 In the previous episode, we learned that particles can spin in two possible ways, but we can never know which one until we measure it, and it changes depending on how we measure it. How can we possibly make sense of such a property, and how can we describe that mathematically? That's it! That's the question we will answer in this video, and in doing so, we will discover that nature follows a symmetry known as SU2. But more importantly, we will discover what object follows that symmetry, and that will be the key for QFT. In 1922, when the Schtenger-Lach experiment was made, scientists had already all the evidence they needed to figure out spin, but they didn't. It took them until 1927. These five years were the era of mistakes, as scientists tried to figure out the mystery of the fourth quantum number. While researching for this video, I found the story of those five years to be so interesting, so fun, and so compelling, that I decided to make an entire video just about it. But for now, let's see how the mystery was solved. Our protagonist today will be Charles Darwin. No, not that one, this one. Yeah, although oddly enough, they were grandson and grandfather. For the past five years, Darwin had seen how people tried to use the quantum numbers to make sense of the Seaman effect and the Schtenger-Lach experiment, and this is where they were at. One number for the orbit, with one value every time. One number for the angular momentum, never larger than the previous one. One number for the orientation, symmetric from side to side. And one mysterious number for an unknown purpose. One number to split the lines. One number to pair the atoms. One number for the magnetic moment and the right factor, somewhere in the atom where the mysteries lie. In many scientific articles of the time, people referred to the fourth quantum number as the angular momentum of the core. But this was a very big term that was never properly defined. Professor Landé, I was wondering, what do you mean by the core? Is that the same as the atomic nucleus? What? No, come on. The nucleus is not the same as the core, otherwise we would have said the nucleus. Oh, so is it like a core made of the electrons in the inner orbits, or something like that? No, the core is the core, like you know the core of the atom, right? The core, like, you know, no, you know, the core, the core. Come on, like, the core, like, the core, the core, like we all know the core, the core. The core is, you just invented a big term for one question, and the core is just the same as the nucleus, because I don't know what the core is. Well, no, no, no, it's fine. I mean, you at least did discover why the magnetic fields of atoms have the strength they have, right? You didn't swept everything you didn't know under the metaphorical rug of a single variable, right? But then, a couple of students who were just learning about these problems for the first time, Chronic, Gudsmith, and Ullenbeck, independently came up with the same solution. Maybe electrons were spinning over their own axes, and this was the fourth quantum number. I have a theory about why a bunch of students without much experience were able to do what the geniuses of the era couldn't, but that will have to wait for the bonus episode. For now, what matters is that with spin as the fourth quantum number, they could finally make sense of the semen effect and the Stenger-Lach experiment, as long as they restricted spin to follow a bunch of rules they have no good reasons for, like only having two allowed values. This is where Darwin comes in, because when he heard this idea that spinning was the fourth quantum number, he probably remembered everything he had learned about angular momentum in school, and it all revolves around the right hand rule. So it goes like this. Let's say that we have an object spinning like this, and yeah, I was never able to just spin the ball in just one finger, I'm sorry, but okay, so we have an object spinning like this, and we want to describe this motion. Well, then we can use the axis of rotation. That's the vector along which this object is rotating, but there are two different ways an object can rotate along an axis of rotation, see? Okay, so this is where the right hand comes in. So if I point my finger, my thumb, like this, and my fingers curl in the direction in which the object is rotating, like this, see? Then we say that this object has spinned up. But if in the contrary, if I point my thumb down, and my fingers curl in the direction in which the object is rotating, like this, well, then we say that this object has spinned down, and we represent that we are another vector. But I want to make clear here that there's nothing special about the right hand, or just like hands in general. Like, we could have chosen to define it the other way around. We could have chosen this to be a vector pointing down, and this a vector pointing up. But like, we had to take to choose something, like we have to choose one of those two options. And so we chose one, and there was a very convenient way to remember it using the right hand. But like, if you don't have a right hand or something, like you can use your left hand, you can use the left hand rule. It is perfectly valid. Just remember that at the end, you have to flip everything so that it matches the convention that the rest of the everyone else is using. And I insist on this because I had a teacher that treated the right hand rule as some sort of dogma that came from on high. And that's not right. Like, physics is full of conventions, full of choices that could have been made the other way around. And like, that's something that we should recognize. Anyway, the point is that now Darwin could use angular momentum as a sort of tool to understand the Steng or Lag experiment. When an object with electric charge, like an electron, moves in circles, this creates a magnetic field. And so now Darwin could understand that, yeah, the electrons were spinning in different directions, and that gave them magnetic fields. And depending on their magnetic field, they would be either attracted or repelled by the magnetic field of the Steng or Lag device, which I'm just going to call the SG device from now on. Okay, so now we can think of like a beam of electrons coming into the SG device. And initially, they have spins in random directions, right? And so that's why you would expect them to be deflected in a range of different directions as they pass through the magnetic field. But that's not what we observe. What we observe is that half of the electrons come out with spin up, and half of them come out with spin down. And if we rotate the magnet, then we see that the beam of electrons is still splitting in half in the direction of each magnetic pole. And this is really weird when you think about it, because what is happening is that we are basically changing the angular momentum of the electrons in the direction in which we measure it. And we change it like in two different ways, and it's like at random. But the key insight of Darwin was realizing that this was very similar to another much simpler device, a light polarizer. The polarization of light is a very interesting subject. And if you let your guard down, I will talk about it all day. But luckily, three blue and brown recently made a couple of views about it, which means I have to do less work. I mean, which means that you can go watch them if you want to learn more. But for now, what matters is this. You have probably seen light represented as something like this, a wave moving up and down, or maybe even two waves perpendicular to each other, one representing the magnetic field and the other representing the electric field. Although if you know one of the waves, you can figure out what the other should be. So we almost always just use the electric part of the wave. And the direction in which the electric field is oscillating is known as the polarization of light. And it can be polarized up and down, side to side, diagonally, even circularly or elliptically. When this was explained to me, I was confused for years thinking that light itself moved a bit up and down as it traveled. But no, light moves in a straight line. And what happens is that as it moves, the strength and the direction of the electromagnetic field changes periodically. And this is what makes light a wave. The problem is that our experience relies so much on physical movement that this doesn't sound like a real wave. But it is. Any quantity that changes periodically over space is a wave, even if that quantity is, for example, the position of air molecules or the position of water molecules or the strength of the electromagnetic field. And one experiment that shows this is actually Fresnel's dog, which we saw in the previous video. You send this light wave and it is blocked by an object. But because of how waves behave, how they propagate, they rejoin behind it, creating a bright spot in the middle of the shadow. Another example is that we can send a light wave through two slits, breaking it into two waves. And so each of these waves is going to contribute a different value to the electromagnetic field at each point in space. In some points, these values are going to cancel out. And in some places, they are going to add up. And this is going to create an interference pattern. But in my opinion, the best way to convince yourself that light is indeed a wave doesn't require any fancy experiments, just a cool rock. This is a calcite crystal. And it has a very strange property. See how it makes us see double? What happens is that the light bouncing off of this object is randomly polarized. The photons have all kinds of different polarizations. And so we call this unpolarized light. But it's not going to stay like that for long, because the atoms in this crystal act in a very special way. They absorb the light wave and they resonate with it in very complex ways we have no time to get into. But what matters is that the crystal rotates the polarization of each photon so that it is either aligned with this special direction of the crystal or it is exactly at 90 degrees from it. But wait, there's more! This crystal also deflects the photons with a second polarization at a slightly different angle, basically splitting the light wave into two waves with opposite polarizations. That's why one of the images moves as I move the crystal. The calcite is deflecting that beam of light at an angle but keeping the other one straight. In the early 1800s, people were beginning to understand all of these aspects about light. And that's why, in 1828, a guy called William Nicholl, this William guy figured out a way to use a calcite crystal to get light polarized in a single direction. He had created the first polarizer. Here's how it worked. William noticed that calcite is transparent, but only if light hits it at a very specific angle, because if the angle becomes too wide, then calcite behaves like a mirror. Not a good mirror, but a mirror nonetheless. And also calcite deflects one polarization of light at an angle. And this gave William an idea. He took a calcite crystal and he caught it at a very specific angle and then he glued it back together with a special Canadian glue. And like, I have no idea why this Canadian glue was so important, but all the sources mention that this glue was Canadian, so it must have been important somehow. Anyway, so he glued it back together and then he sent the beam of light through it. What happened was that, well, calcite did what calcite does and it separated that beam of light into two beams with opposite polarizations. The beam with the first polarization went straight through. No problem. But the beam with the second polarization, when it hit this wall, it hit it at an angle wide enough to be reflected out of the crystal. This means that only half of the light you shine makes it out the other side, but the half that makes it out has only one polarization. Honestly, this sounds to me