Lecture 09. The Morse and "6-12" Potential and Quantization in Two Spatial Dimensions
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Shaka, Athan J.
 This course provides an introduction to quantum mechanics and principles of quantum chemistry with applications to nuclear motions and the electronic structure of the hydrogen atom. It also examines the Schrödinger equation and study how it describes the behavior of very light particles, the quantum description of rotating and vibrating molecules is compared to the classical description, and the quantum description of the electronic structure of atoms is studied. 
 Welcome back to Chemistry 131A. Today we're going to talk about more realistic vibrational potentials, the Morse potential, the 612 potential, and we're also going to extend our vision into at least two spatial dimensions. Rather than talking about a particle in a one-dimensional box, we'll extend it to two spatial dimensions and see what complexities that involves. It turns out for a particle in a box that won't be serious, but depending on the form of the potential, having more than one spatial dimension can make the equations much more difficult to solve. When we last left, I made the remark that we didn't have to do a certain integral and the reason why we didn't have to do it is because the integrand was odd or anti-symmetric and the interval was symmetric. And I'd like to just take a second to show you how that works as a practice problem. So suppose we consider an anti-symmetric function of X. That is that the function of X is equal to minus the function of minus X. Then we have to show that the integral of such a function is zero between symmetric limits. Just the statement of the problem gives us a clue how we're going to have to proceed. We're going to have to make use of the symmetric limits because it certainly isn't true if we integrate from zero to something. That has a finite value. We're going to also have to use the fact that when we change the sign of the function, the argument that the function changes sign. Let's see how we can do this. The trick here, and it's a very good trick to remember, is that we don't actually try to show that the integral is zero. That turns out to be pretty difficult. What we instead do is we show that the integral is equal to minus itself. And then the only number that's equal to minus itself is zero. Oftentimes very tricky things in math rely on not showing that X is two, but showing that X is greater than or equal to two, and that X is less than or equal to two, and then X has to be two. And this is a very good trick to remember if you get stuck on some mathematical problem, is to try to sort of paint it into a corner by inequalities or relationships rather than just trying to get it spot on. Here's what we can do then. The dummy variable that we use in integration, the value of the integral doesn't depend on that. That's just a notational consideration. Therefore, if I call the integral I, which is the integral from X is minus A to A of the function F of X. That's also equal to the integral from U equals minus A to A, F of U, DU, same thing. Now I just make the substitution that U is equal to minus X. So I put in minus X equals minus A for the lower limit, that's U equals minus A. Minus X equals plus A for the upper limit. F of U is F of minus X, D minus X. And then I bring out the minus sign from the D minus X, and I have minus F of X from minus X equals A to minus X equals plus A. And then I use the fact that the function is anti-symmetric, and I change minus F of X to F of X. And then I just say, look, if minus X is equal to A, that means, excuse me, if minus X is equal to minus A, then that means X is equal to A. And so I'm really integrating from X equals A to X equals minus A. But I know that's equal to minus if I swap the limits. And that's equal to minus I. And so by that chain there, I've shown that the integral I, whatever it is, is equal to minus I, and therefore the integral is zero. Okay, let's move on and talk about a more realistic potential. The harmonic oscillator gave us a differential equation that we could solve. It was harder than a particle in a box, but we could solve it. We could with a little mathematical sophistication, write down all the wave functions for all the excited states, and there are an infinite number of them. But it's not very realistic for a real chemical bond, because atoms, when they get too far apart, don't interact very well. We know that from gases and other systems. We can only have a chemical bond with a force constant restoring the atoms to an equilibrium position when they're rather close. Otherwise, we would suddenly be bonding to all sorts of things, which does not happen. But it, so the harmonic oscillator is a very bad approximation, unless we're near the bottom of the potential. If we're near the bottom of the potential, it's quite good. As we get out further, it becomes very bad. And if we're interested in breaking bonds or simulating breaking of bonds somehow, it's extremely bad. Therefore, we need something that's a little bit better. And therefore, the question is, is there an alternative potential, which one is more realistic, and two still allows us to get the exact solutions for the time-independent Schrodinger equation, to get the energy levels and so forth and so on. And the answer to both questions is yes, there is such a potential. And it's called the Morse potential after Philip Morse, who first suggested it in 1929. It did not take long after wave mechanics was invented for people to start working like crazy on this new field and make all sorts of refinements to the simple problems and really go in depth and advance the field. And this can model vibrations much better than the harmonic oscillator, especially for excited states in a molecule. The number of bound states in the Morse potential is finite. And that means that there is a limit after which the chemical bond breaks and the atoms fly apart. That's already a substantial improvement over the harmonic oscillator. And furthermore, even the bottom part of the well, where we haven't gone up very far, is represented much better with the Morse oscillator than the harmonic oscillator. In fact, the energy levels always get closer together with the Morse oscillator as we go up as we'll see. Okay, what's the functional form? Well, it's something that's not very obvious at all. It's the potential as a function of R should be some constant DE with units of energy times the quantity 1 minus E to the minus A times R minus RE quantity squared. That's not a very obvious potential to choose. And it's not at all obvious at first blush that one could solve that potential in the Schrodinger equation and get solutions. It looks to be fairly complex. Here, D is called the well depth and A is called the width parameter. And so we have two things to control. We can control how deep the well is and we can control how wide it is. The depth has to do with the total energy before we dissociate. And the width has to do with the spring force constant and the depth. They're related. When A is small, the parameter in the exponential, what that means is that the well is narrow and there's a larger force constant. Here's a comparison that I've adapted from Wikipedia of the harmonic oscillator which is in green and the Morse oscillator which is in blue. And there are a couple of things to notice. The Morse oscillator is this beautiful function which is asymmetric in pretty much a very realistic way for a real chemical bond. The levels get closer and closer together as we go up. And there's a definite prediction about what the dissociation energy is. The dissociation energy is not the well depth but it's a little bit different than the well depth because of the zero point energy of the oscillator. And we call the dissociation energy D naught and the well depth D e. And there's an equilibrium distance which we've assumed to be identical for the two oscillators in this case, r sub e, the equilibrium distance. Now, the energy levels for the Morse oscillator which we won't solve the equation and go through all that in this course. But just to quote the result, the energy levels of the Morse oscillator get linearly closer as we go up. The spacing between zero and one is one unit. Then the spacing between one and two might be point nine and then point eight and point seven. And that equation fits this functional form. The energy of quantum level V which is an integer number is h bar omega times V plus one half. That's exactly the same thing as what we got for the harmonic oscillator. Except that kept going and going. And then there's a new term which shows you how clever this potential is because it gives you what you had before and then it gives you just the right correction to make it more realistic. Minus h bar omega x e times quantity V plus one half squared. And therefore there's this quadratic term that's lowering the ladder of states as we go up. Clearly there's some value of V where the quadratic term gets so big that the thing would turn around and head back down. And at that value of V where it starts doing that, that's the highest one. After that the bond is broken and we don't consider that energy equation any longer. And the dimensionless parameter x sub e is just called the anharmonicity constant. It's a measure of the distortion away from a harmonic oscillator and to a different kind of motion that hangs out here for a long time and is definitely not a harmonic oscillator in terms of going back and forth evenly. And as I mentioned at some point V max, the energy of V max plus one according to the formula anyway would