Hi there. So today we are going to move on to talk about the energy levels that come out of solutions of the Schrodinger equation for hydrogenic atom systems. We have talked quite a bit already about the, the wave functions that come out. But now we are going to talk about the energy levels just as we did, talked about the energy levels for say our simple system, our particle in a box system. Now the energy levels for a hydrogenic atom system are given by E n, where n is the principal quantum number. And we can simplify it and say that it's given by the negative of R, where r is called the Rydberg constant, times z squared, z remember is the, is the atomic number. And then that's all divided again by, by n squared your, your principle quantum number. So R here is known as the Rydberg constant [BLANK_AUDIO]. And, it's easy to remember this because it corresponds to exactly 13.6 electron volts. What you should also note from up here is the [INAUDIBLE] the energy only depends on n. Remember we had the three quantum numbers. We had the n. The l and the n sub l. But the energies, levels for hydrogenic atom system depend only on the principal quantum number n. So let's now try to plot these energy levels out into a qualitative energy level scheme and see what they, what they look like. So if we move down here. What I've plotted out already is some energy levels, and basically energy is increasing in this direction, in this direction here. Now we know that the first principal quantum number is n equals 1 so this level here in energy, this is the n equals 1 level. Now if we write down our equation again that we wrote just above there to remind ourselves. En is equal to minus 13.6. Times z squared, all divided my n squared. And that's giving the energies in electron, electron volts. So if we consider here this n equals 1 and if we looked at this equation here then n is 1 so n squared is 1, and z squared of course is the, is the atomic number. So if we take simplest system, the hydrogen. So it's going to be 1, so the energy of e 1, is going to be minus 13.6 electron volts. If it was helium from number 2, then of course you would have z squared, so z is 2, so z square is 4, so it'll be 4 times that. Likewise we go into lithium as we go on up the periodic table. If z is 9 so it will be 9 times 13.6. So let's stay with the hydrogen atom here. So what we have on this side therefor we know that the energy corresponds to exactly minus 13.60. Electron-volts. Next level, we know that n is equal to 2. [BLANK_AUDIO]. So, now, we're going to stay with the the hydrogen atom for the, for the moment. So, now we have minus 13.6 z again is 1. So, now n is 2. So, n squared is going to be 4. So, the energy at this level is going to be 13.6 divided by. 4, so if you do that you are going to get minus 3.40 electron volts. The next one as we know is n equals 3, so now you are going to get N squared is going to be 9. So, it's going to be minus 13.6, divided by, by 9. And if you do that, you should get a value of minus 1.51 electron volts. And on up, n equals 4. And now, you have to divide by 16 and that will give you a value of minus let me see what it is, minus 0.85 electron volts I get. And so on up, we are getting a little bit tighter here at the moment, I'll explain that in a minute. And then it becomes minus for n equals 5, minus 0, sorry that's 0.54 sorry. So it's minus 0.54 electron volts. Okay, so I'm just going to do one more here, so let's do n equals 6. And then if you worked that out you're going to get the energy level is m, [INAUDIBLE] divided by 36 or 30 plus 6 plus 6. 13.6 by 6. By, by 36 and you get minus 0.38 electron volts. Now what you have here and what we are trying to illustrate is as you increase n, you divide by n squared all the time so this is getting smaller. And smaller. And of course you can we know from the solutions of the the Schrodinger equations that n can vary from one up to integer values up to n infinity. And what you can keep doing is, you can imagine you come to eventually n equals to infinity, and you get levels now are really coming close. Really coming close together. Coming in to a kind of continuum. And of course when n is equal to infinity, the energy infinity is going to be 0.00 electron volts. So this really is the energy levels that the solutions that Schrodinger equation predict for the hydrogen, hydrogen atom system. And as we say, the energy levels depend only on the principal quantum number. So when we're talking about n equals 1 here. We've already said that this, of course, corresponds to the wave function, psi 1, 0, 0, where 1 is the principal quantum number and the l and the nl are 0. So, there's only one energy level. Or there's only one energy state, for that energy level. However when we go up to n equals 2, we know that we have psi 2 0 0. That's like if you, like the 2 s orbital. But you also have got psi. Let's see if we can fit them in here. So it's psi 2 1 0. Which we would refer to as 2 p z. But we've also got the other one, psi 2 1 1. And last you've got psi 2 1 minus 1. So what you can see here is for each, for the integer level here it's just it's just one state. But for energy level 2. That's four states. Likewise I'm sure you can show that for n equals 3. You're going to get 9. And if