like when people find exploits in video games. Taking advantage of unintended interactions between different mechanics. And I don't know, it gives me the feeling that this William guy would have really loved Minecraft and he would have gone crazy with Redstone. Although today, if we want to polarize light, we don't have to rely on calcite crystals and Canadian glue, because through some arcane ritual, chemists have invented plastics that polarize light, which is baffling. And a little less baffling, we have these photographic lenses that are also polarizers. But in any of these cases, the result is exactly the same. Half of the light comes out polarized and the other half is reflected away. But now the question is, what happens if you sense light that is already polarized through a polarizer? Well, let me show you. Okay, so now here you can see me, you can see me through this polarizer, but now I'm going to put this polarizer in front and you can still see me. But then now you cannot see me. Now you see me? Now you don't. Now you see me? Now you don't. Now you see me? You don't. You see me? You don't. You see me? You don't. What's going on here? How does this work? Okay, so let's say that we have a beam of unpolarized light, which passes through a first polarizer. Half of the light is reflected away and the other half is polarized in a single direction. Okay, so far so good. Then this polarized light reaches a second polarizer. And as you saw, if the polarizers are aligned in the same direction, all of the light will make it through with the same polarization. But if we rotate the second polarizer, some portion of the light will be polarized in this new direction and the rest will be reflected away. But we have no idea how much. All we know is that if we continue rotating the polarizer, at some point, all of the light will be reflected away and none of the photons will make it to the other side. And so the question that we want to answer to understand what's going on here is how much light is going to be polarized and how much is going to be reflected away? To find the answer, first we have to realize that the polarization of light and the orientation of the polarizer are both vectors. And vectors have a couple of very useful mathematical properties. And one of them is that they can always be written as combinations of other vectors. For example, if we have this vector, we can write it as the combination of these two vectors. Like this, it is three times this vector plus four times this other vector. But we can also write it as a combination of these two other vectors because it is three-fourths of this vector plus six point two times this other vector. And we can keep going. We can keep finding other vectors and express the same vector as a combination of like anything we want, really. What matters in the end are the numbers, right? Because these numbers tell us how much of one vector is inside another vector. Or maybe they are telling us like how similar two vectors are to each other or I don't know, something like that. These numbers are called the coefficients. And this whole process is known as a linear combination because we are combining vectors linearly. If you want to find the coefficients for a combination, you are in luck because mathematicians already did most of the work for us. All you have to do is use the cosine function because this function basically measures how similar two vectors are to each other. For example, if you want to know the coefficients of this vector as a combination of these other vectors, you just need to look at the angles between them, calculate their cosines and then multiply by the length of the first vector, which we represent by putting the symbol of the vector between parallel lines. The result will be the coefficients of this combination. If you are not familiar with trigonometry, it might seem crazy that this works, but lucky for you, there is a myriad of videos explaining trigonometry really well. Just make sure that you watch at least one ad in this video before you watch one of those. All of this is important because remember that a polarizer divides light into two beams with polarizations at 90 degrees from each other. One of those directions is accepted and makes it out of the polarizer and the other is rejected and is reflected away. But now what we can do is that we can look at the polarization of light as a combination of the accepted and rejected directions. And guess what? The coefficient of the polarization in the accepted direction is how much light will make it out of the polarizer. Okay, in retrospect I realize this comes out of nowhere. What happens is that there is a certain conservation of polarization. One way to see this is that every photon has some amount of polarization in the accepted and rejected directions and the polarizer cannot create nor destroy this polarization. It can merely redistribute it so that every photon ends up with polarization in only one of those directions. If the beam of light had like 80% of its polarization in the accepted direction, then the polarizer can give 80% of the photons this polarization. And so only 80% of the photons will make it through while the rest are rejected. And what the cosine does is to give us this percentage. The funny thing is that this conservation of polarization is actually just a special case of the conservation of momentum. And if you're curious about how that works, tough luck pretty boy. We won't be covering that in this video. You'll have to learn it on your own. What are you gonna do? Cry? Go to Introduction to Electrodynamics by David J. Griffiths, chapter 9, section 2, subsection 3 titled energy and momentum in electromagnetic waves, or maybe just read chapters 8 and 9 for good measure, or maybe just read the entire book. It's actually pretty amazing. Now we can understand what's going on in this case. We can understand that the amount of light that makes it out of the second polarizer depends on the cosine of the angle between them, or well between their orientations. And so at 90 degrees, all the photons are aligned in the rejected direction. And so none of them make it through, which should be right around here. But in the end, the main mathematical takeaway from all of this is that we can write the polarization of light as a combination of two other polarizations at 90 degrees from each other. By the time of Darwin, all of this was already known. People had a lot of experience working with the polarization of light and making better and better polarizers. In fact, the Seaman effect requires the use of polarizers. In his papers, Darwin mentions the polarization of light many times. But he never mentions how he noticed that it was similar to the Schtenger-Lach experiment for the first time. But I think that it was because of the order and disorder of the vectors. Because in both cases, we begin with vectors that are just like randomly aligned in a bunch of different directions. And as they pass through either the polarizer or magnetic field, these vectors come out order on the other side. The main difference is that with polarizers, you always lose some portion of the light. While the SG device lets all of the electrons through, none of them are rejected or bounced away. So it's not a perfect parallel, but they are certainly similar. And then there was another idea floating around at the time. The idea that all particles were also waves. As you might remember, Einstein had proposed the idea that light was made of photons that were both particles and waves. And initially, people had resisted that idea a lot, even blank. But as the decades went on, quantum mechanics gathered more and more evidence in its favor. And so the haters were finally dying. And young scientists had grown up with these ideas, and they accepted the evidence. Incidentally, this is how we might start solving climate change. We just have to wait for the boomers to die. This is where a young scientist called Louis de Broglie took the next step. Maybe all particles were also waves, not just photons. We are going to learn more about de Broglie and his story in the episode about the Schrodinger equation. But for now, what matters is that using de Broglie's ideas, Erwin Schrodinger was able to find an equation to describe these new matter waves, and this became known as Schrodinger's equation. Using it, it is possible to correctly predict the structure of the hydrogen atom, along with many other observations in quantum mechanics, making it the low quantum mechanics. For now, as you can imagine, many people didn't like the idea of all particles being waves. But good scientists accept the evidence, and the evidence was that Schrodinger's equation certainly worked, and it worked really well. Now imagine that you are derby, and you are thinking that the SG device and polarizers are very similar to each other, because they both align vectors, although different ones. Polarizers align the electromagnetic field, and the SG device aligns angular momentum. But now you hear this new idea that all particles are also waves, and well, waves must be polarized in some way. So if the SG device is like a polarizer, then maybe the polarization of matter waves is the spin. Wait, okay, let's think more carefully about this. Let's think about an individual electron going through the SG device. Okay, so initially this electron is going to have some random spin that we don't know, right? And then it's going to pass through the magnetic field, and it's going to end up with either spin up or spin down. But which one? If we knew the initial spin, then maybe we would be able to predict if it's going to end up with spin up or spin down. But how can we know that initial spin? Because to know something, to know anything, at some point, you have to make some observations, you have to make some measurements. But in this case, when we try to make a measurement, we change the thing we are trying to observe. And so it's impossible. It is impossible for us to know the initial spin. And so if we cannot know the initial spin, we cannot predict the final spin. It is impossible to make predictions. And if we cannot make predictions, can we even do science? Yes, we can. Because as Senku says, if there are rules, it is science. And in this case, we have at least one rule that half of the electrons are going to have spin up, and half of them are going to have spin down. This is like the only thing we know. And we know this will be true even if we rotate the magnetic field to some other orientation. The beam of electrons is still going to split in the direction of either magnetic pole weight. In the case of light, there is no difference between oscillating up and down, and down, and up. It is the same thing. But in the case of spin, spin up and spin down are completely different situations. Sure, like the vectors are aligned in the same direction, but they are describing something very different, even opposite to each other, which means that maybe the SG device isn't quite like a polarizer. It is more like a calcite crystal. The SG device is just like a calcite crystal, because a calcite crystal separates light into two beams with opposite polarizations. The SG device is doing the same thing. It is splitting the electron beam into two beams with opposite polarizations, which means that the polarization of matter waves, yeah, it is the spin, and the two opposite polarizations are not separated by 90 degrees, but by 180, which is kind of weird, because when calcite divides light into two, those two polarizations are always 90 degrees apart. And that's why we can write any polarization of light as a combination of the polarizations of calcite. And we cannot do this in this case with the SG device, because you can combine these two vectors all day long if you want. They are always going to point along the same axis. If the SG device is working like a calcite crystal, then I think we can turn it into a polarizer by just