be less than the energy of V max. The states stop going up. At that point that's the highest level that can be supported in the potential and that allows us to count how many states could be in there and get an idea of that. The wave functions for the Morse oscillator are quite a bit more tedious to calculate than the wave functions for the harmonic oscillator. And the harmonic oscillator was already much harder than the particle in a box. And therefore we're not going to write down the explicit form of the wave functions. What we usually want in chemistry is the energies to get an idea of what kinds of processes can happen if we put in light of a certain energy. Will it be absorbed? Could I get this thing to break a bond and so forth and so on? Occasionally we're really interested in detail and we'd like to have some idea of the electron density but usually it's very, very hard to calculate the wave function and so there are other tricks like density functional theory to calculate the electron density without actually trying to calculate the full blown wave function. For real bonds then just building on this Morse theory one could suppose that you could have another anharmonicity and in this case it's Y e. For all real bonds the energy levels get closer together when they're near the bottom and that's why the first term has a minus sign because chemists like positive numbers. So we like X e to be a positive number so we put a minus sign there so that X e is a positive number. But Y e the next term in v plus 1 half cubed in this power series then in the energy as a function of v is could be plus or minus and it could depend a little bit on how many levels you're trying to incorporate into this equation when you do the fitting. And so for that one we have a plus and we allow it to be Y e to be either plus or minus as the case may be. In chem 131b what you'll find out in IR spectroscopy when you do that as an actual method of analysis is that you'll find out that IR or infrared spectroscopy is very important method to glean all kinds of information about the strengths of bonds and rotational and vibrational states of simple molecules. And in fact it's of course the vibrations of molecules like CO2 that are causing all the problems with climate change and radiative forcing as it's called. As the earth emits infrared radiation which you can think of as heat comes out at certain wavelengths some of it may actually instead of going out into outer space and cooling off like taking cookies out of an oven or putting a pie on a window sill some of it may actually hit CO2, an excited vibration in CO2. And CO2 may vibrate for a while and then it may decide to emit a photon. And when it emits it it has lost memory of all direction. It may have spun around and may have done all kinds of things in between and so it emits it in all directions. In particular it emits it back toward the ground. And what that means is that the rate of escape of radiation is altered and when you change the rate of escape of heat you heat up. That's all you do when you put on a parka. You change the rate of escape of heat from your body and you're much warmer than if you don't have it on. And so if you, we study the vibrational modes of these molecules in the lab we can predict which ones for example might be very bad to release in large amounts into the atmosphere. And unfortunately one of those is exactly the molecule that they put into car air conditioners hydrofluorocarbons which have very great greenhouse gas radiative forcing and could be very bad to release. The force constants that we obtain when we look at these bonds really agree with exactly what we would think would happen when we have a single bond or a double bond or a weak bond or a strong bond. If the bond is strong then the force constant is pretty big and if the bond is weaker it's small. And here are some examples. The force constant here is in newtons per meter. For hydrogen fluoride it's 970, reasonably strong. For hydrogen iodide it's only 320. Both of those are single bonds but the overlap between the orbitals on fluorine and hydrogen is much better than iodine which is a great big thing with diffuse elephant ear orbitals makes a very weak bond. And carbon monoxide has a force constant of around 1860. When we draw the Lewis structure for carbon monoxide we draw three bonds between the carbon and the oxygen. So that makes sense that that would be an extremely strong one. And other ones with double bonds are in between. And so this at least makes very good qualitative sense when we compare what our notional idea of a strength of a bond is with this force constant of this spring and this quantum mechanical model that we're solving. Now there's another potential that is used which you should know about and which is of historical importance mainly because of its simplicity and the way computers worked. And this potential is called the Leonard Jones or 612 potential. It's commonly used for example to simulate liquids. In the early days of computer simulation we couldn't do complex systems like membrane proteins and all these other things that we can do now. In fact what we started out doing was the simplest things like noble gas liquids try to simulate a liquid argon, something not doing too much and see if you can figure out when based on the potential when it will solidify, when it will boil, when it will do this and that. And that can already be an extremely challenging problem with the 18 electrons you have there. But you can make a very approximate model of this because when two molecules, when two atoms are far apart there is a very decent model for how they will attract. If there's any kind of charge fluctuation of the charge cloud of one atom and it's very unlikely that if you have 18 electrons for example that they're all going to be totally symmetrically distributed every which way at every instant. So suppose just for a tiny bit of time the electrons have 10 on one side and 8 on the other. Well then this side looks positive and if anything's nearby that then these electrons may rush over because they may see that there's a positive charge here. And so they tend to attract and that means they always tend to attract. And when you work out the way it works with two dipoles, one over r cubed, you get an attraction that goes like one over r to the sixth. So for a long way away it's flat and then as you come in it starts going down, down, down like one over r to the sixth. But we know when they get too close that they repel. And they repel because when the electron clouds overlap, Polly tells us that we can't have electrons in the same orbital with the same quantum number more than two and then even then they have to have opposite spin. So they repel, they bounce off and there has to be a strong repulsive force. There's not such a good theory however about what that repulsive force should be. And but we know it's very strong. The Morse potential while it gives this nice theory has one thing that is a little bit troubling and that is if you go back and I'll let you do that, go back to the formula and you put in r equals to zero so that the two particles, the two nuclei we think of them as positively charged are right on top of each other or extremely close. We'd expect tremendous repulsion under those circumstances. But the Morse potential gives us some finite value. It doesn't go up to infinity. Infinity may be too big but a finite value like the Morse potential may be much too small especially when r is tiny. And usually we aren't modeling such small values of r anyway but if we have things hitting hard, we want to make sure that they hit hard and that they repel appropriately if we're trying to model that kind of behavior. The 612 potential instead of having a finite value like the Morse potential, we just add a term plus 1 over r to the 12 and that goes up much, much faster than minus 1 over r to the 6 goes down. So we get something that goes down to a minimum and then goes up like crazy when you get too short and then we can adjust by some terms. We can adjust how fast, deep it goes down and what the equilibrium is for the best situation, the minimum energy. So the numbers in 612 refer to exponents, the 1 over r to the 6, 1 over r to the 12. And that it's a very nice functional form. Here it is, v of r's just could be a number like a over r to the 12 minus b over r to the 6. That could be a simple way to put it in. And a and b are your two parameters that allow you to control how deep it goes down and what the equilibrium is for the best distance between. In other words, how big do I have basket balls? Do I have tennis balls? You know, what's the right scale? Usually we tidy it up though, mathematically like this. We say v of r is epsilon, some number with units of energy, times rm over r to the 12 minus 2rm over r to the 6. And rm is the minimum and epsilon is the depth of the energy minimum at r equals rm. And as I said, rm is the minimum value where we're at the equilibrium where it's most like a harmonic oscillator, for example. Now why pick r to the 12? Why not pick something else? And the legacy of that is kind of interesting. In the old days, digital computers were so slow that it was very time consuming to compute this potential. And if I want to simulate a lot of particles moving around and I got to figure out who's attracting whom and how much and who's repelling, then I have to basically compute these v of r over and over and over and over. And computing something like an exponential function or a cosine or sine in the very early days, that was just a lot of machine code instructions and took a very long time to do. And so it