Lecture 01. General Course Information and Introduction to Quantum Mechanics
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. 
 Hi and welcome to Chem 131A, Physical Chemistry. I'm Dr. Shaka and I'll be leading you through these series of lectures on physical chemistry starting with an Adams-Up approach. Quantum mechanics is our first topic, which is kind of a rude introduction to the subject. But here we go. I'll give you some general course information and then an introduction to quantum mechanics as seen through the eyes of a chemist rather than a physicist. We have slightly different viewpoints on some things. So here's a preliminary problem, practice problem one. Here is a jumbled word and the question is what word can you make? And even though there are five letters, you might find it pretty hard. But the answer is you can make tool. But you have to have that word in your vocabulary if you're going to make it. And if you're in the fashion business or you're working with fabrics, you may happen to know that is a type of fabric. But if not, you could play around with that word for a long time. Mathematical equations that we're going to be dealing with are quite similar. We have to rearrange symbols in ways that are legal and we have to somehow see where to go. That means two things. You have to have the basic vocabulary to understand what it means. And you have to have enough background material that you know what a legal move is. And I will assume that you've read in the book the background material, the fundamentals. And if you haven't, then that's your first task right now. So here's some general information. The lecture attendance is optional except on exam days. We'll have one problem do each Friday. And we only have one problem because they're quite hard actually. And we have a website and on it we'll have what's new. But avoid sending me email. Just ask in person after class. And warning, do not start the problem on Thursday evening. Please realize that it's one problem and it's multiple parts and it's really quite difficult to actually complete it. And so have a go, clarify your understanding, and then try again. The TAs and I will go over some of the problems at the end of the chapter and clarify any ambiguous wording of the problem. Homework counts for 20% of the grade in this class. I'm not a big fan of making exams count a lot. There will be two midterms and a final exam. And reading, I can tell you, is not the best way to learn physical chemistry. Reading in chemistry is like tying your shoes to run a race. You have to tie your shoes but that doesn't count. It's not training. What's training in chemistry is practicing, solving problems, visualizing what things look like, and trying to work quickly and accurately. The textbook we're going to use is Quanta Matter and Change by Atkins, DePaula, and Friedman. And we'll cover the first five chapters. But I'll warn you that textbooks are getting a little bit like Amazon.com. They're trying to be all things to all people. It's not necessary that you memorize lots of facts or small details or who did what. What's important is to just try to understand the ideas and develop an intuition. It can be done even in a field like quantum mechanics. So chapter one, quantum mechanics is the study of the small specifically things that we can't see. Though viewed closely, the world appears to be digital. In other words, there are small packets of everything. And atoms are the smallest unit of a particular element. Of course, even the ancient Greeks realized that it might be true that things had a smallest indivisible amount. Now we know that that is in fact true. But they're very, very small and very light. One neutral carbon third carbon 12 atom has a mass of only about 2 times 10 to the minus 23 grams. An electron is much, much smaller, has a mass of 9 times 10 to the minus 28 grams. And these very small things are very unfamiliar to us because we can't see anything that small. So human intuition is guided by things that are around the same size as us. Things that are much, much bigger than us or much, much smaller than we are are hard to understand. And we have to use careful experiments to try to figure out what's going on. And likewise protons and neutrons occur in integer units. You can have one proton or none, but you can't have half. And you can't have half an electron. This is pretty much similar to currency. There's a minimum amount of currency. It's different for every currency, but there's still a minimum amount. In the U.S. system, it's 1 cent. So you can have 1 cent, but you can't have anything smaller and have it be legitimate currency. Light, Newton actually believed that light also had a currency that there was a minimum amount that light was corpuscular because light seemed to travel in straight lines just like a bullet fired from a gun. But later investigations in which light went through narrow slits or pinholes showed interference phenomena very much like water waves. And so the work of Huygens caused Newton really to abandon his corpuscular theory of light because it didn't seem like that theory could explain this kind of wave phenomena. Waves have positive or negative phase, and so they can subtract as well as add. A particle can either be there, be positive, or zero, but it can't be negative in the usual sense. And so we wouldn't expect small numbers of marbles to somehow give interference phenomena if we fire them through slits. So for example, here's a picture of two slightly different wavelengths of a wave. And if we add them up, you can see that there are positions where in space here, because this is wavelength, there is a very small response, and then there's another place where there's a very large response, and then there's a small response, and so on. And this is very much similar to sound waves, for example, where if you tune a violin, you hear the difference when you aren't in tune. You hear this wah wah wah, this sort of beating, and you're seeing in this picture that kind of beating occurring, and it's a universal phenomenon with waves. White light, it turns out, has a mixture of different wavelengths, and that was sort of first surmised when you passed white light through a prism, and you resolve it into a rainbow. But that doesn't necessarily mean that white light is composed of different colors, because it could be that the colors somehow come from the prism. And so it was only really when you took another prism, and then you took the same rainbow and returned it into white light that people became convinced that white light was really a mixture of colors, and there was nothing coming necessarily from the prism. When you have a convex surface near a flat surface, there's a difference in refractive index, and the condition for these beats to add up or subtract depends on the wavelength, and this is a phenomenon called Newton's rings, a pattern of constructive interference that, for example, you can see if you spill oil, which tends to bead up on water, on a wet surface, and we used to always look at that as a kid because cars in Salt Lake City always had a lot of oil leaking out of them in those days, and it rained, and we would spend a lot of time looking at these beautiful patterns. So we can see, here's an example, we can see how the colors vary systematically with wavelength, and much like there were two beating patterns in that graph I showed you, this will repeat more than once depending on the total thickness of the contrasting media. There's another experiment that unfortunately made it difficult to understand light as waves, and that was black body radiation. So this is a classic thing where you think you understand everything, and you have a very simple calculation, and there are some very, very smart people like Lord Rayleigh, and you do the calculation, and then you compare with the experiment, this is the essence of science. If you think you understand something, you ought to be able to predict it or explain why you can't predict it, but at this time there was Maxwell's equations pretty much, people believed that these wave equations really described light in totality, and that light was an electromagnetic wave, and that theory was wildly successful. So we could explain the unification of electricity and magnetism, why the speed of light has its speed, diffraction, reflection, refraction, how lenses work, and so on and so forth. But the two crucial experiments showed that this description of light was incomplete, and the first one was black body radiation. So what is black body radiation? Well, if I take a black body, by which you could just mean a lump of coal or lamp black, and heat it up in a vacuum so that there's no air that doesn't burst into flame, it'll glow, and this glow gives a characteristic spectrum of colors just like white light has a characteristic spectrum of colors, and it was known that the color depended only on the temperature, hence we speak of red hot and things like that. And Rayleigh, with a correction by genes, calculated this spectrum of wave-like light, and what they found is that it would be much, much more likely that you would get a lot of high-frequency radiation because the chance of getting each frequency was equally likely, and there are a lot more high frequencies, than there are low frequencies. And this was called the ultraviolet catastrophe because it basically predicted that if we opened something like a kitchen oven, then outcome a ton of x-rays and very high-frequency light, and kill us basically, and that's obviously not what happens. So we, there was a big, big soul-searching, and there was a lot of thought about what might be going on, and some theories were put forward that were later proven not to be quite right, but it was Max Planck that found the correct solution. What he found that fit the observations, although he didn't really quite believe it himself, but he could follow through the physics, was that light was quantized. That is, there's a minimum amount of light, and the smallest amount was related to the frequency of the light, and once the frequency was chosen, the energy of the light was E equals H nu. H has subsequently been called Planck's constant in honor of his discoverer. So the frequency could apparently be continuous, anything you want, but once you choose it, then there's a minimum amount, and if you don't have the minimum amount, there's no light, and since the minimum amount depends on the frequency, higher frequencies, if there's not enough energy around, can't have the minimum amount to make a single particle or quantum of light, and therefore those frequencies get cut off. So that gets around this problem with the x-rays killing us if we open the oven. So here's an equation that just says the likelihood that you're going to have a certain amplitude of light, or number of relative amount of light per unit frequency given according to the Rayleigh-Jeans law, and you can see that it's basically a parabola in the frequency squared, so as the frequency goes up, the amount of light that's predicted is predicted to go up faster and faster and faster, and that's not what's observed. Now what Planck did instead of this continuous equation is this. He quantized the photon energy, and he obtained this formula which on first blush looks completely different. It has a cube in the enumerator, and then it has this funny exponential of h nu upon kT in the denominator, and we can compare these two formulas on a graph and see what happens. And in this formula, let me remark that k is Boltzmann's constant, h is Planck's constant, and T is the temperature in Kelvin, and of course in physical chemistry or chemistry of any type, you never quote temperature in anything other than Kelvin, because if you do, you're very likely to be wrong if you plug it into a formula. If you quote the temperature in Kelvin and you say it's a balmy day, it's 298 Kelvin, you may be eccentric, but you're never wrong. But in chemistry, if you use any other units, you're very likely going to be wrong, so you have to be careful. So here's a graph. Planck is in the green and Rayleigh jeans in the sort of pink color, and you can see that for a certain temperature of 300 Kelvin, they agree at very, very low frequency where the quantization is very small, and then as the frequency increases, Planck actually follows the observed distribution almost exactly, and Rayleigh jeans diverges more and more and more from it, and would continue to go up. Now, we can make a connection between the two theories, because we know they agree when the frequency is small and the energy is low, and so we can take this parameter H nu upon kT, and we can assume it's much, much less than 1, and we know from calculus that we can expand E to the X in a power series, 1 plus X, and then if X is small, then X squared is very small, so we can throw it away, and that means we can write E to the H nu upon kT as 1 plus H nu upon kT, and then if we make that substitution in, we find that as long as the frequency is not high compared to the temperature kT, that