 All right, so we're still talking about the radial solutions to the hydrogen atom wave function, the radial portion of the solutions, and what we've seen so far is that the function e to the minus some constant times r is a solution to the radial portion of the Schrodinger equation, and we determined that what that constant was is the atomic number divided by this collection of constants that we've called the Bohr radius. So this function e to the minus zr over a naught is a solution to the Schrodinger equation, and in fact, if I stick any constant I want out in front of that equation, it will still be a solution to the Schrodinger equation, and you won't be surprised to learn that that n turns out to be related to what will turn into a normalization constant in the near future. So that's one solution to the Schrodinger equation. You also won't be surprised to hear that there's a large number of solutions, even for the l equals 0 wave function. This was a solution to the l equals 0 portion of the radial wave function. Even for the l equals 0 portion, there's more than just this one solution, and we end up numbering those. So this is when l equals 0, the one solution that we've obtained so far, I'm going to name that the r10 radial function. That looks like n times, let me leave a blank there, e to the minus zr over a naught. That product is a solution to the Schrodinger equation, or the radial portion of the Schrodinger equation. There's a second solution that we can consider, and we want through all the math of confirming that this is in fact a solution, but the second solution to the l equals 0 radial wave function, radial Schrodinger equation that we'll call r20 is a different normalization constant, again multiplied by the same exponential, I'm sorry it's actually multiplied by a slightly different exponential, e to the minus zr over 2 a naught, and also an extra term, a polynomial term that involves r. 2 minus z times r divided by a naught, so that new function exponential times this polynomial also turns out to be a solution to the Schrodinger equation, the radial portion of the Schrodinger equation, and there's an infinite series of these, and there's also a whole different set of solutions, not when l equals 0, but when l equals 1, and when l equals 2 and so on. So if we pause for a second to write out some more of these candidate solutions in just a second, so as I said, there's a whole infinite series of these solutions, when l equals 0, there's an n equals 1, n equals 2, n equals 3 solutions, any one of these, if we plugged it into the radial version of the Schrodinger equation, we could confirm that these equations each solve that differential equation. For the l equals 1 version of the radial Schrodinger equation, there's an n equals 2, n equals 3, and so on, whole set of those, and I've got only one example here for the l equals 2 function, but the point is there's this very large family of solutions to the radial version of the Schrodinger equation, and if we look carefully, we can see some things they have in common. If we look at what this radial solution of the Schrodinger equation looks like, it always has e to the minus z times radius divided by the Bohr radius, but sometimes it's divided by 1 a naught or 2 a naught or 3 a naught, and that's in fact the reason we label them n equals 1, n equals 2, n equals 3, n tells us how many Bohr radiuses we divide by in the exponential, so that's, for example, why we started counting from n equals 2 rather than n equals 1 for these solutions is because the first solution has e to the minus z r over 2 a naught, and the next one has e to the minus z r over 3 a naught, so we call that the n equals 2 solution, the n equals 3 solution, so the exponential always has e to the minus z r over n copies of a naught. There's also always a normalization constant out front. That normalization constant, I've labeled them all n, but if we calculate the actual values of those, those are different normalization constants for every one of these functions, and then the polynomial portions of these functions, which I've written in green, we can see some patterns there also. Sometimes it's a polynomial function like these, sometimes that polynomial function is preceded by a ratio of z r over a naught with some constants thrown in, so that term, if we look carefully, looks like 2 z r over n a naught. Sometimes, so that's the term like this one, that's 2 z r over 2 a naught, 2 z r over 3 a naught, 2 z r over 3 a naught, so it's always 2 z r over the n a naught terms. That's either raised to the first power, 1 l equals 1, or it's raised to the second power, 1 l equals 2, or it's missing entirely 1 l equals 0, because it's raised to the zero power, so that term is raised to the elf power, and then there's this other more complicated polynomial term that involves constants z r over a naught, z over a naught squared, and so on. So for now, we'll just write that as some polynomial function of r, which again is going to depend on the specific values of n and l. We'll talk more about that in the future, but that's some additional polynomial function of r. So this is the general form of each of these radial solutions of the Schrodinger equation. The one thing left that we have to comment on these is what values of n and l we should expect to find solutions for. l, you'll recall, is the exact same l that we've had in the rigid rotor Schrodinger equation, so we know that the value of l can be any non-negative integer, l can be 0, 1, 2, and so on. What we see here is that when l equals 0, the value of n can be 1 or 2 or 3 or larger values. When l equals 2, the smallest value of n we're allowed to have is n equals 2, I'm sorry, when l equals 1, n must be 2 or 3 or some higher value. When l equals 2, then n can be 3 or higher. So that pattern persists. So whatever the value of l is, n can be 1 larger than that or larger integers still, but it can't be as small as l or anything below. Usually we don't think of the rules in this form, we think of them instead in the following form. n, as you see here, n can be any positive integer, it's possible for n to be 1 or 2 or 3, there are no values of n equals 0, and that's because if we had a 0, e to the minus 0 over 0, a naughts, then that would not be possible mathematically. So 1 is the smallest value of n that we're allowed to have, and when n is any one of these values, if we rearrange these values, if n must be l plus 1 or larger, then l can't be any larger than n minus 1. So l can range from 0, 1, 2, etc., all the way up to n minus 1. So that version of the n and l rules may begin to sound familiar from what you've learned about the hydrogen atom before. You know that there's a quantum number that can be 1 or 2 or 3, and another quantum number that can range from 0 up to n minus 1.