Now we have the Schrodinger Equation. The most fundamental equation of master equation. The most important equation in quantum mechanics. Once we have our equation, what shall we do? We would like to have some fun with the equation. Play with this equation. What fun can have with this equation okay? We'd like to study models. What kind of models do we study? There are two criterias. Number one would like to study some models which are interesting in physics. Okay, nontrivial models and number two, I would like to study simple models when it is nontrivial. I would like to study as simple as possible. Okay, so what are the possible nontrivial and simple models do we have, to simplify this equation? The first idea that we'd like to simplify this equation by restricting the study of this equation to a particular kind of state, which is the state with definite energy. Why definite energy is interesting. This is because if we assume the Hamiltonian is time independent and actually, yes, when we were writing down the hamiltonian, we wrote down the time independent one, that means the potential, the particle is moving in a potential. The potential only depends on X doesn't depend on time. So it is indeed the time independent Hamiltonian. And for a time independent Hamiltonian, actually the energy is a conserved quantity. So once we start off with a particle having a definite energy, then the particle continuous to have a definite energy, it will not evolved into other components of energy. So indeed starting a particle with definite energy at the initial time, meaning that the particle is always having the definite energy. Okay, so that makes sense in physics and as energy is so important quantity, it is interesting. And now the question is with this restriction, how do we simplify the equation? Okay, how do we simplify the equation here? What we are doing is considering that the state has a definite energy, so the hamiltonian acting on this state equals to the energy E acting on this state and so does this one since this is an equality. And what's the implication for this mathematically? The implication is, let's take a look at this equation. Let's take a look at this equation. And the implication for this equation is we can solve it. We can solve it. How? Because we notice this equation, although there is a x, there is a variable x in Psi. However, elsewhere in the differentiation operator in the energy there is no reference of x. So we were so in this part of the equation we don't care about x at all. Okay then we can solve it very easily. We just the first divide, divide this psi into here, divide this psi from the right hand side to the left hand side. What do we get? We will get I hbar and then partial T logarithm of psi equals to the energy. And then I will divide this fact of i hbar also into the right hand side. So it will be minus E over age hbar and the fact of i. Okay, so this is the equation that we'd like to study and the solution of this equation is that the logarithm of psi is E equal to this minus E over hbar t plus an integration constant. And what is this integration constant? We don't know at this moment, but what we know is is time independent and what do I mean by time independent in general it could be space dependent. Okay, if it is based dependent, inserting it back into this part of the equation, you will still have it is a solution okay? So in general will plus a space dependent integration constant. And what this means is psi, the solution for psi will be the exponential. So we have psi as a function of X and T. equals to exponential of this guy, and plus this integration constant and plus on the exponential. What that means is we can separated it into two exponentials. And then let me define the e to the c of x. The integration constant part just define it into psi of x. Here I'm not using the best possible notation. So bear with me that I'm using psi denoting two things. The one thing is a function of space and time and the other is only a function of space. There are two different functions okay? There are two different functions bear with me with that. And this psi is simply exponential to that integration constant all right? And now we have solved we have solved this equation. This part of the equation and the solution of this part of the equation is an explicit time dependence very similar to plain wave. In general it could be not plain wave, because here the potential may not be a constant, but it's very similar is the same time dependence as plane wave. And this is what the schrodinger equation tells us for a state with constant energy. And now let's insert this solution to the other part of the equation insert this solution to this part of the equation. And what we get inserting it to here we notice we recall that the Hamiltonian has no time directive in it. Okay, Recall what's the form of the Hamiltonian? Take a look, there is no time directive in the Hamiltonian. as a result in the upper hand side and down and side. We can simply pull out this e to the minus I H T over hbar factor. And this factor can cancel between the upside and the downside of this equation. So we end up with the equation where this factor is canceled, the time dependence factor is canceled in here and the equation that we have will be the Hamiltonian acting on psi of X. There is no longer T in here equals to E Acting on psi of X recall that the psi of X. And psi of x and t are related like this. In other words, we have first solved how this state should depend on time. And then we have reduced the problem into a simpler problem. We have known the time dependence and now what is the space dependence? And this equation is known as the stationary state Schrodinger equation, mathematically, in case you know, a little bit of differential equations is great that we have reduced a partial differential equation into an ordinary differential equations. In other words, the original Schrodinger equation has two kinds of derivatives. One is partial T, and the other is partial X. So we have two different directives, so it is a partial differential equation, and now there is no longer partial T. In the stationary state Schrodinger equation. So there is only partial X, which is an ordinary differential equation, which is usually much simpler to solve and in this case definitely simpler to solve. So if you don't know the theory of differential equation, that doesn't matter that just to notice. Originally we have here I H bar, partial T. And now we have replaced that differentiation into a number intuitively it has to be simpler to solve, okay, so now we have a simple equation with the assumption that the Hamiltonian doesn't explicitly depend on time and then we have considered a state with definite energy, then we have a simple equation to solve. And what's the next step? Okay the next step will be again the same philosophy that we try to find out. What's the simplest possible case of this equation which is yet interesting and nontrivial. Okay, first of all, the simplest possible case is a constant potential. Okay, if the potential in the Hamiltonian everywhere constant, that's simpleest however, is not that nontrivial. Considering that we have already studied plane wave and a constant potential is just a plane wave. We have returned to the stage that equal to where we haven't learned the Schrodinger equation. So it is trivial. And the next nontrivial case is what about some particularly simple but still not a constant, this kind of potential and how this Schrodinger equation behaves in that kind of simple, but not everywhere constant potential. This is what we will do in the next video.