we get exactly the same formula that Rayleigh and jeans got, but if the temperature, if the frequency is high where there was a problem, then it starts to deviate, and so that's very nice because it shows you that you get the same result as the other guys got where their theory seemed to work, and that's what scientists often look for, of course. Now, the physical meaning of kT is that kT is really a measure of the random thermal energy that's available at a temperature T. When something's hot, things are moving, they're colliding, they're banging around, and there's lots of energy available to excite things and to create photons, but if the temperature is very cold, then there's hardly any energy around, and then you just don't have enough energy to make the minimum amount of a photon, and that's why cold things don't glow, but things that are heated up finally do start to emit a glow like an electric element on a stove. So, at high frequency, the problem is we just don't have enough energy to make even a single photon. We just run out before we get there, and so the distribution has to fall off sharply, and the analogy I can give you is that suppose the smallest amount of currency, the smallest coin, were $1,000, then that means that a lot of people are going to have no money at all because they don't have that much. The reason we don't notice the digital nature of light in day-to-day observation, we don't notice that it looks like sand or something like that, is because of the relative size of these two constants, K and H. Boltzmann's constant is 1 times 10 to the minus 23 joules per Kelvin, and Plank's constant is 6 times 10 to the minus 34, and that means there's a factor of about 10 to the 11 or 100 billion between them, and that means that usually at low frequency there are plenty of photons around, but at high frequency we start to notice that H nu, the quantum of light, has a minimum value. The other experiment that really sealed the deal with respect to the corpuscular or quantized nature of light was the photoelectric effect, and it's not quite such a simple experiment as the black body radiation, but nevertheless, it is a pretty simple experiment, and the experiment is this. You evacuate a chamber and you have a clean metal surface, and you shine light on the surface, and what was observed is that electrons would come off the surface, and this is how you make a cathode ray tube, in fact, but electrons would come off the surface of the metal and would come into the vacuum and they would be ejected, and ideally if light were a wave, then what should happen is if you have a wave coming in, it should sort of excite the electrons more and more and more and more and more and then, boom, finally just like pushing a swing, if you push it enough times, you can get the person moving. So that means that when you turn on the light, there should be a delay before the electrons are ejected, and you can measure that by chopping the light, turning it on and off as quickly as you need to, and the electron energy that comes out should depend on the intensity of the light, but what was observed are these three things. First of all, the photo electrons, when they came out, were ejected essentially instantaneously, so there was no delay. The second point is that below a threshold frequency, there were no photo electrons at all, and the third point is that turning up the intensity of a low frequency light made no difference. You still didn't get any photo electrons, and Einstein interpreted this experiment in terms of photons, namely that the particles of light were coming in, and each particle of light could hit an electron, and one particle hits one electron, and if that one particle that hits the electron doesn't have enough oomph to kick the electron out, then the electron doesn't come out, and having a lot of particles, none of which can, on a single hit, hit the electron out, doesn't help you. You need one particle that has enough oomph to hit it in one go, and he found that the particle had an energy exactly in accordance with Planck's formula. So now we have another experiment, which is indicating that light, depending on its frequency, has a quantized energy, and that we call that a photon, and we think of it as a particle. So as I said, if one photon hits an electron, the energy is good enough, it kicks it, otherwise the electron stays in the metal. So here's an idealized view of the experiment. I apologize for the kind of gray of the potassium is hard to see, but this is a potassium metal used because it's very easy to kick electrons out, and there are three wavelengths, and the energies are quoted too, in electron volts. An electron volt is the energy that one electron gets by being dropped through one volt of potential difference, and it's 1.6 times 10 to the minus 19 joules. If we have 700 nanometer light, which is red, we get no electrons. If we shine in green light at 550, we get electrons that come out, and they come out with a maximum speed that we can measure by timing when they hit a detector of about 3 times 10 to the 5 meters per second. And if we use more energetic light toward the violet end of the visible spectrum, then we also get electrons out instantaneously, but now the speed of the electrons is higher so that electrons have more energy. So here's a diagram that shows the energy balance. It takes a certain amount of energy, in this case, for a potassium, two electron volts to get the electron to part ways from the potassium atom, and then whatever energy is left over, since then we believe energy is conserved, even in this crazy realm of quantum mechanics, that energy must be then the kinetic energy of the electron, and so that makes it up to the top. And you can see that if the photon itself doesn't have enough energy to get up to the red bar, then there's no way that the electron is going to be ejected. And the energy phi is called the work function of the metal. It's two volts for potassium, but not for other metals, and the kinetic energy, as I showed, is just the difference. So we can express that in an equation. Kinetic energy of the electron is the difference between the photon energy and the energy to pry it out of the material, and that means that 1 half mv squared is equal to the difference, and so I can solve for v as a square root of 2 times h nu minus phi over the mass of the electron. And what we can see is that the mathematics here gives us a clue that we might have a problem because if h nu is not up to the red bar, if it's less than phi, then we get a negative square root, which would give us an imaginary velocity, which is kind of hard for us to interpret. Just because you get an imaginary number from an equation doesn't mean it's wrong. We're going to see plenty of imaginary numbers, but in this case, when we interpret it as a velocity, we'd have to figure out what an imaginary velocity actually meant in terms of what we would see. Now, it turns out that we can only be sure of the energy of the electron for the potassium atoms that are very near the surface so that the light hits and then ejects the electron. The potassium is a silvery surface almost like a mirror, and so we could guess and we'd be right that the light doesn't really penetrate through like a window and come out the other side. So if we eject an electron from an atom further down, the electron might come up and hit another atom and slow down and heat up the material. So we just look for the maximum energy electrons that come out, and that tells us what it is for the surface. And that incidentally lets us know that this kind of photoelectron spectroscopy is a very good technique to interrogate a solid surface because you will only eject electrons very near the surface of the specimen, and what that means is that you won't see stuff underneath. So in some cases, if you make an alloy or you make a material, what you find out is that the surface tends to be enriched in one kind of element, and the bulk in the center tends to be enriched in another kind. And that could be very important if you're designing parts that are going to fit together, and you think the surface has a certain kind of composition. And in fact, because some atoms prefer to be on the surface because of the way they bond, the composition is much, much different. And in that case, you can use photoelectron spectroscopy and you can do this ancient experiment. And since now all the work functions are known for all the atoms, you can easily figure out who's there. Usually, we know the wavelength of the light. And since the wavelength times the frequency is the velocity which for light is given the special symbol C, we can also write the energy in terms of the wavelength, and here's the equation. The energy of a photon is H times C over lambda. Finally, if you take very energetic photons like gamma rays, then the speed of the electron could approach C, the speed of light. And in that case, we have to use a different formula which I won't derive, but we have to use what's called the total relativistic energy which is given by this formula, E squared equals P squared C squared plus M squared C to the fourth. And I think you can see that if P vanishes, then the rest mass, if you have no other kinetic energy, is E squared equals M squared C to the fourth, and that's where E equals MC squared came from. M naught is just called the rest mass of the electron. It's the mass when it's not moving. And the relativistic momentum P is still just M times V, but M is not the rest mass but rather gets corrected by this formula that includes the ratio of the speed of the particle versus the speed of light. And interestingly, this formula also lets us figure out the momentum of a photon. So we start with the formula, and then we note that a photon has zero rest mass, and therefore, the energy is just E squared equals P squared C squared. And we can take the square root of that and find that the momentum is E upon C, and since light's quantized, that's H nu over C, and since C is the frequency times the wavelength, we just end up with E with, sorry, with P equals H over lambda, which is the momentum. And that means the short wavelength photons with small wavelength have high momentum. And here's an application of this phenomenon. Here's a spaceship that set sail a couple of years ago, and it's a giant solar sail. Of course, you don't have to worry about air resistance if you're out in the middle of space, and you can, with a reflective coating, you can actually use the momentum of the photons from the sun to steer your spaceship around, and you don't need any fuel or anything else, and you could just use the sun itself to push you around, and you can turn things this way and that. It's kind of an interesting application of the photon momentum, and I'll let you speculate about how fast you think that this spaceship could go in open space. Now, there is a thing to take note of, and that is, even if you're going pretty fast, you don't have to worry about relativity unless you're about 10% of the speed of light. If you're 10% of the speed of light or something like that, then you may have to start worrying a little bit about relativity, but normally in chemistry, we don't worry about relativistic corrections. So the only point of introducing this formula was to show that a photon, although it has no mass, has a momentum. So let's do a practice problem. So let's just confirm that the speeds that we quoted on the potassium photoelectric effect diagram are, in fact, the right speeds. So, well, we could use either the wavelength of the light or we could use the energy in electron volts. And as I told you, 1EV is 1.6 times 10 to the minus 19 joules. But since we have the work function for potassium in electron volts, I think that's probably going to be the easier course. So for the 2.2EV photon, the green light, we can set up our energy balance equation that the kinetic energy is the difference between the photon energy and the work function. And then we can solve for the velocity or the speed of the electron that comes out. And you notice that when I solve it, I put in the units. In chemistry, it is very, very important to put in the units and make sure all the units go away except the one that you want to get in the end. So I'm taking 2 times 0.25EV, dividing it by the mass of the electron in kilograms. I'm converting EV to joules. And then I'm remembering that a joule is a kilogram meter squared per second squared. I can remember that because I know that force is mass times acceleration. Acceleration, I can remember, is meters per second squared. So force is kilogram meter per second squared. And a joule is a Newton meter. And I can remember that too. And so I add another meter. And now I see the kilograms go away, the EVs go away, the joules go away, and I have the square root of meters squared per second squared, which is meters per second. Whenever you do a problem in chemistry, you want to analyze it in exactly this way. If you just write down numbers with no units, you're very likely going to have some funny