What is The Schrödinger Equation, Exactly?

Hi guys! I'm Jade, nice to meet you. So the Schro

dinger

equation

is one of those things that comes up a lot in articles, magazines and so on about quantum science, but the journalist usually doesn't go into

what

it means, which is fair because it's a pretty complex topic. So today I just wanted to share with you

what

it really means so that the next time you read it in an article you can have a better idea of ​​what it's all about. So the short version is the Schro

dinger

equation

which tells us everything we can know about a quantum system.

It is basically the F=ma of the quantum world. If you throw a ball and solve F = ma, you can predict its position and momentum at any time. Once you have these two, you can derive basically everything you can know about it, speed, energy, etc. But when we get to the land of particles, Newton's equations no longer work. If we put a particle in a box and want to know where it is, F=ma is simply not enough. The Heisenberg uncertainty principle says that we cannot know both the exact position and momentum of quantum objects, but we can know other things, such as energy levels and wave function, which we will explore in this video.

But all that information is inside the Schrodinger equation and with some deep math we can unravel it. Now for the long version! So before we start, I must say that this is the time-independent version, meaning it doesn't involve time. And if at any point in the video you get lost, don't worry, not even Schrodinger knew

exactly

what his equation meant. So let's say we have a quantum system: an electron in a box. We want to know everything we can about this electron so we can make predictions, where it might be and what energy level it might be at.

All of these answers are buried within the Schrodinger equation. So first let's start with this guy, the gallows. This is the Greek letter psi and it means what is called a wave function. It tells you where the electron is likely to be. But not where it will be. Quantum objects are sneaky because you can't predict

exactly

where they will be until you measure them. You can only predict where they are likely to be. Say there's a guy you know. You've grown up with them your whole life. They are locked in their room with their homework, a Playstation and a bed.

If you had to guess where they were, you'd say there's about an 80% chance they're on the PlayStation, a 19% chance they're in bed, and a 1% chance they're doing their homework. When you open the door you will know for sure. But you were able to make these predictions because you know this guy. What if you had to guess where this electron is? You don't know this electron. Well, that's what the wave function tells us. It gives us the probabilities of where it is likely to be. But a big difference is that while the type is only in one place at a time the electron is in a superposition of all possible places at the same time.

You may have heard the famous thought experiment of Schrodinger's cat, which is in a superposition of being alive and dead until the box is opened and it is forced to choose a state. It's the same deal here. The act of not knowing where the electron is allows its probability distribution to spread out over a large space, like a wave. Different types of waves can represent different probabilities of where they are likely to be, hence the name wave function. It is a function that describes the waveform of the electron probability distribution. Oh yeah, and when you open the door and measure where it is this wave probability cloud function collapses and the electron becomes a particle again.

No wonder Schrodinger was confused. That's what the gallows means, the wave function tells us where our electron is likely to be. Now let's take a look at this E. It represents the energies that the electron is allowed to have. Now, before we get into what that means, I just want to point out that the way this equation is set up, these values ​​are what we're trying to solve for, so that tells us that if you do all of this, you'll be able to figure it out. The energy levels of the electron wave functions! And if we know these two things, we can derive everything else we can know about the particle, such as the position and momentum of the ball.

But let's back up for a second, what do I mean when I say energy levels that the electron is allowed to have? Like if it were an adult electron, it can have whatever energy levels it wants, right? Well, no. In the normal world we see around us, energy can rise and fall smoothly and continuously, but this is not the case in the quantum world and the reason comes from the wave nature of probability distributions. Because our particle is inside the box, it has a zero probability of being inside or outside the wall, so this means that the wave function must always be zero there, otherwise there is some probability that the electron may be outside the box, which we know it is not. .

That means that the electron can only have certain frequencies associated with it. This frequency is allowed since the wave function is zero on both edges, and this frequency is not, this frequency is allowed and this one is not. Then Einstein discovered that energy is actually proportional to frequency through this relationship E=hf where E is the energy, f is the frequency, and this h here is Planck's constant. Don't worry too much about that h for now, all you need to know is that it is a constant, meaning its value doesn't change. So if only certain frequencies are allowed inside the box and this is a constant, then it follows that certain energy levels are also allowed inside the box.

This property of discrete or quantized values ​​is where quantum mechanics gets its name. Things that can take on continuous values ​​in the normal world, such as energy levels, can only take on certain quantized values ​​on the quantum scale. Now let's look at the other side of the equation. We know what we are solving for: energy levels and wave functions. But how will all this help us get there? Well, general energy is made up of kinetic energy and potential energy. If a skater is on a skating ramp, he will travel at a certain speed and have some kinetic energy, but when he stops at the top he will still have energy, it's just transformed into a different type: potential energy.

All of the energy in the system is just kinetic energy plus potential energy, and while it can move back and forth between those two states, its total value is always conserved. Sometimes the potential energy is written as V, so this term is the potential energy of the wave function. So if this is the potential energy, that must mean that this term here is the kinetic energy, I know that doesn't actually look like any kind of kinetic energy equation that we've seen before, so here's the derivation if you don't. you believe me. So if we can solve for the potential and kinetic energy of our quantum system, this will tell us the allowed energy levels, and that's all there is to know about our little electron!

So what would some typical Schrodinger solutions look like? Well, for this particular problem, all wave function solutions take these two forms and the energy equation that emerged was this. Great Jade, what the hell does that mean? Well, the first thing to keep in mind is that each term in this expression is a constant or an integer: h-bar is a constant, 2 is obviously a constant, m the mass of the electron is a constant, pi is a constant, and L, the length of the box is a constant, and n represents the different states of the electron and they are all integers, 1, 2, 3, etc.

So the energy E can only have certain values. It is quantified. But what about the wave function? Where is the electron? Well, let's look at this guy when the electron is in its first energy state, when n is equal to 1. We understand this. That's one of the wave functions of the electron and if we square it we get the probability distribution, also known as where the electron is likely to be. We can see that there is a high probability that it is located in the middle here, but a zero probability that it is located right on the edges.

Below are some more wavefunctions and probability densities for other energy states. See how the wavefunction is always 0 right at the edges. This took me an entire semester of a physics degree to understand and what really helped me was solving a lot of problems and taking the time to develop a solid intuition. Brilliant.org has an entire course dedicated to quantum mechanics that begins with the experiments that first discovered quantum behavior and leads to the derivation of the Schrodinger equation. It has examples that you can work through at your own pace. I actually went through it to refresh my memory and learned some things I didn't know and definitely understood some things better.

They also have many other courses mainly on physics, mathematics and computer science and they are always adding more. The first 200 people to click the link below and register will get a 20% discount. Just go to shiny.org/UpandAtom. The link is on the screen and in the description. And if you're wondering why I didn't include the math behind the solutions to the Schrodinger equation it's because it would have taken me about an hour to write probably another five weeks to explain it, but for those of you who are especially curious, I've posted my final exam essay on quantum physics in description.

It includes the derivation of Schrodinger's equation, as well as solutions and some other interesting stuff, like random questions for my teacher. So, guys, be honest: do you understand the Schrodinger equation a little better now? I hope you do, but if not, please feel free to give me feedback on how I can improve my explanations for future videos. Quantum physics is cool and I really don't want anyone to miss it. So yeah, that's all from me. Bye bye! -