Thanks to Brilliant for sponsoring this episode. Hey Crazies. I’m going to be straight with you. I hate quantum mechanics. It hurts my brain. But misery loves company, right? So, let’s take a trip down the quantum brick road. First, this is not a video about the double-slit experiment. I did that a few months ago. Feel free to check it out later if you haven’t seen it. It’s an overused example. Plus, in my experience, any specific experiment is going to distract from the main point today, which is: Quantum wave functions: what’s actually waving? Yeah, what’s actually waving? Probability. Huh? I’m going to explain! Remember, we’re taking a trip down the quantum brick road. It’s a journey, crazies! Anyway, the name has three words in it: Quantum, Wave, and Function. We better know what those mean separately before we go mixing them together. A quantum is just the smallest piece of some physical property that could be used during a particle interaction. Like when an electron in an atom drops down energy levels, a photon is usually emitted. That photon is a quantum of light energy. When we realized stuff like this could happen, we had to invent a new type of mechanics to describe it. Naturally, we called it quantum mechanics. And lots of names for things in quantum mechanics start with the word “quantum.” Kind of like how a certain company starts everything with the letter “i”. Alright, next up: wave. A wave is anything that has a back-and-forth shape. They’re often curvy and regular like this, but they don’t have to be. They could be irregular, or boxy, or even triangular. They just need to have that back-and-forth shape. So, yes, while wave is a verb, it’s also a noun. Waves are things! Get on board, people! Last, but certainly not least, function. By that, we mean a mathematical function, which is a relation or map between sets of numbers, but that’s a bit abstract. It’s easier to think of a function like a little mathematical machine. You put numbers in, like maybe time or something, and you get numbers out that depend on that input, like the motion of a squirrel. We’d say that motion is a function of time or just “v of t” for short. So, this function tells you how fast a squirrel is falling at any given moment. It’s just a way of showing that some measurements depend on other measurements. Now let’s start combining words. A wave function is a function that looks wavy. Surprise! The falling squirrel function isn’t wavy though. It’s just a slanted line. There’s no back-and-forth shape. We need a back-and-forth motion. Let’s attach the squirrel to a spring and give it a little bounce. Put some moving graph paper behind it and bam! We’ve got a wave. The height of the squirrel depends on time, so we say it’s a function of time. Since its shape is wavy, we call it a wave function and it’s written using some kind of sine or cosine because we know those look wavy. Bouncing squirrels aren’t the only waves in nature though. There are waves on strings. There’s sound, water waves, AC voltage waves on oscilloscopes screens, and even light. All of them involving some kind of sine or cosine. That’s all there is to a wave function. It’s adding the word quantum on the front that makes things weird. But we can handle it. Remember, quantum brick road. It’s a journey. OK, a quantum wave function is a wave function for quantum things. It shows dependence between two measurements that has some kind of back-and-forth pattern and is about tiny particles. What two measurements? What’s waving? What? What? What?! Dude, calm down. Everything is going to be OK. Let’s go deeper. If a quantum wave function is like any function, then it’s a little mathematical machine. You put numbers in and you get numbers out. The input is some measurable property about a particle: where it might be, what it might be doing, when something might happen, that sort of thing. The output? Well, that’s the tricky part. See, with normal wave functions, the output is pretty clear. It’s some kind of distance, or field strength, or something else that’s directly measurable. Quantum wave functions are a bit more abstract than that. In fact, they’re unobservable by definition. It might help to look at how they’re used. I think it’s time for an example. The simplest one I can think of is the particle in a box. For now, let’s keep the box one-dimensional. We don’t want things to get too complex. Get it? Complex? You’ll get the joke later. Inside the box, there is a single particle. Maybe it’s an electron, but it doesn’t matter. Just go with it, OK? That particle has properties we might be interested in knowing something about: Where is it? What is it doing? When will it decay into other particles? Of course, this is quantum mechanics, so we can’t know anything exactly. But we can predict the probabilities of what those properties might be. That’s where the quantum wave function comes in. Say we want know where our particle might be inside the one-dimensional box. It could be just about anywhere, but has a better chance of being in some places than others. Its quantum wave function gives us some idea of where. Just by looking at it, we can see this particle has a better chance of being in the center of the box than on the edges. That’s just a general idea though. It doesn’t give us an actual probability. To make matters worse, this isn’t even really
waving up and down. It’s actually rotating through both real