units left over like the square root of EV over joule or something else. And without the units there to let you know that they didn't all fold up like a hat trick, you just get the wrong numerical answer. And if you write the wrong numerical answer on an exam or submit it in a report or build a bridge with it and it falls down, nobody is interested in why that happened. They're only interested in the mistake. So we went through this and you can see that the way I quote it is I keep a lot of digits and then I put the ones that I don't think are significant into parentheses. And then I can round it to give a 3 times 10 to the 5 meters per second. And the same thing with the violet light, you just put in a slightly different number. And again, you get 6.2 times 10 to the 5 meters per second. Always retain insignificant digits in case you want to continue the calculation further on. And never, ever round in the middle of a calculation. Your calculator will hold 12 digits, 12 digits. It will hold 15 digits. Everything's 15 digits. Never round and you can see in my examples, I take the time to write 1.602, et cetera. I look up the exact value because if I do a lot of calculation and I start rounding things here and there and everywhere, by the time I get to the end, my accuracy is poor. And sometimes if I'm unlucky, it can be very poor. So at least keep all the digits. People kill themselves trying to get those digits on those numbers, that was years of work. And to just say, well, I can't even be bothered to punch them into my calculator is really almost a crime. So in many experiments, light behaves like a wave. And the question is, if it behaves like a particle in these experiments, but it behaves like a wave with two slits and other things like that, then which is it? Is light a particle or is it a wave? And the answer is, it apparently depends on the nature of the experiment. Light itself seems to have both qualities at once. And even though we think of them as completely different kinds of things, apparently these two qualities are not mutually exclusive. But if light is a particle, then the thought is that if I shoot one particle at a time with the two slits, that the interference phenomenon would have to go away. Because the reason we were getting the interference phenomenon is we were getting all these waves going through together and then they would add and subtract. But if they aren't there simultaneously, if there's only one particle going through at a time, pick, pick, pick, pick, then we would see just two piles. You could either go through this slit, you get a pile here, go through this slit, you get a pile there. But interestingly enough, this fails. And so this two-slit experiment where you get an interference pattern, you get exactly the same pattern even if you can verify that you're shooting photons one at a time. So you shoot one photon, another, another, you detect where they end up. And at the end of the day, you get an interference pattern. And the only way you can really try to explain that is it seems like the photon, which you're claiming is a particle when it suits you, is now the particle is somehow slipping through both slits. In other words, the particle can interfere with itself. This is a very, very foreign idea in terms of what we understand in the physical world, where if we have a particle and we, the particle goes through one slit, we know it went through that slit. And it doesn't somehow break up and then reconstitute itself on the other side. And this was one of the most discomforting things about this new theory of quantum mechanics because it seemed to suggest something that was very foreign to our physical intuition and even foreign to our common sense notion about what a particle is. So if we fire one photon at a time and do photon counting, what we find is we have to fire a lot of photons to get good statistics. But when we do, if we have one slit, we get the pattern on the left. And if we have two slits, we get the pattern on the right. And we get the pattern on the right with two slits, whether we fire the photons one at a time and take forever to do it or whether we fire a bunch of them at once and they all go through. And that means that whatever the wave nature of light is like, it doesn't seem to have much to do with water waves. Even stranger, an electron has a certain amount of charge. It has a certain mass. And I've never seen half an electron. But if we fire electrons now from an electron gun, like an electron microscope, and we fire them one at a time, and then we look with two slits at what happens, we would expect to get, again, two piles of shot. If the electron went through here, it would go here and it could have a certain angle. We get a big pile. And then we'd get a pile over here for the ones that apparently went through this slit because we can't control exactly like a marksman where the electrons go. So they may go through. But whichever ones go through the slits, we should get two lumps. If you do this and you fire them one at a time again, what you find is that you do not get two lumps. So what you find is shown on the next slide. This is a brilliant experiment that was done at Hitachi using an electron microscope. And what you can see here is just how interesting it is because when you have 10 electrons, you have just 10 spots. And it looks to me like they may have slightly miscounted or there may be a glitch because you can see 11 if you look closely. But anyway, you get 10 spots and you notice that when the electrons hit the screen, you get a spot as if it were a very tiny particle. And the slits are much farther apart than any kind of width of these dots. And then when you do 200 electrons, you get a shotgun pattern. And then when you do a lot more, you start to see ridges like a wave. And then when you finally do hundreds of thousands of electrons, you see this clear kind of tin roof appearance of the pattern of intensity which even though you shot the electrons through one at a time, seems to be indicating that each electron goes through both slits. This is even worse than the photon because I don't have any particular picture about a photon. It was a mathematical thing that came up. That doesn't bother me maybe too much. But I certainly do have a picture of an electron. And if the electron is interfering with itself, the question you might ask is, well, which slit did the charge go through? Which slit did the mass go through? Why is it that whenever I look at an electron, I see exactly the same mass and exactly the same charge. And then when I fire them through these two slits, I do not. And the short answer to that is that if you look at which slit the electron goes through, you get two lumps of shot. So if you try to intercept the electron and you try to see it by looking, then you get two lumps of shot and the electron says, okay, you're going to look at what I'm doing. I'm going to go through with the left slit or the right slit. That's it. But if you don't look, which of course they were not looking, they had two slits set up, they fire electrons. If you look, normally we think, well, if I want to see the rug, I want to see the monitor, I want to see whatever, I simply look at it. But in fact, what's happening is we've got light on. We've got light on and because I'm so heavy and the light's so light, it's not moving me around. It's not doing anything to me. But in fact, if I've got an electron going through a slit, I can't just see it. It's too tiny. I need to shine some light on it. And when I shine light on it, I know the photon can even kick an electron out of a metal. And so the photon interacts with the electron and changes it. And so by trying to observe where it is, I actually change the nature of the experiment. And so it's very frustrating because when I don't look, it does something incredible that I can hardly believe. And when I do look, it behaves exactly the way I would have thought that it would be behaving. And what we'll do then is we'll close up there for this lecture. And in the next lecture, what I want to talk about is the connection that a man by the name of De Broglie made between the wavelength of these particles and the wavelength of light, which seemed to be a major advance. And it's kind of interesting that that's the one thing he did. And then he did that, and that was great. And then he never did much else after that. Okay, thanks very much.
 Hi and welcome to Chem 131A, Physical Chemistry.  I'm Dr. Shaka and I'll be leading you through these series  of lectures on physical chemistry starting  with an Adams-Up approach.  Quantum mechanics is our first topic,  which is kind of a rude introduction to the subject.  But here we go.  I'll give you some general course information  and then an introduction to quantum mechanics as seen  through the eyes of a chemist rather than a physicist.  We have slightly different viewpoints on some things.  So here's a preliminary problem, practice problem one.  Here is a jumbled word and the question is what word can you make?  And even though there are five letters,  you might find it pretty hard.  But the answer is you can make tool.  But you have to have that word in your vocabulary  if you're going to make it.  And if you're in the fashion business or you're working  with fabrics, you may happen to know  that is a type of fabric.  But if not, you could play around with that word  for a long time.  Mathematical equations that we're going to be dealing  with are quite similar.  We have to rearrange symbols in ways that are legal  and we have to somehow see where to go.  That means two things.  You have to have the basic vocabulary  to understand what it means.  And you have to have enough background material  that you know what a legal move is.  And I will assume that you've read in the book the background  material, the fundamentals.  And if you haven't, then that's your first task right now.  So here's some general information.  The lecture attendance is optional except on exam days.  We'll have one problem do each Friday.  And we only have one problem  because they're quite hard actually.  And we have a website and on it we'll have what's new.  But avoid sending me email.  Just ask in person after class.  And warning, do not start the problem on Thursday evening.  Please realize that it's one problem and it's multiple parts  and it's really quite difficult to actually complete it.  And so have a go, clarify your understanding,  and then try again.  The TAs and I will go over some of the problems at the end  of the chapter and clarify any ambiguous wording  of the problem.  Homework counts for 20% of the grade in this class.  I'm not a big fan of making exams count a lot.  There will be two midterms and a final exam.  And reading, I can tell you, is not the best way  to learn physical chemistry.  Reading in chemistry is like tying your shoes to run a race.  You have to tie your shoes but that doesn't count.  It's not training.  What's training in chemistry is practicing,  solving problems, visualizing what things look like,  and trying to work quickly and accurately.  The textbook we're going to use is Quanta Matter and Change  by Atkins, DePaula, and Friedman.  And we'll cover the first five chapters.  But I'll warn you that textbooks are getting a little bit  like Amazon.com.  They're trying to be all things to all people.  It's not necessary that you memorize lots of facts  or small details or who did what.  What's important is to just try to understand the ideas  and develop an intuition.  It can be done even in a field like quantum mechanics.  So chapter one, quantum mechanics is the study  of the small specifically things that we can't see.  Though viewed closely, the world appears to be digital.  In other words, there are small packets of everything.  And atoms are the smallest unit of a particular element.  Of course, even the ancient Greeks realized  that it might be true that things had a smallest  indivisible amount.  Now we know that that is in fact true.  But they're very, very small and very light.  One neutral carbon third carbon 12 atom has a mass  of only about 2 times 10 to the minus 23 grams.  An electron is much, much smaller, has a mass of 9 times 10  to the minus 28 grams.  And these very small things are very unfamiliar to us  because we can't see anything that small.  So human intuition is guided by things  that are around the same size as us.  Things that are much, much bigger than us or much,  much smaller than we are are hard to understand.  And we have to use careful experiments to try  to figure out what's going on.  And likewise protons and neutrons occur in integer units.  You can have one proton or none, but you can't have half.  And you can't have half an electron.  This is pretty much similar to currency.  There's a minimum amount of currency.  It's different for every currency,  but there's still a minimum amount.  In the U.S. system, it's 1 cent.  So you can have 1 cent, but you can't have anything smaller  and have it be legitimate currency.  