it's 4 you're going to get 16. So if you like to we call this. There's Force four-fold degeneracy for n equals 2. Nine-fold degeneracy for n equals 3. So the degeneracy and energy levels is equal to, is equal to n squared. At this stage your, I suppose a valid question. Free to ask is, how do we know that this is, is, is true? Is there any experimental evidence to support these energy levels that the Schrodinger equation predicts? And the answer is that there, there is indeed, and this evidence was presented well before quantum theory was presented. And what I show down here is both the, what we call the absorption, sorry this is the absorption spectrum. And this is the emission spectrum, of the hydrogen atom. Now this is in the visible, just talking about the visible region of the electromagnetic spectrum. And you should be know of the spectrum of colors going from, from red. Up to violet, this is the well known visible region spectrum. So what's happening here, we aren't going to go into any of the experimental details of how this is done, but this is when you say you shine, visible light. And you pass through hydrogen atoms. What you find is what comes out is you have your, your visible spectrum again, but you notice these lines in the spectrum, these dark lines in the spectrum. And the red region here going into the blue region, and then into the, into the violet region. Now you can also perform this on what we call an emission spectrum, and this is where you have an excited state of an atom, in this case, and it emits energy. And what you you'll find is it's the opposite here to the absorption spectrum. You get a line. A method in the red region. Also two in the blue region. And then again two in the indigo. In the indigo violet region. So these as I say were known well before people knew about quantum, quantum mechanics. And they were an intriguing. Intriguing spectrum and intriguing mystery if you like. Now what I've done here is, this is in the visible region. So I've actually given the exact values that people measure for these lines. So the red line is 656. The first blue line here, blueish green line is 486. 434 410, and so forth for, for the other lines. So what's interesting, though, if you go back up to the, spectrum or the energy levels for, for, for hydrogen. As I said for the particle in a box, what quantum mechanics is telling us is that these are the only energy levels allowed for this system. You can't get intermediate levels, there a ladder as I said before of, of energy levels. You can't have intermediate, intermediate levels. So when the, the hydrogen atom the will exist in it's. Lowest energy level, which is n equals 1. But to get up to these other energy levels that are also with the quantum mechanics that is also available for it, then it will require energy. To get there. Now what's intriguing is, if you, you can work out, so you can work out the energy difference between these different levels. Now what's intriguing from what we've been talking about here is that if you do measure the energy level, say between n equals 2. And n equals 3, so that's simply a difference between 3.4 and 1.51 electron volts. What that comes out to be is exactly equal to the energy, of this line here, the 656 nanometer line. Because this stuff corresponds exactly. So the difference between 3.40 and 1.51 electron volts. So what we can do is we can, let's just draw a line in there. So our red line corresponds to exactly the difference in energy. Predicted by the Schrodinger equation for the n equals 2 and the n equals 3 levels. Likewise you find that for the light-blue, bluish-green line, it corresponds exactly to the energy difference between the n equals 4 and the n equals 2 level and the blue line corresponds to the difference between the n equals 5. And n equals 2 line, and the last line in the indigo corresponds to the n equals 6 and the n equals 2, 2 level. So, this is pretty impressive evidence that the Schrodinger equation is predicting. These experimentals transitions exactly, because the energies here that are measured experimentally correspond exactly to the predicted values given by the Schrodinger equation. Now, over here, this series of lines, as I said, this was originally known well before quantum theory was known. And, this series of lines was called the, the, the Balmer series after the person who first noticed them. There's also other lines that people, line spectrum people observe, and these are more into the, in this region here, as you go into UV and also as you decrease in energy. Long wavelength. You go into the infra-red. And these, actually. Again, if you can show that these correspond to transitions between the one. From the 1 level. What from the 1 level absorption or to the 1 level in the emission spectra. So you have 1 to 2, 1 to 3, 1 to 4, and so forth. And then they're called [UNKNOWN] series again after the scientist who discovered 'em. And you'll also find other transitions, from the n equals 3 levels [UNKNOWN] and these of course are lower in energy so they're going to fall in the infrared region of the spectrum. And again, you can find the exact correspondence between the values predicted. From the solutions of the Schrodinger equation and the experimental measurements. [BLANK_AUDIO]