Something Strange Happens When You Trust Quantum Mechanics

[0:00]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 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, 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 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 degrees Celsius, 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 new 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 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 black body 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 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 Rayleigh-Jeans 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 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 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's it, this 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's decided to do is discretize the orbital angular momentum for no good reason. He says, let me slap that on and say, angular momentum electrons 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 2pi come in, so it's really n h over 2pi, what we now call n 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. So I think that really was compelling, and to take some quantity with units of action and apply some again, kind of ad hoc just discretization of 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 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 particle's 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 2pi r must be equal to some multiple n times the wavelength. We can sub in de Broglie's expression for the wavelength to get that 2pi r equals nh over mv. But we can rearrange this to get that mvr, the angular momentum, is equal to nh over 2pi. 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 of 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, etcetera. 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 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, 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 sun and back. I 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. Feynman'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 geoblocks 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. 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