blocking one of the beams. Because if we do, all the electrons that come out are gonna have the exact same spin. They are all polarized in the same direction. And yeah, I'm sure we lose half of the electrons, but that's the exact same thing that is happening with light polarizers. They make us lose half of the light. So yeah, it is the exact same thing. But then, if we can put one polarizer in front of another, what would happen if we put one SG device in front of another one? And guess what? People have been wanting to do that experiment from day one. Like when Einstein heard about the Stenger-Lach experiment, he first sat down, he thought about it, he read the reports, and then he made his own proposal. He proposed what he called a multi-stage SG device. And so the idea was that we would have the electrons coming out, and then we would send them through a second magnetic field to see what would happen. The problem was that this experiment was just way too difficult. Like blocking one of the beams and making sure the other beam goes through a second magnetic field, it required too much precision. And so in 1922, they didn't have the technology to do that. And in fact, they wouldn't have it until like the early 1930s, nearly 10 years after the original experiment. And for that reason, back in like 1926, 1925, Darwin didn't know what would be the result of such an experiment, but he could still try to imagine it. Okay, so let's imagine a beam of electrons passing through an SG device. The beam is going to split in two, but we are going to block one of the beams to make the whole thing work like a polarizer. And then we are going to send the second beam through a second SG polarizer. It's going to split in two again, but we are going to block one of the beams to make the whole thing work as a second polarizer. The question is, how many electrons are we going to get in the end? And the answer, it's not going to be as simple as just the cosine of the angle between the magnets, because even if electrons are waves of some kind, they are not exactly like light because with light, the two different polarizations are at 90 degrees from each other. But with electrons, they seem to be at 180 degrees. Okay, so right now we don't know how many electrons we are going to end up with. But at the very least, we know that it cannot be more electrons than the ones we started with, right? Because the alignment between the magnets is not going to create electrons by itself. So at the very most, we can have 50% of the electrons, right? Because the first polarizer blocked 50% of the electrons. And so that's the maximum amount that we could possibly have at the end. Okay, so let's say that by pure random chance, we got lucky. And the two magnets are aligned in such a way that we get 50% of the electrons at the end. Okay, then if we change that alignment, that amount can only go down, right? Because it cannot go up. And so it goes down. Okay, okay. This looks very similar, like, yeah, we have two polarizers and they begin aligned and all of the light is making it through. But as they begin to be more and more misaligned, less light makes it through. But wait, no, this is worse. This is worse, don't you see? Because before, if all the electrons that made it through the first SG device make it through the second SG device, it means that the beam is not splitting, right? But then if we change the alignment, and less electrons make it through, it must mean that the beam is splitting. And this is a huge problem because we finally know the initial conditions. Like before, we couldn't know the initial spin, right? It was impossible to know. But now we've measured, we know what the initial spin of these electrons is. We know that for a fact. And yet when they pass through the magnetic field, they are affected in two opposite ways. We finally know the initial conditions and it is still impossible to predict the final conditions. This goes against the promise of Newton and Galileo. They said, hey, guys, if you know the initial conditions of a system, you can use these equations to predict the final conditions. But in this case, that's not true. It is impossible to know the final conditions. I guess the best we can do is see we can see the proportion in which the beam is splitting. And that can tell us the probability that an electron will go either way. But that's the only thing we know. The only thing we can be sure about is the probability. And as we rotate the magnet, that probability changes. It changes in ways we should be able to calculate. The probability changes as we change the alignment between the magnets. The probability changes as we change the alignment between the polarizers. It was never spin. It was never spin. Spin is not the polarization of the electron wave. The polarization of the electron wave is the probability of having a certain spin. Okay, so I guess we need a way to develop this idea a little more. Okay, so electrons are waves. And the polarization of this wave is the probability that the spin will be measured in a certain way. Okay, so we need a way to visualize this polarization. Okay, so let's say that the x-axis is the probability that the spin will be measured down. And the y-axis is the probability that it will be measured up. And let's call this whole thing probability space. Every point in probability space represents the different probabilities an electron has to be measured in a certain way. So for example, this point represents that an electron has 50% probability to be measured up and a 97% probability to be measured down. Now, that cannot be true. Okay, so new rule. All the probabilities must add up to 100%. And that means that the only valid combinations of probabilities are in this line. That said, all the polarizations of probability should be somewhere in this line. But when you think about it, there's nothing special about the number 100. We just needed a number to represent certainty, but we could have chosen any other number like 5 or 3.2 or 9001 or whatever. So let's choose a more convenient number. And what number is more convenient than one? Okay, so from now on, a probability of zero means that something is impossible. It cannot happen. A probability of one means that something is guaranteed to happen, like you're liking and subscribing. A probability of one over two means that something will happen one half of the time. A probability of one over five means that it will happen one fifth of the time. And so on, you get the idea. Before reaching the SG device, the probability vector of each electron can be pointing anywhere in probability space, just like on polarized light coming from a lamp. But after they reach the SG device, after they are measured, the probability wave is polarized in one of these two directions. And that means that we are finally certain about the direction of the spin vector. Darwin published his ideas about these vector waves of probabilities for the first time in February of 1927. And Pauli did not like it in this lightest. We know that they were aware of each other's work, but I'm not sure if they ever met in person to discuss these ideas. But if they did, I imagine it went something like this. So what do you think? I hate it. And I hate you personally. Now, we've talked so much about spin that maybe I gave you the impression that the probability wave is only about spin. I'm sure in this video, spin is going to be our main focus. But the probability wave is also about the position of the particle, the speed of the particle, its energy, and anything else that can be known about the particle. All the information of the particle is in the probability wave. And a good example of this is the double slit experiment. We mentioned this experiment earlier when we were talking about light. The idea is that if you send a light wave through these two slits, it divides into two waves. And each wave contributes in different ways to the electromagnetic field at each point in space. In some points, those contributions cancel each other out. And this is known as destructive interference. But in some other places, they help each other out. And this is known as constructive interference. And this creates a very recognizable interference pattern. So if particles were really waves, they should also create an interference pattern if we make them pass through two slits. And guess what? They make an interference pattern if we make them pass through two slits. The difference here is that the waves are telling us the probability of finding an electron at that point in space. But guess what? If you set up some kind of measurement device to detect the electron as they pass through the slits, what you are doing is that you are polarizing the probability wave in that direction. And for that reason, the waves don't interact with each other, and there is no interference pattern. This basically proves that particles are, indeed, waves of probability. Although, of course, there are some other interpretations, like the pilot wave theory. But in this house, in this YouTube channel, we follow the Copenhagen interpretation, and there will be no pilot wave theorist on the right roof! No, no, I'm just joking. Like, debate in the comments. But, like, I'm promoting the Copenhagen interpretation, because it seems to me that it's the one that is best supported by evidence. It seems to me like it makes the most sense. But if you make your own research and you debate, and you end up being convinced by the pilot wave theory, it will be difficult for me. But I will always love you. You will always be my subscriber. Like, I hope, I guess you can unsubscribe. Okay, I'm taking this metaphor way too far. Let's move on. Another famous experiment about how measurements polarized probability waves is known as Schrodinger's Cat. And yeah, I know, I know, you've heard about it a million times, but I'll give it a twist. Let's say that we put a cat in a box with an SG device, and this SG device sends an electron through the magnetic field once every minute. And if it ever detects three electrons in a row with a spin up, it will release a poisonous gas that will kill the cat. Say that we let, like, an hour go by. Now the question is whether the cat in the box is dead or alive. And normally this would be like an extremely simple question to answer. But now it's not so simple, because the state of the cat depends on the probability waves of the electrons. And so the probability wave of the electrons and the probability wave of the cat have become, like, intertwined or entangled in a way, because the polarization of one wave affects the polarization of the other waves. And so in a sense they have become like a single giant probability wave. For this reason, as long as the box remains sealed, it doesn't make sense to say that the cat is dead or alive, because it's not that we don't know the answer. The answer doesn't even exist, because the probability wave of the cat has not been polarized in the probability space of dead and alive. When we finally open the box, we will have polarized the probability wave along the dead or alive axis. And we will know the answer with certainty. But opening the box is not our only option. For example, we could also smell the box. And trying to find the smell of the poisonous gas or the rotten cat, depending on how long we leave this running. And if we smell the box, and it does smell bad, then we have polarized the probability wave of the experiment along the stings, dozen stink axes. And maybe now the probability vector is closer to the dead axis, but it's still not polarized in that direction. There is still a probability that the cat is alive, because who knows, maybe the cat pooped in the box. And that's why it smells bad. That's why it smells bad. Like the answer doesn't exist. The answer will not exist until we open the box. The summary of all of this is that measuring quantum objects is weird. Quantum mechanics is weird. Who'd have thought? Although