 Okay, we know how to write down the wave function for a three-dimensional particle in a box, a particle confine in a box where there is no potential energy inside the box and is confined to not leave the box. Or that is we almost know how to write down the wave function. We haven't yet talked about the value of the constant a. And the way we get that constant, as we did for the one-dimensional particle in a box, is by normalizing the wave function. Recall what normalization means is the particle must exist. It must exist somewhere. It has 100% chance of being found somewhere in space or for the particle in a box, 100% chance of being found somewhere in the box. So if I integrate the probability everywhere, I have to get 100% and the probability is the wave function squared in this particular quantum mechanical way that we do squaring. So I just need to take this wave function, square it. I don't have to worry about complex conjugation because there's no i's and there's no imaginary numbers in this wave function. So I just need to square the wave function, integrate it, make sure it comes out equal to one, and figure out the value of a that makes that be true. So writing down what that integral is equal to, the wave function is this long thing. If I square it, I'm going to get a squared, sign squared, of n pi x over a, sign squared of a different n pi y over b, sign squared of a still different n pi z over c. So I need to ask myself what I'm integrating with respect to. I'm not just integrating with respect to x because I have a three-dimensional function, so I have x, y, and z. I need to integrate with respect to x and y and z. So this is actually a triple integral. I could write a triple integral sign here. Let me go ahead and do that. So I've got a triple integral over x and y and z. So that's the integral I have to perform. The a, of course, can come out of the integral. The triple integral is actually a product of three separate integrals. I have a piece that involves x, a separate piece that involves y, a separate piece that involves z. So I can write this as the integral of sign squared n pi x over a dx. That's our x integral. Integral of sign squared n pi y over b. That's the y integral. And a separate integral of sign squared n pi z over c. And that's the z integral. All I have left to do now is remind myself, what are the limits of these integrations? If I've confined my particle to a box that runs from 0 to a in the x direction, 0 to a box length b in the y direction, and 0 to a box length c in the z direction, and those box lengths match the box links that make sure our wave function hits 0 at the edges of the box, then the first integral I have to do, integral of sign squared n pi x over a from 0 to a, that's exactly the same integral we had to do for the one-dimensional particle in a box. So I'll just remind you what the answer was when we talked about that integral. Previously, that was a over 2, box length over 2. So this whole integral is a over 2. For exactly the same reasons, integral of sign squared constant times pi times y over a box length integrated from 0 up to that box length, that's the exact same mathematical problem. So that will also be box length over 2. But the box length isn't a in this case. The box length is b. And so that's our second integral. The third integral, the result of that integral is going to be c over 2. So it's the same problem we did for one-dimensional particle in a box. We just do it three times and multiply them together. That product a squared times a times b times c in the numerator, 2 times 2 times 2 in the denominator, must equal 1. So since we're interested in solving for the value of a, a squared needs to be 8 over box length in x, box length in y times box length in z. Length times width times height of our box, that's just the volume of the box. So the constant a, the normalization constant, is the square root of either one of these quantities. I can write it as the square root of 8 over a, b, c. I can write it as the square root of 8 over v. In fact, I can write it as, if I remember that 8 is 2 times 2 times 2, that's where the 8 came from. I can write this as square root of 2 over a, square root of 2 over b, square root of 2 over c. The reason it's worth writing it out in that more complicated form is just to remind us that that's the normalization constant for a one-dimensional particle in the x direction, times the normalization constant for a one-dimensional particle in a box in the y direction, and then multiplied by a normalization constant in the z direction. So any, any one of these is an appropriate way to write the normalization constant for the special case where we have a cubic box, where the length of the box is the same in the x and the y and the z directions. Then each of these a's and b's and c's is the same. And in that case, the normalization constant simplifies a little further to maybe 8 over a cubed, if a times b times c is the same as a times a times a. Or we can also write that as 2 over a, not raised to the one-half power, but 2 over a square root times another square root of 2 over a times another square root of 2 over a, that gives us 2 over a to the three-halves power. So any one of these is the normalization constant for a cubic box. Any one of these is the normalization constant for a non-cubic box. And that's the constants we would insert here if we want to write down the full complete wave function for the three-dimensional particle in a box.