 So while we're talking about the one-dimensional particle in a box, let's point out one additional important feature of these wave functions, which is that we can use them to construct other functions. We can use them as a basis to construct other functions, and that word will make more sense in just a few minutes. So you might be familiar with Fourier series, that's something you might have heard of before, which is the idea that if I have a periodic function, I can use sine waves and cosine waves to construct any periodic function I want. And if you haven't heard of Fourier functions, Fourier series before in another course, that's okay. But if you have, you can use those as an analogy for what we're about to talk about, which is the fact that if I have some wave functions, like let's say the particle in a box wave functions, I can write any function I want, any function I choose to write down, at least any function that obeys the same boundary conditions as the one-dimensional particle in a box wave function, I can construct that function, I can build that function up out of a linear combination of the wave functions. So I can take wave functions size of one plus size of two plus size of three, instead of just adding them together, I add them together with some different coefficients in front of them, and by taking a combination of all these wave functions together, I can build up any other function that I like. So to give you an example of what I mean by that, let's do a concrete example. Let's say my function is a parabola. I'm going to take the parabola negative x squared plus ax, and if we put up a graph of what that function looks like on the screen, then that's a parabola that points downwards. That's the graph of this parabola. And what I've claimed is that I can construct that function by adding up wave functions. So for the 1D particle in a box, those wave functions are sine waves, sine of n pi x over a. So a sine function is not a parabola, it has some similarities, sine functions go up and then they come back down. So it's not unreasonable to say I could make something that looks a little bit like a parabola by using a sine function, and that's what we can do if we put up the next figure on the screen. This orange graph right here is a graph of sine of 1 pi x over a, where a, the box length is where this function hits zero, one of the zeros of this function. So you can see here that the sine wave, the orange curve, is similar in the sense that it has a peak in the middle and zero on the two edges, but it's not exactly the same mathematical function. Certainly the orange function is not the same as the white function. Sine of n pi x over a is not the same as a parabola. But I didn't say I could do that with just one wave function. I said if I took a sum of multiple wave functions, and in fact if I add to this, let's see, and let's do that in a different color. If I take and add sine of 3 pi x over a, and then we'll pull up the next graph on top of this one, that's what I've got in this yellow curve, is sine of pi x over a plus the sine of 3 pi x over a with some coefficients in front. So a different coefficient multiplying the sine and the 3 sine. And you can see why I needed sine 3 pi x over a. If we look back at the orange curve, it kind of undershoots the parabola for the first part, and then it overshoots for a while, and then it undershoots again for a while. So what I needed to add to the orange curve was something that was a little bit positive over this range, and then negative over this range, and then positive over this range. So in other words, something that looked a little bit like positive, negative, positive. So that's something like sine of 3 pi x over a. So I'm using this sine of 3 pi x over a to fix the deficiencies in the orange curve. And when I add this sine 3 pi x over a with a small coefficient to the orange curve, then I get the yellow curve. And you can see here, unless you look very closely at the screen, there's a few places where you can see the yellow curve and the white curve are not exactly the same, but I've made sine of pi x over a and sine of 3 pi x over a, the sum of those two, very, very close to a parabola. And if I wanted to make it even better, I could continue and add sine of different n pi x's over a. And the more of those I add, the closer and closer I can get to approximating this function. So that's what I mean when I say I can add up wave functions and construct or build any other function that I want to. That's the idea of using these wave functions as a basis. These individual functions themselves, we can call those basis functions. So the functions that I'm using to construct this other function are these wave functions. So I'm using these wave functions as basis functions and taken together. We say they are a basis for constructing another function. So that leaves only one question, which is perhaps I've convinced you with an example that this might be possible, but it's only useful if we can have a recipe for how to calculate what these coefficients are. So I can tell you, for example, that when I constructed this example, I used 0.182 times the sine wave and 0.007 of a sine wave with three times the frequency. So how did I know to use those particular coefficients? How do I know in general how to find the c sub n? And it turns out that's not very difficult to find what the c sub n should be as long as our wave functions are orthonormal, as we've seen that the 1d particle in a box wave functions are. So let's go back to this expression. So the function is the sum of a bunch of wave functions multiplied by some coefficients that we're interested in finding the values of. To find out what those values are, essentially to isolate the coefficients in this expression, what I can do is I can choose the coefficient that I want to know the answer to. Maybe I'm interested in c sub 3 or c sub 5 or c sub 7. I'm interested in some particular coefficient. So if I take the coefficient I'm interested in, let's call that the k-th coefficient, coefficient number k. If I multiply both sides of this equation by that wave function. So I'm going to multiply on the left by psi k with a star, complex conjugate, on the right. This is a sum, so each term of that sum I need to include psi sub k with a star. So again, I multiplied from the left-hand side and the reason I wanted to multiply from the left-hand side is because the next thing I'm going to do is now that I've got a wave function with a complex conjugate times a different wave function. I want to integrate both sides of this expression. So if I just throw an integral sign on both sides of the expression, this looks familiar. This is an overlap integral of the type that we have talked about. And so this integral is the overlap integral between wave function k and wave function n. Remember that these exist inside a sum. So this sum, if I write down what this sum, this whole thing looks like, it looks like the first one times the overlap integral of k with 1. And then let's go ahead and write that out, overlap integral of k with 1. And then the second one, overlap integral of k with 2, and so on. So what we know about these overlap integrals since the wave functions are orthonormal, if k is equal to 1, then this thing will be equal to 1. Otherwise, it will be equal to 0. And whichever one of these terms is equal to 1, all the other ones have to be 0. k is equal to exactly one of these n's. It's either equal to 1, or it's equal to 2, or it's equal to 3. So this overlap integral, they will all be 0s, except for one of them. And the one that will not be 0 is the one where the k and the n are the same as each other. So what that does is it grabs out the particular value of c sub k. So this sum looks like a 0 and a 0 and a 0. And then eventually there's a 1. And the 1 comes along with the coefficient c sub k. So the net result of this whole sum of all these overlap integrals is that this is equal to c sub k. And on the left side, I still have complex conjugate of psi sub k times the function integrated. So if I want to know what is the value of the k-th coefficient, coefficient c sub 1 or c sub 3, I just take my original function that I'm interested in, maybe the parabola or something else, multiply it by the k-th wave function with the complex conjugate, and take the integral. And that's how we find the value of these individual coefficients. So that's a recipe for finding what the coefficients are that I need to build up my arbitrary function out of these individual wave functions. So with that, we know how to use one-dimensional particle and unbox wave functions as a basis just to recap. We know we've seen that one-dimensional particle and unbox wave functions are normalized. They're mutually orthogonal to one another. So those two things together make them orthonormal. They're an orthonormal set of wave functions. And because they're orthonormal, I can use them to construct other functions. So the next thing we'll do with that piece of information is to see how to use these one-dimensional particle and unbox wave functions to describe some problems that actually have relevance in the real world.