I should clarify that in real life, it would be really difficult to do the experiment of Schrödinger's cat. Partly because ethics committees are a bunch of cohorts, but also because the information would sort of leak out of the box. I mentioned smells earlier, but there would also be sounds or changes in the center of mass of the box as the cat moves around or stops moving around. And so all of these little pieces of evidence over time as we get more and more of them would allow us to deduce what happened inside the box without having to open it. Like it's like little by little the probability vector moves towards one direction and eventually becomes polarized without us having to open the box. And so for the experiment to work, we would need to isolate the box from the rest of the universe, which I don't know, maybe it's possible, but it would be really, really difficult. But when you are working with atoms or particles, it's actually kind of easy because as long as they are not interacting with something else, they are already isolated from the rest of the universe. So maybe just like make sure there isn't too much dust in the air or even better make your experiment in a vacuum. In fact, Schrödinger Lach made their experiment in a vacuum. So they didn't have to worry about interactions affecting their results. Instead, they worried because back then Germany was suffering from hyperinflation. So there was no money for experiments. But someone believed in them and held them out. Can you imagine who? Well, none other than my favorite motherfucker, Albert Einstein. He gave them 10,000 rice marks, which is around $2,500 in today's money, or about nine months of my student debt payments. And I am in a very similar situation to Schrödinger Lach. Not because I'm planning an experiment that is going to change the world, but because I do need money. Like I spent, I don't know how many months working only on this video, like first researching and writing and then recording. And then I have to do the editing and the special effects. And like, I need this time and effort to be worth it. And so if you want to help me, you can send me some one time donations through PayPal or buy me a coffee or coffee, which is a different one. Or you can just join my Patreon for as little as $1. And then if you want, you can edit your pledge to give me whatever amount of money you want each month. And if you cannot spare any money, but you still want to help me, well, you can. You can subscribe to the channel. You can like this video. You can comment. And you can share this video with your friends. Like if you have only five friends, but each of your friends watches this video a trillion times, I would be sad. I wouldn't ask for money in videos ever again. Anyway, let's go back to spin and apply everything we've learned so far. What we want is to find an equation that helps us predict the results of any measurement we could do on the spin. Let's imagine that we used an SG device in polarizer mode to create a beam of electrons all with spin down. And then we measure those electrons with spin down along the right left direction. What would happen? Let's look at probability space. These axes represent spin up and spin down. In physical space, these directions are 180 degrees apart. But in probability space, they are only 90 degrees apart. It seems that angles are halved between physical space to probability space. In physical space, left and right are halfway between up and down. So the same should be true in probability space, which means that in probability space, spin right should be here. But then right is 180 degrees apart from left in physical space, which means that in probability space, they should be 90 degrees apart. Huh, this is weird. Now we have this right and left probability space that has a bit outside or initial up and down probability space. So that would mean that if a particle had these probabilities to be left or right, it would have no probabilities to be up and down. And that's not true. I mean, we can always measure spin in any direction we want, and we will get a result. So yeah, this is wrong. We need to fix this. On the right side, it seems that the vector representing spin down, which is the spin that we know these electrons have, seems to be aligned with the vector that represents a 50% probability to be either left or right. Which seems to mean that if we take these electrons with spin down, and we measure them along the left-right direction, then the beam will split in half, because each of them has a 50% chance to be left or right. That way, we couldn't know that this would be the result of the experiment, because remember, it wouldn't be made until the 1930s. But he could still predict it, and his predictions would be later confirmed. But for now, we have a couple of big problems here. The first one is that each direction of measurement creates a different probability space, and these probability spaces don't match with each other. And the probability vectors don't match with each other either, but they can assure, right? An electron with 100% probability to be measured down has a 50-50 probability to be measured left or right. But it is the same electron. It is the same state of affairs, and yet it is represented with two different vectors in each measurement space. I think I know where all of our problems are coming from. Look, these are all the possible orientations of spin in physical space. As you can see, they form a circle. But in probability space, we are trying to squeeze this circle into a triangle. But wouldn't it be easier? Wouldn't it be beautiful even if we had a circle in probability space? I know this idea is weird, and if you start thinking about it, you are going to notice some quirks. But we are going to address them, and we are going to explain everything. But let me show you just one reason why this is a good idea. And that is that the vector doesn't change size as it moves. That's actually really nice. But, okay, so it has one problem, right? And it's that before the coefficients added up to 1, and 1 was the total probability, and now they do not add up to 1. But you know what? Like, if we look at this vector move, we can notice that it is always the long side of our right triangle, right? And so if we use Pythagoras' theorem, we can see that the length of the long side is always equal to the sum of the square of the short sides, right? And the short sides are the coefficients. And so we can say that the square of the coefficients is the probability of each result. And that's nice, except if the probability is the square of the values of the coefficients, then what happens if we remove the square? Like, what is this new quantity? Well, we call it the amplitude. What? Probability space is evolving! Congratulations! Your probability space has evolved into an amplitude space! Much like Todoroki, amplitude space has two quirks, and you might have already noticed the first one. We have negative numbers here. But this isn't like an immediate problem, because to get the probability, well, we have to square the amplitude, which means that even if the amplitude is negative, the probability is always positive. But I still don't trust this, and neither should you. So let's keep an eye on this, as we learn about the second work. Much like Todoroki! Let's say that we have the spin vector in spin-up, and the probability vector also in spin-up. Or, well, I guess it doesn't make sense to call it the probability vector anymore. We could call it the amplitude vector, but a better name for it is the state vector, because it tells us the state of the particle. Okay, so let's say that we have the spin vector in spin-up, and the state vector also in spin-up. And then we move the spin vector to spin down. Well, the state vector also moves to spin down, and we see that the spin vector moved 180 degrees. But the state vector only moved 90 degrees. We already knew this, that angles are kind of halved between physical space and amplitude space. But check this out. Now, if the spin vector gives a full rotation around physical space, then the state vector has only gone halfway. So if we want the state vector to give a full revolution, we have to move the spin vector like twice. It has to give to revolution. Yeah, this is really weird. And what this means is that for every point in physical space, we have two points in amplitude space. In mathematical terms, we would say that amplitude space is the double cover of physical space. You may be thinking that this is artificial, that this is just a consequence of the mathematics that we chose to use to represent these physical phenomena. And that if we chose to use some other mathematics, maybe we would have some other quirks, or maybe there would be no quirks at all. Maybe there's a better mathematical framework that we could use. But no. This has a real physical meaning. Look at these two waves. They have the same polarization, certainly, but they are not the same. Where one goes up, the other one goes down. Where one is positive, the other one is negative. And this is a real difference that can have real consequences. In fact, this is the reason for the interference pattern in the double slit experiment. Each wave has to travel a different distance to reach the same spot in the wall. It takes them a different amount of cycles. And for that reason, they arrive out of sync with each other, or as mathematicians would call it, out of phase. This is what the negative numbers in the amplitude are representing. They flee peaks and valleys, indicating that, sure, two waves may have the same polarization, but they are out of phase in one or both components. But while writing this episode, I came up with my own experiment to see the effect of the phase without needing to use any obstacles. We would need electrons with spin left, because their state vector is one minus one. So then if we measure them along the up and down direction, those that end up with spin up would have a state vector of zero minus one, which is out of phase with a normal spin up that is just zero one. So then if we had two beams of electrons with these two different states, and we saw somehow mix them, we should get an interference pattern without having to use any physical obstacles. Of course, the problem would be mixing these two beams of electrons without changing their spins too much. You know, like maybe it can be done, but maybe not. Maybe it is impossible. I don't know. Or maybe I'm wrong and this experiment wouldn't work the way I'm thinking for some reason. I don't know. Please let me know in the comments. And actually, in fact, if you can come up with your own experiment to see the face of the probability wave, also let us know. But in the rest of this video, we're going to continue focusing mostly on the polarization of the probability wave. But keep in mind that the face exists and it is important because it's going to become relevant later. But anyway, remember that the whole reason we started going through this rabbit hole is because we wanted to find an equation that would tell us how many electrons would end up with a certain spin after being measured with two SG devices. And now that we have amplitude space, maybe we can finally do that. Maybe we can import the mathematical formalism we have for life and use it here. Okay, so for example, let's say that first we measure the spin of an electron up, right? And then we put a second SG device that is at 45 degrees from the first one. And we want to know the probability that the electrons are going to have a spin aligned in that direction at 45 degrees. Well, how does that look in amplitude space? Well, in amplitude space, the angles from the physical world are halved, remember? So those 45 degrees become only 22.5 degrees in amplitude space. And now we can compare the polarization of the probability wave in amplitude space with this vector that represents the measurement. And then we can use the