Transcription
This single equation helped unlock the modern world. It gave us computer chips, electron microscopes, atomic clocks, GPS, high-speed internet, the list goes on.
But where did this equation come from? Well, according to Richard Feynman, it comes from "nowhere." Out of man's imagination, struggles with the details of experiment and all kinds of mysteries. That man was Irvin Schrödinger. He derived it in 1925 while vacationing in the Swiss Alps with his mistress.
So I have two questions. First, how can we intuitively build this equation ourselves from scratch? But secondly, why is an equation with such real impact built using an imaginary number? What's "i" doing there? If you're ready, let's find out.
It all starts in 1853 when the physicist Anders Enström finds that hot hydrogen gas gives out very specific colors of light. We soon figured out it's not just hydrogen. Every hot element gives out its own signature spectrum. And suddenly this became a powerful tool to discover brand new elements just by looking at their light. This is how we discovered helium for the very first time in the sun. So in his honor, the unit for measuring these wavelengths was named the Enström. But nobody knew why these elements gave out those specific colors of light.
The first clue actually came a few decades later from a Swiss math teacher named John Balmer. Balmer was obsessed with numbers and patterns. And he finds a surprisingly simple formula for the hydrogen spectrum just by trial and error. And the formula predicts there should be more lines. And we actually confirm it. But nobody had any clue what this formula meant. Why did it work?
To answer this, we needed a good theory of light. By now, we knew light is a wave, confirmed by the interference pattern. And Maxwell showed that light is basically a ripple in the electromagnetic field produced by accelerating charges. So wiggling charges produce light. Wiggling it slowly gives us low-frequency light and wiggling it faster gets us high-frequency light. And these EM waves or light could wiggle other charges, which meant wireless communication. It was a breakthrough in communication technology, but it couldn't explain the hydrogen spectrum.
See, according to Maxwell, a hot glowing gas has billions of randomly jiggling charges from very low frequencies to very high frequencies, which meant they should be giving out every color of light, but they didn't. So something was horribly wrong with Maxwell's theory. But it got worse. Pretty soon we discovered that atoms had a positive nuclear core. So we thought the negative electrons must be orbiting this nucleus, making the atoms stable. But if they did that, they would be constantly accelerating, and accelerating charges radiate EM waves. So they would lose energy and collapse. So now we couldn't even explain why atoms were stable. Physics seemed to be in crisis.
But a few years later, everything changed. If you shine UV light on zinc, for example, electrons come out. That makes sense. Electrons receive energy from the EM waves. But if you shine a brighter visible light, no electrons came out. That didn't make any sense. I mean, you're pumping in more energy. So, we would expect electrons to come out with more energy. So, why didn't they? Why did the color matter and not the brightness?
To explain this, Albert Einstein proposed something radical. What if light doesn't deliver energy continuously but in discrete chunks called photons? And what if the energy of each photon depends only on its frequency? Then a bright visible light will deliver a lot of photons per second, but each one's too weak to budge an electron. It's like throwing ping pong balls at a bowling ball. But on the other hand, a dim UV light will deliver fewer photons per second, but each one is strong enough to knock it off like a single cannonball. So this explained the photoelectric mystery beautifully, and Einstein won the Nobel Prize for it. And this constant is, you probably know, the Planck's constant.
But a few years later, a Danish physicist named Niels Bohr pushed this idea even further. Bohr wondered if light is absorbed in chunks, it must also be emitted in chunks, right? Which meant electrons had to transition from a higher energy level straight to a lower energy one to release these energy chunks. So he postulated that electrons orbit the nucleus only at some special energy levels, nowhere in between. And in these special orbits, he said they don't radiate energy. It seemed like he was just making stuff up. But look what happens now. When you heat up a gas, electrons jump to a higher allowed level. But when they fall back down, they release the energy difference as a photon of a specific frequency. This explained the hydrogen spectrum. And the photon's energy is basically the energy lost by the electron. From this, he derived an expression for the frequency and got Balmer's formula. So his postulate explained the hydrogen spectrum, the Balmer's formula, and atomic stability all at once. This was huge. Later, in an interview, he says, "As soon as I saw Balmer's formula, the whole thing was immediately clear to me." He was probably showing off, but there were many unanswered questions here. I mean, why were electrons restricted to those specific orbits? And why don't they radiate while they are there? Bohr basically said, "Trust me."
