian so you get the right action and out come the laws of physics. So you've got a separate Lagrangian for classical mechanics, for special relativity, for electrodynamics, 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 is this Lagrangian that can spit out all of the laws of physics in this universe? That's really what they're asking. 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.