slowed everything down. The beauty of this 612 potential is the fluctuations go like r to the minus 6. And then once I've got r to the minus 6, so I know what that number is, I just square it, I just multiply it by itself and I get the repulsive part r to the plus 12. And that's just one multiplication. And that's why it became so popular because it let you simulate things for a lot longer than anybody else could with some other more realistic potential that might involve an exponential function or something else. And then it let you do some interesting numerical experiments on the computer. It's just amazing how much more powerful computers are today than they were when people were doing landing on the moon and things like that. But the computer I used as a graduate student had 8k of memory. And I remember quite clearly when they upgraded it to 32k, I thought, what am I going to do with all that massive amount of RAM? How could I possibly use it all? And when we wrote programs, we used three variables, X, Y and Z, because you didn't want to run out of memory. And because you didn't want to run out of disk, you basically never put any comments on because that would take more space on the disk. And at the end of the day, you had to be extremely careful to note yourself what you were doing because nothing had any name that made any sense. It was all X, Y and Z, I, J and K and so on. And nothing had any comments. And therefore, if the program went south and you made an error, it could be extremely difficult to debug. Okay. As I mentioned, r to the plus 12 apparently, r to the minus 12, excuse me, plus some number over r to the minus 12 has no theoretical justification. But it was just easy to compute. There are other forms and nowadays, Leonard Jones is still used quite commonly because it's still fast. But there are other forms and computer power is so much higher that it really depends on what you're doing, how accurately you want to compute these potentials, what kind of form you use, what you're trying to match in terms of physical properties. Okay. Let's move now to simple quantum systems in 2D and 3D and talk a little bit about degeneracy which is quite important in real systems. Ultimately, to do anything with real atoms or molecules, we have to be able to solve in at least three spatial dimensions, X, Y and Z. But as a stepping stone, let's try solving in two first and get the math down for that. And then we can extend it to three. How easy or hard it is to solve these equations really comes down to the potential. If the potential has a simple form, then it's easy. If it has terms like X times Y in it, it can be very, very difficult to do. Usually the best case is no potential. That's a particle in a two-dimensional box. That's what we're going to do first. Nothing can go wrong there. And after that, some kind of potential that's additive separately in X, Y and Z but doesn't have any cross terms. Doesn't have anything X times Z or anything like that. Okay. Suppose we have a particle and it's confined to a 2D region. So here's what we're going to say. The potential energy is zero if X is between zero and LX. And Y is between zero and LY. And otherwise, if you're outside of the box, the potential is infinite. We know by our argument of a one-dimensional box that means the wave function has to vanish at the edges both with X and Y. Otherwise, the total energy is infinite. The particle cannot have infinite energy. Now we have a two-dimensional equation. Luckily, and psi of X, Y, we don't know what the form of psi of X, Y is. But we have just the kinetic energy minus H bar squared over 2M, second derivative with respect to X. We've got to use partial derivatives because we have to keep clear what we're keeping constant and what we're letting be a variable. Plus the same kind of term, second derivative with respect to Y, psi of X, Y is E psi of X, Y. And as I mentioned, we have to use partial derivatives because we have to be absolutely clear what we're keeping constant and not when we have both X and Y in the wave function. The first thing you always try whenever you have an equation like this, even if you don't know anything about it, is you try taking a product. You try saying, I think that psi of X, Y is something, function in X, F of X times G of Y. Now that doesn't encapsulate all possibilities by any means. But what it does do is it means that the, something like this, the energy in X and the energy in Y are separate from each other. And they're just going to add up separately. And so that's the simplest assumption, is that they have nothing to do with each other. And fortunately for the simplest kinds of atoms like the hydrogen atom, that's really good. And we can figure out all the beautiful wave functions and make these beautiful electronic orbitals which we'll get into later on in this course. If this assumption that psi is a product, F of X times G of Y, works, then it solves the problem immediately and you're done. And it doesn't take too long to see if it's going to work. If it doesn't work, that means that you've got a bit of a nasty equation, then you consult an expert. That's why we have experts. They're in their offices there with all their papers piled up and they can solve a lot of mathematical problems or you go to a reference book on differential equations or you take a more advanced course in differential equations. And all those things are good things to be able to do so that when you hit a problem like that, you can solve it. If you can't solve difficult problems quickly, the thing is you lose your train of thought. And you may hit a problem and you may say, gee, if I could solve this, I could figure out this other thing and so forth. But if you keep hitting it and you can't solve it and you don't know where it's going and you can't make progress, it's like reading a novel letter by letter and it takes you a long time to recognize you've got a word. The problem is if you go so slowly like that, you can't even get the train of the story because by the time you get halfway through the chapter, you've forgotten what the plot was. And it's a little bit like that in these branches of science. If you can't see things quickly enough by learning how to do it, it takes too long and then you get lost. And then the problem is that you don't even glean the chemical, the physical chemistry knowledge out of it because you get lost in this labyrinthine maze of mathematics. Okay. Now, if you can't solve it and there's no expert, what you can do is you can try to throw away certain things from the Schrodinger equation that are crossing you up and then you can treat them as a perturbation because at least we know how to do that. So we can take any nasty term that prevents us from factorizing it and put that aside and then we can calculate a correction based on perturbation theory. And that can also be a good way to go if you want to figure out something quickly, whether something is going to have a certain effect or not. How do we know that it's worked? Well, when the 2D equation separates into two 1D equations. 1D equation is some equation that has derivatives with respect to x and blah, blah, blah, x, x, x, x, nothing else. And then another equation that has everything with respect to y and nothing else. But there's no psi of x, y left. If you can separate it like that, it's worked. If you cannot, then you're going to have to either treat it as a perturbation or get an expert to help you. It's commonplace to use capital X of x rather than F of x and capital Y of y rather than G of y. But it's the same thing. Capital Y of y is a function that depends only on the argument y. It has no x's in it and x is likewise. They can have constants, of course, but they can't have x. Now our equation looks like this. The energy operator, kinetic energy operator. And now rather than operating on psi of x, y, it operates on x of x times y of y. And it gives back e times x of x times y of y. So we're just putting this in as a guess. And in the partial derivative with respect to x, we can take y of y, which doesn't have any x's, as a constant. And we can move it through the partial derivative operator and put it out in front as a constant. And vice versa. On the other derivative, we can move the part that we don't have to worry about out in front. And then after we've done these derivatives, we now have y of y times the second derivative of x of x plus x of x times the second derivative of y of y, all times minus h bar squared over 2. And that's equal to the energy. Now there's one more trick. And the one more trick is we divide both sides by x of x times y of y. And if we do that, then the y of y in the first term goes away. And we end up with 1 over big x of x times the second derivative of x with respect to x. And the other term has 1 over y times the second derivative of y with respect to y. And that's now equal to e, some number. Now the first term has nothing to do with y. So if we look at the first term, it has no y's in it. The second term has a bunch of y's in it. And so if I change little y in the second term, it's going to be different values. And that means that it can't add up to the energy. So if I pick a particular value of x and let y move around, that can't be a constant unless x itself, that first term is a constant, separately, some constant. And the second term, which has only y in it, is also some constant. And that's the argument that allows us to separate the equation into two. They have to separate because when I hold one of them fixed and let the other one fluctuate around, the answer doesn't change. Therefore, they each separately have to come to some