Light, Newton actually believed that light also had a currency  that there was a minimum amount that light was corpuscular  because light seemed to travel in straight lines just  like a bullet fired from a gun.  But later investigations in which light went  through narrow slits or pinholes showed interference phenomena  very much like water waves.  And so the work of Huygens caused Newton really  to abandon his corpuscular theory of light  because it didn't seem like that theory could explain this kind  of wave phenomena.  Waves have positive or negative phase,  and so they can subtract as well as add.  A particle can either be there, be positive, or zero,  but it can't be negative in the usual sense.  And so we wouldn't expect small numbers of marbles  to somehow give interference phenomena  if we fire them through slits.  So for example, here's a picture  of two slightly different wavelengths of a wave.  And if we add them up, you can see that there are positions  where in space here, because this is wavelength,  there is a very small response, and then there's another place  where there's a very large response,  and then there's a small response, and so on.  And this is very much similar to sound waves, for example,  where if you tune a violin, you hear the difference  when you aren't in tune.  You hear this wah wah wah, this sort of beating,  and you're seeing in this picture that kind of beating occurring,  and it's a universal phenomenon with waves.  White light, it turns out, has a mixture  of different wavelengths, and that was sort of first surmised  when you passed white light through a prism,  and you resolve it into a rainbow.  But that doesn't necessarily mean that white light is composed  of different colors, because it could be  that the colors somehow come from the prism.  And so it was only really when you took another prism,  and then you took the same rainbow and returned it  into white light that people became convinced  that white light was really a mixture of colors,  and there was nothing coming necessarily from the prism.  When you have a convex surface near a flat surface,  there's a difference in refractive index, and the condition  for these beats to add up or subtract depends on the wavelength,  and this is a phenomenon called Newton's rings,  a pattern of constructive interference that, for example,  you can see if you spill oil, which tends to bead up on water,  on a wet surface, and we used to always look at that as a kid  because cars in Salt Lake City always had a lot of oil leaking  out of them in those days, and it rained,  and we would spend a lot of time looking  at these beautiful patterns.  So we can see, here's an example,  we can see how the colors vary systematically with wavelength,  and much like there were two beating patterns in that graph  I showed you, this will repeat more than once depending  on the total thickness of the contrasting media.  There's another experiment that unfortunately made it difficult  to understand light as waves, and that was black body radiation.  So this is a classic thing where you think you understand everything,  and you have a very simple calculation,  and there are some very, very smart people like Lord Rayleigh,  and you do the calculation, and then you compare  with the experiment, this is the essence of science.  If you think you understand something, you ought to be able  to predict it or explain why you can't predict it,  but at this time there was Maxwell's equations pretty much,  people believed that these wave equations really described light  in totality, and that light was an electromagnetic wave,  and that theory was wildly successful.  So we could explain the unification of electricity  and magnetism, why the speed of light has its speed,  diffraction, reflection, refraction, how lenses work,  and so on and so forth.  But the two crucial experiments showed that this description  of light was incomplete, and the first one was black body radiation.  So what is black body radiation?  Well, if I take a black body, by which you could just mean a lump  of coal or lamp black, and heat it up in a vacuum  so that there's no air that doesn't burst into flame,  it'll glow, and this glow gives a characteristic spectrum  of colors just like white light has a characteristic spectrum  of colors, and it was known that the color depended only  on the temperature, hence we speak of red hot and things  like that.  And Rayleigh, with a correction by genes,  calculated this spectrum of wave-like light,  and what they found is that it would be much,  much more likely that you would get a lot  of high-frequency radiation because the chance  of getting each frequency was equally likely,  and there are a lot more high frequencies,  than there are low frequencies.  And this was called the ultraviolet catastrophe  because it basically predicted that if we opened something  like a kitchen oven, then outcome a ton of x-rays  and very high-frequency light, and kill us basically,  and that's obviously not what happens.  So we, there was a big, big soul-searching, and there was a lot  of thought about what might be going on,  and some theories were put forward  that were later proven not to be quite right,  but it was Max Planck that found the correct solution.  What he found that fit the observations,  although he didn't really quite believe it himself,  but he could follow through the physics,  was that light was quantized.  That is, there's a minimum amount of light,  and the smallest amount was related  to the frequency of the light,  and once the frequency was chosen, the energy  of the light was E equals H nu.  H has subsequently been called Planck's constant  in honor of his discoverer.  So the frequency could apparently be continuous,  anything you want, but once you choose it,  then there's a minimum amount,  and if you don't have the minimum amount, there's no light,  and since the minimum amount depends on the frequency,  higher frequencies, if there's not enough energy  around, can't have the minimum amount  to make a single particle or quantum of light,  and therefore those frequencies get cut off.  So that gets around this problem  with the x-rays killing us if we open the oven.  So here's an equation that just says the likelihood  that you're going to have a certain amplitude of light,  or number of relative amount of light per unit frequency given  according to the Rayleigh-Jeans law,  and you can see that it's basically a parabola  in the frequency squared, so as the frequency goes up,  the amount of light that's predicted is predicted  to go up faster and faster and faster,  and that's not what's observed.  Now what Planck did instead of this continuous equation is this.  He quantized the photon energy, and he obtained this formula  which on first blush looks completely different.  It has a cube in the enumerator,  and then it has this funny exponential of h nu upon kT  in the denominator, and we can compare  these two formulas on a graph and see what happens.  And in this formula, let me remark that k is Boltzmann's constant,  h is Planck's constant, and T is the temperature in Kelvin,  and of course in physical chemistry or chemistry of any type,  you never quote temperature in anything other than Kelvin,  because if you do, you're very likely to be wrong  if you plug it into a formula.  If you quote the temperature in Kelvin and you say it's a balmy day,  it's 298 Kelvin, you may be eccentric, but you're never wrong.  But in chemistry, if you use any other units,  you're very likely going to be wrong,  so you have to be careful.  So here's a graph.  Planck is in the green and Rayleigh jeans in the sort  of pink color, and you can see that for a certain temperature  of 300 Kelvin, they agree at very, very low frequency  where the quantization is very small,  and then as the frequency increases,  Planck actually follows the observed distribution almost exactly,  and Rayleigh jeans diverges more and more and more from it,  and would continue to go up.  Now, we can make a connection between the two theories,  because we know they agree when the frequency is small  and the energy is low, and so we can take this parameter H nu  upon kT, and we can assume it's much, much less than 1,  and we know from calculus that we can expand E to the X  in a power series, 1 plus X, and then if X is small,  then X squared is very small, so we can throw it away,  and that means we can write E to the H nu upon kT  as 1 plus H nu upon kT, and then if we make  that substitution in, we find that as long as the frequency is  not high compared to the temperature kT,  that we get exactly the same formula that Rayleigh and jeans got,  but if the temperature, if the frequency is high  where there was a problem, then it starts to deviate,  and so that's very nice because it shows you  that you get the same result as the other guys got  where their theory seemed to work,  and that's what scientists often look for, of course.  Now, the physical meaning of kT is that kT is really a measure  of the random thermal energy that's available at a temperature T.  When something's hot, things are moving, they're colliding,  they're banging around, and there's lots of energy available  to excite things and to create photons,  but if the temperature is very cold,  then there's hardly any energy around,  and then you just don't have enough energy  to make the minimum amount of a photon,  and that's why cold things don't glow, but things  that are heated up finally do start to emit a glow  like an electric element on a stove.  So, at high frequency, the problem is we just don't have  enough energy to make even a single photon.  We just run out before we get there,  and so the distribution has to fall off sharply,  and the analogy I can give you is that suppose the smallest amount  of currency, the smallest coin, were $1,000,  then that means that a lot of people are going  to have no money at all because they don't have that much.  The reason we don't notice the digital nature of light  in day-to-day observation, we don't notice that it looks  like sand or something like that, is because of the relative size  of these two constants, K and H. Boltzmann's constant is 1  times 10 to the minus 23 joules per Kelvin,  and Plank's constant is 6 times 10 to the minus 34,  and that means there's a factor of about 10 to the 11  or 100 billion between them, and that means that usually  at low frequency there are plenty of photons around,  but at high frequency we start to notice that H nu,  the quantum of light, has a minimum value.  The other experiment that really sealed the deal with respect  to the corpuscular or quantized nature  of light was the photoelectric effect,  and it's not quite such a simple experiment  as the black body radiation, but nevertheless,  it is a pretty simple experiment, and the experiment is this.  You evacuate a chamber and you have a clean metal surface,  and you shine light on the surface, and what was observed is  that electrons would come off the surface,  and this is how you make a cathode ray tube, in fact,  but electrons would come off the surface of the metal  and would come into the vacuum and they would be ejected,  and ideally if light were a wave,  then what should happen is if you have a wave coming in,  it should sort of excite the electrons more and more and more  and more and more and then, boom,  finally just like pushing a swing,  if you push it enough times, you can get the person moving.  So that means that when you turn on the light,  there should be a delay before the electrons are ejected,  and you can measure that by chopping the light,  turning it on and off as quickly as you need to,  and the electron energy that comes out should depend  on the intensity of the light,  but what was observed are these three things.  First of all, the photo electrons,  when they came out, were ejected essentially instantaneously,  so there was no delay.  The second point is that below a threshold frequency,  there were no photo electrons at all,  and the third point is that turning up the intensity  of a low frequency light made no difference.  You still didn't get any photo electrons,  and Einstein interpreted this experiment in terms of photons,  namely that the particles of light were coming in,  and each particle of light could hit an electron,  and one particle hits one electron,  and if that one particle that hits the electron doesn't have  enough oomph to kick the electron out,  then the electron doesn't come out,  and having a lot of particles, none of which can,  on a single hit, hit the electron out, doesn't help you.  You need one particle that has enough oomph to hit it in one go,  and he found that the particle had an energy exactly  in accordance with Planck's formula.  So now we have another experiment,  which is indicating that light, depending on its frequency,  has a quantized energy, and that we call that a photon,  and we think of it as a particle.  So as I said, if one photon hits an electron,  the energy is good enough, it kicks it,  otherwise the electron stays in the metal.  So here's an idealized view of the experiment.  