cosine just as we were doing with light. Of course, this is still amplitude, so we need to transform it into a probability. But that's easy, just square it. And there you have it. The probability that an electron with a given spin will end up with another spin after a measurement is equal to the square of the cosine of half of the angle between the current spin and the final spin. This equation is the Stenger-Lach experiment. It tells you everything you could possibly know about the results of this experiment. And that's why it's not very useful. Like, I mean, what did you expect? Like, this is a very artificial situation we invented to help us understand spin, but we're not going to find SG devices lying around in forests or beaches. No. If we want to understand how spin behaves in nature, we are going to need to develop better mathematical tools than this equation. Besides, in this case, we got lucky because there's a very clear connection between the angles in amplitude space and physical space. But that's not always going to be the case. For example, think about the amplitude space of Schrodinger's cat. In this space, the angle tells us the probability that a cat is dead or alive. But how do you relate that angle to anything in physical space? The answer is that you don't. Or at least I cannot imagine how you would do that. And you might be thinking that the problem here is that dead and alive are not physical quantities. They are abstract concepts. So maybe that's the problem. But consider energy. Energy is not abstract. It is very concrete. It is very real. And when we measure the energy of an object, the result depends on the probability wave. So we have an amplitude space for energy. Well, how do you relate the angles in this amplitude space to anything in the physical world? Because energy doesn't have a direction. Energy is not relative to angles. Energy just is. In mathematical terms or physical terms, we would say that energy is a scalar quantity, not a vector quantity. And in fact, do you remember the emissions spectra that we saw in the previous episode? Well, each of these lines is one possible result you could get when measuring energy. So you don't have just like a two-dimensional amplitude space. No, no, no. You have one dimension for every possible value of energy. And for some cases, you have many possible values. So yeah, these are huge amplitude spaces in very large dimensions that we humans cannot really visualize. How are we going to calculate the cosine between two vectors that we cannot even visualize? Well, you will be very pleased to know that once again, mathematicians have already done all of the work for us by discovering the dot product, which allows us to calculate the cosine between vectors in any number of dimensions. But I'll show you how it works using just two, using the amplitude space that we have already been working with. This is how you do the dot product. First, take the vector that represents your measurement and turn it sideways. This is known as a row vector. And then right next to it, the vector that represents the state vector. And this is known as a column vector. This is how we write the dot product in the same way that addition has the plus sign or multiplication has the multiplication sign. This is how we represent the dot product. Okay? And this is what it means. First, take the first number in each vector and multiply them together. And then take the second number in each vector and multiply them together. And then add both of those numbers. This is the dot product between these two vectors. And the great discovery of mathematicians was that this was equal to the length of the first vector multiplied by the length of the second vector multiplied by the cosine of the angle between these two vectors. And remember that to write a vector as a combination of another vector, we just have to multiply the length of the first vector by the cosine of the angle it has with the second vector, which is very close to what we have now. For this reason, you would think that we just have to divide by the length of the measurement vector. But in this case, it's even easier because all of our vectors have a length of one. So we don't have to divide the cosine of the angle is all that we needed. This is why we chose to use one to represent the total probability like half an hour ago to simplify calculations like this one. I am four parallel universes ahead of you. The great thing about this is that once we have the vectors that represent the state and the measurement in amplitude space, we no longer need to know the angle between them in amplitude space. We don't even need to know what that angle would be in physical space. It doesn't even matter if that angle doesn't exist in physical space. It also doesn't matter if those vectors have two, three or a thousand different dimensions. We just have to multiply the components of the vectors in order and we will find the amplitude for that result. And then we just square it to get the probability. Of course, the one thing we do need is to find a way to represent any measurement as a vector in amplitude space. But now that we have the dot product, we can finally do that. And I'm going to continue using the spin amplitude space as an example. But remember that this applies in any amplitude space for any measurement. Let's say that we have an electron with this spin state and that we want to figure out the probability that it will be measured with spin up. Well, all we need is the vector that represents the measurement for spin up. And in this case, this is really easy because there's a very clear connection between amplitude space and physical space. So this vector is just one zero. So then we can calculate the dot product between these two vectors and then we get the probability that it will have spin up. Or, well, the amplitude, but you know what I mean. Okay, but check this out. When we do the dot product, the zero in this vector gets multiplied by the amplitude for spin down, cancelling it out so that in the end we are left only with the amplitude for spin up. And this serves as a template of how we can represent measurements as vectors, because these vectors need to have a one in the place for the result we are measuring and a zero everywhere else. But admittedly, this invites the question of what's happening with spin right and spin left, because those vectors don't have zeros anywhere. I'm sure they don't, but the reason is that we are expressing them as linear combinations of spin up and spin down. And when you think about it, we are expressing spin up and spin down as linear combinations of themselves. But that's not the only way we can do this. We can also represent all of these vectors as linear combinations of spin right and spin left. And that looks kind of like rotating the entire grid. And sure, this changes the numbers. It changes the coefficients we are using to represent the vectors. But it doesn't change the vectors themselves. The vectors stayed in the same place. They didn't move. Everything else moved. The vectors didn't. And this is something really cool about the dot product. It doesn't care how you are representing your vectors. As long as you are representing the same vectors, you will get the same result even if the specific numbers are different. But all of this is very abstract, very mathematical. How would we actually do this in real life? Okay, so let's imagine that we have a bunch of atoms of the same element and that we know it's a mission spectra, right? So these are the possible results we can get. And we are hitting these atoms so they emit light. And then let's say that we are detecting all the photons that come out and that we know how many photons we get each second. But we don't know the energies of these photons. Okay, so what we need is some kind of device that can block all the photons and only let through photons with one energy. How would we do that in real life? I don't know. I figured it out. It's a prism. It's just a simple prism. When the beam of light passes through the prism, each color will be deflected at a different angle. Then we can easily block all the colors except for the one we want to measure. This is basic spectroscopy. I can't believe I forgot about it. Anyway, let's move on. Okay, so then we use this device to block all the photons except those with one energy. And then we detect those. And now we can see a difference before we are getting like, I don't know, like a billion atoms every second. But now we are getting only like 50,000. Well, now we know that the probability of an atom emitting a photon with this energy is like 50,000 over a billion. It's, I don't know, like a number. Okay, now we did that measurement. That's the result of this dot product. So what happened is that by only letting photons of one energy through, we basically polarized the probability wave. We said to the polar, the probability wave, hey, the only way you are gonna be detected is if you are polarized in this one direction. And that only happens some percentage of the time. And that's what we just measured. And from this result, we can start reconstructing the state vector. Okay, so now let's say that we recalibrate our device. So only allow photons of a different energy. So it blocks all the photons with other energies, and only lets through photons with this new energy. Okay, so we now detect a different amount of photons, giving a different probability. And we can keep going, right? So that we know all the possible energies. And so we recalibrate our device each time. And that polarizes the probability wave in a different direction. And so little by little, we reconstruct the entire state vector of the atoms in our experiment. So there you have it. Now you can see how we can work in more abstract amplitude spaces that are not directly related to directions in physical space. And the craziest part is that none of this was in the script. Well, I was rehearsing this video, and while doing that, I thought, oh, people are gonna have questions about this. I should add a little section about it. And here we are. But anyway, let's go back to spin, because that's, you know, the actual topic of this video. We have these two dot products which calculate the probabilities of spin right and spin left. But instead of doing two calculations separately, let's build a mathematical machine that does both calculations at the same time. So this is how we're gonna do it. First, write the first dot product. And then below the first measurement vector, write the second measurement vector. And so this block of measurement vectors is gonna be known as a matrix. And this whole process is not gonna be called like a double dot product or something like that is gonna be known as a matrix multiplication. But it is two dot products. And so we are gonna get two amplitudes. But wait, we give it two amplitudes in the form of the state vector. And it gives us two new amplitudes. So it's only natural that we put them in another vector. So look, this is a machine that you give it one vector, and it gives you another one with the probabilities of these measurements. And this is important because there's another way to look at this situation. Like let's say that you have a vector written as a linear combination of spin up and spin down. And if you want to know the probabilities of spin right and spin left, you are basically asking how to write this vector as a combination of right and left, which remember is just like rotating the grid under the vector. But then if the only thing that matters are the coefficients, if the only thing that matters are these numbers, well, this is the exact same as just rotating the vector in the opposite direction. You see, like all the matters are the numbers. And so, and here's the amazing thing, here's the beautiful thing. This rotation is what this matrix is doing. Like when you look at the initial vector and the final vector you get, it is the same