But a French PhD student named Louis de Broglie came up with an answer. He pushed Bohr's idea to the limit. Light moves as a wave but interacts with stuff like a particle, right? Dual nature. So he wondered, what if matter behaves the same way? What if matter too interacts like a particle but moves as a wave? Then electrons inside an atom could form standing waves. And just like a guitar string can only vibrate with, say, three loops or four loops or five, but nothing in between. Electron waves can only vibrate with three or four or five loops, but nothing in between. This explained why electrons can only exist at those specific distances from the nucleus, as Bohr said. And it also explains why they don't radiate while sitting at those energy levels because they're not orbiting particles. They're not accelerating. They're stationary waves now. They only radiate a photon when they transition from a higher to a lower level. This was so radical. Matter behaving as waves. And that's not all. For his PhD thesis, he derived an expression for its wavelength using special relativity. He had found the wavelength of matter. It seems so bizarre. His examiners only approved it after confirming it with Einstein himself.
Finally, in the following year, a professor at the University of Zurich gave a seminar on this very topic, and after the talk, a colleague in the audience asked, "If matter is a wave, where is the wave equation?" The professor was Irvin Schrödinger, and he took that question seriously. So he went on a vacation to the Swiss Alps over Christmas and came back with the wave equation. But how did he do it?
I went through Schrödinger's original derivation and I couldn't understand a thing. In fact, he himself called it unintelligible later on. So, I looked up Feynman's derivation, and that was also quite mathy. But after a lot of searching, I found a modern version which seemed pretty intuitive. It starts by asking, "What rule should matter waves obey in general?" The most fundamental one we know is energy conservation, right? Total energy equals kinetic energy plus potential energy. Okay, that makes sense. Um, kinetic energy is half mv squared. So if you multiply the top and the bottom by m, we can write it as momentum squared over 2m. And I'm like, "Perfect. This is all starting to make sense." And in the final step, it says, "To make it quantum, replace energy with energy operator." Wait, what? "Kinetic energy with kinetic energy operator." What? What's going on? And they act on the wave function psi, giving us the Schrödinger's equation. What just happened?
After calming down a bit, I realized I just had to answer two questions. First of all, why do we need these operators? And second of all, how do I build them myself intuitively? So, let's start with the first question. We know E equals hf. And we also know p = h / lambda (de Broglie). Why not just substitute them directly into this equation, right? That was what I was thinking. Well, the answer is actually right in front of us. You see, that would only work for an infinitely long pure sine wave because, look, only then it would have one single definite wavelength and frequency. But a general wave doesn't have a single wavelength or frequency. It can have a whole range, all mixed together. Now, for ordinary waves, that's not a big deal. But for matter waves, think about what it means. This means the particle doesn't even have a single value of momentum or energy. At this point, we don't even know how to interpret that. I mean, what does it even mean for an electron to not have a definite energy? Well, let's keep that question aside. We'll come back to it later. But for now, what's important is because matter waves don't have a single value of energy or momentum, we can't substitute directly. Instead, we need something that extracts the entire mixture. That's what these operators do. The energy operator over here pulls the entire range of energy hiding in the wave. Similarly, the kinetic energy operator pulls the entire range of kinetic energy. That's why in quantum mechanics we always talk about operators, right? Because matter waves don't have single values for energy or momentum or whatever. They have a whole range. Okay, so first question answered.
On to the second one, and the most important one. How do I build these operators myself? Here's a powerful problem-solving principle. When you're trying to solve a hard problem, first see if you can create a simpler version of that problem and try to solve that. So in our case, the hard problem is to build operators that work in general. The simpler version would be to try and build these operators for the simplest wave possible, a sine wave. So maybe if I can build an operator for a sine wave, I can then use that intuition to generalize it.
