 So now we understand a little bit about Schrodinger's equation, whether we write it down somewhat explicitly like this, or in this extremely compact notation using operator notation. We can write down Schrodinger's equation. It's important to recognize what type of equation this is, namely that it's a differential equation, not an algebraic equation. So in an expression like this, what we're trying to do is we're trying to solve for the function psi of x, we're trying to solve for the wave function that obeys this equation. We're not trying to solve for some value of x, some numerical value for some variable like x. So that's going to be a little bit difficult to remember for many of you, because you've been doing algebra for so long that when you see an equation you're tempted to isolate x and solve for x. But that's not what we're doing when we're solving Schrodinger's equation. So I'll show you what I mean by that with an example. So let's, as the simplest example we can think of, let's back down to the one-dimensional Schrodinger equation. So we don't have to worry about x and y and z, we'll just do a one-dimensional version of Schrodinger's equation. So I still have the same constants, minus h squared over 8 pi squared m. The derivative in one dimension is just d squared dx squared acting on the wave function. So this is the one-dimensional Schrodinger equation. So to make it as simple as possible, we can't solve Schrodinger's equation, we can't talk about the solutions to Schrodinger's equation without knowing what this potential energy means. And that's going to be different for every problem that we study. Maybe the potential energy is gravitational potential energy, or maybe it's electrostatic potential energy, or maybe some equation describes what the potential energy looks like. So we have to describe what the potential energy is before we can describe the equation in detail. For this simple example, just to keep it as simple as possible, let's suppose that the potential energy is zero. I'm choosing to solve a problem where the potential energy is zero, so that the potential energy term completely disappears. So Schrodinger's equation, one-dimensional version of Schrodinger's equation with no potential energy just simplifies to this form. Second derivative of a wave function multiplied by these negative constants is equal to energy times the wave function. The thing to keep in mind is when we're asking ourselves when a function does or doesn't solve this equation, when I plug the wave function into the left-hand side of the equation, does it or does it not equal what I get when I plug it in on the right side of the equation? So I can say, let's just pick a function, x squared multiplied by some constant. I can say does this function solve Schrodinger's equation or not? In other words, when I insert this candidate wave function on both sides of Schrodinger's equation, is the equal sign appropriate or are they not equal to each other? So in order to plug it on the left side, I'm going to need the second derivative of this wave function. So the first derivative, if I take the first derivative of kx squared, I'll get 2 times k times x. The second derivative, take another derivative, derivative of 2kx is just 2k. If I insert those into our simplified Schrodinger's equation, I'm asking minus h squared over 8 pi squared m times the second derivative times 2k equals e times wave function. Wave function is kx squared. So that's what Schrodinger's equation becomes for this particular wave function. Now stop and ask yourself what the next thing I'm going to do is as I solve Schrodinger's equation here or as I check if this wave function is in fact a solution or not. If what you said is rearrange this equation and solve for x, then you've fallen into the trap of thinking of it as an algebraic equation. We're not solving for x. What we're solving for is does this function, does the stuff on the left equal the stuff on the right? What we have on the left is a bunch of constants, h's and pi's and m's and k's and 2's and a negative sign. That collection of constants is some number. Is that number equal to some other constants, e and k times x squared? If you solve for x, then what you're doing is finding the specific value of x squared that makes the value on the right equal to the value on the left. What we want to be true is for all values of x, for every value of x, the thing on the left has to be equal to the thing on the right. There is no way that these collection of constants equals e times k times x squared for all values of x. It's not always true that e k x squared is equal to this collection of constants. The result is because I have some x's over here, I don't have x's over there. The function on the left, which is just a constant, is not equal to the function on the right, which is quadratic. I can't make those two functions equal to each other, so this wave function does not solve Schrodinger's equation. The answer to both these questions is no. This wave function is not a valid solution to the Schrodinger equation. To show you what it means, in a case that does actually solve Schrodinger's equation, we can do another example. Take with the same one-dimensional Schrodinger equation, and now if I ask what about the function sine of x, does the function sine of x solve Schrodinger's equation? Second derivative, we're going to need the second derivative. The first derivative of sine of x looks like cosine. Take another derivative, so the second derivative, derivative of cosine becomes negative sine. When I plug those, both the second derivative in here and the function in there, then what I find is on the left, I've got minus h squared over 8 pi squared mass. Second derivative is negative sine of x. That's equal to, perhaps. I'm asking myself, is that equal to some constant e times the wave function? Wave function is sine of x. Now the question has become, is this collection of constants times sine of x equal to this constant e times sine of x? Now the answer is maybe. If this collection of constants minus h squared over 8 pi squared m times a negative sign, if those are equal to e, then the constants times sine x match the constants times sine x. It's possible. And the answer to the question is yes, as long as e is equal to negative times negative h squared over 8 pi squared m. So we've done two things, actually. When I ask does this function, sine of x, solve Schrodinger's equation, solve this version of Schrodinger's equation, the answer is yes. And in the process of getting that answer, we find that not only is the answer yes, but the energy that we get in order to make that function a solution to Schrodinger's equation, we also solve for the value of e that makes that equation true. So we've both verified that the wave function is a solution, and we've obtained the value of the energy at the same time. So what we've done is made some progress. We haven't actually solved Schrodinger's equation. I've given you some candidate solutions. You now know how to check whether a function is a solution to Schrodinger's equation or not. So we're one step closer to being able to actually solve Schrodinger's equation. But before we do that, there's one more interesting feature of Schrodinger's equation that we're going to point out first.