 All right, so we've seen a little bit about wave particle duality, which is the feature of quantum mechanical objects, both light and a little more confusingly material objects like atoms and molecules and electrons, that all quantum mechanical properties both behave like a particle sometimes and like a wave at other times. So in order to understand how these quantum mechanical particles can act like waves, we need to understand the wave function. The wave function is exactly that function that describes how quantum mechanical particles exhibit wave-like properties. So we usually use the term psi to describe the wave function, and that's the goal now is to understand a little more about what this thing, a wave function is. You may or may not have heard of wave functions before, possibly you know a little bit about them but not very much. So we want to understand what this function is that describes the wave-like particles of quantum mechanical objects. So first of all, we're calling it a wave function. It's the function that describes the wave-like properties of a quantum mechanical object. So it is a function. It depends on some argument. And what the wave function is a function of is position. So if I know the position of a particle, and I know this function, I can insert the position of the particle in order to calculate the wave function. And the way I've written it down here is with just one argument, the x position more generally for molecules that exist in a three-dimensional space, that position might be a three-dimensional vector. I could write psi of x and y and z. For example, for the Cartesian coordinates of a particle, or I can write it as a vector r, or I can think of it in spherical coordinates, maybe r theta phi. Point is the wave function always depends on the position, maybe in one dimension or maybe in three dimensions of the particle that we're talking about. But the meaning, the physical meaning of the wave function, is often good to keep in mind when you get confused on a quantum mechanical problem and you're asking yourself, what does this actually mean? It's always useful to step back and recall that what the physical meaning of the wave function is. And what the physical meaning of the wave function is, is that the wave function tells us the probability of finding a particle at some position, but not the wave function itself. If I square the wave function, take the wave function and I square it. That tells me the probability that I'm going to find the particle at that position. And I'll point out that the, what look like absolute value signs, these magnitude signs that I've put around the wave function before I square it, that's there to tell us that we don't just square the wave function. That particular flavor of calculating the magnitude of the square is I need to actually take the wave function times the complex conjugate of the wave function. And for now, we're going to ignore that. We'll run into cases where we do have to worry about this complex conjugate. But if the wave function is a real function, if it only has real numbers, not imaginary or complex numbers, then we don't have to worry about the complex conjugate. So that's in there for later when we run across complex or imaginary wave functions. But for now, we don't have to think about that too much. And we can just think of the wave function squared rather than worrying about the complex conjugate. So squaring the wave functions perhaps with this complex conjugate tells us how likely we are to find the particle at some position. And I've written these equations in one dimension. We could write the same equations for a three dimensional particle just using the r vector instead of x for the position of the particle. So that tells us how likely we are to find the particle either in some one dimensional coordinate or in some three dimensional coordinate. One other addition I have to make to these equations is actually to be correct about these definitions. What we are actually defining is not the probability of finding the particle, but the probability density, meaning the probability per unit volume in three dimensions, sorry, d cubed r for the three dimensional coordinate. And in just a second, I'll stick those in an integral and they'll make a little bit more sense. But it's not telling us the probability that finding the particle at a specific position, probability of finding any particle at any individual very specific position is going to be very small. But in order to find the, let's say, the way we use this fact, let's say I have a particle and I want to know what is the probability that its position is between two values, between a and b, then the way I answer that question is to say it's got some probability of being at position x. And if I integrate that everywhere from a up through b, then that tells me the probability of it being somewhere in between those two. So that's why it's necessary to think about probability density. I've got some density of finding the particle at any point in space that I integrate over some region of space or equivalently in three dimensions. If I want to know what's the probability that my position vector is, let's say in some volume of space, some region of space, maybe the left side of a box or maybe inside some sphere, I define that volume. And then if I integrate, I'll have to do a triple integral. If I integrate the probability over x, y, and z or r, theta, phi, whatever my three-dimensional coordinates are, and if I integrate only over the specific volume that I'm interested in, that tells me how likely I am to find the particle in that region in one dimension or in three dimensions. And then if we use the fact that probability is equal to the square of the wave function, each of these expressions, so the triple integral might become wave function squared or the single integral up here. So the reason we calculate the wave function many times is in order to be able to calculate the probability of finding the particle somewhere. And in order to calculate the probability of finding the particle not just at a point, but in a region is to integrate the square of the wave function over that region of space. So that's the physical meaning of the wave function. That's often what we use the wave function for. Another very important property of the wave function is the wave function happens to satisfy an equation called the Schrodinger equation that you may or may not have heard of. And that's the next task we have is to understand what that Schrodinger equation is and how the wave function relates to it.