First of all, this could be a traveling wave, or you know, it could be a standing wave like, you know, de Broglie described. I like standing waves. The math feels slightly more intuitive. So let's go with that. Let's draw a couple of axes. You have the x-axis and you have psi, which is, you know, Schrödinger's own notation. So the first question is, what is the equation for psi? Well, let's pause the animation. We can write psi = sin(x). I mean, we could also write psi = cos(x). It's just a matter of where we put the origin. But let's stick to sin(x). That's the simplest equation, right? But guess what? Psi's height swings between +1 and -1. I want our height to be slightly more general. Let's call it A. So how do we do that? Well, we scale this by A. That's the amplitude. Now I can control the amplitude. Perfect. But sine always resets after 2 pi. Which means right now our wavelength is locked exactly at 2 pi. I don't want that. I want to be able to have any wavelength. I want lambda. So what do we do? Well, we multiply x by 2 pi over lambda. I mean, think about it. Now when x equals lambda, the lambda cancels out, and the argument becomes 2 pi, and the wave resets. So now lambda has become our wavelength.
But this isn't a wave yet. It's a frozen picture. A standing wave means the amplitude itself changes over time periodically. So A itself needs to be some periodic function of time. Again, we'll choose the simplest function, psi, but just like before, psi has a period of 2 pi. So right now our, you know, time period is locked at 2 pi. So we want the period to be, let's say, capital T. So we use the same trick as before. We multiply this by 2 pi over capital T. And we are done. We just need a bit of cleaning up over here. 1/T is frequency. And physicists hate writing 2 pi over and over again. So we will define 2 pi f as a new variable omega, and similarly 2 pi over lambda as a new variable kappa. We're only doing this so that we don't have to write two pies over and over again. Okay. If we substitute it, boom, we have built the equation for our standing wave. But this is still a generic wave. There's nothing quantum about it.
To make it quantum, we bring in Einstein and de Broglie's equation. Now, since this is a pure sine wave of one specific frequency and wavelength, we can directly substitute over here. But before we do that, we have to write this in terms of omega and kappa as well. So, let's quickly do that. To do that, we'll just multiply and divide by 2 pi everywhere. And now look at what we get. h over 2 pi, we'll call that as h-bar. We call this the reduced Planck constant. And 2 pi f is omega. And 2 pi over lambda. Well, we have kappa. And this cleaning up actually makes things much more beautiful. I mean, think about it. What exactly is omega over here? Omega basically tells us the number of waves per second, right? So we can call it the temporal frequency. What about kappa? Well, kappa tells us the number of waves per meter. Look at this. Per meter. So that is the spatial frequency, which means for matter waves, the temporal frequency encodes the total energy. It lives in the time domain, and the spatial frequency encodes the momentum. It lives in the space domain.
When I saw this, a light bulb went off. I mean, think about it. You probably know that in special relativity, space and time are not two separate things. They're just shadows of the underlying four-dimensional spacetime, right? So, we could guess that energy and momentum aren't separate things. Maybe they are just two components, shadows of something much deeper, a four-dimensional object. And that's exactly what we have in special relativity. It's called four-momentum. So, in relativity, we don't think of energy and momentum separately. We just think of them as two components of the underlying object called four-momentum. I know this is a tangent, but oh my god, like that connection is beautiful. Anyways, we substitute now for omega and kappa, and boom, we have built our very first matter wave equation.