I apologize for the kind of gray of the potassium is hard to see,  but this is a potassium metal used  because it's very easy to kick electrons out,  and there are three wavelengths, and the energies are quoted too,  in electron volts.  An electron volt is the energy that one electron gets  by being dropped through one volt of potential difference,  and it's 1.6 times 10 to the minus 19 joules.  If we have 700 nanometer light, which is red,  we get no electrons.  If we shine in green light at 550, we get electrons that come  out, and they come out with a maximum speed that we can measure  by timing when they hit a detector of about 3 times 10  to the 5 meters per second.  And if we use more energetic light toward the violet end  of the visible spectrum, then we also get electrons  out instantaneously, but now the speed  of the electrons is higher  so that electrons have more energy.  So here's a diagram that shows the energy balance.  It takes a certain amount of energy, in this case,  for a potassium, two electron volts to get the electron  to part ways from the potassium atom,  and then whatever energy is left over,  since then we believe energy is conserved,  even in this crazy realm of quantum mechanics,  that energy must be then the kinetic energy of the electron,  and so that makes it up to the top.  And you can see that if the photon itself doesn't have enough  energy to get up to the red bar, then there's no way  that the electron is going to be ejected.  And the energy phi is called the work function of the metal.  It's two volts for potassium, but not for other metals,  and the kinetic energy, as I showed, is just the difference.  So we can express that in an equation.  Kinetic energy of the electron is the difference  between the photon energy and the energy to pry it  out of the material, and that means  that 1 half mv squared is equal to the difference,  and so I can solve for v as a square root of 2 times h nu  minus phi over the mass of the electron.  And what we can see is that the mathematics here gives us a clue  that we might have a problem because if h nu is not  up to the red bar, if it's less than phi,  then we get a negative square root,  which would give us an imaginary velocity,  which is kind of hard for us to interpret.  Just because you get an imaginary number  from an equation doesn't mean it's wrong.  We're going to see plenty of imaginary numbers,  but in this case, when we interpret it as a velocity,  we'd have to figure out what an imaginary velocity actually meant  in terms of what we would see.  Now, it turns out that we can only be sure of the energy  of the electron for the potassium atoms  that are very near the surface so that the light hits  and then ejects the electron.  The potassium is a silvery surface almost like a mirror,  and so we could guess and we'd be right  that the light doesn't really penetrate through like a window  and come out the other side.  So if we eject an electron from an atom further down,  the electron might come up and hit another atom and slow down  and heat up the material.  So we just look for the maximum energy electrons  that come out, and that tells us what it is for the surface.  And that incidentally lets us know that this kind  of photoelectron spectroscopy is a very good technique  to interrogate a solid surface  because you will only eject electrons very near the surface  of the specimen, and what that means is  that you won't see stuff underneath.  So in some cases, if you make an alloy or you make a material,  what you find out is that the surface tends to be enriched  in one kind of element, and the bulk  in the center tends to be enriched in another kind.  And that could be very important if you're designing parts  that are going to fit together,  and you think the surface has a certain kind of composition.  And in fact, because some atoms prefer to be on the surface  because of the way they bond,  the composition is much, much different.  And in that case, you can use photoelectron spectroscopy  and you can do this ancient experiment.  And since now all the work functions are known  for all the atoms, you can easily figure out who's there.  Usually, we know the wavelength of the light.  And since the wavelength times the frequency is the velocity  which for light is given the special symbol C,  we can also write the energy in terms of the wavelength,  and here's the equation.  The energy of a photon is H times C over lambda.  Finally, if you take very energetic photons like gamma rays,  then the speed of the electron could approach C, the speed of light.  And in that case, we have to use a different formula  which I won't derive, but we have to use what's called the total  relativistic energy which is given by this formula,  E squared equals P squared C squared plus M squared C to the fourth.  And I think you can see that if P vanishes, then the rest mass,  if you have no other kinetic energy,  is E squared equals M squared C to the fourth,  and that's where E equals MC squared came from.  M naught is just called the rest mass of the electron.  It's the mass when it's not moving.  And the relativistic momentum P is still just M times V,  but M is not the rest mass but rather gets corrected  by this formula that includes the ratio of the speed  of the particle versus the speed of light.  And interestingly, this formula also lets us figure  out the momentum of a photon.  So we start with the formula, and then we note  that a photon has zero rest mass, and therefore,  the energy is just E squared equals P squared C squared.  And we can take the square root of that and find  that the momentum is E upon C, and since light's quantized,  that's H nu over C, and since C is the frequency times  the wavelength, we just end up with E with, sorry,  with P equals H over lambda, which is the momentum.  And that means the short wavelength photons  with small wavelength have high momentum.  And here's an application of this phenomenon.  Here's a spaceship that set sail a couple of years ago,  and it's a giant solar sail.  Of course, you don't have to worry about air resistance  if you're out in the middle of space, and you can,  with a reflective coating, you can actually use the momentum  of the photons from the sun to steer your spaceship around,  and you don't need any fuel or anything else,  and you could just use the sun itself to push you around,  and you can turn things this way and that.  It's kind of an interesting application  of the photon momentum, and I'll let you speculate  about how fast you think that this spaceship could go  in open space.  Now, there is a thing to take note of, and that is,  even if you're going pretty fast, you don't have to worry  about relativity unless you're about 10% of the speed of light.  If you're 10% of the speed of light or something like that,  then you may have to start worrying a little bit  about relativity, but normally in chemistry,  we don't worry about relativistic corrections.  So the only point of introducing this formula was  to show that a photon, although it has no mass,  has a momentum.  So let's do a practice problem.  So let's just confirm that the speeds that we quoted  on the potassium photoelectric effect diagram are,  in fact, the right speeds.  So, well, we could use either the wavelength of the light  or we could use the energy in electron volts.  And as I told you, 1EV is 1.6 times 10 to the minus 19 joules.  But since we have the work function for potassium  in electron volts, I think that's probably going  to be the easier course.  So for the 2.2EV photon, the green light,  we can set up our energy balance equation  that the kinetic energy is the difference between the photon  energy and the work function.  And then we can solve for the velocity or the speed  of the electron that comes out.  And you notice that when I solve it, I put in the units.  In chemistry, it is very, very important to put in the units  and make sure all the units go away except the one  that you want to get in the end.  So I'm taking 2 times 0.25EV, dividing it by the mass  of the electron in kilograms.  I'm converting EV to joules.  And then I'm remembering that a joule is a kilogram meter squared  per second squared.  I can remember that because I know  that force is mass times acceleration.  Acceleration, I can remember, is meters per second squared.  So force is kilogram meter per second squared.  And a joule is a Newton meter.  And I can remember that too.  And so I add another meter.  And now I see the kilograms go away, the EVs go away,  the joules go away, and I have the square root  of meters squared per second squared,  which is meters per second.  Whenever you do a problem in chemistry, you want to analyze it  in exactly this way.  If you just write down numbers with no units,  you're very likely going to have some funny units left  over like the square root of EV over joule or something else.  And without the units there to let you know  that they didn't all fold up like a hat trick,  you just get the wrong numerical answer.  And if you write the wrong numerical answer on an exam  or submit it in a report or build a bridge with it  and it falls down, nobody is interested in why that happened.  They're only interested in the mistake.  So we went through this and you can see that the way I quote it  is I keep a lot of digits and then I put the ones  that I don't think are significant into parentheses.  And then I can round it to give a 3 times 10  to the 5 meters per second.  And the same thing with the violet light,  you just put in a slightly different number.  And again, you get 6.2 times 10 to the 5 meters per second.  Always retain insignificant digits in case you want  to continue the calculation further on.  And never, ever round in the middle of a calculation.  Your calculator will hold 12 digits, 12 digits.  It will hold 15 digits.  Everything's 15 digits.  Never round and you can see in my examples,  I take the time to write 1.602, et cetera.  I look up the exact value because if I do a lot of calculation  and I start rounding things here and there and everywhere,  by the time I get to the end, my accuracy is poor.  And sometimes if I'm unlucky, it can be very poor.  So at least keep all the digits.  People kill themselves trying to get those digits  on those numbers, that was years of work.  And to just say, well, I can't even be bothered to punch them  into my calculator is really almost a crime.  So in many experiments, light behaves like a wave.  And the question is, if it behaves like a particle  in these experiments, but it behaves like a wave  with two slits and other things like that,  then which is it?  Is light a particle or is it a wave?  And the answer is, it apparently depends on the nature  of the experiment.  Light itself seems to have both qualities at once.  And even though we think of them  as completely different kinds of things,  apparently these two qualities are not mutually exclusive.  But if light is a particle, then the thought is  that if I shoot one particle at a time with the two slits,  that the interference phenomenon would have to go away.  Because the reason we were getting the interference phenomenon  is we were getting all these waves going through together  and then they would add and subtract.  But if they aren't there simultaneously,  if there's only one particle going through at a time,  pick, pick, pick, pick, then we would see just two piles.  You could either go through this slit, you get a pile here,  go through this slit, you get a pile there.  But interestingly enough, this fails.  And so this two-slit experiment  where you get an interference pattern,  you get exactly the same pattern even if you can verify  that you're shooting photons one at a time.  So you shoot one photon, another, another,  you detect where they end up.  And at the end of the day, you get an interference pattern.  And the only way you can really try to explain that is it seems  like the photon, which you're claiming is a particle  when it suits you, is now the particle is somehow slipping  through both slits.  In other words, the particle can interfere with itself.  This is a very, very foreign idea in terms  of what we understand in the physical world,  where if we have a particle and we, the particle goes  through one slit, we know it went through that slit.  And it doesn't somehow break up and then reconstitute itself  on the other side.  And this was one of the most discomforting things  about this new theory of quantum mechanics because it seemed  to suggest something that was very foreign  to our physical intuition and even foreign  to our common sense notion about what a particle is.  So if we fire one photon at a time and do photon counting,  what we find is we have to fire a lot of photons  to get good statistics.  But when we do, if we have one slit,  we get the pattern on the left.  And if we have two slits, we get the pattern on the right.  And we get the pattern on the right with two slits,  whether we fire the photons one at a time and take forever  to do it or whether we fire a bunch of them at once  and they all go through.  