 Alright, so we're talking about the hydrogen atom and solving the Schrodinger equation for the hydrogen atom and the wave function that we're trying to solve looks like this. It's a three-dimensional wave function and depends on the radial position of the electron. It's a distance away from the nucleus and also two angular variables that describe the angular position of the electron with respect to the nucleus. And the full Schrodinger equation is this long equation I've written here. There's a kinetic energy term, which is this term in brackets, and a potential energy term that describes the Coulomb energy of the electron interacting with the nucleus. And you can see I've written the terms that depend on the radial coordinate of the electron in green and the terms that depend only on the angular coordinates of the electron in a different color in this orange color. So the question is how do we solve this Schrodinger equation? We're looking for a wave function that if we take these various derivatives, take the r-derivative, multiply by r squared, take another r-derivative, divide by r squared, do another set of things with theta derivatives, do another set of things with phi derivatives, add in potential energy times the wave function, somehow we need to get back just the original wave function itself multiplied by a constant that we call the energy. So solving this differential equation is not trivial, but it's certainly possible, so that's what we'll spend some time doing. So the first step, we can see that these r-squares show up in nearly every one of these terms. So the first thing we can do is multiply through on both sides of this equation by r-squared. So I can do that relatively easy by just taking out the 1 over r-squared here, get rid of this 1 over r-squared, get rid of this 1 over r-squared, multiplying by r-squared converts this 1 over r to an r, and on the right side of the equation, we also need to multiply it by an r-squared there. So that equation already looks a little bit better. It almost looks like we've got it separated into terms that only involve radial terms and completely separate terms that only involve angular terms. The only thing that's not completely separable about this equation as it stands so far is the wave function itself. Where I've got the wave function written, it depends both on the radial coordinates and on the angular coordinates. So to get around that problem, we can say that the wave function itself, which depends on radial and angular coordinates both, we can hypothesize at least we can consider the fact that that might be a product of one function that purely depends on radial coordinates and another function that depends on angular coordinates. So if the wave function has this form, a piece that looks like r, some function of r, multiplied by some piece that looks like a function of theta and phi, then that turns out that will let us separate the variables in this equation. So we've called the radial part of the wave function capital R of r and the angular part of the wave function we've labeled with a capital Y. So what that means is these derivatives that we need to take, the derivative of the wave function with respect to r or with respect to theta or respect to phi, those take the form, I guess I should write this one in green, so that r derivative affects only the radial part of the wave function. Likewise, if I take the theta derivative of the wave function, it doesn't affect the r portion and the theta derivative only affects the angular part of the wave function and the phi derivative of the wave function is the radial part multiplied by the derivative of the angular wave function with respect to phi. So we can then take those derivatives and substitute them back into the original Schrodinger's equation and again, I'll attempt to do that without rewriting the entire equation by erasing, so where I see derivative of the wave function with respect to r, I'll write dr du r times y. So this derivative of psi looks like derivative of r times y and because the r derivatives don't affect the y term, I can write that anywhere I like. Likewise, the derivative of the wave function with respect to theta becomes derivative of y with respect to theta times an r. Second derivative of the wave function with respect to phi becomes the second derivative with respect to, second derivative of y with respect to phi multiplied by an r and I suppose this r should be green and over here the terms that involve the wave function, I can write those as simply r times y and r times y. Alright, so now I've separated the variables, I've written my wave function as a product of two functions that involve the radial and the angular terms, rewritten the Schrodinger equation to involve those terms and now we can see what I meant when I said the wave function itself involves both r terms and y terms. So I've got a mix of the green terms and the orange terms in the individual terms in this equation. But now I can solve that problem. As the next step what I can do is divide through by r times y, divide by the wave function on both sides. So a term where that's easiest to see is right here, if I take this term and I divide by r and y, the r and the y disappear. As in this term if I divide by r and y, the r and the y both disappear. If I take these terms if I divide by r and y, dividing by y makes this term go away, dividing by r, brings in a 1 over r. Second term I can divide by r and then divide by y, the third term in brackets I can divide by r and then again divide by y. So now we can see that we've completely successfully separated the variables. I've got a purely radial term, a purely angular term, a purely angular term, a purely radial term and then this is also a purely radial term. So as one last step let's bring this er squared term over to the left hand side of the equation. So now I've got a sum of a bunch of radial terms and then a sum of a bunch of angular terms that must sum to zero. So the only way that all the radial terms can always add up with all the angular terms to zero is if all the radial terms sum up to one number and all the angular terms sum up to the opposite of that number. So if I write down an equation form what that means, the radial half of the Schrodinger equation will look like this set of constants times 1 over r, d dr of r squared dr dr, d big r d little r. The next radial term I have is the potential energy term so that looks like minus charge on the nucleus, charge on an electron squared, 4 pi epsilon knot times r. If I subtract from that er squared I must get not zero but some number, I'll write that as minus alpha. The angular portion of the Schrodinger equation then, if I write the terms that involve the angular terms, I've again got the typical constants out in front of the Schrodinger equation. The first angular term looks like 1 over y, 1 over sine theta, d d theta, sine theta, d y d theta. The next term looks like 1 over y, 1 over sine squared phi, I'm sorry that should be sine squared theta, no sine squared should have been a sine squared theta all along. Sine squared theta, d squared y, d phi squared and that's the end of my angular terms. Those must sum to some particular value, I don't know what that value is yet but I can call that alpha. What I do know is that if I sum this green equation and this orange equation together they must give me zero so whatever the radial term summed to that must be the negative of whatever the angular terms summed to. So what we've succeeded in doing at this point is we've broken down the Schrodinger equation into a form into two separate parts. We've separated the radial parts and the angular parts. What we have here is two different differential equations now. If we solve this differential equations for this function r of r and if we solve the second differential equation for the angular function y of theta and phi then we can recombine those two to find the total wave function.