But remember, what our actual goal was here was to build energy and kinetic energy operators. So let's start with kinetic energy. Since it has momentum squared in it, our question would be, how do we pull momentum out of this equation? We can differentiate it with respect to x, right? Well, actually, we need to do a partial derivative. Then this part becomes a constant. And now look, p over h pops out in front, and psi turns to cos. But we don't want just momentum. We want momentum squared because our goal is to get kinetic energy. So what do we do? Well, we differentiate again. On the left-hand side, we get a second derivative. And over here, p over h pops out one more time. And cos becomes negative sine. And if you look carefully, look, this is our original function psi. So we have got psi back. And if we rearrange, we have built our very first operator, the momentum squared operator. Look, when you do this operation on psi, you get momentum squared times psi. So momentum squared has been extracted. So this is the momentum squared operator. For the last step, I need kinetic energy, right? So kinetic energy is just momentum squared by 2m. So let me just divide by 2m. And this is the kinetic energy. So I have found how to extract kinetic energy. So this must be the kinetic energy operator. Wow. We've built it all by ourselves. And if we compare it to the actual Schrödinger's equation, oh my god, it's exactly the same thing. Whoa.
All right, let's calm down. But what does it actually say? Well, second derivative is basically curvature, right? So this says if your wave has more curvature, then it will have more kinetic energy. And that makes sense because we already saw that, you know, from de Broglie's equation that shorter wavelength means more momentum means more kinetic energy. Shorter wavelength means more curvature. But our understanding has upgraded because the idea of wavelength only works for pure sine waves, right? But the idea of curvature can be defined at every single point. So it is a general way of thinking about it. Kinetic energy is encoded in the curvature. Oh man, that is awesome.
But this brings up an annoying problem. See, we derived this operator for a special case, a pure sine wave, right? But it turns out the operator works in general for any shape. Now, that is awesome. I'm not complaining because this means we can do the same thing for building the energy operator and then we can finish the Schrödinger equation and fulfill our destiny. But I won't be able to sleep at night because although we have some intuition for why this should work in general, I can't actually convince myself mathematically why something we derived for one specific special case perfectly works in general. I was stuck here for a while until I met a man who was so obsessed with heat that apparently he kept his room at blazing temperatures and he sat inside them wrapped in a blanket during hot summers. His name, John Baptist Joseph Fourier. Fourier believed heat had magical healing powers. But historians think it's probably because, you know, he developed extreme cold sensitivity during his time in the Egyptian desert. But whatever it is, what's important for us is that he was a mathematical genius. And so obviously he wanted to mathematically model how heat flows. And he did that. And while doing so, he developed an incredible principle. He found that any wave or any shape at all can be written as a sum of sines and cosines waves. Here's what I mean. Take this square wave. According to Fourier, you can write this as just sums of sines and cosines. If you just use one sine wave, that doesn't look like much. But if you add a second one with a slightly different height and width, look, it gets closer. Add a third and a fourth and a fifth, and you keep going, and look, look, the shape converges towards a perfect square wave. Now, of course, technically we need infinitely many, but look, I mean, even with a few, we can get remarkably close, right? Here's another example. Again, by adding multiple sine waves of just the right width and height, we can construct this shape too. We can construct any shape. And Fourier showed that this works in general. And the idea is called the Fourier series, or more generally, it's called the Fourier transforms. This idea was so radical that even the top mathematicians back then just couldn't believe it would be true. But today, we have a very elegant proof for it. Let me know if you want me to make a video on that. But for now, we'll just accept that.
So, according to Fourier, our matter wave can be written as the sum of lots and lots of pure sine waves. So now, what will happen if we apply that operator we built to this general wave? Well, derivatives are linear, meaning derivative of A plus B equals derivative of A plus derivative of B. That means this operator gets applied to every single component. For each component, it spits out the component's kinetic energy multiplied by the component's wave function. And then it adds up all of those and spits this entire sum back. Which means, look, the operator, when working on a general wave, spits out the full mixture of the kinetic energy as a weighted sum. And so Fourier helps us understand why if an operator works for a sine wave, it should work for any wave in general. Oh man, Fourier beauty. Imagine if Fourier knew that his obsession with heat would one day unlock the framework for just, you know, modeling the quantum world. Oh my god, how would he be feeling? Oh my god.
