 So, we've managed to come up with a whole family of solutions for the wave functions that solve the harmonic oscillator Schrodinger equation. They all have this form of some Hermite polynomial multiplying a Gaussian. We've seen in particular that for the ground state wave function, the simplest of these functions with just a constant out front of the Gaussian, that when we plug that wave function into the Schrodinger equation, the energy turned out to be one-half times h times the vibrational frequency of this oscillator, the harmonic oscillator that we're talking about. We've not plugged any of the others into the Schrodinger equation, and we won't do that now. That would be a good problem either. Very likely you'll work on those as some homework problems at some point, or they'd be good exercises now if you want to between videos take any one of these wave functions and plug it into the harmonic oscillator Schrodinger equation. Some terms will cancel, the terms that don't cancel will end up telling you the energy of the wave function multiplied by the original wave function. And we'll see that if I plug this function with the linear polynomial out front of the Gaussian into Schrodinger's equation, I don't get one-half times h times nu as the energy of the wave function. I get three-halves times h times the frequency. Similarly, if I plug the psi two wave function into the harmonic oscillator Schrodinger equation, I'll get the energy of the n equals two wave function, and that will turn out to be five-halves times h times nu. And this pattern repeats. Notice that the energies are one-half, three-halves, five-halves. They're going up by two-halves every time, so saying that in other words the energy is one-half for the n equals zero wave function, one-and-a-half for the n equals one wave function, two-and-a-half for the n equals two wave function, and for the nth wave function with this general form, the energy is n-and-a-half, or n plus a-half factors of h times nu. So what that means if I draw an energy ladder for the energy levels of a harmonic oscillator? So here's the bottom. There's zero energy. The ground state, the lowest energy wave function, is this one with energy e naught, which is equal to one-half times this collection of constants, h times nu. The next one up is three times larger, three-halves, so one-half, two-halves, three-halves. Somewhere around here will be the energy of the n equals one wave function, and then an equal amount higher will be the n equals two wave function, and so on. So notice that the gap between the e zero and e one wave function, three-halves minus one-half, is one factor of h nu. That's the difference in energy between these two successive states. Likewise, the difference between e two and e one, that energy difference is also five-halves minus three-halves is also one factor of h times nu. And this continues as we go up the ladder. Every pair of states increases its energy. Every successive pair of states has a difference of energy of h times nu because every energy level increases by an increment of exactly one times h nu as we climb the ladder. So there's two very important features of the harmonic oscillator energies that we've just written down. First of all, delta e is h nu between every pair of successive states. In other words, as I climb the ladder, the gap between states remains exactly the same. That's different than it was for the particle in a box or for the rigid rotor where that spacing changed as I climbed the energy ladder. So here the energy gaps remain constant as I climb the ladder. The other important point worth mentioning is that the ground state energy, the energy for the n equals zero state, we've seen that that's equal to one-half times h nu. The relevant fact here is that number is not zero. It's bigger than zero. The lowest energy is a positive energy. So there's no way to lower the energy of the system any lower than one-half h nu. In particular, I can't get the energy all the way to zero. So no matter what energy level the system's in, it cannot have zero energy. It must at least have one-half h nu worth of energy and often more than that. So we say that the system has a zero point energy, often abbreviated ZPE, saying the system has a zero point energy. Essentially it just means that if we bring it down to its lowest, its zero energy level, it has some non-zero amount of energy when it's at that zero point. No matter how cold we make the system, how low the state it's occupying, it always has some non-zero energy. We can never fully remove all the energy from the system because we can't drop below the zero energy level. We can't drop the energy below one-half h nu. So that's what we mean when we say this system has a zero point energy of one-half h nu, or if the system has a zero point energy in general, it just means that has some energy that we can't fully get rid of in the system. So those two features, the fact that our harmonic oscillator has a zero point energy and the fact that the gaps between these energy levels remains constant as I climb the ladder, those will combine to have some pretty important consequences when we describe the properties of a harmonic oscillator, the properties of a diatomic molecule vibrating and acting as a harmonic oscillator. In particular, that will affect their spectroscopy, the colors of light that the molecule will absorb, as well as their thermodynamic properties. But before we go on and talk about those properties of the molecules, since we've written down now both the energies as well as the wave functions, we can talk a little more about how those two properties of the harmonic oscillator interact and what that tells us about the harmonic oscillator.