 So let's talk about the energy levels of the rigid rotor, which we now know we can write in this form, collection of constants times some integers for any particular wave function with quantum number L, which must be some non-negative integer. There's a variety of different values of m that are also possible. And the energy doesn't depend on the value of m. Any value of any of these allowed values of m has the same energy. If we know the value of L, we can calculate L times L plus 1 and use that to get this energy. So we've already seen, for example, but now we can use this formula to double check. If L is equal to 0, the energy of the 0th wave function, the 0,0 wave function, L equals 0, m equals 0, if I plug in L equals 0 into these two places, 0 times 1 is 0, and that 0 times these constants gives me 0. So what we'll do is over here, I'll write an energy ladder. So here are the energies of the different states of the rigid rotor. And down at the bottom where 0 is, that is the energy of the ground state, E naught. And that's equal to 0. L equals 1, the next step up this ladder, the quantum number L equals 1. If I plug in an L equals 1 here, and then L plus 1 is equal to 2. So 1 times 2 is 2. So h squared over 8 pi mu squared r squared, h squared over 8 pi squared mu times r squared. That's multiplying 1 and 2. The energy of this state, the equals 1 state, or the L equals 1 state, that's twice these constants, h squared over 8 pi squared mu r squared. But there's one other thing to pay attention to with this energy level, the L equals 1 energy level, the n values. This is the energy of E10, or E11, or E1, negative 1. Those are the three possible values of m, m equals negative 1, or 0, or 1, that obey this rule for what the magnetic quantum number can be. So there's three different states, three different wave functions, that all have the same energy. So this one energy level that has this particular value has three states that are associated with it. So there's what we could number these however we want, but the L equals 0, I'm sorry, L equals 1, m equals 0, m equals 1, m equals minus 1. Those three states all have the same energy as each other. Their degeneracy, the degeneracy of the L equals 1 level is 3. There's three states with that energy. The degeneracy of the 0th level is just 1. There's only one state that has that energy. If we continue investigating the degeneracy as we climb this ladder, when L equals 2, 2 times 3 is equal to 6. So here's twice this collection of constants somewhere up here around 6 times that collection of constants. I've got another set of states and I've drawn five of them because the number of states that are degenerate that have the same energy as each other. When L is equal to 2, m can be anywhere between negative 2 and positive 2. So it can be negative 2, negative 1, 0, 1, or 2. So there's 1, 2, 3, 4, 5 states that all have five wave functions, five states that all have the same energy as each other. So the degeneracy of the L equals 2 level is 5. The energy of the L equals 2 level is constants times 6. One more, just to add one that we haven't seen yet. When L equals 3, 3 times 4 is equal to 12. And we can begin to see instead of writing them all out, how many states there are that are going to have the same energy. When L is equal to 3, I could have negative 3 up through positive 3. So negative 3, negative 2, negative 1, plus a 0, and then 1, 2, 3. So there's seven different states. So there's anywhere from E3, negative 3, all the way up to E3, 3. There's seven integers from negative 3 up to positive 3. So at 12 times our collection of constants, so about twice as high as that one, so somewhere up here, there's a seven-fold degenerate state, a seven-fold degenerate energy level. The degeneracy of the L equals 3 level is 7. Energy is 12 times these constants. And of course, we could keep going as long as we want. In general, as this equation tells us, the L energy level has an energy of h squared over 8 pi squared mu r squared times L and L plus 1. The degeneracy of that level, because m can be anywhere from negative L up through 0 all the way to positive L, there's L numbers that are negative. There's L of these numbers that are positive. And the 0 in the middle is an extra one. So the total degeneracy is twice L plus 1. So just for example, when L equals 0, the degeneracy twice 0 plus 1 is 1. And then the degeneracy climbs as odd numbers. It goes from 1 to 3 to 5 to 7, increasing by 2 for every step up the ladder we take. So a couple of important things to point out about this energy ladder that we've begun to construct here. Number one, let's ask ourselves what happens to the gaps between these energy levels. So the gap between energy level and energy level zero and energy level one is smaller than the gap between energy levels one and two, which is smaller still than the gap between energy levels two and three. So that gap increases from two times these constants to four times the constants to six times the constant. That gap keeps increasing as we climb the ladder. So that gap between energy levels increases as the quantum number increases, as we get more and more excited on this rotational energy ladder. Similarly, the degeneracy of the L level is also increasing as we climb the ladder. The degeneracy goes from 1 to 3 to 5 to 7, climbing again. Every rung up this ladder we go, the more states there are that have the same energy, the higher the degeneracy of that rung on the ladder. So those two facts, the fact that the energies keep getting higher more and more quickly as we climb the ladder, but also we get more and more states that exist at each rung in the ladder, the combination of those two facts ends up having some important consequences on what the rotational properties are of real molecules. And that's what we'll talk about next.