Quantum Wavefunction | Quantum physics | Physics | Khan Academy

So when people first showed that matter particles like electrons can have wavelengths, and when de Broglie showed that the wavelength is Planck's constant over the momentum, people were like: "Cool, it's pretty sweet." But, you know, someone was like: "Wait a minute! If this particle has wave-like properties and it has a wavelength, what exactly is waving? What is this wave we're even talking about?" Conceptually, it's a little strange. I mean, a water wave, we know what that is; it's a bunch of water that's oscillating up and down. A wave on a string, we know what that is; the string itself is moving up and down and it extends through space.

But it's hard to imagine how this electron is having a wavelength, and what is the actual wave itself. So physicists were grappling with this issue, trying to conceptually understand how to describe the wave of the electron. They wanted to do two things: they wanted a mathematical description for the shape of that wave, and that's called the wave function. This wave function gives you a mathematical description for what the shape of the wave is. So different electron systems are going to have different wave functions. This is, this is psi; it's the symbol for the wave function.

So this is psi, the psi symbol—it's a function of x. At different points in x, it may have a large value; it may have a small value. This function would give you the mathematical shape of this wave, so that was one of the things they were trying to determine. But they also wanted to interpret it; like, what does this wave function even mean? So we’ve got two problems: we want a mathematical description of the wave, and we want to interpret what does this wave even mean.

Now the person that gave us the mathematical description of this wave function was Erwin Schrödinger. So Schrödinger is this guy right here. Schrödinger’s right here; he wrote down Schrödinger’s equation, and his name now is basically synonymous with quantum mechanics, because this is arguably the most important equation in all of quantum mechanics. There are a bunch of partial derivatives in here and Planck's constants, but the important thing is that it’s got the wave function in here.

Now, if you’ve never seen partial derivatives or calculus, it’s okay. All you need to know for our purposes today in this video is that this equation is a way to crank out the mathematical wave function. What is this function that gives us the shape of the wave as a function of x? You can imagine plotting this on some graph. So once you solve for this psi as a function of x, you could plot what this looks like; maybe it looks something like this. And who knows? It could do all kinds of stuff; maybe it looks like that. But Schrödinger’s equation is the way you can get this wave function.

So Schrödinger gave us a way to get the mathematical wave function, but we also wanted to interpret it. What does this even mean to say that this wave function represents the electrons? It’s still strange. What does that mean? Schrödinger tried to interpret it this way: he said, "Okay, maybe this electron really is like smudged out in space, and its charge is kind of distributed in different places." Schrödinger wanted to interpret this wave function as charge density. I mean, it’s kind of a reasonable thing to do; the way you get a water wave is by having water spread out through space, so maybe the way you get an electron wave is to have the charge of the electron spread out through space.

But this description didn’t work so well, which is kind of strange. Schrödinger invented this equation; he came up with this equation, but he couldn’t even interpret what he was describing correctly. It took someone else—it took a guy named Max Born—to give us the interpretation we go with now for this wave function. Max Born said, "Nah, don’t interpret it as charge density. What you should do is interpret this psi as giving you a way to get the probability of finding the electron at a given point in space."

So Max Born said this: "If you find your psi, like he said, go ahead and use Schrödinger’s equation, use it, get the psi. Once you have psi, what you do is you square this function. So take the absolute value square it, and what that’s going to give you is the probability of finding the electron at a given point." Now, technically it’s the probability density, but for our purposes, you can pretty much just think about this as the probability of finding the electron at a given point.

So if this was our wave function, in other words, Max Born would tell us that points where it’s zero—these points right here, where the value is zero—there is a zero percent chance you’re going to find the electron there. Points where there’s a large value of psi, be it positive or negative, there’s going to be a large probability of finding the electron at that point. We could say the odds of finding the electron at a given point here are going to be largest for this value of x right here, because that’s the point for which the wave function has the greatest magnitude.