constant. Otherwise, that could not be true. So the sum is a constant and that lets us write these two equations. Now I've said, okay, the energy for the x part is e sub x, the energy for the y part is e sub y. And now because I don't have any other variables except x and one and y and the other, I can also switch from the funny looking d for partial derivative to the regular d. And I end up with the same equation that I already had, which was the equation for a one dimensional particle in a box. One of them uses the variable x and is from 0 to l sub x. The other uses the variable y, big deal, it's from 0 to l sub y. And we know what the solutions are because we worked those out. Big x of x is the square root of 2 over lx times the sine of nx pi x over lx. n is an integer, 1, 2, 3, etc. And big y of y is the same thing, just instead of x we use y. And our solution then is the product of those. So rather than one sine function this way in one dimension, we've got one this way and then we've got one this way as well. And that means that at the corners where they're both small, the particle really avoids the corners. The opposite of a mouse really stays away from the corners, stays in the center of the box in both dimensions in the ground state. So our final solution for the whole thing when we put in the energy and so forth, I've written out here in gory detail. The energy e depends on two quantum numbers, it depends on nx, which is the amount of excitation in the x direction. You can think of that as something to do with how the particle is moving that way. And then n sub y, which is the amount of energy in the y direction, you can think of that as how the particle is moving in the y direction, which is independent of x. And instead of just one, it's just h squared over 8m times quantity nx squared over lx squared plus ny squared over ly squared. And the wave function is the product. And we've normalized it so that the integral over the whole box in both dimensions is equal to 1. We can clearly see the effect of confinement by looking at real quantum systems. It wouldn't surprise you that if I take a 3D box and then I make the assumption that the wave function is big x of x times big y of y times big z of z and I go through the same machinations, I find it works. And I find the energy has another term that has nz squared over lz squared. And therefore, if I make particles of certain sizes that are otherwise the same, well, it could have an electron in the particle that's basically rattling around a so-called quantum dot, then if I look at the light emitted from this electron moving between levels like some level n equals 2 to 1 and emitting light, which can happen, I can get an idea about the size of the box by looking at the light emitted. Now, how do I get the electron to go up to some high level? Where I put in a photon with enough energy. Remember, with the photoelectric effect, we could kick electrons up by using light of a certain energy. We showed potassium there. And if we put in light with a certain number of electron volts of energy per photon, we could kick an electron up. And so we can put in invisible light, UV light, which we can't see. We can excite electron up. It rattles around. It comes down. It emits a photon. And the photon has a certain frequency and wavelength. And if it's in the visible region, we can just see it. And this has a practical application. So here I've shown this is a beautiful illustration, which I've adapted, again, from Wikipedia, that shows when you have little boxes of different sizes that you get different wavelengths of light. And you can actually choose what you want by choosing the size of the box. Well, that's going to be extremely useful if you've got an application for a display. And you want the color to be red. Or you want this or that. And so this has immediate applications in all kinds of fields. This ability to just change the size of the box, not change the material necessarily, but just change the size by how long you let it go or how you heat it or various other tricks of colloidal synthesis. And just change the appearance of the final thing. Okay. Let's do a practice problem now on degeneracy, because degeneracy comes in in this two-dimensional box. Let's go ahead and consider a 2D quantum box. And let's ask the following question. Under what circumstances would there be more than one distinct wave function with the same energy? That's what degeneracy means. It means that there's more than one wave function that has the same energy as another one, but is a different function. Usually what happens is if you've got degeneracy, it means that you've got some kind of symmetry. If you've got a symmetry in the problem, then oftentimes you end up with degeneracy. And likewise, degeneracy can be a clue that you have symmetry. There's also something called accidental degeneracy, which just happens that two things just happen to be close. But that's not likely. That's not likely to happen. Usually it's symmetry related. So let's have a look what would be a symmetry. Well, if we look at the quantized energy, we see it has Nx squared over Lx squared plus Ny squared over Ly squared. We can easily see, for example, the easiest case is suppose we choose Lx and Ly to be the same. So instead of a rectangle, so we have a square. Then if Nx is 1 and Ny is 2, that's the same energy as Nx2 and Ny1. And they're related by symmetry. Now this is maybe not the only condition. I'll let you puzzle about how you might discover all the conditions and it's kind of an interesting problem. But let's just have a look at this symmetry. What I've done here is plot what it looks like for Lx equals Ly. And to make it more interesting, I picked Nx equals 2 and Ny equals 3. And then we'll compare that with Ny equals 2 and Nx equals 3. If we plot them as contour plots where light color is the wave function pointing up and dark color is the wave function pointing down, you can see that when Ny is 3, we've got 3 lobes up, down, up in the Y direction. And when Nx is equal to 2, we've got 2 lobes, which is just down, up on the other way. And they're multiplying each other. So you get this kind of egg carton pattern of light and dark areas indicating where the wave function is positive and negative. And if we look at the other condition where we swap the quantum numbers, all we do is we change whether there's 3 going this way or 3 going up. And it turns out that that's the same exactly as just taking the whole box and just rotating it, which is a symmetry operation. And so we can see that the fact that they have the same exact energy is a fact that you can take one wave function and you can just grab the whole thing and rotate it in space and just change what you're calling X and Y. It's the same shape. It's going to have the same energy, of course. And this is a very important point is that symmetrical systems always have degeneracy. And whenever you've got energy levels that are close together, that means that perturbation can have a big effect. Because now anything that changes the length of one dimension or does something can make one of them slightly lower than the other one. And oftentimes that happens. So in other cases, there may be individual tiles rather than the whole thing rotating, for example. I could have a situation where subsections of it rotate like gears and make a new pattern. And that would still have the same energy or could. So it needn't be such a simple thing as just overall thing, but it could be some more complex relationship between the things. And you can explore the situation, for example, in which one dimension is twice the other one. And you could have different situations. And in 3D, of course, because we have this extra term, NZ squared over LZ squared, now the possibility for degeneracy is higher because now we have a third dimension and if any of them are equal or there are multiples or various other conditions, we could have energy levels that are close to the same energy. And so there are more chances for degeneracy. So in closing, if there's any perturbation to the quantum system, it will usually lift the degeneracy. And what happens is if we have two energy levels that have the same energy, but we only have enough particles to fill them up halfway, what may happen is that something may change so that one energy level is lower than the other. And then both the particles, let's say two electrons, go into the lower state and that spontaneous symmetry breaking can cause some complexes to get distorted so they change from their ideal shape. And that's a very interesting area of study, for example, in inorganic chemistry of some complexes. And the reason why that happens is just because there's a difference in the number of particles versus the number of energy levels that are available. And so, of course, it may try to adjust, move slightly, change the length of the box, so to speak, so that one drops down and then that one's occupied. And once that happens, that's the lower energy state and it just stays like that. I'll close there. And next time what we're going to do is do more realistic two-dimensional problems. We're going to start with a pseudo one-dimensional problem, which will be kind of interesting, the particle on a ring. We mentioned that when we were talking about the de Broglie wavelength, but I want to go back to it. And then a two-dimensional problem, which is a subset of a three-dimensional problem, which is the particle on a sphere. And these will be our stepping stones to get up to understanding how atomic orbitals are formed in atoms like hydrogen. So we'll do that next time around.
