Physics can be really confusing what with all the weird looking maths and all that jazz Also, how on earth do these maths equations even link to real life in the first place? How do they describe the world around us? In this series, I'm going to break down some of the most important equations in physics So hopefully anyone can understand them... EVEN YOU! And don't worry, you don't need to know any math for this video I'm gonna try and break it down and explain it in as simple terms as possible using just good old English In this episode we're covering an equation that forms the basis of the world of very very small things Quantum mechanics. The equation in question is known as the Schrodinger equation That's right Schrodinger of dead or alive cat fame. So without further ado, let's get into it - roll the intro What's up you lot my name is Parth and I make fun physics videos though I don't have to try too hard because physics is already fun :D if you enjoy this video Please leave a thumbs up below and also don't forget to subscribe to my channel Hit that Bell button, cuz I like to hear those bells ringing. And without further ado, let's get into it So like I said earlier, we're looking at the Schrodinger equation today and what an equation it is It forms the cornerstone of quantum mechanics and it comes in various forms. However in its simplest form, it looks something like this. AHHHH The maths the symbols it hurts. Well, don't worry by the end of this video You'll have at least a basic understanding of what each one of these symbols represents and what the Schrodinger equation is trying to tell us So let's start with this symbol here the bit in the middle is the Greek letter "psi". The line and the pointed bracket or pointed parenthesis if you're from across the pond is put around the "psi" to let us know that we're looking at a quantum state. In quantum mechanics we often look at very small things such as atoms or even subatomic particles that are particles that make up those atoms such as electrons protons neutrons as well as stuff like photons which are packets of energy in the form of light And this is just the beginning. There's a lot more to it The point is is that whatever objects we're studying in every given scenario is what we call our quantum system "But Parth, What on earth even is a quantum system?" I hear you ask. Good question. I reply. The point is best illustrated with a few examples. The first example: if you are studying how a hydrogen atom behaves then your hydrogen atom is your quantum system as well as any extra information that we need such as "Where is the hydrogen atom?" "What's happening in its surroundings?" that kind of thing. If we place the hydrogen atom in a magnetic field then a quantum system is the hydrogen atom in that magnetic field. Simple as. Another example is if we look deeper at the hydrogen atom and study the proton and electron and sometimes neutrons that make up that hydrogen atom. In that case, those particles are our quantum system So coming back to this weird 1/2 bracket thing that we've called the quantum state The quantum state is just a mathematical description of our quantum system It's a mathematical expression that gives us as much information as we know about the quantum system So in essence the quantum state just tells us the state of the quantum system Who would have thought?! Now that we know this fact we can see that the rest of the things in the Schrodinger equation are going to kind of be affecting the quantum state in some way this means that what we're studying is the behavior of the quantum state and the Schrodinger equation tells us exactly how this quantum state should behave Now, let's look at this bit the i h bar, d over dt bit This part is fairly simple to understand firstly it's important to note that both i and h bar are just constants They're just numbers. Very important numbers but they're just numbers This is kind of like how we're used to the value PI in like the circumference of a circle or something like that Pi is just a number. Similarly in quantum mechanics h-bar is a very important number But it's just number. i, on the other hand, is the square root of -1. Now if you haven't studied complex numbers at school then you'll think this is not something that's possible. We can't take the square root of any negative number. Well, that's not technically true We can but let's not delve too deep into that If you want to learn more check out complex numbers or like youtube or something There are lots of tutorials and it's very well explained already. Instead, let's look at this bit It's pronounced d by dt. This notation you'll have seen if you know some basic calculus. Basically d by dt just measures how fast something changes with respect to time So, yes, the "t" in the d by dt bit represents time I can give you another example that might be helpful: speed We know that the speed of something is how far it travels in a given unit of time or how much distance it covers every second or every minute or every hour because we measure speed in Kilometers per hour, miles per hour, meters per second, that kind of thing So it's a distance divided by a time. More specifically however, when an object is not traveling at the same speed all the time, then we need calculus to help us out with that we write the speed as dx by dt where x is the distance that it travels and t is the time taken for it to travel. Again, we won't go into the nuances of that, but basically d by dt just measures how fast something changes with respect to time So i h-bar d by dt of the quantum state. All this means is that we've got some number, i h-bar multiplying d by dt of the quantum state. We've covered most of the equation now but we need to come to the juicy bit. The H with a hat on top. This is known as the Hamiltonian operator Hence the H. But why the hat? Well, in quantum mechanics there are these things known as "operators", such as the Hamiltonian operator and we write them with a hat on top of their symbol They basically "operate" on quantum states - they do things to this quantum state. Hence, they're called operators The Hamiltonian operator is the Big Daddy of the quantum operators because it's linked to the total energy of the system It often takes into account the kinetic energy of the system The potential energy all of these terms that you might have heard when you are at school - all of these different types of energy So essentially all that's happening in the Schrodinger equation is that the quantum state psi is changing with time in a very specific way Depending on the total energy of the system this equation This equation therefore allows us to predict how a quantum system should behave if we can mathematically write down all of the energy terms of the system. Sounds simple enough, right? However, there is a problem. For systems larger than a few particles, this becomes impossible really really quickly It's quite easy to find a quantum system where the Schrodinger equation cannot be solved analytically Basically what this means is that it's really easy to find a system where we can't solve the Schrodinger equation using just pen and paper and trying to work out a solution. Rather, we need numerical methods we need computers doing it for us But at this point in time, even our fastest computers cannot handle systems that are larger than a few dozen atoms And so this is the limitation we face in our day and age with the computing power that we have Let me give you an example: a helium atom this is one of the simplest atoms that there is. The only atom that simpler is a hydrogen atom. Now already we're gonna have to make some assumptions. Let's assume that the nucleus is at the centre of the atom and that it doesn't move. In other words all the particles in the nucleus are stationary - they're not moving. It's only the electrons that are moving around the nucleus. In reality of course, this is not true The nucleus does move and the particles in the nucleus do jiggle about but let's not go into that just yet. Even in this massively simplified situation, the Hamiltonian, which consists of all the energy terms in the system looks something like this it contains the following terms: the kinetic energy of the first electron. The kinetic energy of the second electron. The electrostatic attraction of the nucleus to the first electron. The same thing with the second electron. The electrostatic interaction between the two electrons. And remember this model is so oversimplified that it doesn't even begin to take into account anything that's happening in the nucleus. As well as this we haven't considered the electron spins the proton spins and blah blah blah So even though we have this ridiculously powerful equation that could give us deep insights into how the universe works We have to come up with increasingly clever and complicated ways of dealing with this equation without expecting even our best Supercomputers to crash or need millions of years to do a calculation. In many cases there are workarounds to this problem However, that's a story for another video So I hope you've learned something here about the Schrodinger equation and what it means essentially What we're trying to do is to predict the behavior of a quantum system and that quantum system can be anything on the small scale If you have enjoyed this video, please leave a thumbs up Don't forget to subscribe and hit that Bell button. If you've got any questions, leave them down below I'll try and answer them if I can and I hope I've been clear in this video I hope I've not just been rambling on and on about something that doesn't make sense But if that is the case Feel free to ask me and I'll try and clarify also If you like this kind of video check out a video I did previously where I analyzed the world of Harry Potter using physics They'll be linked up here lastly Lastly, tell me in the comments below what equation from physics you want me to cover next or what aspect of physics interests you most so I can cover it in a future video. Let me know and I'll see you next time buh buh, buh, buh. Bye
