 So, so far we've talked about the one-dimensional particle in a box. That's our one example we've considered so far of solving a quantum mechanical problem, solving Schrodinger's equation. That's given us a little bit of a taste of quantum mechanics, so we have a little bit of experience under our belt, but before we move any farther, let's talk about the, what are called the postulates of quantum mechanics, some of the basic foundations for doing quantum mechanics that will help us understand a lot better what the answers mean when we solve more complicated problems than the particle in a box. So it turns out there's going to be five of these postulates of quantum mechanics. The first one refers to the properties of what we've been calling the wave function. So this first postulate says the state of a system, a quantum mechanical system, is completely described by its wave function. So there's two important words that deserve a little bit of discussion there. What do I mean by the state of the system? What do I mean by the wave function? And we'll also talk about what it means that it's completely described. So the first, we'll start with the wave function. We've talked about wave functions before. The one thing I need to tell you about wave functions that we haven't talked about before is that there's one property that these wave functions have to satisfy that we haven't, we've only briefly touched on so far. And that's that these wave function must be square integrable. And that's a technical term, so let me explain what that means. So if I have a wave function psi, if I calculate this integral that we've seen several times before, if I take the wave function multiplied by itself with a complex conjugate, that wave function integrated everywhere, that wave function must be finite. So for something to be an actual bona fide wave function, it has to be square integral. It's square in this sense integrated has to be a finite number. And that's just to guarantee that we can normalize the wave function. If you remember when we talked about the free particle, a sine wave that oscillated forever in all directions all the way out to infinity, the square of that wave function integrated was going to give us an infinite answer and that's one of the reasons we didn't like that wave function. And we had to start talking about particles confined to boxes so that the integral of their probability or the square of the wave function would be finite. So wave function is the type of thing we've seen before with the additional requirement that it has to be square integrable. So by state of the system, what I mean is the same as if I were to describe the state of a problem back when we were doing lattice models, for example. If I describe all the properties of the system, if I've given you all the details of the system, I've described the state of the system completely. So what this postulate tells us is that anything that it's possible to know about a quantum mechanical system, I can obtain from its wave function. Somehow, if I know the wave function, there's a way of extracting enough information to describe everything that I can describe about the system. So that's what this postulate says. So what do I mean when I say that I can extract the properties from the wave function? We have seen just one example of this so far. The wave functions for the two-dimensional, sorry, one-dimensional particle in a box have these wave functions, sine of n pi x over a with a normalization constant out front. The one property we know how to ask questions about for this wave function for a particle in a box is probability. Remember the probability that I'm going to find an electron or a particle between two different limits, A and B. That's just the integral from A to B of the squared wave function. And for example, let's take a particle in a box, maybe, let's take the, here's a box, zero to A. Let's take the psi two wave function. So the wave function looks like that. The square of the wave function looks like that. And if I were to ask, for example, what's the probability that the molecule is in the left half of the box? That it's in this range between zero and A over two. In other words, if I integrate from zero to A over two, the wave function squared, so two over A, sine of two pi x over A squared. And the square root has become squared. If I do that integral, again, we could do the integral mathematically. Ask ourselves what that is, or we could just look at the graph and say that I know that the whole area under this curve is one, because I've normalized my wave function. And I know that the left half is symmetric with the right half. So by symmetry, I know that if I did the work, that integral is going to come out to be equal to one half. So there's a 50% probability that the electron is in the left half of the box, and it exists at different positions in the box with different probabilities given by the wave function. So that's an example of how I can find out something about the state of the system. Where is the particle with what probability does it occupy some regions or others? I can obtain that from the wave function. I have to know what the wave function is, and then there's a recipe for how to find the question I'm interested in. Notice that when I say the state of the system is completely described by the wave function, if I have two systems that have the exact same wave function as each other, I'm always going to get the same properties when I ask questions about the state of those systems. So if two systems have the same wave function, they're identical in every way. So that's a consequence of the fact that the state is completely described by the wave function. So all we know how to do so far is answer questions about the probability using the wave function. If we want to answer questions about other properties other than the probability, we need more information than this to tell us how to extract those properties out of the wave function. And that's the topic of the next few postulates of quantum mechanics.