 So we're mainly interested in using the harmonic oscillator model to treat the vibration of diatomic molecules, but anytime we have a potential energy that looks like some variable squared, that ends up being called the harmonic oscillator model, so it's worth spending a few minutes explaining why this model is called a harmonic oscillator and what it looks like when you're not solving quantum mechanical problems with it. So this expression, potential energy, looks like the square of some variable. Let me just draw what that potential energy would look like in graphical form. So there's some variable, which for us might be a bond length, but in general might be many different things. The potential energy is going to look like the square of that variable, so it increases symmetrically on either side of zero for that variable like a parabola. And that, any variable that behaves like this whose potential energy behaves like the square of that variable, that's going to act essentially like a classical Hooke's law spring. So let's say for example I have a spring, object at the end of a spring, maybe attached to a wall, if I stretch that spring and let it go, it will snap back and then oscillate back and forth. If I compress the spring, it'll snap forwards and then oscillate back and forth. And the reason for that on this potential energy curve is if I move the value of x to some positive number where the energy is large, it will be able to reduce that positive energy by moving the x-coordinate smaller and the particle will slide down this potential energy well, sliding back up to a point on the other side where the potential energy is equal to where it started. Likewise if I compress it, it'll roll back and forth. So I can think of it as a spring, I can think of it as placing a ball inside of a parabolic bowl that will roll back and forth, or I can think of it as a diatomic molecule which is a lot like a spring. The reason we call it a Hooke's law spring, you may remember that a Hooke's law spring is one whose the more I stretch it, the larger the restoring force gets and the restoring force is proportional to the amount of stretch. Mathematically what we're saying is the force, if I stretch this spring, the force that the spring feels, a force is negative derivative of potential energy with respect to position. So since I know what v of x is, I can write negative derivative of one half kx squared is equal to twice one half is just one and then a k and then reducing the power of x by one, the force is equal to minus kx. So this is our Hooke's law spring. If I double the stretch in the spring, I've doubled the restoring force in the negative direction that the particle feels. So that's our Hooke's law spring. The other thing, and I guess we should use some terminology to point out that this constant k, when we're talking about springs, we call that a force constant. A strong stiff spring one that's very hard to stretch because the energy rises very quickly as I stretch the bond length as a large value of k or a large force constant, a very soft spring with a low force constant would be one that's very soft to very easy to stretch like a slinky, for example. So that variable is called the force constant. The other thing we know about the behavior of a system like this is we have started talking about force. We started talking about equations that you maybe have last seen in a physics class. The other physics equation that you'll be familiar with, the relates to force is f equals ma, mass times acceleration, or if we prefer mass times the second derivative of position with respect to time. And actually, since the variable we use for mass when we're talking about harmonic oscillator is not m, but we typically use the reduced mass of our object. So we'll actually write this as mu times the second derivative of x with respect to time. So this is our version of Newton's second law, f equals ma, with the reduced mass instead of the actual mass of the object. So we know two equations for force. Force for this potential energy must be minus kx. And force is also equal to the mass of reduced mass times the second derivative of x. So if I equate those two terms, minus kx is equal to mu d squared x dt squared. Actually, I'd rather have them on the opposite side. So let me put d squared x dt squared on the left, x on the right. And then let's put all the constants on the right. So there's a minus k in front of the x. And if I bring this mu over to this side, I've got a k over mu. So there's a negative sign in front of that fraction. So what must be true about any system with this quadratic potential energy is the second derivative of a position, the acceleration, has to be equal to negative the force constant divided by mass times the original x. And this x that we're talking about, this is the x that we're taking the time derivative of. x is a function of time. What we mean by that is when I have illustrated by stretching the spring to this point and letting it go, I stretch it to some initial value of x where it has some initial value of the potential energy. But when I let it go, the force on the spring will cause its position to change. And it will go back to its initial position, compress, stretch, compress, stretch. So that oscillation back and forth is changing the value of x as a function of time. So that x will change over time in some way that will oscillate. And the derivative, second derivative of that function with respect to time, has to be equal to the original function itself, multiplied by some negative constants. That equation should look a little familiar, should seem a little familiar. That's exactly the same differential equation we solved when we considered the one-dimensional particle in a box. We were looking for some wave function whose second derivative is negative the original function. There we were taking x derivatives and we called the function psi. But the differential equation is exactly the same. So building off of either what you've done in a physics class or a differential equations class or what we talked about when we did the one-dimensional particle in a box, we can say that this x as a function of t, this function, the only functions we can think of whose second derivatives are the negative of the original functions, those are cosines and sines. So maybe some amount of cosine plus some amount of a sine function, those depend on t, but I can also have some variable inside the sine function along with them. So I'll say cosine of some variable omega times t, where omega is an angular frequency. So cosines and sines, both if I take second derivatives, will return to me the negative of the original values, pulling out some factors of omega as I take those derivatives. And in fact, if I solve for what the omega has to be, such that if I take two derivatives, pulling out an omega and then a minus omega, that omega squared has to be minus k over mu. So omega squared has to be k over mu. So omega has to be square root of k over mu. So the angular frequency with which this, if I draw how the position of this particle, after I've stretched it to some positive position and let it go, it's going to go negative, positive, negative, positive. It's going to oscillate back and forth like a cosine and or a sine wave. The angular frequency with which it does that oscillation is going to be square root of k over mu. We're often interested in not the angular frequency, but the frequency, the difference between those. This is how many radians per unit time, the frequency in radians per unit time. The frequency is the number of round trips per unit time. So the only difference between an angular frequency and a frequency of the type that we'll usually use is this factor of 2 pi radians per complete round trip in a circle. So if I want to convert this angular frequency into a regular frequency, I just divide by 2 pi. So I can write 1 over 2 pi square root of k over mu. And let me decorate this frequency when we use, just like we've used mu, thinking ahead for the harmonic oscillator, because we'll be talking about the reduced mass of a diatomic molecule. The frequency that we'll talk about for diatomic molecules will be the equilibrium vibrational frequency that we typically label with a subscript e. So this nu sub e is the equilibrium vibrational frequency. Not only for a harmonic oscillator, but for any system that has this potential energy where we can do these classical physics equations to determine the behavior of the molecule. So let me go ahead and put this equation in a box. So the vibrational frequency, 1 over 2 pi square root of k over mu, square root of the force constant divided by the reduced mass of the diatomic molecule if we're thinking about harmonic oscillators. So this is, to explain the name harmonic oscillator, we've explained now why the molecule oscillates. Harmonic just means it's oscillating as a pure sine or cosine wave with a single frequency, sort of a music-related term. The harmonic oscillation is oscillating with a single pure frequency given by this expression. So this has all been a classical, purely classical, Newtonian physics description of the oscillation of this system with a quadratic or parabolic or harmonic potential energy function. The next step will be to plug that into Schrodinger's equation and see how it behaves in a quantum mechanical system so we can describe vibration of diatomic molecules.