thing as if you have rotated that vector. This is one of the most important ideas in this video, that this is what measurements do. This is what measurements are. Measurements rotate state vectors in amplitude space. Like before, at the beginning of this video, that sentence would have been meaningless. It was gibberish. But now, well, I don't expect you to be like an expert on quantum mechanics, but hopefully you can see how that makes sense. So hopefully that sounds reasonable. And yeah, it makes so much sense because think about it. Before, before you do a measurement, you don't know anything because the answers don't even exist yet. And so a measurement is basically a situation in which we are forcing a probability wave to be polarized in one direction that we can know. We are forcing it to have an answer. We are forcing the answer to exist. And that means that the probability wave has to be polarized in one direction. And this polarization looks as a rotation in amplitude space. This is extremely important because we have just discovered that this is the mathematics of quantum mechanics. And it is known as linear algebra. And of course, there's so much more to learn about this branch of mathematics. But as you can imagine, three blue, one brown already has some excellent videos going over it. He doesn't mention quantum mechanics in those videos, but everything he says is applicable to quantum mechanics. And who knows, maybe you already knew linear algebra because linear algebra is very useful. So if you have learned like electronics or robotics statistics, you have probably already learned a little bit or a lot about linear algebra. But maybe you haven't. Maybe this is the first time you are being exposed to linear algebra. You learned quantum mechanics before learning linear algebra. And if that's a case for you, congratulations, because the same thing happened with Heisenberg. It's a really funny story. Let me tell you. Okay, so what happened was that Heisenberg was studying quantum mechanics with a lot of intensity, right? There were a lot of new ideas, new experiments. And he realized that he couldn't understand all of this. No one could because he realized that they needed a new kind of mathematics to understand the quantum mechanics. And so he was working at a university. He asked for a vacation and then he took all of his money and he went to a deserted island. Oh well, he wasn't at a deserted island. He had like a hundred people living in it, but a mostly deserted island. And so he was in this island alone. He started furiously working, trying to develop this new branch of mathematics. And well, he stopped for like eating and sleeping and swimming, maybe talking with some of the locals. But anyway, so he spent like months working on this island, developing this new branch of mathematics. And he did it. He succeeded. And then he went really excited to his friend Max Born. He was like, Max, Max, develop this new branch of mathematics that is going to help us to understand quantum mechanics. And then Max Born started reading. He was like, oh, it's incredible. This looks a lot like linear algebra. And then he was like, what do you mean linear algebra? Max was like, didn't you learn linear algebra in school? And he was like, no, what is it? And Max Born was like, well, this is it. This is mathematics we have here. This is linear algebra. And then Heisenberg was like, oh, well, it's a very new branch of mathematics, right? Very unknown. It's not been very developed. And Max was like, no, it was this comet like a thousand years ago in China. And Heisenberg was like, well, but he arrived very recently in Europe, right? And Max was like, no, he arrived like in the 16th Congress. And Max, I know Heisenberg was like, I spent like three months in a night working on this. Developing from scratch. And Max was like, yeah, it shows. It shows everything you have here is so convoluted, so messy. We have easier ways to do everything. All of this. This is way too difficult. We have easier ways to do all of it. Finally, Heisenberg was like, well, I'm gonna read my room, reading it with linear algebra. Thank you very much, Max. Poor Heisenberg probably had thought how he was gonna call this branch of mathematics. And he had names for all of the theorems. I don't know if it was already known. But it wasn't a complete waste. No, it wasn't a complete waste. A little longer than a few minutes later. It wasn't a complete waste of time because discovering that linear algebra was the correct mathematics to use with quantum mechanics. That was an important step. So yeah, it wasn't a complete waste. Oh, and also, while Heisenberg was in that island, he also discovered the uncertainty principle, which was extremely important. And we are going to learn more about that in future episodes. But yeah, it wasn't a waste of time, but it was extremely funny. Now let me show you something. Cancer cells divide really quickly, and they do everything in a rush. And for this reason, radiation is their kryptonite. If you showed radiation at a healthy cell and a cancer cell, both are damaged. But the healthy cell can check its blueprints and repair itself. While the cancer cell has corrupted blueprints, and it is in a rush to divide, so it doesn't repair itself correctly, or it doesn't repair itself at all. And for that reason, instead of dividing, it dies. This is something so fundamental to the nature of cancer that it can never be changed. Cancer can never develop resistance to radiation, because to be resistant to radiation, it would have to be a healthy cell. And so the same reason that makes cancer a problem is also its worst weakness. This is why we use radiotherapy so much. These machines accelerate particles, almost always electrons, at extremely high speeds and then shoot them at the tumors of the patients to destroy the cancer. Either that or we make those electrons crush with pieces of metals, which release x-rays that we then shoot at the tumors inside the patients. But either way, the point is that we have to accelerate particles at extremely high speeds. And the way these machines do that is with magnetic fields. Can you see where I'm going with this? Radiotherapy machines are basically extremely advanced versions of the Stanger-Lach experiment. This is technology that has saved millions of lives, and it will continue to save millions more. These are human souls that we get to share this world with, or that we get to be in the first place, all because people like Darwin and Pauli sat down to think really hard about this stuff until they figured it out. And I just need you to keep this in mind as we go to the next section, because this is where it gets hard. We've been treating spin as if it was a vector in a plane, but it's not. It is a vector in three dimensions. It can be pointing anywhere in physical space, and we need to represent this complexity in amplitude space. The first thing we notice is that all the possible spin orientations in physical space form a sphere, just like they form a circle earlier. This means that if we want to expand amplitude space to represent all the possible spin states, we need to have a sphere in amplitude space. In our original plane, we could represent spin up, spin down, spin right, and spin left. But in physical space, we also have four words and backwards. So you might be thinking that all we have to do is just add one extra axis to amplitude space and call that four words and backwards. But it's not so simple, of course not, because in physical space, right is between up and down, right? And it is still between them in amplitude space. So it ends up at an angle. Well, up is between four words and backwards in physical space. So when we have the angles in amplitude space, they end up like this. Okay, so now that we know the general shape of amplitude space, we need a way to represent any point in this sphere. And you might be thinking that since we are working in three dimensions, we are going to need three numbers, x, y, and c. But that's not how spheres work. Now, I don't want to brag, but I'm an expert on spheres. I live nearly my entire life in one of them. And if there's one thing I know is that to represent any point in a sphere, you only need two numbers. Those cartographer guys call them latitude and longitude, but I call them phi and alpha. They represent the angle with the x-axis and the angle with the x-y plane. That is all you need. In mathematical terms, we say that the sphere has two degrees of freedom, as in, you are free to be anywhere you want in the sphere, but I'm always going to need only two numbers to say where you are. In fact, all surfaces have two degrees of freedom. That's kind of like what it means to be a surface, like the circle or this triangle. The difference is that here we have a surface of two dimensions in three dimensions. And that may sound a bit weird, but think about it. Lines have one dimension, and we have no problem imagining them in three dimensions or two dimensions or one dimension. So yeah, sure. We can have surfaces in two dimensions or three dimensions or four or six, or any other number. I mean, I didn't say six for any particular reason. It was random. It was random. Okay. I was born in the 90s. Very random. Like, rawr, moustache, hot topic, Evangelion. Does anyone remember Megas Accelerate? Okay, so it's settled. We can represent any point in this sphere using only two numbers, like pheonalpha. Except I don't want to use pheonalpha. The whole point of amplitude space is working directly with the amplitude so that we can see how the probabilities change. And I will find a way to work with amplitude here. Let's think about our two base vectors, one zero and zero one. What we need is to write a third vector that is somehow perpendicular to both of them, but without using any new degrees of freedom. And this seems impossible because if we write a vector with any two numbers in it, it's always going to end up somewhere in this plane. It's never going to come out of the plane no matter what numbers you put here. So it seems like we would need a new kind of number. Wait, that's it. A new kind of number. What we need is a new kind of number. A number that itself is perpendicular to the other numbers. Okay, so this is the number line. And all of these numbers represent quantities of something. But what about this point here? What does it represent? Well, it cannot represent a quantity because all the possible quantities are already represented. But it seems to be like a position. Yeah, a position. Yeah, like who says that numbers have to represent quantities? A number can represent a position. Sure. And this number represents this position. So can we do this? Can we just invent the numbers? Yes, we can. And there is no power human nor divine who can stop us. And to prove that, I'm going to give this number a name. And I'm going to give it like a random name. I'm a night escape. You know, we are so random. So I'm just going to randomly choose to call it i. Yeah, I'm going to represent this new number with the letter i. Okay, so we have our new number i. And we should be able to use it like we use any other number. Like we should be able to add it to other numbers or to multiply them. And in fact, you know what, let's say that we take the vector one zero. What if we multiply each number in this vector by i? Well, then one times any number is always that other number. So this should be i. And zero times any number is always zero. So this should be zero. So now we have a vector called i zero. And this vector is nowhere in the plane that we had already been using. It's nowhere else. So maybe this is our new vector. And but wait, now it looks as if we have the vector one zero, we multiply it by i, and we ended up with a new vector i zero that is at 90 degrees from the original. It is almost as if multiplying by i had the effect of rotating vectors by 90 degrees. Okay, so then let's say that we continue, let's