 Okay, so for the three-dimensional particle in a box, we've seen that the energy levels for the particle in a box are determined by these three quantum numbers. If we have three different positive integers, n sub x, n sub y, n sub z, then the energy of a particle in that state for the 3D particle in a box is given by this expression here. What that means is we're now in a position where we can use those energies to calculate partition functions and then eventually some thermodynamic properties. So as a reminder, for a quantum mechanics problem, if we start by defining what the potential energy is, we can solve Schrodinger's equation to obtain the wave functions and the energies. Once we have the energies, what we're about to do next, once we have energies, we can use the energies to get a partition function. And that's on the road to calculating thermal properties. So if we'd like to know what is the partition function for any problem, it's the sum of the Boltzmann factors, e to the minus energies over kt, summed up over all the different states that the system can have, all the different energies of each different state. So in this case, our states are described by these constants n sub x, n sub y, n sub z. So we need to know what is the sum, n sub x can be any value from 1 to infinity, n sub y can be any value from 1 to infinity, n sub z can be any value from 1 to infinity. And then the energy is this expression. So if I put that in for the energy, e to the minus h squared over 8m, nx squared over a squared, ny squared over b squared, and nz squared over c squared. Okay, so that's the sum we need to compute in order to figure out what the partition function is, and that looks like a lot of work. It looks like an infinite amount of work three different times. So that's slightly challenging, but we can make use of a shortcut that we've talked about before. If we notice that this energy, the energy of the 3D particle in a box, is the sum of three separate terms, one term for the x portion of the energy. So these constants times nx squared over a squared, that's the energy in the x direction. Another term, constants times ny squared over b squared for the y direction, and a third term, constants times nz squared over c squared for the z direction. So if we notice that 3D particle in a box energy can be written as the sum of one piece for x and one piece for y and one piece for z. And so this is the x contribution, the y contribution, the z contribution. We know that if the energy is the sum of the energies of several different sub-system energies or sub-component energies, then the partition function is going to be the product of the partition functions for those three different subsystems. So this 3D particle in a box partition function is the product of three different one dimensional particle in a box partition functions. The one we obtained from the x energies, the y energies, the z energies. And just to make that clear, what I mean by k sub x here is the sum of the Boltzmann factors, e to the minus energy over kt, where the energy that I'm talking about here is not the full three dimensional particle in a box energy, but just the piece that comes from the x component. So it's the constants times the n sub x, the quantum number in the x direction, and a, the box length in the a direction. So that would be e to the minus h squared over 8m, nx squared over a squared, all divided by kt in the exponent. Sum that for every possible state, n sub x is one running through infinity. That sum, still an infinite number of terms in that sum, but that's only a single sum that gives us kx. There's very similar terms for qx, very similar terms for qy and qz. The only differences are, I'm summing over n sub y. My dummy variable has a different name, although that's not terribly important. So I'm summing over ny squared. What is important is that I'm dividing by the box size in the y direction, dividing n sub y squared by b squared because the box length may be different in the y direction than it is in the x direction. And lastly, there's a term for qz, summing from one to infinity, e to the minus h squared over 8m, nz squared over c squared, all divided by kt. So we've made some progress. We've converted our partition function for the 3D particle in a box, which involved this triply infinite sum into just the product of three different partition functions, each of which now only involves a one dimensional, a single infinite sum. So it's still going to be a little bit of work to figure out what this infinite sum is equal to, and that's what we'll do in the next video lecture. But if we take the product of those three slightly easier problems, that'll give us the value for our 3D particle in a box partition function.