And that means that whatever the wave nature  of light is like, it doesn't seem to have much to do  with water waves.  Even stranger, an electron has a certain amount of charge.  It has a certain mass.  And I've never seen half an electron.  But if we fire electrons now from an electron gun,  like an electron microscope, and we fire them one  at a time, and then we look with two slits at what happens,  we would expect to get, again, two piles of shot.  If the electron went through here, it would go here  and it could have a certain angle.  We get a big pile.  And then we'd get a pile over here for the ones  that apparently went through this slit  because we can't control exactly like a marksman  where the electrons go.  So they may go through.  But whichever ones go through the slits,  we should get two lumps.  If you do this and you fire them one at a time again,  what you find is that you do not get two lumps.  So what you find is shown on the next slide.  This is a brilliant experiment that was done  at Hitachi using an electron microscope.  And what you can see here is just how interesting it is  because when you have 10 electrons, you have just 10 spots.  And it looks to me like they may have slightly miscounted  or there may be a glitch because you can see 11  if you look closely.  But anyway, you get 10 spots and you notice  that when the electrons hit the screen, you get a spot  as if it were a very tiny particle.  And the slits are much farther apart than any kind of width  of these dots.  And then when you do 200 electrons,  you get a shotgun pattern.  And then when you do a lot more, you start to see ridges  like a wave.  And then when you finally do hundreds of thousands  of electrons, you see this clear kind of tin roof appearance  of the pattern of intensity which even though you shot the  electrons through one at a time,  seems to be indicating that each electron goes  through both slits.  This is even worse than the photon  because I don't have any particular picture about a photon.  It was a mathematical thing that came up.  That doesn't bother me maybe too much.  But I certainly do have a picture of an electron.  And if the electron is interfering with itself,  the question you might ask is, well,  which slit did the charge go through?  Which slit did the mass go through?  Why is it that whenever I look at an electron,  I see exactly the same mass and exactly the same charge.  And then when I fire them through these two slits, I do not.  And the short answer to that is that if you look  at which slit the electron goes through,  you get two lumps of shot.  So if you try to intercept the electron and you try  to see it by looking, then you get two lumps of shot  and the electron says, okay, you're going to look  at what I'm doing.  I'm going to go through with the left slit or the right slit.  That's it.  But if you don't look, which of course they were not looking,  they had two slits set up, they fire electrons.  If you look, normally we think, well,  if I want to see the rug, I want to see the monitor,  I want to see whatever, I simply look at it.  But in fact, what's happening is we've got light on.  We've got light on and because I'm so heavy  and the light's so light, it's not moving me around.  It's not doing anything to me.  But in fact, if I've got an electron going through a slit,  I can't just see it.  It's too tiny.  I need to shine some light on it.  And when I shine light on it, I know the photon can even kick  an electron out of a metal.  And so the photon interacts with the electron and changes it.  And so by trying to observe where it is,  I actually change the nature of the experiment.  And so it's very frustrating because when I don't look,  it does something incredible that I can hardly believe.  And when I do look, it behaves exactly the way I would have  thought that it would be behaving.  And what we'll do then is we'll close up there for this lecture.  And in the next lecture, what I want to talk about is the  connection that a man by the name of De Broglie made  between the wavelength of these particles and the wavelength  of light, which seemed to be a major advance.  And it's kind of interesting that that's the one thing he did.  And then he did that, and that was great.  And then he never did much else after that.  Okay, thanks very much.
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Hi and welcome to Chem 131A, Physical Chemistry. I'm Dr. Shaka and I'll be leading you through these series of lectures on physical chemistry starting with an Adams-Up approach. Quantum mechanics is our first topic, which is kind of a rude introduction to the subject. But here we go. I'll give you some general course information and then an introduction to quantum mechanics as seen through the eyes of a chemist rather than a physicist. We have slightly different viewpoints on some things. So here's a preliminary problem, practice problem one. Here is a jumbled word and the question is what word can you make? And even though there are five letters, you might find it pretty hard. But the answer is you can make tool. But you have to have that word in your vocabulary if you're going to make it. And if you're in the fashion business or you're working with fabrics, you may happen to know that is a type of fabric. But if not, you could play around with that word for a long time. Mathematical equations that we're going to be dealing with are quite similar. We have to rearrange symbols in ways that are legal and we have to somehow see where to go. That means two things. You have to have the basic vocabulary to understand what it means. And you have to have enough background material that you know what a legal move is. And I will assume that you've read in the book the background material, the fundamentals. And if you haven't, then that's your first task right now. So here's some general information. The lecture attendance is optional except on exam days. We'll have one problem do each Friday. And we only have one problem because they're quite hard actually. And we have a website and on it we'll have what's new. But avoid sending me email. Just ask in person after class. And warning, do not start the problem on Thursday evening. Please realize that it's one problem and it's multiple parts and it's really quite difficult to actually complete it. And so have a go, clarify your understanding, and then try again. The TAs and I will go over some of the problems at the end of the chapter and clarify any ambiguous wording of the problem. Homework counts for 20% of the grade in this class. I'm not a big fan of making exams count a lot. There will be two midterms and a final exam. And reading, I can tell you, is not the best way to learn physical chemistry. Reading in chemistry is like tying your shoes to run a race. You have to tie your shoes but that doesn't count. It's not training. What's training in chemistry is practicing, solving problems, visualizing what things look like, and trying to work quickly and accurately. The textbook we're going to use is Quanta Matter and Change by Atkins, DePaula, and Friedman. And we'll cover the first five chapters. But I'll warn you that textbooks are getting a little bit like Amazon.com. They're trying to be all things to all people. It's not necessary that you memorize lots of facts or small details or who did what. What's important is to just try to understand the ideas and develop an intuition. It can be done even in a field like quantum mechanics. So chapter one, quantum mechanics is the study of the small specifically things that we can't see. Though viewed closely, the world appears to be digital. In other words, there are small packets of everything. And atoms are the smallest unit of a particular element. Of course, even the ancient Greeks realized that it might be true that things had a smallest indivisible amount. Now we know that that is in fact true. But they're very, very small and very light. One neutral carbon third carbon 12 atom has a mass of only about 2 times 10 to the minus 23 grams. An electron is much, much smaller, has a mass of 9 times 10 to the minus 28 grams. And these very small things are very unfamiliar to us because we can't see anything that small. So human intuition is guided by things that are around the same size as us. Things that are much, much bigger than us or much, much smaller than we are are hard to understand. And we have to use careful experiments to try to figure out what's going on. And likewise protons and neutrons occur in integer units. You can have one proton or none, but you can't have half. And you can't have half an electron. This is pretty much similar to currency. There's a minimum amount of currency. It's different for every currency, but there's still a minimum amount. In the U.S. system, it's 1 cent. So you can have 1 cent, but you can't have anything smaller and have it be legitimate currency. Light, Newton actually believed that light also had a currency that there was a minimum amount that light was corpuscular because light seemed to travel in straight lines just like a bullet fired from a gun. But later investigations in which light went through narrow slits or pinholes showed interference phenomena very much like water waves. And so the work of Huygens caused Newton really to abandon his corpuscular theory of light because it didn't seem like that theory could explain this kind of wave phenomena. Waves have positive or negative phase, and so they can subtract as well as add. A particle can either be there, be positive, or zero, but it can't be negative in the usual sense. And so we wouldn't expect small numbers of marbles to somehow give interference phenomena if we fire them through slits. So for example, here's a picture of two slightly different wavelengths of a wave. And if we add them up, you can see that there are positions where in space here, because this is wavelength, there is a very small response, and then there's another place where there's a very large response, and then there's a small response, and so on. And this is very much similar to sound waves, for example, where if you tune a violin, you hear the difference when you aren't in tune. You hear this wah wah wah, this sort of beating, and you're seeing in this picture that kind of beating occurring, and it's a universal phenomenon with waves. White light, it turns out, has a mixture of different wavelengths, and that was sort of first surmised when you passed white light through a prism, and you resolve it into a rainbow. But that doesn't necessarily mean that white light is composed of different colors, because it could be that the colors somehow come from the prism. And so it was only really when you took another prism, and then you took the same rainbow and returned it into white light that people became convinced that white light was really a mixture of colors, and there was nothing coming necessarily from the prism. When you have a convex surface near a flat surface, there's a difference in refractive index, and the condition for these beats to add up or subtract depends on the wavelength, and this is a phenomenon called Newton's rings, a pattern of constructive interference that, for example, you can see if you spill oil, which tends to bead up on water, on a wet surface, and we used to always look at that as a kid because cars in Salt Lake City always had a lot of oil leaking out of them in those days, and it rained, and we would spend a lot of time looking at these beautiful patterns. So we can see, here's an example, we can see how the colors vary systematically with wavelength, and much like there were two beating patterns in that graph I showed you, this will repeat more than once depending on the total thickness of the contrasting media. There's another experiment that unfortunately made it difficult to understand light as waves, and that was black body radiation. So this is a classic thing where you think you understand everything, and you have a very simple calculation, and there are some very, very smart people like Lord Rayleigh, and you do the calculation, and then you compare with the experiment, this is the essence of science. If you think you understand something, you ought to be able to predict it or explain why you can't predict it, but at this time there was Maxwell's equations pretty much, people believed that these wave equations really described light in totality, and that light was an electromagnetic wave, and that theory was wildly successful. So we could explain the unification of electricity and magnetism, why the speed of light has its speed, diffraction, reflection, refraction, how lenses work, and so on and so forth. But the two crucial experiments showed that this description of light was incomplete, and the first one was black body radiation. So what is black body radiation? Well, if I take a black body, by which you could just mean a lump of coal or lamp black, and heat it up in a vacuum so that there's no air that doesn't burst into flame, it'll glow, and this glow gives a