Anyways, this means all we have to do is repeat the same process for extracting energy, and we are done. It will be a great idea to pause the video over here and see if you can try this yourself. Please do that. Moment of truth. All right. So to extract energy, look, I have to differentiate with respect to time. This time, this term is a constant. So nothing happens to it. So when I differentiate sine, well, again, E over h pops out, and sine turns into cos. We have our energy. So let's just rearrange. And wait, there's a problem. I'm not getting my function back because sine turned into cos. So I can't write this as psi. Wait, why did it work last time? Oh, last time it worked because we took a double derivative, right? Because I wanted momentum squared. And so when I took a second derivative, well, psi turned to cos, and cos turned back to psi. And I was able to cleanly write this as psi and build my operator. Oh, but this time I just need a single derivative because I already found energy. I don't want energy squared. So I can't take a second derivative. But I can't write this as psi because the sine has turned into cos. Oh man, we were so close. What? What do we do?
Let's just think mathematically. For this to work, to get our psi back, this function of time has to be such that its first derivative must be itself. But a psi or a cosine or any periodic function, for that matter, none of them give us that. There's only one function in this entire multiverse with that property. It's the exponential function. I mean, think about it for a second. Imagine that this function was exponential in time. Now, if we differentiate it with respect to time, again, E over h-bar pops out. And this time, because the derivative of exponential is itself, look, we get our function psi back. We can now rearrange and build our operator just like before. And the math would work out. But the problem is this is not a wave. For a wave, the amplitude needs to be some kind of a periodic function of time. But exponentials are not periodic functions. Exponentials will just keep blowing up forever, right? Or if you, if you put a negative sign, for example, then yeah, then the exponential would just keep decaying forever. Whatever it is, this is not a wave. So we have a problem. I mean, for the physics to work, we need this function to be periodic. But when you use a periodic function, the math breaks down because I cannot extract energy and build an operator. For that to happen, I need that function to be exponential. But then the math works, but the physics doesn't work because it's not a wave. So to make both of them work, we somehow need an exponential function of time that's also periodic. But that's impossible. By definition, exponential functions are exponential. So, how in the world can we build a function that's both exponential and periodic?
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Okay, so coming back, how do we create a function that's both exponential and periodic? The answer actually comes from a bookkeeper and an accountant from Paris. His name, John Robert Argand. Argand asks, "Mahesh, why do exponential functions in general blow up or decay?" Well, I say that's because that's literally what exponential is. There's no deeper explanation, right? Well, he says, "Think about it physically. The derivative of an exponential is proportional to itself, right? So, think of this as velocity. The velocity in this particular case is proportional to the position and it's in the same direction as the position. So a little time later, the position grows, but the velocity grows as well. So now the position grows even faster. The velocity grows even faster, and that's how we get a runaway effect. The whole thing blows. If you had a negative exponent, then you would have the same effect, except now the velocity would be in the opposite direction because of the negative sign. So now a little time later, the position reduces because of that, the velocity also reduces. So the position reduces slower, the velocity reduces even slower, and you get a decay. But Argand asks, "What if somehow we could make the velocity perpendicular to the position?" This time, a little later, the position neither grows nor shrinks, but it just shifts sideways, which means the velocity magnitude stays the same as well. And it continues to stay perpendicular, meaning it keeps pushing it sidewards forever. That is a uniform circular motion. That will make our exponential periodic. I mean, sure, it's not oscillating, but it, that's okay. It's periodic. That's what I want. But Argand, how do we control the direction of this um velocity? And how do we make it perpendicular? Well, Argand says, "Think about this. When we multiply the exponent by i, the velocity points in the same direction as position." That's a 0-degree rotation of the velocity. When we multiply by -1, it points in the opposite direction. So this rotates velocity by 180 degrees, which means we need to multiply the exponent with some number that rotates the velocity by 90 degrees. How do we find that? Well, say it's simple. Whatever number this is, if you multiply it by itself one more time, well, it would rotate again by 90 degrees, meaning 180 degrees, but we already know that is negative one. So, in other words, whatever this number is, it multiplied by itself should give me -1, or the square of that number should give me -1, or that number which rotates by 90 degrees is the square root of -1. The "i" has entered the room. Oh, that's, that's where it, ah, that's where it comes from. I'm sorry. Let's calm down.