 Welcome back to Chemistry 131A. Today we're going to talk  about more realistic vibrational potentials,  the Morse potential, the 612 potential, and we're also going  to extend our vision into at least two spatial dimensions.  Rather than talking about a particle  in a one-dimensional box, we'll extend it  to two spatial dimensions and see what complexities  that involves.  It turns out for a particle in a box that won't be serious,  but depending on the form of the potential,  having more than one spatial dimension can make the equations  much more difficult to solve.  When we last left, I made the remark that we didn't have  to do a certain integral and the reason why we didn't have  to do it is because the integrand was odd or anti-symmetric  and the interval was symmetric.  And I'd like to just take a second to show you how  that works as a practice problem.  So suppose we consider an anti-symmetric function of X.  That is that the function of X is equal to minus the function  of minus X. Then we have to show that the integral  of such a function is zero between symmetric limits.  Just the statement of the problem gives us a clue how we're  going to have to proceed.  We're going to have to make use of the symmetric limits  because it certainly isn't true if we integrate from zero  to something.  That has a finite value.  We're going to also have to use the fact  that when we change the sign of the function, the argument  that the function changes sign.  Let's see how we can do this.  The trick here, and it's a very good trick to remember,  is that we don't actually try to show  that the integral is zero.  That turns out to be pretty difficult.  What we instead do is we show that the integral is equal  to minus itself.  And then the only number that's equal to minus itself is zero.  Oftentimes very tricky things in math rely on not showing  that X is two, but showing that X is greater than or equal  to two, and that X is less than or equal to two,  and then X has to be two.  And this is a very good trick to remember if you get stuck  on some mathematical problem, is to try to sort of paint it  into a corner by inequalities or relationships rather  than just trying to get it spot on.  Here's what we can do then.  The dummy variable that we use in integration,  the value of the integral doesn't depend on that.  That's just a notational consideration.  Therefore, if I call the integral I, which is the integral  from X is minus A to A of the function F of X. That's also  equal to the integral from U equals minus A to A, F of U,  DU, same thing.  Now I just make the substitution that U is equal to minus X.  So I put in minus X equals minus A for the lower limit,  that's U equals minus A. Minus X equals plus A  for the upper limit.  F of U is F of minus X, D minus X. And then I bring  out the minus sign from the D minus X, and I have minus F  of X from minus X equals A to minus X equals plus A. And then I  use the fact that the function is anti-symmetric, and I change  minus F of X to F of X. And then I just say, look,  if minus X is equal to A, that means, excuse me,  if minus X is equal to minus A, then that means X is equal to A.  And so I'm really integrating from X equals A  to X equals minus A. But I know that's equal to minus  if I swap the limits.  And that's equal to minus I. And so by that chain there,  I've shown that the integral I, whatever it is, is equal to minus  I, and therefore the integral is zero.  Okay, let's move on and talk about a more realistic potential.  The harmonic oscillator gave us a differential equation  that we could solve.  It was harder than a particle in a box, but we could solve it.  We could with a little mathematical sophistication,  write down all the wave functions for all the excited states,  and there are an infinite number of them.  But it's not very realistic for a real chemical bond,  because atoms, when they get too far apart,  don't interact very well.  We know that from gases and other systems.  We can only have a chemical bond with a force constant restoring the atoms  to an equilibrium position when they're rather close.  Otherwise, we would suddenly be bonding to all sorts of things,  which does not happen.  But it, so the harmonic oscillator is a very bad approximation,  unless we're near the bottom of the potential.  If we're near the bottom of the potential, it's quite good.  As we get out further, it becomes very bad.  And if we're interested in breaking bonds or simulating breaking  of bonds somehow, it's extremely bad.  Therefore, we need something that's a little bit better.  And therefore, the question is, is there an alternative potential,  which one is more realistic, and two still allows us to get the exact solutions  for the time-independent Schrodinger equation,  to get the energy levels and so forth and so on.  And the answer to both questions is yes, there is such a potential.  And it's called the Morse potential after Philip Morse,  who first suggested it in 1929.  It did not take long after wave mechanics was invented for people  to start working like crazy on this new field and make all sorts of refinements  to the simple problems and really go in depth and advance the field.  And this can model vibrations much better than the harmonic oscillator,  especially for excited states in a molecule.  The number of bound states in the Morse potential is finite.  And that means that there is a limit after which the chemical bond breaks  and the atoms fly apart.  That's already a substantial improvement over the harmonic oscillator.  And furthermore, even the bottom part of the well,  where we haven't gone up very far, is represented much better  with the Morse oscillator than the harmonic oscillator.  In fact, the energy levels always get closer together with the Morse oscillator  as we go up as we'll see.  Okay, what's the functional form?  Well, it's something that's not very obvious at all.  It's the potential as a function of R should be some constant DE  with units of energy times the quantity 1 minus E to the minus A times R minus RE  quantity squared.  That's not a very obvious potential to choose.  And it's not at all obvious at first blush that one could solve that potential  in the Schrodinger equation and get solutions.  It looks to be fairly complex.  Here, D is called the well depth and A is called the width parameter.  And so we have two things to control.  We can control how deep the well is and we can control how wide it is.  The depth has to do with the total energy before we dissociate.  And the width has to do with the spring force constant and the depth.  They're related.  When A is small, the parameter in the exponential,  what that means is that the well is narrow  and there's a larger force constant.  Here's a comparison that I've adapted from Wikipedia of the harmonic oscillator  which is in green and the Morse oscillator which is in blue.  And there are a couple of things to notice.  The Morse oscillator is this beautiful function which is asymmetric  in pretty much a very realistic way for a real chemical bond.  The levels get closer and closer together as we go up.  And there's a definite prediction about what the dissociation energy is.  The dissociation energy is not the well depth but it's a little bit different  than the well depth because of the zero point energy of the oscillator.  And we call the dissociation energy D naught and the well depth D e.  And there's an equilibrium distance which we've assumed to be identical  for the two oscillators in this case, r sub e, the equilibrium distance.  Now, the energy levels for the Morse oscillator which we won't solve the equation  and go through all that in this course.  But just to quote the result, the energy levels  of the Morse oscillator get linearly closer as we go up.  The spacing between zero and one is one unit.  Then the spacing between one and two might be point nine and then point eight and point seven.  And that equation fits this functional form.  The energy of quantum level V which is an integer number is h bar omega times V plus one half.  That's exactly the same thing as what we got for the harmonic oscillator.  Except that kept going and going.  And then there's a new term which shows you how clever this potential is  because it gives you what you had before and then it gives you just the right correction  to make it more realistic.  Minus h bar omega x e times quantity V plus one half squared.  And therefore there's this quadratic term  that's lowering the ladder of states as we go up.  Clearly there's some value of V where the quadratic term gets so big  that the thing would turn around and head back down.  And at that value of V where it starts doing that, that's the highest one.  After that the bond is broken and we don't consider that energy equation any longer.  And the dimensionless parameter x sub e is just called the anharmonicity constant.  It's a measure of the distortion away from a harmonic oscillator  and to a different kind of motion that hangs out here for a long time  and is definitely not a harmonic oscillator in terms of going back and forth evenly.  And as I mentioned at some point V max, the energy of V max plus one according  to the formula anyway would be less than the energy of V max.  The states stop going up.  At that point that's the highest level that can be supported in the potential  and that allows us to count how many states could be in there and get an idea of that.  The wave functions for the Morse oscillator are quite a bit more tedious  to calculate than the wave functions for the harmonic oscillator.  And the harmonic oscillator was already much harder than the particle in a box.  And therefore we're not going to write  down the explicit form of the wave functions.  