characteristic spectrum of colors just like white light has a characteristic spectrum of colors, and it was known that the color depended only on the temperature, hence we speak of red hot and things like that. And Rayleigh, with a correction by genes, calculated this spectrum of wave-like light, and what they found is that it would be much, much more likely that you would get a lot of high-frequency radiation because the chance of getting each frequency was equally likely, and there are a lot more high frequencies, than there are low frequencies. And this was called the ultraviolet catastrophe because it basically predicted that if we opened something like a kitchen oven, then outcome a ton of x-rays and very high-frequency light, and kill us basically, and that's obviously not what happens. So we, there was a big, big soul-searching, and there was a lot of thought about what might be going on, and some theories were put forward that were later proven not to be quite right, but it was Max Planck that found the correct solution. What he found that fit the observations, although he didn't really quite believe it himself, but he could follow through the physics, was that light was quantized. That is, there's a minimum amount of light, and the smallest amount was related to the frequency of the light, and once the frequency was chosen, the energy of the light was E equals H nu. H has subsequently been called Planck's constant in honor of his discoverer. So the frequency could apparently be continuous, anything you want, but once you choose it, then there's a minimum amount, and if you don't have the minimum amount, there's no light, and since the minimum amount depends on the frequency, higher frequencies, if there's not enough energy around, can't have the minimum amount to make a single particle or quantum of light, and therefore those frequencies get cut off. So that gets around this problem with the x-rays killing us if we open the oven. So here's an equation that just says the likelihood that you're going to have a certain amplitude of light, or number of relative amount of light per unit frequency given according to the Rayleigh-Jeans law, and you can see that it's basically a parabola in the frequency squared, so as the frequency goes up, the amount of light that's predicted is predicted to go up faster and faster and faster, and that's not what's observed. Now what Planck did instead of this continuous equation is this. He quantized the photon energy, and he obtained this formula which on first blush looks completely different. It has a cube in the enumerator, and then it has this funny exponential of h nu upon kT in the denominator, and we can compare these two formulas on a graph and see what happens. And in this formula, let me remark that k is Boltzmann's constant, h is Planck's constant, and T is the temperature in Kelvin, and of course in physical chemistry or chemistry of any type, you never quote temperature in anything other than Kelvin, because if you do, you're very likely to be wrong if you plug it into a formula. If you quote the temperature in Kelvin and you say it's a balmy day, it's 298 Kelvin, you may be eccentric, but you're never wrong. But in chemistry, if you use any other units, you're very likely going to be wrong, so you have to be careful. So here's a graph. Planck is in the green and Rayleigh jeans in the sort of pink color, and you can see that for a certain temperature of 300 Kelvin, they agree at very, very low frequency where the quantization is very small, and then as the frequency increases, Planck actually follows the observed distribution almost exactly, and Rayleigh jeans diverges more and more and more from it, and would continue to go up. Now, we can make a connection between the two theories, because we know they agree when the frequency is small and the energy is low, and so we can take this parameter H nu upon kT, and we can assume it's much, much less than 1, and we know from calculus that we can expand E to the X in a power series, 1 plus X, and then if X is small, then X squared is very small, so we can throw it away, and that means we can write E to the H nu upon kT as 1 plus H nu upon kT, and then if we make that substitution in, we find that as long as the frequency is not high compared to the temperature kT, that we get exactly the same formula that Rayleigh and jeans got, but if the temperature, if the frequency is high where there was a problem, then it starts to deviate, and so that's very nice because it shows you that you get the same result as the other guys got where their theory seemed to work, and that's what scientists often look for, of course. Now, the physical meaning of kT is that kT is really a measure of the random thermal energy that's available at a temperature T. When something's hot, things are moving, they're colliding, they're banging around, and there's lots of energy available to excite things and to create photons, but if the temperature is very cold, then there's hardly any energy around, and then you just don't have enough energy to make the minimum amount of a photon, and that's why cold things don't glow, but things that are heated up finally do start to emit a glow like an electric element on a stove. So, at high frequency, the problem is we just don't have enough energy to make even a single photon. We just run out before we get there, and so the distribution has to fall off sharply, and the analogy I can give you is that suppose the smallest amount of currency, the smallest coin, were $1,000, then that means that a lot of people are going to have no money at all because they don't have that much. The reason we don't notice the digital nature of light in day-to-day observation, we don't notice that it looks like sand or something like that, is because of the relative size of these two constants, K and H. Boltzmann's constant is 1 times 10 to the minus 23 joules per Kelvin, and Plank's constant is 6 times 10 to the minus 34, and that means there's a factor of about 10 to the 11 or 100 billion between them, and that means that usually at low frequency there are plenty of photons around, but at high frequency we start to notice that H nu, the quantum of light, has a minimum value. The other experiment that really sealed the deal with respect to the corpuscular or quantized nature of light was the photoelectric effect, and it's not quite such a simple experiment as the black body radiation, but nevertheless, it is a pretty simple experiment, and the experiment is this. You evacuate a chamber and you have a clean metal surface, and you shine light on the surface, and what was observed is that electrons would come off the surface, and this is how you make a cathode ray tube, in fact, but electrons would come off the surface of the metal and would come into the vacuum and they would be ejected, and ideally if light were a wave, then what should happen is if you have a wave coming in, it should sort of excite the electrons more and more and more and more and more and then, boom, finally just like pushing a swing, if you push it enough times, you can get the person moving. So that means that when you turn on the light, there should be a delay before the electrons are ejected, and you can measure that by chopping the light, turning it on and off as quickly as you need to, and the electron energy that comes out should depend on the intensity of the light, but what was observed are these three things. First of all, the photo electrons, when they came out, were ejected essentially instantaneously, so there was no delay. The second point is that below a threshold frequency, there were no photo electrons at all, and the third point is that turning up the intensity of a low frequency light made no difference. You still didn't get any photo electrons, and Einstein interpreted this experiment in terms of photons, namely that the particles of light were coming in, and each particle of light could hit an electron, and one particle hits one electron, and if that one particle that hits the electron doesn't have enough oomph to kick the electron out, then the electron doesn't come out, and having a lot of particles, none of which can, on a single hit, hit the electron out, doesn't help you. You need one particle that has enough oomph to hit it in one go, and he found that the particle had an energy exactly in accordance with Planck's formula. So now we have another experiment, which is indicating that light, depending on its frequency, has a quantized energy, and that we call that a photon, and we think of it as a particle. So as I said, if one photon hits an electron, the energy is good enough, it kicks it, otherwise the electron stays in the metal. So here's an idealized view of the experiment. I apologize for the kind of gray of the potassium is hard to see, but this is a potassium metal used because it's very easy to kick electrons out, and there are three wavelengths, and the energies are quoted too, in electron volts. An electron volt is the energy that one electron gets by being dropped through one volt of potential difference, and it's 1.6 times 10 to the minus 19 joules. If we have 700 nanometer light, which is red, we get no electrons. If we shine in green light at 550, we get electrons that come out, and they come out with a maximum speed that we can measure by timing when they hit a detector of about 3 times 10 to the 5 meters per second. And if we use more energetic light toward the violet end of the visible spectrum, then we also get electrons out instantaneously, but now the speed of the electrons is higher so that electrons have more energy. So here's a diagram that shows the energy balance. It takes a certain amount of energy, in this case, for a potassium, two electron volts to get the electron to part ways from the potassium atom, and then whatever energy is left over, since then we believe energy is conserved, even in this crazy realm of quantum mechanics, that energy must be then the kinetic energy of the electron, and so that makes it up to the top. And you can see that if the photon itself doesn't have enough energy to get up to the red bar, then there's no way that the electron is going to be ejected. And the energy phi is called the work function of the metal. It's two volts for potassium, but not for other metals, and the kinetic energy, as I showed, is just the difference. So we can express that in an equation. Kinetic energy of the electron is the difference between the photon energy and the energy to pry it out of the material, and that means that 1 half mv squared is equal to the difference, and so I can solve for v as a square root of 2 times h nu minus phi over the mass of the electron. And what we can see is that the mathematics here gives us a clue that we might have a problem because if h nu is not up to the red bar, if it's less than phi, then we get a negative square root, which would give us an imaginary velocity, which is kind of hard for us to interpret. Just because you get an imaginary number from an equation doesn't mean it's wrong. We're going to see plenty of imaginary numbers, but in this case, when we interpret it as a velocity, we'd have to figure out what an imaginary velocity actually meant in terms of what we would see. Now, it turns out that we can only be sure of the energy of the electron for the potassium atoms that are very near the surface so that the light hits and then ejects the electron. The potassium is a silvery surface almost like a mirror, and so we could guess and we'd be right that the light doesn't really penetrate through like a window and come out the other side. So if we eject an electron from an atom further down, the electron might come up and hit another atom and slow down and heat up the material. So we just look for the maximum energy electrons that come out, and that tells us what it is for the surface. And that incidentally lets us know that this kind of photoelectron spectroscopy is a very good technique to interrogate a solid surface because you will only eject electrons very near the surface of the specimen, and what that means is that you won't see stuff underneath. So in some cases, if you make an alloy or you make a material, what you find out is that the surface tends to be enriched in one kind of element, and the bulk in the center tends to be enriched in another kind. And that could be very important if you're designing parts that are going to fit together, and you think the surface has a certain kind of composition. And in fact, because some atoms prefer to be on the surface because of the way they bond, the composition is much, much different. And in that case, you can use photoelectron spectroscopy and you can do this ancient experiment. And since now all the work functions are known for all the atoms, you can easily figure out who's there. Usually, we know the wavelength of the light. And since the wavelength times the frequency is the velocity which for light is given the special symbol C, we can also write the energy in terms of the wavelength, and here's the equation. The energy of a photon is H times C over