 Alright, so we're making progress in understanding the radial portion of the wave function for the hydrogen atom. And what we've seen so far is that radial portion of the wave function looks like some normalization constant, a collection of r over a naught raised to some power that depends on the l value of the radial wave function that we're talking about, some additional polynomial in r that is the main purpose that we'll talk about right now, and then an exponential. So you can see these n's and l's that show up throughout this wave function for various values of n and l, we can write down different wave functions. So it turns out that the polynomial portion of this wave function, this function we've called f, can be determined from the value of something called a Laguerre polynomial, I'll write that down in just a second. So there's this family, I believe it's n minus l plus 1, I'll double check, no it's n minus l minus 1. So this well-known family of polynomials called the generalized Laguerre polynomials, so that I'll define in just a second. If I know, if I want to know the n-elf version of this polynomial that shows up in the regular wave function, I just need to figure out one of these generalized Laguerre polynomials that has a superscript of 2l plus 1 and a subscript of n minus l minus 1. So what those Laguerre polynomials are, I'll give you the first few. So it turns out that instead of using this complicated argument, I'll just describe the Laguerre polynomials as a function of some simpler variable x. If the subscript is 0, it doesn't matter what the superscript is, 0 for subscript in any value, I've written alpha here for the subscript, that just has the value 1. So the 0 family of Laguerre polynomials are very simple, they're just 1. If there's a 1 as a subscript, they're a little more complicated but not very much. It's 1 plus alpha minus x. So this 1 plus the superscript value, whatever that superscript value is, minus x and that's the family with the subscript of 1. And then once you've got those, it might not surprise you to hear that we have a recursion formula that we can use. If I want to know the k plus first Laguerre polynomial, again with any arbitrary exponent up top, then I can calculate that using the two previous ones. So that is a collection of constants, let me double check. That multiplied by the previous one. So 2k plus 1 plus alpha minus x times the kth Laguerre polynomial with the same superscript and then I subtract, again let me double check the constants, k plus alpha times the k minus 1th Laguerre polynomial and that whole thing is divided by k plus 1. So there's the recursion formula that I can use. If I know the 0th and the first, then I can use this to calculate the second and then bootstrap my way up to the third and the fourth and so on. You can also evaluate these Laguerre polynomials a different way if you wish. You can evaluate them directly without using the recursion formula with the following formula. What we can do is we can take this function x to the minus alpha times e to the x over k factorial and multiply that by the kth derivative of this function e to the minus x x to the k plus alpha. So you can double check if you want. If we plug that in with a k equals 0, we'll get 1. If we plug it in with a k equals 1, we'll get this family of functions. If you don't want to bootstrap your way all the way up to the, let's say the 10th l sub 10 Laguerre polynomial, then you could do it directly from this formula with the caveat that you'd have to take a 10th derivative of this function in order to find it. So these functions do get more complicated as the subscript gets larger. So again, these are useful because if I want to know what the polynomial piece of the radio wave function looks like, I need to look up or calculate what the n sub l f function is, which is related to one of these generalized Laguerre polynomials. Just to see how that works, let's do some cases that we've seen already. So previously we've seen that the polynomial portion of the n equals 2, l equals 0 wave function has the form, we saw that was 2 minus zr over a naught. So we can double check and make sure that that works out to be true. So this 2, 0 polynomial is a Laguerre polynomial with n equals 2 and l equals 0. So if I plug in n equals 2 and l equals 0, 2 minus 0 minus 1 is 1. l equals 0, so 2 times l plus 1, I have a 1 for a subscript. So in that Laguerre polynomial, the argument of that Laguerre polynomial is 2zr over n a naught and n for us is 2. So I need to figure out what this Laguerre polynomial is, l sub 1, superscript 1. That's one of the ones that is relatively easy that I've started this out with. l 1 1 is 1 plus alpha, alpha has the value 1, and then I subtract the value of x, which is the argument of the function, so I subtract 2zr over 2 a naught, so I'm just going to subtract zr over a naught, so what I'm left with is 2 minus zr over a naught, which is exactly what we have seen previously. That one was easy enough that it didn't require us to use the recursion formulas, so let's do one more example. When n equals 3 and l equals 0, so that was a case where we've seen previously that the 3-0 polynomial looked like 3 minus twice zr over a naught plus two-ninths of zr over a naught squared, and I'll close my brackets. So we can double check that when we plug in n equals 3 and l equals 0 into this recursion formula, we're going to get the same polynomial function here just as an example of how that recursion formula works. So now what we're looking for, the f 3-0 function is going to be l 3 minus 0 minus 1, so the subscript is 2, superscript still with an l equals 0, superscript 1, and again evaluated for 2zr over n, which is 3 a naught. So to calculate what that is, now with a superscript, subscript 2, we do need to use the recursion formula. We know what l naught and l 1 are, we don't know what l 2 is without using the recursion formula. So to calculate l sub 2, I can use that as, calculate that as sum multiple of the l 1 minus sum multiple of the l 0 function, all divided by k plus 1. So using that recursion formula, our value of k here. So let's see, the k plus first, so here I'm using k plus 1 equals 2, so the value of k is going to be 1. So 2 times k, 2 times 1 is 2, plus 1 is 3, plus alpha. Adding 1 gives me 4, so I've got 4 minus the value of x. So let's leave, yeah let's go ahead and say 2zr over 3 a naught. I'm multiplying that by l sub k, so remember k is 1, so l sub 1 is this function, 1 plus alpha or 1 plus 1 minus x, so I'm subtracting the value of x, which is 2zr over 3 a naught. That takes care of the first piece of the recursion formula. Now I need to subtract k plus alpha, so remember k is 1, alpha is 1, so k plus alpha is 2. And I multiply that by the l sub k minus 1, so this was l sub 1, this one is l sub 0. So l sub 0 is the value of 1, so I just need to subtract 2 from that first product. And when I'm all done, I divide by k plus 1, k plus 1 is 2, so I take this whole thing and I divide by 2. So now what I'm left with is this little bit of algebra to multiply this 4 minus 2zr over 3 a naught times 2 minus 2zr over 3 a naught. What I find when I do that multiplication, 4 times 2 is 8. These two terms are both the same, so 4 times a negative 2zr over 3 a naught and 2 times a negative 2zr over 3 a naught gives me a total of minus 6, 2zr over 3 a naught. And lastly, the r terms squared, so I've got a negative which cancel and I'm left with two thirds times two thirds, which is four ninths, zr over a naught squared. And that whole thing, I've also forgotten to subtract 2, I've got a minus 2 and then that whole thing is divided by 2. So we're almost there, 8 minus 2 is 6, when I divide by 2 I get 3, 6 divided by 2 is 3, which cancels the 3 in the denominator, so what I'm left with is a negative 2zr over a naught for this term. And then 4 ninths divided by 2 gives me 2 ninths multiplying zr over a naught squared and that's exactly the polynomial term that we saw that shows up in the 3, 0 solution to the radial hydrogen atom Schrodinger equation. So what we've seen here is just an example of how to use these generalized Laguerre polynomials to figure out what this polynomial term is in the radial wave function. So now that we understand where that comes from, if we have any n, any l value, then this equation will tell us exactly how to calculate that radial portion of the wave function.