What we usually want in chemistry is the energies to get an idea of what kinds  of processes can happen if we put in light of a certain energy.  Will it be absorbed?  Could I get this thing to break a bond and so forth and so on?  Occasionally we're really interested in detail and we'd like to have some idea  of the electron density but usually it's very, very hard to calculate the wave function  and so there are other tricks like density functional theory  to calculate the electron density without actually trying  to calculate the full blown wave function.  For real bonds then just building on this Morse theory one could suppose  that you could have another anharmonicity and in this case it's Y e.  For all real bonds the energy levels get closer together when they're near the bottom  and that's why the first term has a minus sign  because chemists like positive numbers.  So we like X e to be a positive number so we put a minus sign there  so that X e is a positive number.  But Y e the next term in v plus 1 half cubed in this power series then  in the energy as a function of v is could be plus or minus and it could depend a little bit  on how many levels you're trying to incorporate into this equation when you do the fitting.  And so for that one we have a plus and we allow it to be Y e to be either plus  or minus as the case may be.  In chem 131b what you'll find out in IR spectroscopy when you do that as an actual method  of analysis is that you'll find out that IR or infrared spectroscopy is very important method  to glean all kinds of information about the strengths of bonds and rotational  and vibrational states of simple molecules.  And in fact it's of course the vibrations of molecules like CO2 that are causing all the  problems with climate change and radiative forcing as it's called.  As the earth emits infrared radiation which you can think of as heat comes  out at certain wavelengths some of it may actually instead of going out into outer space  and cooling off like taking cookies out of an oven or putting a pie on a window sill some  of it may actually hit CO2, an excited vibration in CO2.  And CO2 may vibrate for a while and then it may decide to emit a photon.  And when it emits it it has lost memory of all direction.  It may have spun around and may have done all kinds of things in between  and so it emits it in all directions.  In particular it emits it back toward the ground.  And what that means is that the rate of escape of radiation is altered  and when you change the rate of escape of heat you heat up.  That's all you do when you put on a parka.  You change the rate of escape of heat from your body and you're much warmer  than if you don't have it on.  And so if you, we study the vibrational modes of these molecules  in the lab we can predict which ones for example might be very bad to release  in large amounts into the atmosphere.  And unfortunately one of those is exactly the molecule that they put  into car air conditioners hydrofluorocarbons  which have very great greenhouse gas radiative forcing  and could be very bad to release.  The force constants that we obtain when we look at these bonds really agree  with exactly what we would think would happen when we have a single bond  or a double bond or a weak bond or a strong bond.  If the bond is strong then the force constant is pretty big  and if the bond is weaker it's small.  And here are some examples.  The force constant here is in newtons per meter.  For hydrogen fluoride it's 970, reasonably strong.  For hydrogen iodide it's only 320.  Both of those are single bonds but the overlap between the orbitals on fluorine  and hydrogen is much better than iodine which is a great big thing  with diffuse elephant ear orbitals makes a very weak bond.  And carbon monoxide has a force constant of around 1860.  When we draw the Lewis structure for carbon monoxide we draw three bonds  between the carbon and the oxygen.  So that makes sense that that would be an extremely strong one.  And other ones with double bonds are in between.  And so this at least makes very good qualitative sense  when we compare what our notional idea of a strength of a bond is  with this force constant of this spring  and this quantum mechanical model that we're solving.  Now there's another potential that is used which you should know about  and which is of historical importance mainly because of its simplicity  and the way computers worked.  And this potential is called the Leonard Jones or 612 potential.  It's commonly used for example to simulate liquids.  In the early days of computer simulation we couldn't do complex systems  like membrane proteins and all these other things that we can do now.  In fact what we started out doing was the simplest things  like noble gas liquids try to simulate a liquid argon,  something not doing too much and see if you can figure out when based  on the potential when it will solidify, when it will boil,  when it will do this and that.  And that can already be an extremely challenging problem  with the 18 electrons you have there.  But you can make a very approximate model of this  because when two molecules, when two atoms are far apart there is a very  decent model for how they will attract.  If there's any kind of charge fluctuation of the charge cloud of one atom  and it's very unlikely that if you have 18 electrons for example  that they're all going to be totally symmetrically distributed every which way  at every instant.  So suppose just for a tiny bit of time the electrons have 10 on one side  and 8 on the other.  Well then this side looks positive and if anything's nearby  that then these electrons may rush over because they may see  that there's a positive charge here.  And so they tend to attract and that means they always tend to attract.  And when you work out the way it works with two dipoles, one over r cubed,  you get an attraction that goes like one over r to the sixth.  So for a long way away it's flat and then as you come in it starts going down,  down, down like one over r to the sixth.  But we know when they get too close that they repel.  And they repel because when the electron clouds overlap,  Polly tells us that we can't have electrons in the same orbital  with the same quantum number more than two and then even then they have  to have opposite spin.  So they repel, they bounce off and there has to be a strong repulsive force.  There's not such a good theory however  about what that repulsive force should be.  And but we know it's very strong.  The Morse potential while it gives this nice theory has one thing  that is a little bit troubling and that is if you go back and I'll let you do that,  go back to the formula and you put in r equals to zero so that the two particles,  the two nuclei we think of them as positively charged are right on top  of each other or extremely close.  We'd expect tremendous repulsion under those circumstances.  But the Morse potential gives us some finite value.  It doesn't go up to infinity.  Infinity may be too big but a finite value  like the Morse potential may be much too small especially when r is tiny.  And usually we aren't modeling such small values of r anyway  but if we have things hitting hard, we want to make sure that they hit hard  and that they repel appropriately if we're trying to model that kind of behavior.  The 612 potential instead of having a finite value like the Morse potential,  we just add a term plus 1 over r to the 12 and that goes up much,  much faster than minus 1 over r to the 6 goes down.  So we get something that goes down to a minimum and then goes up like crazy  when you get too short and then we can adjust by some terms.  We can adjust how fast, deep it goes down and what the equilibrium is  for the best situation, the minimum energy.  So the numbers in 612 refer to exponents, the 1 over r to the 6, 1 over r to the 12.  And that it's a very nice functional form.  Here it is, v of r's just could be a number like a over r to the 12 minus b over r to the 6.  That could be a simple way to put it in.  And a and b are your two parameters that allow you to control how deep it goes down  and what the equilibrium is for the best distance between.  In other words, how big do I have basket balls?  Do I have tennis balls?  You know, what's the right scale?  Usually we tidy it up though, mathematically like this.  We say v of r is epsilon, some number with units of energy, times rm over r to the 12 minus 2rm  over r to the 6.  And rm is the minimum and epsilon is the depth of the energy minimum at r equals rm.  And as I said, rm is the minimum value where we're at the equilibrium  where it's most like a harmonic oscillator, for example.  Now why pick r to the 12?  Why not pick something else?  And the legacy of that is kind of interesting.  In the old days, digital computers were so slow that it was very time consuming to compute this potential.  And if I want to simulate a lot of particles moving around and I got to figure out who's attracting whom  and how much and who's repelling, then I have to basically compute these v of r over and over  and over and over.  And computing something like an exponential function or a cosine or sine  in the very early days, that was just a lot of machine code instructions and took a very long time to do.  And so it slowed everything down.  The beauty of this 612 potential is the fluctuations go like r to the minus 6.  And then once I've got r to the minus 6, so I know what that number is,  I just square it, I just multiply it by itself and I get the repulsive part r to the plus 12.  And that's just one multiplication.  