Hello, today we're going to talk a bit about real atoms. It turns out the hydrogenic atom model is good for beginning to learn about atoms. But in the case of real atom things are a tad more complicated. And unfortunately, that complexity is not very well described by the simple theory. There's a list here on this slide of some of the things that contribute to actual atomic structure. There are what are called higher order effects. I guess higher-order would imply small effects, but these are not necessarily small effects, especially the first item on the list spin-orbit coupling. There are relativistic effects, nuclear spin can contribute, there's something called the Zeeman effect. And then if you have multiple electrons, which you do when all the atoms except hydrogen, the electrons themselves, the electromagnetic fields associated with the electrons can interact makes things more complicated. And then finally, we have the quantum mechanics of identical particles, electrons being identical particles and that leads to the Pauli exclusion principle. So perhaps the most important of these effects, of the higher order effects, I should say, it's been orbit coupling. Electrons have spin and that's angular momentum as does orbital motion of the electrons. And those two emotions generate magnetic fields that can interact. And so there is some total angular momentum and the total angular momentum can be expressed as the sum of the two angular momentum's orbital and spin. Well, it turns out you can show that total orbital angular momentum is what becomes quantized or is quantized and that introduces a new quantum number lowercase j. And the total angular momentum is then equal to the square root of j times j plus 1 times h bar. And in addition like orbital angular momentum, the z component is quantized and so that introduces yet another quantum number m sub j. And there's a restriction on the value of m sub j. Clearly, it can't be bigger than j since it's just one component of j. So it can take on the values of minus j to j an integer increments. So if l is equal to 0, then all we have is the electron spin contribution and so j will be equal to 1/2 lowercase j, the quantum number will be 1/2. And then if l is greater than 0 j will be equal to l plus or minus 1/2. So as a result, m sub l and m sub s cease to be so-called good quantum numbers and are replaced by j and m sub j. So the good quantum numbers become n, l, j, and m sub j. So the spin-orbit effect, the coupling between orbital momentum and an electron spin momentum causes quantum states that would otherwise have the same energy to split. And that changes the the degeneracy or can change the degeneracy that you have to deal with as well. So this particular diagram, it looks a little confusing, the lower left hand branch and this diagram is done for n equals 2. And so for n equals 2, the total degeneracy would be 2n squared which is eight and it's shown there in the diagram g sub n is equal to 8. The lower branch that drops below the n equals 2 line shows the choices without splitting and what leads to the degeneracy of 8. You have the l equals 0 branch and you have one value of m sub l and two values of m sub s and then you have the l equal 1 possibility. So there you have three branches because you can have three different values of m sub l and each of those involve two values of m sub s. So that leads to a total of degeneracy of 8, but it doesn't lead to splitting. The right hand side of the figure going to the right illustrates the three branches that the energy is split into. The first one at the top is n equals to l equals 1 and j equals three-halves. The middle line is n equals 2, l equals 0, j equals one-half, and the bottom line n equals 2, l equals 1, j equals one-half wand then also listed are the values of m sub j that would go with those lines. And if you add up all the possibilities again, you get 8. Now is this splitting significant? You bet it is, here's the energy level diagram for sodium. And what I want you to observe is that the lowest energy state labeled three on the left end of the horizontal line, that's the ground state. But if you go up to the next levels, the two levels side by side, they also are n equals three and they have the same principle quantum number but different angular momentum quantum numbers. Well, then you can see that there's even a higher lying line that has n equal three. And so that is three energy levels that have or three quantum states that have the same principle quantum number and very, very different energies. And then you can see there are a number of levels with principal quantum number 4, with very different energies, etc. So the spin-orbit effect can be quite substantial and it cannot be neglected. There's some other notation on this chart across the top which we will discuss later. So that's it for this video, thanks for listening, have a great day.