lambda. Finally, if you take very energetic photons like gamma rays, then the speed of the electron could approach C, the speed of light. And in that case, we have to use a different formula which I won't derive, but we have to use what's called the total relativistic energy which is given by this formula, E squared equals P squared C squared plus M squared C to the fourth. And I think you can see that if P vanishes, then the rest mass, if you have no other kinetic energy, is E squared equals M squared C to the fourth, and that's where E equals MC squared came from. M naught is just called the rest mass of the electron. It's the mass when it's not moving. And the relativistic momentum P is still just M times V, but M is not the rest mass but rather gets corrected by this formula that includes the ratio of the speed of the particle versus the speed of light. And interestingly, this formula also lets us figure out the momentum of a photon. So we start with the formula, and then we note that a photon has zero rest mass, and therefore, the energy is just E squared equals P squared C squared. And we can take the square root of that and find that the momentum is E upon C, and since light's quantized, that's H nu over C, and since C is the frequency times the wavelength, we just end up with E with, sorry, with P equals H over lambda, which is the momentum. And that means the short wavelength photons with small wavelength have high momentum. And here's an application of this phenomenon. Here's a spaceship that set sail a couple of years ago, and it's a giant solar sail. Of course, you don't have to worry about air resistance if you're out in the middle of space, and you can, with a reflective coating, you can actually use the momentum of the photons from the sun to steer your spaceship around, and you don't need any fuel or anything else, and you could just use the sun itself to push you around, and you can turn things this way and that. It's kind of an interesting application of the photon momentum, and I'll let you speculate about how fast you think that this spaceship could go in open space. Now, there is a thing to take note of, and that is, even if you're going pretty fast, you don't have to worry about relativity unless you're about 10% of the speed of light. If you're 10% of the speed of light or something like that, then you may have to start worrying a little bit about relativity, but normally in chemistry, we don't worry about relativistic corrections. So the only point of introducing this formula was to show that a photon, although it has no mass, has a momentum. So let's do a practice problem. So let's just confirm that the speeds that we quoted on the potassium photoelectric effect diagram are, in fact, the right speeds. So, well, we could use either the wavelength of the light or we could use the energy in electron volts. And as I told you, 1EV is 1.6 times 10 to the minus 19 joules. But since we have the work function for potassium in electron volts, I think that's probably going to be the easier course. So for the 2.2EV photon, the green light, we can set up our energy balance equation that the kinetic energy is the difference between the photon energy and the work function. And then we can solve for the velocity or the speed of the electron that comes out. And you notice that when I solve it, I put in the units. In chemistry, it is very, very important to put in the units and make sure all the units go away except the one that you want to get in the end. So I'm taking 2 times 0.25EV, dividing it by the mass of the electron in kilograms. I'm converting EV to joules. And then I'm remembering that a joule is a kilogram meter squared per second squared. I can remember that because I know that force is mass times acceleration. Acceleration, I can remember, is meters per second squared. So force is kilogram meter per second squared. And a joule is a Newton meter. And I can remember that too. And so I add another meter. And now I see the kilograms go away, the EVs go away, the joules go away, and I have the square root of meters squared per second squared, which is meters per second. Whenever you do a problem in chemistry, you want to analyze it in exactly this way. If you just write down numbers with no units, you're very likely going to have some funny units left over like the square root of EV over joule or something else. And without the units there to let you know that they didn't all fold up like a hat trick, you just get the wrong numerical answer. And if you write the wrong numerical answer on an exam or submit it in a report or build a bridge with it and it falls down, nobody is interested in why that happened. They're only interested in the mistake. So we went through this and you can see that the way I quote it is I keep a lot of digits and then I put the ones that I don't think are significant into parentheses. And then I can round it to give a 3 times 10 to the 5 meters per second. And the same thing with the violet light, you just put in a slightly different number. And again, you get 6.2 times 10 to the 5 meters per second. Always retain insignificant digits in case you want to continue the calculation further on. And never, ever round in the middle of a calculation. Your calculator will hold 12 digits, 12 digits. It will hold 15 digits. Everything's 15 digits. Never round and you can see in my examples, I take the time to write 1.602, et cetera. I look up the exact value because if I do a lot of calculation and I start rounding things here and there and everywhere, by the time I get to the end, my accuracy is poor. And sometimes if I'm unlucky, it can be very poor. So at least keep all the digits. People kill themselves trying to get those digits on those numbers, that was years of work. And to just say, well, I can't even be bothered to punch them into my calculator is really almost a crime. So in many experiments, light behaves like a wave. And the question is, if it behaves like a particle in these experiments, but it behaves like a wave with two slits and other things like that, then which is it? Is light a particle or is it a wave? And the answer is, it apparently depends on the nature of the experiment. Light itself seems to have both qualities at once. And even though we think of them as completely different kinds of things, apparently these two qualities are not mutually exclusive. But if light is a particle, then the thought is that if I shoot one particle at a time with the two slits, that the interference phenomenon would have to go away. Because the reason we were getting the interference phenomenon is we were getting all these waves going through together and then they would add and subtract. But if they aren't there simultaneously, if there's only one particle going through at a time, pick, pick, pick, pick, then we would see just two piles. You could either go through this slit, you get a pile here, go through this slit, you get a pile there. But interestingly enough, this fails. And so this two-slit experiment where you get an interference pattern, you get exactly the same pattern even if you can verify that you're shooting photons one at a time. So you shoot one photon, another, another, you detect where they end up. And at the end of the day, you get an interference pattern. And the only way you can really try to explain that is it seems like the photon, which you're claiming is a particle when it suits you, is now the particle is somehow slipping through both slits. In other words, the particle can interfere with itself. This is a very, very foreign idea in terms of what we understand in the physical world, where if we have a particle and we, the particle goes through one slit, we know it went through that slit. And it doesn't somehow break up and then reconstitute itself on the other side. And this was one of the most discomforting things about this new theory of quantum mechanics because it seemed to suggest something that was very foreign to our physical intuition and even foreign to our common sense notion about what a particle is. So if we fire one photon at a time and do photon counting, what we find is we have to fire a lot of photons to get good statistics. But when we do, if we have one slit, we get the pattern on the left. And if we have two slits, we get the pattern on the right. And we get the pattern on the right with two slits, whether we fire the photons one at a time and take forever to do it or whether we fire a bunch of them at once and they all go through. And that means that whatever the wave nature of light is like, it doesn't seem to have much to do with water waves. Even stranger, an electron has a certain amount of charge. It has a certain mass. And I've never seen half an electron. But if we fire electrons now from an electron gun, like an electron microscope, and we fire them one at a time, and then we look with two slits at what happens, we would expect to get, again, two piles of shot. If the electron went through here, it would go here and it could have a certain angle. We get a big pile. And then we'd get a pile over here for the ones that apparently went through this slit because we can't control exactly like a marksman where the electrons go. So they may go through. But whichever ones go through the slits, we should get two lumps. If you do this and you fire them one at a time again, what you find is that you do not get two lumps. So what you find is shown on the next slide. This is a brilliant experiment that was done at Hitachi using an electron microscope. And what you can see here is just how interesting it is because when you have 10 electrons, you have just 10 spots. And it looks to me like they may have slightly miscounted or there may be a glitch because you can see 11 if you look closely. But anyway, you get 10 spots and you notice that when the electrons hit the screen, you get a spot as if it were a very tiny particle. And the slits are much farther apart than any kind of width of these dots. And then when you do 200 electrons, you get a shotgun pattern. And then when you do a lot more, you start to see ridges like a wave. And then when you finally do hundreds of thousands of electrons, you see this clear kind of tin roof appearance of the pattern of intensity which even though you shot the electrons through one at a time, seems to be indicating that each electron goes through both slits. This is even worse than the photon because I don't have any particular picture about a photon. It was a mathematical thing that came up. That doesn't bother me maybe too much. But I certainly do have a picture of an electron. And if the electron is interfering with itself, the question you might ask is, well, which slit did the charge go through? Which slit did the mass go through? Why is it that whenever I look at an electron, I see exactly the same mass and exactly the same charge. And then when I fire them through these two slits, I do not. And the short answer to that is that if you look at which slit the electron goes through, you get two lumps of shot. So if you try to intercept the electron and you try to see it by looking, then you get two lumps of shot and the electron says, okay, you're going to look at what I'm doing. I'm going to go through with the left slit or the right slit. That's it. But if you don't look, which of course they were not looking, they had two slits set up, they fire electrons. If you look, normally we think, well, if I want to see the rug, I want to see the monitor, I want to see whatever, I simply look at it. But in fact, what's happening is we've got light on. We've got light on and because I'm so heavy and the light's so light, it's not moving me around. It's not doing anything to me. But in fact, if I've got an electron going through a slit, I can't just see it. It's too tiny. I need to shine some light on it. And when I shine light on it, I know the photon can even kick an electron out of a metal. And so the photon interacts with the electron and changes it. And so by trying to observe where it is, I actually change the nature of the experiment. And so it's very frustrating because when I don't look, it does something incredible that I can hardly believe. And when I do look, it behaves exactly the way I would have thought that it would be behaving. And what we'll do then is we'll close up there for this lecture. And in the next lecture, what I want to talk about is the connection that a man by the name of De Broglie made between the wavelength of these particles and the wavelength of light, which seemed to be a major advance. And it's kind of interesting that that's the one thing he did. And then he did that, and that was great. And then he never did much else after that. Okay, thanks very much.
then an introduction to quantum mechanics as seen training. What's training in chemistry is quantum mechanics is the study of the small frequency is not high compared to the temperature particles of light were coming in, and each energy and the energy to pry it out of the equation that the kinetic energy is the with itself. This is a very, very foreign idea in