multiply i zero by i. So if that rotates vectors by 90 degrees, we should end up here with the vector minus one zero. So weird is that it's almost as if i squared was equal to minus one. Okay, so then we can do that again. And then we multiply by i again, and we should have the vector minus i zero, which is in the other side of i zero. Okay, and then if we multiply by i one last time, then we come back to the beginning. Huh, we multiply by i four times and we came back to one. Yeah, the imaginary numbers. Yeah, these are imaginary numbers. But the reason I introduced them this way is because imaginary numbers tend to feel very esoteric, like they come out of nowhere, they charm you, they solve all of your problems, and they are really weird, just like me. So I wanted to motivate their introduction so that it felt natural and necessary and useful. Did it work? Please let me know. Now we can represent spin forwards and spin backwards in amplitude space as the vectors i one and minus i one. And in fact, we can represent any other possible spin state by using combinations of real numbers and imaginary numbers, which are known as complex numbers. And yeah, that's a very good name because they are certainly complex. But what about the probability? Since i times i is equal to negative one. Now when we square the amplitude, sometimes we are going to get negative numbers, negative probabilities. And initially that doesn't seem to make much sense, which would lead you to believe that maybe our understanding of amplitude is wrong and you would be right. You're correct. Wait, what? You are absolutely right. Our understanding of amplitude so far has been wrong. But that doesn't mean that we have to abandon the concept of amplitude. No, it means that we have to improve it. First, we got to invent a way to get positive numbers out of complex numbers in a way that resembles squaring them. Squaring, squaring, squaring. Anyway, so the way we're going to do this is that take your number and then just flip the sign of the imaginary part. This is known as the complex conjugate. And if I could go back in time, I would kill the person who invented this name because it's meaningless. It tells you nothing. It's just so obtuse. Anyway, we are stuck with it. When you multiply a number by its complex conjugate, you will always get a positive number. And that positive number is known as the magnitude. And I like that name. That's a good name. Now, instead of saying that the amplitude is the square root of the probability, we can flip it around to say that the probability is the magnitude of the amplitude. But wait, before if we had a state vector and then we squared each of the amplitudes and we added them together, we would always get one because one was a total probability. How does that work now? Well, the dot product comes to the rescue because remember that the dot product between two vectors is equal to the product of their lengths times the cosine of the angle between them. But what if we take the dot product of a vector with itself? Well, in that case, and also remember that our vectors have a length of one. Okay. And the angle between a vector with itself is zero. And the cosine of zero is one. So we should get one when we take the dot product of a vector with itself. The problem is that now that we have complex numbers, sometimes we could get minus one, or we could get I or something like that. And so we have to improve the definition of the dot product. And this is how we're going to do it. Instead of just turning one of the vectors sideways, we are going to turn it sideways. And then we are going to take the complex conjugate of each of the numbers inside it. And now it is guaranteed that if we take the dot product of any vector with itself, even if it has complex numbers, we will always get a positive number. Now, this amplitude space has only two dimensions. But as I mentioned earlier, we can be working with the amplitude spaces of many more dimensions and things can get really complex. So to simplify things, in 1939, Paul Dirac invented the bracket notation. It works like this. So this is a ket and it represents a normal column vector. This is almost always going to be a state vector. And then this is a bra. But instead of putting your boobs in it, you put a row vector made of the complex conjugate of each number in what would otherwise be a normal column vector. And then the dot product is written like this. I really like this notation because it does simplify things. It looks really nice. Dirac had an eye for aesthetics. But my favorite reason is that it forces very serious professors to say the word bra a lot in rooms full of young adults. And the best part is that the word bra apparently started being used around the 1930s. So it is completely possible that Dirac did this on purpose. Like, sure, it could have been a coincidence Dirac might have been oblivious to this, but I choose to believe. I choose to believe Dirac did it on purpose. This is all nice and good. But what about the vector zero i? What? I'm just saying, if we have i zero, then we must also have like zero i, right? Like, there's there's no rule that we can only have imaginary numbers in one part of the vector. Don't be stupid. We were almost done. We don't need this. Are you scared? Are you a coward? No, then we will allow this and we will deal with what it means, whatever it is. It seems that in our attempt to add a third dimension and still use amplitude as our two degrees of freedom, we accidentally ended up adding a fourth dimension into this mess. And because of the limitations of human perception, it's literally impossible for me to show you how that looks like. And so I guess I'm just gonna have to keep using 3d plots and you guys are just gonna have to use your imagination. And maybe this fourth dimension feels wrong or unnecessary or meaningless, like it has no physical meaning, right? But no, it is the entire opposite, all because of face. We were so focused with polarization that we nearly forgot that amplitude tells us one more thing about the wave, its face. And since every wave can be in face or out of face, this adds two more degrees of freedom to the two we already have for the polarization, for a total of four degrees of freedom. You need four numbers to completely describe the state of a wave. But like, if you look at it as a vector, this still describes the surface of a sphere just in fourth dimensions. But in a way, if you look at real numbers and imaginary numbers as separate degrees of freedom, well then this describes, I don't know, some other shape in fourth dimensions. Look, mathematics is weird. But the weirdest part is that this complexity was here from the start, from the very beginning. Like, look, when we realized that negative amplitude represented waves that were out of face, well, we only could represent waves that were in face with positive numbers and out of face with negative numbers. But of course, those are not the only two options, right? Waves could be in any face with respect to each other. And we have no way to represent that, because we only have one and negative one. We needed i and minus i to represent all the possible waves that waves can be in face or out of face. And in fact, we can use imaginary numbers and real numbers to represent complex interactions between the polarization and the face that results in waves that actually spin as they move. And this is real, this is known as circularly polarized light. And for example, take this beetle. When light reflects off of this beetle, it acquires a left-handed circular polarization. That means that if you point your thumb in the direction the light is moving, the electromagnetic field is actually turning in this direction. What just happened is amazing. We forgot about the face. But the mathematics didn't let us down. They forced us to put back those two degrees of freedom we had forgotten about or that we had not even realized they were there in the first place. And look, I know this isn't a very in-depth explanation of circular polarization, because it is very tangential for this topic and we have to move on to other stuff, but I just have to mention it in case some of you may get curious and want to learn even more about polarization. And now we are finally ready to understand what Darwin initially found, because all the things I've shown you in this video, Darwin didn't find them. They were developed by Pauli, Heisenberg, and others based on Darwin's ideas, but what he found was initially very different. He describes it in his paper, The Electron as a Vector Wave, and he says that to describe this strange spin wave, he needed six different dimensions. And you might be thinking, six dimensions? You need four, and that's weird enough already. Where is he getting six dimensions from? Well, Darwin was just as confused. He talked about how difficult this is to visualize, and it is clear that he doesn't fully understand what this is, but today we can understand what was going on. Look, you have spin up, spin down, spin right, spin left, spin forwards, and spin backwards. For each of the three dimensions of physical space, we have two different spin states for a total of six different spin states, which we can represent using six different vectors in amplitude space. If you wanted, you could represent any spin state as a linear combination of these six vectors. I mean, we don't have to, and we are certainly not going to do that, but you could. And this is what Darwin found. And it might seem crazy that he arrived at this beat of mathematical truth without fully understanding it. But what happened is that he got here by filling his way through the darkness, stumbling with many conceptual roadblocks, standing up and going on and continuing until he arrived here. And when he did, he didn't fully understand what this was or what this meant, but he could see that it was important and that it was useful. And that's why he needed the help of people like Pauli to help him make sense of all of this. And now the road here, the path here, is paved in beautiful and clear mathematics, and the way is lighted by the people who have explained it to us. And now you are ready. You are ready for the last part of this journey. You are ready to find the SU2 symmetry. How can we represent measurements in this four-dimensional amplitude space? That's it. That's the question we will answer in what is left of this video. And there are a couple of things we already know. Number one, measurements are rotations of the state vector in amplitude space. That's still true that hasn't changed. Number two, to rotate a vector with two elements, you need a 2x2 matrix. And number three, these 2x2 matrices are now going to have to have imaginary numbers. Okay, so we know that any measurement we do on the spin can be represented using a 2x2 matrix. But this doesn't mean that all the possible 2x2 matrices represent measurements. Only some of them do. But which ones? At this point, we don't know. So if this was like a mystery novel, or suspects are everyone, all the 2x2 matrices. And so we need a way to start crossing them out. Something we can do is choose a couple of random matrices and see what effects they have on the state vector. And here we get our first clue. Because some matrices not only rotate the vector, they also make it longer or shorter. And this doesn't make sense for a measurement, because the total probability must never change. It must always be equal to one. And so we can reject all the matrices that change the length of the vector. But which ones do that? Well, there is an easy test. If you have a matrix with entries a, b, c and d, just take a times d and subtract b times c. And if the result is exactly one, then this matrix doesn't change the length of vectors. But if it is anything else, it does change their size. This is known as the determinant of the matrix. And if you want to understand why this works, I'm going to outsource that