[BLANK_AUDIO]. So we're now going to move on to talk about the the Schrödinger Equation. This equation was first presented by Erwin Schrödinger in 1926 and in some ways you can consider as a development from the De Broglie relationship. De Broglie as we've into previously showed that all moving particles have wave like characteristics. And Schrödinger in a certain fashion used the wave equations that were available at that time and combined this with Toboga's ideas and derived his famous equation. So if we have a look at this first it's a rather firmer looking equation but we will write it out in a form that you will commonly it's minus h bar squared over 2 m. Delta squared psi plus v psi equal to e psi. So. The idea is, or the rumor goes, that Schrödinger finally formulated this equation when he was on a skiing trip. So that may be, may be the case, but this is the equation that he gave here. Now, as I've said, again this is a rather formidable looking equation, but what I'm hoping to do in this presentation, is to break it down, if you like. And take, hopefully take some of the mystique away from it. And I'll show that by using fairly, some simple mathamat, mathematical, manipulations we can actually come up with this, come up with this equation. Now the first place I want to start is how we define how you know from school how you define energy and for moving particles, say, you can define the energy as the kinetic energy. Which is the energy due to movement. So, that's a kinetic energy. And, you also have the energy due to position which we call the potential energy. And total energy then is given by the sum of these two, so these, add these two together, and you get the total energy. So let's put in some symbols here. So you have we usually define the kinetic energy, or we usually notate with the symbol E sub k. And then you have the potential energy, which we designate as V and the total energy we'll give as, as E. What you also know from the school physics is that the kinetic energy, EK. Classically is equal to one half m v squared. Where m is the mass of the particle and v is it's velocity. So we're going to first of all work out the Schrödinger equation for a one dimensional system. We're going to call that along the x direction. And we may as well say in the. So we want [UNKNOWN] in the x direction. So, we know that velocity is a, is a vector. So we should properly write that Ek is equal to one half m, and we're talking about velocity in the x direction so it's v sub x squared. Now we can re-write that a little bit or we can rearrange that and we can say that we'll be at a half in v sub x squared and that's equal to m squared v x squared, all over 2 m. So that's multiplying a cross by by m. And what we can also do is we define. What we call, you may have heard of this again in a Basic Physics class Px is the momentum along the x direction, and that's defined as the mass, which is m, times the velocity in that direction. We can also now say that P sub x squared is equal to m squared e sub x squared. Okay so now we can come across here. Let's get a different different color. So we can Put that in here. So we therefore know that the E k, the kinetic energy, is equal to P sub x squared all over 2 m. So there for if we go back up above here, the total energy is the sum of the kinetic energy plus the potential energy, so therefore we can say that P sub x squared all over 2 n plus v, which is our potential energy, is equal to, is equal to E. [BLANK_AUDIO]. Now, in, in wave mechanics, what we do is we can replace some of our physical observables by what were called, are called operators. And one of the main ones we can do, this end is for our momentum. So, let's put down so in Q m, quantum mechanics. We can define what we call a momentum operator, a linear momentum operator. And we usually define the the operator by putting a little hat on it like that, or a carrot symbol. Now, what you define that as, is h bar. Over i, d by dx. Now, again from your mathematics you should know that i is a, called an imaginary number, it's the square root of minus 1, h bar. You may have come across that as well, h bar is just Planck's constant h divided by 2pi. So what we can now say. We can have our [COUGH] linear momentum operator and we can square that. And that's going to be h bar squared. All over i squared. D 2 by the dx square. Now we know i is equal to the squared route minus 1. So therefore we can say that, that is equal to minus h barred squared. D2 by dx squared. So what we can now do is we can define our quantum mechanical linear operator, and we can go back up to our equation here. Where we define the total energy, and we can substitute this form in for the for the P x value. So if we do that. We're going to get minus h bar squared over 2 m. D 2 by dx squared. Plus V, and that's equal to E. So what we done here is we substituted [UNKNOWN] equation. We substituted Px by minus h bar squared d2 divide by dx squared and it was px squared over 2m, so this is where we get our 2m. Okay. So, what we do now [SOUND]. Remember we are trying to get to that shorter equation. If we multiply that across by what we call function, sign of x, what you get is what we call a way fiction. So you'd then if we multiply, and we're this across by this. And you therefore get minus h bar squared, all over 2 m, d 2 by dx squared, plus V, psi of x, is equal to E psi of x. So we can actually then expand that out a little bit. Let's just use a different color. So if we just expand that equation out you're going to get minus h bar squared all over 2 m. D2 by dx squared, psi of x plus V, psi of x is equal to E psi of x. That's the next term. Kay. So what this equation here actually corresponds to, doesn't exactly look like the one that we started off with in the beginning, but this is Schrödinger's equation. [BLANK_AUDIO]. Equation in one dimension. It's actually [INAUDIBLE] exactly right. It's the Schrödinger's time independent equation in, in, in one direction. And again we, we called a psi of x. This is called a wave function. [SOUND] Okay. So, what have up here as well, if we sort of move back up here and we highlight this bit here. [INAUDIBLE] down here. Now this bit, if we move it on down, this is called a, the Hamiltonian. That's called, and the symbol for that it's an operator, it's called a Hamiltonian, or a Hamiltonian Hamilton operator. So, you will often see again, this H here so you'd often see the Schrödinger equation written as H bar psi 6, and the x direction again is equal to E sigh of x. So let's say very much more abbreviated form of the Schrödinger equation that you'll see. Okay so what we've been talking about so far is for one dimension. And let's just write it down again. Our, our Schrödinger equation in one dimension. We had it up there, let's put it down here. To be, to be a bit clearer. So we had minus h bar squared over 2 m, we had d 2 by dx squared, that was psi of x, and that was V times psi of x is equal to E psi of x. Okay. Sorry about the x's. Psi of x. So, this is one dimension, and this derivative here, the first to the, the x direction, and if you want to extend that to three dimensions, what you need to put in is a second derivative for the x direction, the y direction, and the zed direction. And mathematically correct, you put they called them partial derivatives, so lets just write it out for you. So you, three dimensions, you have minus h bar square, over 2m, del 2, del x squared plus del 2 del y squared is del, functions and entities are partial derivatives, plus del 2 del zed squared. And now the wave function, it's a function of the three courses. This time, it's a function of x, y and z. And then you have plus V psi of x, y and zed and that's equal to. We'll just go down here. E times psi x, y and zed. Now, what you usually call this function here, let's get a color for it, this bit here, is often given the symbol del squared. It's called the Laplacian, Laplacian operator, Laplacian operator squared. And this, and this occasion. So therefore, if we go down here, we can now write the Schrödinger equations is minus h bar squared all over 2m, del squared, psi. Now I'm going to leave out the x, y, and z because it'll be implied by this del operator here. Plus V psi, you can leave out the x, y, z. And that's equal to E psi. So if we look that, let's put the, highlight it here. This is the time independent Schrödinger equation in three dimensions, and if we go back up to start, see where we started. When we first presented this form Schrödinger equation it should be yes here we are, with Schrödinger thinking about there. This is the same form that we've just worked out. So you can see that it is a it's a very formidable equation. [COUGH] But if you break it down in terms of energy, then it's, you can explain it fairly well. And hopefully, the good thing about this is, now that you can understand at least what the different parts are, hopefully you'll be able to remember it better, and then you'll be able to, to apply it later on. [BLANK_AUDIO]