But what is the "i" doing over here? The "i" in the exponent produces a 90-degree rotation, making our exponential function spin, making it periodic. Now, of course, this rotation is not in real space. This is a complex plane. So, this is the real axis, and this is the, um, imaginary axis. Now, because of Argand's beautiful geometric insight into these complex exponentials, we call this complex plane the Argand diagram. So, just to recap, we started with microwaves oscillating up and down, and we found that the math didn't work. We couldn't cleanly extract the energy. Now, to make the math work, we needed exponentials, but they aren't periodic. Argand gave us a way out: complex exponentials. Now, the math works because I can cleanly extract energy because it's an exponential function. But the physics works too because the function has become periodic. This means now we have to accept that our microwave is rotating in some kind of an abstract complex plane. But that's okay. At least we found a way out. So, let's run with it now. So, what we're going to do is, let's use this complex exponential as our basic wave. Okay? Now, the first question is, does that change anything that we did so far? Well, remember, we are taking a partial derivative with respect to x, and when we do that, the time part is treated as a constant. So whether we have a sine here or, you know, exponential over here, it doesn't matter. So everything stays the same, and so nothing changes over here. So that's awesome. I don't have to do any more work over here.
But now it's time to build the energy operator. So if you differentiate with respect to time, we get energy extracted, and we get our function back. So I get my psi cleanly. So it's time to rearrange and isolate energy over here. And if I multiply numerator and denominator by i, we have done it. We have cleanly extracted energy. This is the energy operator. But wait, when I saw this, I was like, "Wait a second, why is it a negative sign?" I mean, I know that the original Schrödinger's equation, as we will see, doesn't have a negative sign. Well, it turns out it's a small convention thing. We chose our matter wave spinning in the clockwise direction as seen from here. Well, it turns out physicists like to choose anticlockwise or counterclockwise as the convention. So physicists love to choose the negative sign over here for their exponentials over here. So this would also be negative, and so you'll have a negative popping out that cancels with this one. And so our energy operator wouldn't have a negative sign over here. That's just a matter of convention.
Anyways, thanks to Fourier, I know that this operator works in general, which means we can now build our Schrödinger's equation. So we start by energy conservation: Total energy equals kinetic plus potential. And then for total energy, we substitute the energy operator. For the kinetic energy, we substitute the kinetic energy operator, and they're working, and they're operating on the wave function psi, and we have our Schrödinger's equation in its full glory.
So can we now answer our original question intuitively? Why is there an "i" over here? Well, because our wave function itself is complex. Why should it be complex? Because it needs both an exponential and a periodic function. Why does it need to be an exponential function? Well, because that's the only way I can extract energy cleanly. I can build an energy operator because I need the first derivative of my function to be itself. Right? That doesn't that make sense? Okay. Now, here's another question. What would happen if I were to consider the same equation without the "i"? Now, the solutions would be real exponentials in time. They will no longer rotate, which means they would just blow up or decay forever. Would it represent anything physical? Yes. Say the vertical axis was temperature and the horizontal axis represented a rod, which means we now have a temperature distribution. And as time passes, the hot regions cool down and the cool regions heat up. So the graph shrinks. But as the temperature gets closer to each other, it gets slower over time. Meaning we get exactly a decay. In other words, this now represents the heat equation. How temperature changes over time. But of course, it can have different constants, and we would expect it to not have any potential energy term. So we can now intuitively see why the Schrödinger's equation looks so similar to the heat equation, except for the "i". It's the "i" that turns an exponential into a spinning exponential, periodic.
At this point, I'm really satisfied with the Schrödinger equation, where it comes from, and why there is an "i". And I think I have a really good intuition behind it. But a part of me still feels that it's all still mathematical. It kind of feels like, you know, there might be some kind of a mathematical trick that we just haven't thought of yet, using which we can get rid of that "i". And that's exactly what Schrödinger was trying to do for months after publishing his equation. And when he couldn't, he wrote a letter to Hendrik Lorentz saying, "What is unpleasant here and indeed directly to be objected to is the use of complex numbers. Psi is surely fundamentally a real function." So mathematically, it makes sense why the "i" must be there. But what is the physical reason behind it?