learning to Khan Academy, because that's where I learned it. But let's say that you have a matrix U, whose determinant is indeed one. But you still don't trust it. You want to make sure that this matrix is not changing the length of your vector. Well, then something you can do is, well, take the matrix U, multiply it by your vector, and then you're going to get a new vector, right? And then you know that the dot product of your original vector with itself is equal to one, right? That's the total probability. And well, the same thing should be true with the new vector you got, right? Because the length shouldn't change. But here's the thing. You start thinking that this ket is just equal to U times your original vector, right? And so then maybe this bra here is equal to a bra of the original state vector multiplied by some other matrix. And would you believe me if I said that that matrix is equal to U, except everything has been turned sideways and complex conjugated? Well, you should believe me because that's the truth. We call this whole thing the transpose conjugate or the adjoint or their mission. If look what happens is that all of those words have strict have different meanings in different areas of mathematics. But in this one case, they just happen to be the same thing. And it's a mess. And so for I know the difference, okay, but I'm just going to call it the transpose conjugate. And we're going to move on from that. Okay, I don't want to see anyone in the comments arguing the difference between adjoint and the transpose conjugate. I don't care. I don't care. Okay, so we have the bra of the original vector. And then we have the conjugate transpose of U. And then we have U. And then we have the original vector. And all of this must be equal to one. Here we can start abusing the properties of linear algebra, because it turns out that we can multiply this matrix by this matrix to get a third matrix and then multiply that third matrix by the by the ket to get a new ket and then multiply that ket by the bra. And that should still be equal to one. But wait, if the dot product of these two vectors is also one, then this third matrix didn't have an effect, right? It must not have had an effect. And in fact, the matrix that has no effect is known as the identity matrix. And it is always written like this. It has one single diagonals and zeros everywhere else. And it is the only way to write this. This is the only matrix that has these properties of having no effect. Okay, so this is actually a new clue. This is actually a new property. We need matrices that when we take their transpose conjugate and then we multiply by the original matrix, we get the identity matrix. And in order to never say conjugate transpose again, we call this property unitarity. So we say that we are looking for matrices that are unitary and have a determinant of one. Oh, and just in case you were wondering, we can have matrices that are unitary, but don't have a determinant of one or matrices with a determinant of one that are not unitary. And here we have examples of both. I know that was a lot of mathematics, but the good news is that we are done with these two properties. We can completely define all the matrices that represent measurements of spin. And now that we have these matrices, we can start looking at how they interact with each other. For example, let's say that we have a vector like this, and then we do one measurement and rotate it like this. And then we do another measurement and we rotate it like this. Well, that's the same as if we had just done one measurement that rotated the vector like this. And the same would be true if we have done like three rotations or four or five or any number. No matter how many measurements or how many rotations you make, there is always one measurement, one rotation that has the effect of all of them combined. This property is known as closure, because the idea is that no matter how we combine these matrices, we are always going to get another matrix that also represents a measurement. Kind of like how, for example, if you have a plane, you can combine any vectors in this plane, and you are always going to get another vector in this same plane. They never come out of the plane. It is closed. That's kind of like the idea behind the name. Another interesting property is that if you make a measurement, there must exist some other measurement that rotates the vector in the opposite direction on doing the effect of the first rotation. This is known as the inverse, and every rotation must have an inverse. Also, you could choose to not do any measurements, leaving the state vector in the same place. This is known as the identity, and in fact, it is represented by the identity matrix. And finally, we have that the order in which we do the rotations matters, which you can see with a phone or a book. I mean, you could see it with a pencil, but it would be really hard to see. So for example, here I have the ERC quartet by Ursula Leuwin. So if I rotate this book first to the right and then to the front, it ends up like this. But if it's starting from the same position, I first turn it to the front and then to the right, it ends up completely differently. But things are different if we add a third rotation. I wanted to show you with the book as well, but it was too difficult. So I'm going to leave this for the animations department. I mean, I don't have an animations department. I'm not that big, but I just hired someone to help me with animations because this part was like too difficult to keep track in my own head. Okay, so we have these three rotations here. But remember that whenever we have two matrices, we can replace them with their equivalent, right? Like there must be a matrix that is the equivalent of the second and third rotations. And similarly, there must be a matrix that is the equivalent of the first and second rotations. And if we replace these matrices by their equivalent, then the effect is the same. Nothing changes. And okay, I know maybe this is like difficult to understand, even when you are looking at the animations. This is actually easier to understand when you look just at the mathematical symbols like this. So what is in parentheses is what you are going to replace by a single matrix. And both of these give you the same result. This is known as associativity. Look at the properties we have found for these matrices. They are very mathematical, sure, but they all come from the physical world. A determinant one means that the total probability never changes. Unitarity comes from probability being the magnitude of the amplitude. And the amplitude comes from the properties of polarization and phase of the probability wave. And then a closure inverse identity and associativity all come from how rotations work. And here's a question for you. Who do all of these properties belong to? Like we represent them with matrices, yeah, but what physical object do they come from? Do you know? I know, I know, you know, I know, you know, let's say in the count of three, one, two, three, the probability wave, they come from the probability wave. This is the most important part of this video. This is the thing I want you to learn. Everything else I said in this video was meant to help you understand this one idea. The probability wave contains all the information about the spin. And these matrices describe how that information changes when we measure it. Because information is a physical quantity as real as energy, as real as temperature, as real as light. This is the most important part of this video. This is the idea I want you to learn. Everything else I said in this video was meant to help you understand this idea, that the information transforms in specific ways when we measure it. And when it comes to spin, the way this information transforms has a single explanation. It comes from a single place, a symmetry known as SU2. So let's talk a bit about symmetry. Look at this pentagon. If we rotate it by 72 degrees, it keeps the same orientation. It remains symmetric. And yet a change did happen, which we can see by labeling the corners. This is what symmetry is. A change that leaves some property unchanged. In the case of the pentagon, it is its orientation. But in the case of the state vector, it is its size. Because the total probability never changes regardless of how we rotate it in amplitude space. In the case of the pentagon, to preserve the symmetry, we have to rotate it by some multiple of 72 degrees. But if we rotate it by some other amount, the symmetry is broken. But the state vector has a symmetry that is preserved by all possible rotations. As far as we know, this symmetry is a feature of reality itself. It is as if we have lifted the hood of nature to look at its engine, and this is one of the pieces. But notice that we didn't set out to find a symmetry. In fact, I didn't even mention symmetries for most of this video. We just wanted a way to describe the results of the Stenger-Lach experiment. And step by step, little by little, we just happened to find a symmetry. We asked a question, and this is the answer. Such an important symmetry of information itself needed a name. And in my opinion, they gave it a pretty good one, because every part of this name means something important. The two means that this symmetry is represented by two by two matrices. The u means that they are unitary, and the s means that they are special, meaning that their determinant is one. And so this means the symmetry represented by all the two by two special unitary matrices, or just SU2, among friends. You did it. Now you understand that particles are described by probability waves, and that these waves can be polarized by measurements in different directions, which represent all the possible results. And that the way the spin polarization happens always preserves a symmetry known as SU2. And you know the matrices we can use to represent this symmetry. This is what Pauli discovered in May of 1927, when he found the mathematics to describe the ideas that Darwin had first published in February. Pauli did all of this between February and May. It takes me it takes me longer to make this video than it took Pauli to develop this whole branch of physics. At the time, this use of symmetry seemed to be limited to spin. But soon enough, Heisenberg would use it to describe the difference between protons and neutrons. And then a young, unemployed, failed physicist on whom I am not projecting would hear about these ideas and develop them to find two new forces of nature. But we will learn more about Yukawa in a future episode. At the time of re... What? You want to know about Pauli matrices? Okay, sure. You know how if you have a plane, you have infinitely many vectors in this plane. But they are all linear combinations of just two vectors. Well, similarly, we have infinitely many matrices in SU2. But we can represent all of them as linear combinations of just three matrices. These are the Pauli matrices. And sure, right now they seem like artificial consequences of the mathematics we chose to use. But no, they have a very deep physical meaning. The problem is that their meaning is mostly related to measuring the amount of spin, the amount of angular momentum. And in this episode, we've been focusing almost exclusively in measuring the orientation of spin. So we haven't built the background to understand their physical meaning. But we will in future episodes. At the time of recording this, to me, you are in a shoddinger box in the future. But if you're hearing this, it must mean that your probability wave was polarized in such a way that you reached the end of the video. Oh my God! Thank you so much. This video is so long and you watch until the end. People like you motivate me to keep doing this, for real. Thank you. And thanks to my producers in Patreon, who helped me immensely to continue doing this. And special thanks to... Thanks. A thousand times, thanks. A million times.