And that's why it became so popular because it let you simulate things for a lot longer  than anybody else could with some other more realistic potential  that might involve an exponential function or something else.  And then it let you do some interesting numerical experiments on the computer.  It's just amazing how much more powerful computers are today than they were when people were doing landing  on the moon and things like that.  But the computer I used as a graduate student had 8k of memory.  And I remember quite clearly when they upgraded it to 32k, I thought,  what am I going to do with all that massive amount of RAM?  How could I possibly use it all?  And when we wrote programs, we used three variables, X, Y and Z,  because you didn't want to run out of memory.  And because you didn't want to run out of disk, you basically never put any comments on  because that would take more space on the disk.  And at the end of the day, you had to be extremely careful to note yourself what you were doing  because nothing had any name that made any sense.  It was all X, Y and Z, I, J and K and so on.  And nothing had any comments.  And therefore, if the program went south and you made an error,  it could be extremely difficult to debug.  Okay. As I mentioned, r to the plus 12 apparently, r to the minus 12, excuse me,  plus some number over r to the minus 12 has no theoretical justification.  But it was just easy to compute.  There are other forms and nowadays,  Leonard Jones is still used quite commonly because it's still fast.  But there are other forms and computer power is so much higher  that it really depends on what you're doing, how accurately you want  to compute these potentials, what kind of form you use,  what you're trying to match in terms of physical properties.  Okay. Let's move now to simple quantum systems in 2D and 3D and talk a little bit  about degeneracy which is quite important in real systems.  Ultimately, to do anything with real atoms or molecules, we have to be able to solve  in at least three spatial dimensions, X, Y and Z. But as a stepping stone,  let's try solving in two first and get the math down for that.  And then we can extend it to three.  How easy or hard it is to solve these equations really comes down to the potential.  If the potential has a simple form, then it's easy.  If it has terms like X times Y in it, it can be very, very difficult to do.  Usually the best case is no potential.  That's a particle in a two-dimensional box.  That's what we're going to do first.  Nothing can go wrong there.  And after that, some kind of potential that's additive separately in X, Y and Z  but doesn't have any cross terms.  Doesn't have anything X times Z or anything like that.  Okay. Suppose we have a particle and it's confined to a 2D region.  So here's what we're going to say.  The potential energy is zero if X is between zero and LX.  And Y is between zero and LY.  And otherwise, if you're outside of the box, the potential is infinite.  We know by our argument of a one-dimensional box that means the wave function has to vanish  at the edges both with X and Y.  Otherwise, the total energy is infinite.  The particle cannot have infinite energy.  Now we have a two-dimensional equation.  Luckily, and psi of X, Y, we don't know what the form of psi of X, Y is.  But we have just the kinetic energy minus H bar squared over 2M,  second derivative with respect to X. We've got to use partial derivatives  because we have to keep clear what we're keeping constant  and what we're letting be a variable.  Plus the same kind of term, second derivative with respect to Y, psi of X, Y is E psi of X, Y. And  as I mentioned, we have to use partial derivatives because we have  to be absolutely clear what we're keeping constant and not  when we have both X and Y in the wave function.  The first thing you always try whenever you have an equation like this,  even if you don't know anything about it, is you try taking a product.  You try saying, I think that psi of X, Y is something, function in X, F of X times G of Y.  Now that doesn't encapsulate all possibilities by any means.  But what it does do is it means that the, something like this, the energy in X and the energy  in Y are separate from each other.  And they're just going to add up separately.  And so that's the simplest assumption, is that they have nothing to do with each other.  And fortunately for the simplest kinds of atoms like the hydrogen atom, that's really good.  And we can figure out all the beautiful wave functions and make these beautiful electronic  orbitals which we'll get into later on in this course.  If this assumption that psi is a product, F of X times G of Y, works,  then it solves the problem immediately and you're done.  And it doesn't take too long to see if it's going to work.  If it doesn't work, that means that you've got a bit of a nasty equation,  then you consult an expert.  That's why we have experts.  They're in their offices there with all their papers piled up and they can solve a lot  of mathematical problems or you go to a reference book on differential equations  or you take a more advanced course in differential equations.  And all those things are good things to be able to do so that when you hit a problem  like that, you can solve it.  If you can't solve difficult problems quickly, the thing is you lose your train of thought.  And you may hit a problem and you may say, gee, if I could solve this,  I could figure out this other thing and so forth.  But if you keep hitting it and you can't solve it and you don't know where it's going  and you can't make progress, it's like reading a novel letter by letter  and it takes you a long time to recognize you've got a word.  The problem is if you go so slowly like that, you can't even get the train of the story  because by the time you get halfway through the chapter, you've forgotten what the plot was.  And it's a little bit like that in these branches of science.  If you can't see things quickly enough by learning how to do it, it takes too long  and then you get lost.  And then the problem is that you don't even glean the chemical, the physical chemistry knowledge  out of it because you get lost in this labyrinthine maze of mathematics.  Okay. Now, if you can't solve it and there's no expert, what you can do is you can try  to throw away certain things from the Schrodinger equation that are crossing you up  and then you can treat them as a perturbation because at least we know how to do that.  So we can take any nasty term that prevents us from factorizing it and put that aside  and then we can calculate a correction based on perturbation theory.  And that can also be a good way to go if you want to figure out something quickly,  whether something is going to have a certain effect or not.  How do we know that it's worked?  Well, when the 2D equation separates into two 1D equations.  1D equation is some equation that has derivatives with respect to x and blah, blah,  blah, x, x, x, x, nothing else.  And then another equation that has everything with respect to y and nothing else.  But there's no psi of x, y left.  If you can separate it like that, it's worked.  If you cannot, then you're going to have to either treat it as a perturbation  or get an expert to help you.  It's commonplace to use capital X of x rather than F of x and capital Y of y rather than G of y.  But it's the same thing.  Capital Y of y is a function that depends only on the argument y.  It has no x's in it and x is likewise.  They can have constants, of course, but they can't have x.  Now our equation looks like this.  The energy operator, kinetic energy operator.  And now rather than operating on psi of x, y, it operates on x of x times y of y.  And it gives back e times x of x times y of y.  So we're just putting this in as a guess.  And in the partial derivative with respect to x, we can take y of y,  which doesn't have any x's, as a constant.  And we can move it through the partial derivative operator and put it out in front as a constant.  And vice versa.  On the other derivative, we can move the part that we don't have to worry about out in front.  And then after we've done these derivatives, we now have y of y times the second derivative  of x of x plus x of x times the second derivative of y of y, all times minus h bar squared over 2.  And that's equal to the energy.  Now there's one more trick.  And the one more trick is we divide both sides by x of x times y of y.  And if we do that, then the y of y in the first term goes away.  And we end up with 1 over big x of x times the second derivative of x with respect to x.  And the other term has 1 over y times the second derivative of y with respect to y.  And that's now equal to e, some number.  Now the first term has nothing to do with y.  So if we look at the first term, it has no y's in it.  The second term has a bunch of y's in it.  And so if I change little y in the second term, it's going to be different values.  And that means that it can't add up to the energy.  So if I pick a particular value of x and let y move around, that can't be a constant unless x itself,  that first term is a constant, separately, some constant.  And the second term, which has only y in it, is also some constant.  And that's the argument that allows us to separate the equation into two.  They have to separate because when I hold one of them fixed  and let the other one fluctuate around, the answer doesn't change.  Therefore, they each separately have to come to some constant.  Otherwise, that could not be true.  So the sum is a constant and that lets us write these two equations.  Now I've said, okay, the energy for the x part is e sub x,  the energy for the y part is e sub y.  