Well, the breakthrough came as a footnote, actually, in a paper published by Max Born. Born says, "Let's go back to the double-slit." We are now coming full circle to where we started. Awesome. This time, if we send only one photon's worth of energy through those slits, it has to land somewhere at one specific spot on the screen, right? Mahesh, can you tell me where? And I say, "I have no idea." And Born says, "Neither do I, but I can tell you where it's more likely to land. The chances of landing in these dark regions is almost zero because no photons have landed so far. And the brightest spot in the center, the chances is very high there because lots of photons have already landed there before. So the dimmer the regions, the lower the chances." So even though I can't say exactly where one photon lands, I can talk about the probability, and that probability is proportional to the brightness or intensity, which is proportional to the amplitude squared. This is how Einstein's idea can be interpreted probabilistically. And Born says, "Well, I just thought that we can try and apply the same thing to matter waves. If you replace light with a source of electrons, they too travel as waves, and then we should get the exact same result." Which means the probability of finding now the electron in any spot is the square of the wave function at that point. This is today called the Born rule. This is what he wrote down as a casual footnote, and it gave us a way to interpret the matter wave. It's a probability wave.
And remember at the beginning, we asked a question of how matter waves carry mixtures of energy and momentum values. How do you make sense of that? Well, that mixture is really a probability distribution. So, for example, when you measure its energy or momentum, there's some probability of getting each particular value. And that probability distribution is hidden in that wave function.
So, coming back to our original question, how does the Born rule give us a physical meaning to "i"? Suppose our matter wave just oscillates back and forth on the real axis. Let's say we found some mathematical trick to make this work. Okay? Then the psi squared over here would also fluctuate, right? And since psi squared is the probability of finding the particle somewhere, the total area under this curve should represent the total probability of finding this electron anywhere in the universe. But in this particular case, that total probability fluctuates. I mean, at some moment, it can even go to zero, for example. That makes no sense, right? I mean, the total probability of finding the electron somewhere in the universe has to be 100%. Right? But how can it be zero? It doesn't make any sense. The probability, for example, can shift from place to place, but the total should never change. So you can clearly see a simple up and down standing wave can't give us a probability wave, a matter wave. There's no way to make it work.
But a wave that is rotating in a complex plane has no such problem. Look here, at every point in the complex plane, the height of this "i" stays exactly the same because it's just spinning. So what happens to psi squared? The psi squared stays fixed, which means the total probability now stays fixed. Now, of course, for more complicated matter waves, the probability will change with time, but the total probability will still stay the same. However, if you were to model matter waves using just up and down oscillating motion in just the real axis, there's no way for that to happen. Even the simplest wave, you can't model it. So, it's the "i" that makes the wave function spin and conserves the total probability. It's the square root of -1, the imaginary number, that makes the physics real.
Schrödinger solved his equation for the hydrogen atom using the 1/r potential function. And not only did he get the hydrogen spectrum, he also predicted the relative brightness of each line and showed how adding electric or magnetic fields splits these lines. But more importantly, the equation predicted the three-dimensional probability distributions of the electrons in the hydrogen atom, the orbitals. In short, the equation was a complete, radical breakthrough. It became the F=MA of quantum mechanics. As a result, Schrödinger shared the 1933 Nobel Prize in Physics with Paul Dirac. And today, scientists and engineers have done extraordinary things with it. We've found ways to focus electron waves and build electron microscopes powerful enough to see individual atoms. We've calculated approximate energy levels inside heavier atoms like cesium, for example, to build atomic clocks. And when multiple atoms come close together, we found that those energy levels turn into bands. And by controlling the band gaps, we've built ultra-tiny transistor switches that make up every single microchip. Our real modern world, brought to you by the imaginary number.