But you won’t necessarily find the electron there. If you repeat this experiment over and over, you may find the electron here once, you may find it over here, you may find it there next time. You have to keep taking measurements, and if you keep taking measurements, you’ll get this distribution where you find a lot of them here, a lot of them there, a lot of them here, and a lot of them here. Always where there are these peaks, you get more of them than you would have at other points where the values were smaller. You build up a distribution that’s represented by this wave function.

So the wave function does not tell you where the electron is going to be; it just gives you the probability. And technically, the square of it gives you the probability of finding the electron somewhere. So even at points down here where the wave function has a negative value, I mean, you can’t have a negative probability. You square that value, that gives you the probability of finding the electron in that region.

So in other words, let’s get rid of all this. Let’s say we solved some Schrödinger equation, or we were just handed a wave function and we were told it looks like this, and we were asked: "Where are you most likely to find the electron?" Well, the value of the wave function is greatest at this point here, so you’d be most likely to find the electron in this region right here. You’d have no shot of finding it right there; you’d have pretty good odds of finding it right here or right here, but you’d have the greatest chance of finding it in this region right here.

So you’d have to repeat this measurement many times. In quantum mechanics, one measurement doesn’t verify that you’ve got the right wave function, because if I do one experiment and measure one electron—boop!—I might find the electron right there. That doesn’t really tell me anything. I have to repeat this over and over to make sure the relative frequency of where I’m finding electrons matches the wave function I’m using to model that electron system.

So that’s what the wave function is—that’s what it can do for you. Although, if I were you, I’d still be unsatisfied. I’d be like: "Wait a minute! Okay, that’s fine and good. The wave function can give us the probability or the probability density of finding the electron in a given region, but we haven’t answered the question: What is waving here? And what exactly is this wave function? Is this a physical object, sort of like a water wave or even an electromagnetic wave, or is this just some mathematical trickery that we’re using that has no physical interpretation other than giving us information about where the electron's gonna be?"

And I’ve got good news and bad news. The bad news is that people still don’t agree on how to interpret this wave function. Yes, they know that the square of it gives you the probability of finding the electron in some region, but people differ on how they’re supposed to interpret it past that point. For instance, is this wave function the wave function of a single electron, or is this wave function really the wave function of a system—an ensemble of electrons all similarly prepared—that you’re going to do the experiment on? In other words, does it describe one electron or only describe a system of electrons? Does it not describe the electron at all, but only our measurement of the electron? And what happens to this wave function when you actually measure the electron? When you measure the electron, you find it somewhere, and at that moment, there’s no chance of finding it over here at all.

So does the act of measuring the electron cause some catastrophic collapse in this wave function that’s not described by Schrödinger's equation? These and many more questions are still debated and not completely understood. That’s the bad news. The good news is that we don’t really need to understand that to make progress. Everyone knows how to use the wave function to get the probabilities of measurements. You can have your favorite interpretation, but luckily, pretty much regardless of how you interpret this wave function, as long as you’re using it correctly to get the probabilities of measurements, you can continue making progress, testing different models, and correlating data to the measurements that people make in the lab.

Now, I’m not saying that interpretations of this wave function are not important. People have tried cracking this nut for over a hundred years, and it’s resisted. Maybe that’s because it’s a waste of time, or maybe it’s because the difficulty of figuring this out is so great that whoever does it will go down in history as one of the great physicists of all time. It’s hard to tell right now. But what’s undebatable is that for about a hundred years now, we’ve been able to make progress with quantum mechanics, even though we differ on how exactly to interpret what this wave function really represents.

So recapping, the wave function gives you the probability of finding a particle in that region of space. Specifically, the square of the wave function gives you the probability density of finding a particle at that point in space. This almost everyone is agreed upon. Whether the wave function has deeper implications besides this, people differ; but that hasn’t yet stopped us from applying quantum mechanics correctly in a variety of different scenarios.