And now because I don't have any other variables except x and one and y and the other,  I can also switch from the funny looking d for partial derivative to the regular d.  And I end up with the same equation that I already had,  which was the equation for a one dimensional particle in a box.  One of them uses the variable x and is from 0 to l sub x.  The other uses the variable y, big deal, it's from 0 to l sub y.  And we know what the solutions are because we worked those out.  Big x of x is the square root of 2 over lx times the sine of nx pi x over lx.  n is an integer, 1, 2, 3, etc.  And big y of y is the same thing, just instead of x we use y.  And our solution then is the product of those.  So rather than one sine function this way in one dimension, we've got one this way  and then we've got one this way as well.  And that means that at the corners where they're both small,  the particle really avoids the corners.  The opposite of a mouse really stays away from the corners,  stays in the center of the box in both dimensions in the ground state.  So our final solution for the whole thing when we put in the energy and so forth,  I've written out here in gory detail.  The energy e depends on two quantum numbers, it depends on nx,  which is the amount of excitation in the x direction.  You can think of that as something to do with how the particle is moving that way.  And then n sub y, which is the amount of energy in the y direction, you can think  of that as how the particle is moving in the y direction, which is independent of x.  And instead of just one, it's just h squared over 8m times quantity nx squared  over lx squared plus ny squared over ly squared.  And the wave function is the product.  And we've normalized it so that the integral over the whole box  in both dimensions is equal to 1.  We can clearly see the effect of confinement by looking at real quantum systems.  It wouldn't surprise you that if I take a 3D box  and then I make the assumption that the wave function is big x of x times big y  of y times big z of z and I go through the same machinations, I find it works.  And I find the energy has another term that has nz squared over lz squared.  And therefore, if I make particles of certain sizes that are otherwise the same,  well, it could have an electron in the particle that's basically rattling  around a so-called quantum dot, then if I look at the light emitted  from this electron moving between levels like some level n equals 2 to 1 and emitting light,  which can happen, I can get an idea about the size of the box by looking at the light emitted.  Now, how do I get the electron to go up to some high level?  Where I put in a photon with enough energy.  Remember, with the photoelectric effect, we could kick electrons up by using light  of a certain energy.  We showed potassium there.  And if we put in light with a certain number of electron volts of energy per photon,  we could kick an electron up.  And so we can put in invisible light, UV light, which we can't see.  We can excite electron up.  It rattles around.  It comes down.  It emits a photon.  And the photon has a certain frequency and wavelength.  And if it's in the visible region, we can just see it.  And this has a practical application.  So here I've shown this is a beautiful illustration, which I've adapted, again,  from Wikipedia, that shows when you have little boxes of different sizes  that you get different wavelengths of light.  And you can actually choose what you want by choosing the size of the box.  Well, that's going to be extremely useful if you've got an application for a display.  And you want the color to be red.  Or you want this or that.  And so this has immediate applications in all kinds of fields.  This ability to just change the size of the box,  not change the material necessarily, but just change the size by how long you let it go  or how you heat it or various other tricks of colloidal synthesis.  And just change the appearance of the final thing.  Okay. Let's do a practice problem now on degeneracy,  because degeneracy comes in in this two-dimensional box.  Let's go ahead and consider a 2D quantum box.  And let's ask the following question.  Under what circumstances would there be more  than one distinct wave function with the same energy?  That's what degeneracy means.  It means that there's more than one wave function that has the same energy as another one,  but is a different function.  Usually what happens is if you've got degeneracy,  it means that you've got some kind of symmetry.  If you've got a symmetry in the problem, then oftentimes you end up with degeneracy.  And likewise, degeneracy can be a clue that you have symmetry.  There's also something called accidental degeneracy, which just happens  that two things just happen to be close.  But that's not likely.  That's not likely to happen.  Usually it's symmetry related.  So let's have a look what would be a symmetry.  Well, if we look at the quantized energy, we see it has Nx squared over Lx squared plus Ny squared  over Ly squared.  We can easily see, for example, the easiest case is suppose we choose Lx and Ly to be the same.  So instead of a rectangle, so we have a square.  Then if Nx is 1 and Ny is 2, that's the same energy as Nx2 and Ny1.  And they're related by symmetry.  Now this is maybe not the only condition.  I'll let you puzzle about how you might discover all the conditions  and it's kind of an interesting problem.  But let's just have a look at this symmetry.  What I've done here is plot what it looks like for Lx equals Ly.  And to make it more interesting, I picked Nx equals 2 and Ny equals 3.  And then we'll compare that with Ny equals 2 and Nx equals 3.  If we plot them as contour plots where light color is the wave function pointing up  and dark color is the wave function pointing down,  you can see that when Ny is 3, we've got 3 lobes up, down, up in the Y direction.  And when Nx is equal to 2, we've got 2 lobes, which is just down, up on the other way.  And they're multiplying each other.  So you get this kind of egg carton pattern of light and dark areas indicating  where the wave function is positive and negative.  And if we look at the other condition where we swap the quantum numbers,  all we do is we change whether there's 3 going this way or 3 going up.  And it turns out that that's the same exactly as just taking the whole box  and just rotating it, which is a symmetry operation.  And so we can see that the fact that they have the same exact energy is a fact  that you can take one wave function and you can just grab the whole thing and rotate it  in space and just change what you're calling X and Y. It's the same shape.  It's going to have the same energy, of course.  And this is a very important point is  that symmetrical systems always have degeneracy.  And whenever you've got energy levels that are close together,  that means that perturbation can have a big effect.  Because now anything that changes the length of one dimension  or does something can make one of them slightly lower than the other one.  And oftentimes that happens.  So in other cases, there may be individual tiles rather  than the whole thing rotating, for example.  I could have a situation where subsections of it rotate like gears  and make a new pattern.  And that would still have the same energy or could.  So it needn't be such a simple thing as just overall thing,  but it could be some more complex relationship between the things.  And you can explore the situation, for example,  in which one dimension is twice the other one.  And you could have different situations.  And in 3D, of course, because we have this extra term,  NZ squared over LZ squared, now the possibility for degeneracy is higher  because now we have a third dimension and if any of them are equal  or there are multiples or various other conditions,  we could have energy levels that are close to the same energy.  And so there are more chances for degeneracy.  So in closing, if there's any perturbation to the quantum system,  it will usually lift the degeneracy.  And what happens is if we have two energy levels  that have the same energy, but we only have enough particles  to fill them up halfway, what may happen is that something may change  so that one energy level is lower than the other.  And then both the particles, let's say two electrons,  go into the lower state and that spontaneous symmetry breaking can cause some complexes  to get distorted so they change from their ideal shape.  And that's a very interesting area of study, for example,  in inorganic chemistry of some complexes.  And the reason why that happens is just because there's a difference  in the number of particles versus the number of energy levels that are available.  And so, of course, it may try to adjust, move slightly,  change the length of the box, so to speak, so that one drops down  and then that one's occupied.  And once that happens, that's the lower energy state  and it just stays like that.  I'll close there.  And next time what we're going to do is do more realistic two-dimensional problems.  We're going to start with a pseudo one-dimensional problem,  which will be kind of interesting, the particle on a ring.  We mentioned that when we were talking about the de Broglie wavelength,  but I want to go back to it.  And then a two-dimensional problem, which is a subset  of a three-dimensional problem, which is the particle on a sphere.  And these will be our stepping stones to get up to understanding how atomic orbitals are formed  in atoms like hydrogen.  So we'll do that next time around.
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