 So, this is a good point to step back and look at the big picture and see what we understand about the harmonic oscillator so far. So, as a quick recap, we started by saying if the potential energy is quadratic, one half kx squared, we can use that to solve Schrodinger's equation and obtain these wave functions. These wave functions have this particular set of energies. So, let's draw some pictures and see what that tells us about the properties of this harmonic oscillator. So, let's start with the energy. So, if I draw an energy ladder, as we've seen, so here's the energies. I'm going to draw the energy of each one, each level. So, there's an energy at one half h nu, somewhat higher. There's an energy of, let me draw these way out to the side and then an equal amount higher. There's an energy of five halves h nu and that ladder continues with equal spacing as I climb up. So, that's what we know about the energies. Let's, since this is a graph of energies, let's draw the potential energies on the same graph. So, the potential energy is also a function of x. So, if I make this a two-dimensional graph with energy or potential energy as a function of x, then that parabola, that quadratic function is zero when x equal to zero and increases parabolically or quadratically as x increases. So, I've drawn a poor parabola, but that's supposed to be a parabolic function. So, this is the graph of what the potential energy looks like. If I now, on top of the same graph, attempt to superimpose the graph of what the wave functions themselves look like. We can use that composite graph to understand a little bit about the harmonic oscillator. So, let me do that with a different color. So, we have different wave functions at every different energy level. The zeroth level has wave functions psi naught. The first energy level has wave functions psi one. This one has wave functions psi two. So, I can draw several different wave functions. If I treat this as the baseline for this wave function, I know what the zeroth wave function looks like. It looks like a Gaussian. We've drawn that one before. So, the wave function looks like that. Squaring the wave function looks like a Gaussian just with a different exponent. So, this one is psi naught. I'll draw psi one as well. Remember psi one looks like x e to the minus alpha x squared over two. If I draw that wave function and I'll superimpose it on top of the n equals one energy level. That one starts out negative, goes positive and dies as x is large in either direction. So, that's the wave function with a single node in it at x equals zero. That's what the psi one wave function looks like. Psi two is the same thing with two nodes. So, here's a different baseline. I'll draw psi two starting positive, going negative, and then positive and decaying at the edges. So this shows us several things. First of all, I'll make sure you understand what we've drawn here. I've drawn three different things on the same graph. I've drawn the energy levels themselves. I've drawn the shape of the potential energy function. And now with this last curve, the blue curve, I've drawn the shape of the wave functions. And in particular, those wave functions are all at different baselines. This is psi equals zero for the psi two wave function. This level is psi equals zero for the psi one wave function. This is psi equals zero for the psi not wave function. So I've drawn the x axis essentially at different heights for each of these three wave functions to keep them from sitting on top of each other and make them easier to see. But the other thing that shows us is remembering what the wave function means, the wave function tells us where I'm likely to find the particle. In particular, I have to square the wave function to find out where I'm likely to find the particle. But in the psi not wave function, I'm most likely to find the particle near the center, near x equals zero, near the places where the potential energy is zero. So that makes sense. I'm most likely to find the wave function where the potential energy is small. As I switch to more and more excited states, I'm able to find the wave function, able to find the particle in other regions, not just where the wave function is large because the potential energy is low. But also in some regions like here or here where the potential energy is considerably higher. So the higher the excited state of the harmonic oscillator, the more likely I am to find the x value far away from the origin. Or thinking about what that means for an oscillating diatomic molecule. I don't just find it at the equilibrium distance, I also find it at stretched or compressed values of the bond as well. So when I excite the harmonic oscillator quantum mechanical molecule, I find that it's more commonly found, more frequently found, at small or large values of the bond displacement. There's also some similarities between the wave functions we've just drawn for the harmonic oscillator. So this is all for the harmonic oscillator model. If we think back to what the similar graph would have looked like, we never drew that graph. But if I draw a similar graph for the particle in a box, the one dimensional particle in a box, and I'll just draw those very quickly. Remember particle in a box, the particle was found between 0 and A. The potential energy for the particle in a box, I've done that in green. So the potential energy is 0 when I'm inside the box, and it's infinitely large when I'm outside the box. So the graph goes up to infinity as I reach the edges of the box. So actually that's what I should have drawn in pink. That's the energy of the particles, the energy levels, the e sub n. So those were e1, e2, e3. Those were things like h squared over 8ma squared times 1 or times 4 or times 9 or so on. So there were different energy levels for the particle in a box. And then the wave functions, I'll draw in blue. Those had perhaps one node, or perhaps two nodes, or perhaps three nodes, and so on, depending on which state of the particle in a box we were talking about. Those were functions that look like sine of 3 pi x over a, for example. So now that I've drawn both of those, we can see there's some similarities and some differences. In particular, the wave functions tend to oscillate, both the harmonic oscillator wave function and the particle in a box wave function tend to oscillate. So the third wave function, not the ground state, but the second excited state has three individual regions. It goes positive, then negative, then positive. The same thing is true for the harmonic oscillator. Positive, then negative, then positive. As a side effect of that, the wave functions have nodes. If I'm switching from positive to negative, I have to have a node. I have to have a region where the wave function reaches zero. And there's two nodes in this wave function for the harmonic oscillator, just like there's two nodes in the third wave function for the particle in a box as well. But there's some things that are quite distinctly different for the harmonic oscillator in the particle in a box. The reason for the oscillation or the mathematical form of the oscillation is a little bit different in these two cases. I've got an e to the minus alpha x squared for the harmonic oscillator and sine functions. Sine of n pi x over a for the particle in a box. So there's clearly a difference between the trigonometric oscillations and the exponential Gaussian shaped oscillations. The nodes in this case are formed by the zeros of the polynomial that sits out in front of the wave function, the Cermit polynomial. The zeros for the particle in a box are caused by the trigonometric functions themselves. Another difference. What other differences do we have? Another difference is the boundary conditions. For the particle in a box, you remember we spent quite a bit of time guaranteeing that the wave function reached zero at the edges of the box. Because the potential energy was becoming infinite, the probability of finding the particle at the edge of the box had to reach zero. So we had to make sure the wave function reached zero at the edge of the box. That is quite clearly not happening for the harmonic oscillator. These Gaussian functions don't reach zero when they hit the potential energy wall of this box. They asymptotically decay to zero, and they do so very, very gradually, and never quite reach zero until x reaches negative infinity or positive infinity. So that's an important difference between the harmonic oscillator, which in principle can have any amount of bond stretch or bond compression in the particle in a box where the particle is not allowed to escape the box. Another, I guess, similarity rather than difference between these two models is they both have some zero point energy. We've talked about how the harmonic oscillator has some energy even in the ground state. The energy of 1 half h nu is above zero. The same thing is true for the particle in a box. The ground state, the E1 state has an energy that's larger than zero. So both those models do have some zero point energy. And then the last point that we'll point out between these two models is the types of energy that we're talking about. Remember that the particle in a box, the potential energy was zero for the particle in a box, which meant that the only type of energy that model had was kinetic energy. In the harmonic oscillator, we do have some potential energy. So the harmonic oscillator has kinetic energy as well as potential energy. And that will turn out to be important when we talk about the thermodynamic properties of these models. Thermodynamic properties that we'll get eventually from the partition function from the models. And now that we have the energies of the harmonic oscillator energy levels, 1 half h nu, 3 halves, 5 halves, h nu and so on, we have enough information to plug those into the equation and obtain the partition function for the harmonic oscillator. So that's what we'll do now.