and imaginary numbers. The combination of which we call complex space. Hence that complex joke from earlier: We don’t want things to get too complex. Get it? Complex? Ugh, I’m such a dork. The point is the quantum wave function isn’t even completely real, at least in the mathematical sense. But that doesn’t mean we can’t get something real out of it. If you multiply a complex number by what we call its conjugate, that’s just a sign swap on the imaginary part, then you’ve done something called a complex square. The imaginary parts cancel and you’re left with only real parts. [Censor] just got real! [Laughing] Jokes aside, now that all we have are real
parts, we can actually make physical sense of this. Instead of looking at the quantum wave function itself, we look at its complex square, something we call the probability density. In this case, it’s a probability per unit length measured along our one-dimensional box. This view of the wave function is called the “Born Rule” named after Max Born who came up with it in 1926. This realization was ground-breaking. So, we’d say this is the probability density for position. It’s best to think of it like a bunch of separate numbers along the box. The height of each of these bars is the probability density at that location. There are a lot of possible locations for
this particle though, so there are a lot of vertical bars. So many, in fact, that it all just looks like a shaded area. The total area covered by all the bars should be equal to 1 or 100%. Because there’s a 100% chance of finding the particle somewhere in the box. I mean, we put it in there ourselves, right? But let’s say we only want to know the probability of finding the particle between here and here. All we do is add up all the skinny rectangles between those two points and the shaded area is the probability. It’s that easy. You can clearly see that some areas have a higher probability than others. What if I’m interested in something other than position? Then you make that thing your input instead. This was in terms of one-dimensional position, so it tells you probabilities in position. You want to know about momentum instead? Just transform the wave function. Any areas you shade now will give you probabilities in momentum, as long as the total area is... Whoa! Wait! Whoa! Whoa! That wasn’t supposed to happen. It appears our probability density blows up to infinity in a couple places. What’s that mean?! Eh, nothing profound. It’s just that our example is a bit too simplified. In other words, I was lazy. When you make a particle box that’s infinitely deep, infinity is bound to show up in a few results. I just happened to find one. That doesn’t invalidate the whole process though. It works if you chose a model that’s a little more realistic. Here’s a more accurate probability density for the electron in a hydrogen atom. The input is the distance from the center of the atom. Just like before, any shaded area gives you the probability of finding the electron within that range of distances. If you transform it so that momentum is your input, you can find the probabilities for momentum instead. See. No infinities this time. You can write the wave function in terms of anything you want to know: Position, energy, linear momentum, angular momentum, whatever. It’s the same wave function just set up to accept different inputs. So what’s a quantum wave function? It’s a mathematical entity that has no physical meaning. But, if you take its complex square, you give it meaning as a probability density. The total area shaded by it should be 1 or 100%, so any smaller area you shade will give you the probability of finding those values when you measure that property. If you’re interested in another property, you just transform the wave function, but the same rules apply. Shaded areas are probabilities. A single quantum wave function contains all the probabilities we might want to know about any property of a quantum particle and, ultimately, that’s all we get to know. So, did this help you understand quantum mechanics a little better? Let us know in the comments. Thanks for liking and sharing this video. Don’t forget to subscribe if you’d like to keep up with us. And until next time, remember, it’s OK to be a little crazy. Becoming great at math and science doesn’t have to be dull. Brilliant is a problem solving website and app with a hands-on approach. There are over 50 courses full of storytelling, interactive challenges, and problems to solve. If you watched this video, you’d probably like their course on quantum objects. The course covers everything from concepts to the unique mathematical notation used in quantum mechanics. They even get deeper into those graphs I spent most of this video talking about. Brilliant is built for ambitious and curious people, who want to excel at problem solving and understanding the world. It’s a great complement to watching educational videos, with well-curated sequences of problems that help you master all sorts of technical subjects. If this sounds like a service you’d like to use, go to brilliant dot org slash Science Asylum today. The first 200 subscribers will get 20% off
an annual subscription. Several of you pointed out the ridiculously slow drift velocity of charges in a current. And you’re right. It’s on the order of centimeters per hour, give or take. But relativistic effects occur at all speeds. It’s just a matter of whether or not it’s measurable and that depends on more than just the speed. Anyway, thanks for watching.
