Lecture 01. General Course Information and Introduction to Quantum Mechanics
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Lecture 01. General Course Information and Introduction to Quantum Mechanics
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Lecture 01. General Course Information and Introduction to Quantum Mechanics

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Lecture 01. Quantum Principles: General Course Information and Introduction to Quantum Mechanics

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1

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28

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2014

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English

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UCI Chem 131A Quantum Principles (Winter 2014) Instructor: A.J. Shaka, Ph.D Description: 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. Index of Topics: 0:05:31 Light 0:12:05 Quantization 0:19:10 The Photoelectric Effect 0:28:59 Photon Momentum

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hi and welcome to can 131 a physical chemistry I'm Dr. shocker and I'll be leading you through the series of lectures on physical chemistry starting with an Adams up approach of quantum mechanics is our 1st topic which is kind of a rude introduction to the subject but here we go again some general cost 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 the here's a preliminary problem practice problem 1 here is a jumbled word and the question is what word can you make indeed another 5 letters you might find it pretty hard but the answer is you can make to 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 are you 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 the quite similar we have to rearrange symbols in ways that are legal and we have to somehow seek where to go that means 2 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 moving and I will assume that you've Fred in the book the background material the fundamentals and if you haven't done that your 1st task right we hear some general information the lecture in attendance is optional except on exam days will have 1 problem do each Friday and we only have 1 problem because they're quite hard actually and we have a website and on it will have what's new but avoid sending me email just ask person after class and warnings do not start the problem on Thursday evening please realize that 1 problem and its multiple parts and it's really quite difficult actually completed and so have a go clarify your understanding and then try again the TAC and I'll go over some of the problems at the end of the chapter and clarify any ambiguous wording of the problem all were counts for 20 per cent of the great in this class I'm not a big fan of making exams count a lot Doby to 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 quantum matter and change by Atkins DePaul and Friedman and will cover the 1st 5 chapters but I 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 of who did what was 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 1 quantum mechanics is the study of the small specifically things that we can't see mobutu closely the world appears to be digital in other words the small packets of everything and Adams are the smallest unit of a particular element of course even the ancient Greeks realized that it might be true that things had 1 the smallest indivisible amount now we know that that is in fact true but they're very very small and very light 1 neutral carbon carbon 12 Adam has a massive only about 2 times minus 23 grams electron is much much smaller has a massive 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 much human intuition is guided by things that 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 1 proton or not but you can't have passed and you can't have Hassan electron this is pretty much similar to currency there's a minimal amount of currency it's different for every currency but there's still a minimum amount in the U.S. system that's 1 cent so you can have 1 cent but you can't have anything smaller and have been legitimate currency light heat Newton actually believed that light also had a currency that there was a minimum amount of light was corpus because light seem 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 Hoy against cousin really to abandon his cough muscular 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 at a particle can either be there be positive or 0 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 freefire them through so for example here's a picture of 2 slightly different wavelengths of 0 4 0 away and if we add them up you can see that their positions where In space here because this is wavelength there is a very small response and then there is another place where there's a very large response and then there's a small response and and this is very much similar to each but sound waves for example where the tune of violin and you hear the difference when you aren't in tune you hear this wall while while sort of beating and you're seeing in this picture that kind of beating occurring and it's a universal phenomenon with weight white light it turns out has a mixture of different wavelengths and that was sort of 1st 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 some have come from the president and so is only really when you took another prism and then took the same rainbow and really
08:01
turned it into a white light that people became convinced that white light was really a mixture of colors and there was nothing coming necessarily from the prison but when you have a convex surface near of flat surface there is a difference in refractive index and the condition for these deeds 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 spilled oil which tends to beat up on water on a wet surface and we used to always look about as a kid because cars in Salt Lake City always have 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 is an example we can see how the colors very systematically with wavelength and much like there were to be eating patterns In the end that graphite showed you this will repeat more than once depending on the total thickness of of by contrast the media there's another experiment that unfortunately made it difficult to understand why this waves and that was a black body radiation this is a classic thing where you think you understand everything and you have a very simple calculations and there are some very very smart people like Lord Rayleigh and you do the calculation and many compare with the experiment this is the essence of science if you think you understand something you ought to be able to predicted or explain why you can't predicted but at this time there was Maxwell's equations pretty much people believe that these wave equations really described life in totality and that light was an electromagnetic wave and that theory was wildly so we could explain the unification of electricity and magnetism why the speed of light hazards Speed diffraction reflection refraction how lenses and so on and so forth but the 2 crucial experiments show that this description of light was complete incomplete and the first one was black body radiation so what is black body radiation well if I take a black body by which he could just mean a lump of coal a lamp black and heated up in a vacuum so that there's no error doesn't burst into flames it'll and this global 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 redhot and things like that and greatly with the correction by genes calculated the spectrum of waiflike life and what they found is that it would be much much more likely that you would get a lot of highfrequency 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 other than outcome a town of Xrays and very high frequency light and kill us basically and that's obviously not what happened so we thought there was a big big enough soulsearching and there was a lot of thought about what might be going on in some theories were put forward that were later proven not to be quite right but it was Max plant that found the correct solution but what what he found that 50 observations although he didn't really quite believe it himself but he could follow through physics was the light was quantized that is there is 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 equal stage New Age has subsequently been called plunks constant honor it was discovered so the frequency could apparently be continuous anything you want but what she chooses then there is a minimum amount and if you don't have the minimum amount there is no light and since the minimum amount depends on the frequency higher frequencies if there's
13:08
not enough energy around can't have the minimum amount to make a single particle for Quantum of light and therefore those frequencies get cut off so far that gets around this problem with the Xrays killing us if we open the other so here is an equation that just says the likelihood that you're going to have a certain amplitude of light on a number of relative amount of light perunit frequency given according to the really Gene's law and you can see but it's basically a parabola In the frequency squared so as the frequency goes up the amount of light that's predicted predicted to go up faster and faster and faster and that's not what's observed and what plant did I'm in instead of this continuous equation Is there the quantized photon energy and he obtained this formula which on 1st blush looks completely different it hasn't hue In the numerator and has this funny exponential of each new upon T In the denominator and that we can compare these 2 formulas an autograph and see what happens and in again this formula leverage let me remarked that case Boltzmann constant pages plants constant and tedious the temperature Calvin and of course in physical chemistry or chemistry of any time you never "quotation mark temperature in anything other then Calvin because if you do you're very likely to be wrong if you plug it into a formula is a "quotation mark the temperature and Calvin and you say it's a balmy days 298 Kelvin you may be eccentric but you're never wrong but in chemistry if use any other units you're very likely going to be wrong so you have to be careful we hear is a graph part is in the green and rarely James In the sort of pink color and you can see that 4 a certain temperature of 300 Calvin they agree had very very low frequency where the quantise Asian is very small and is the frequency increases plant actually follows the observed distribution almost exactly and really genes diverges mourned warned more from and would continue to go up now we can make a connection between the 2 theories because we know they're greedy when the frequency is is small and the energy slow and so we can take this parameter H. new upon Katie and we can assume it's much much less than 1 and we know from calculus that we can expand the to the accident power series 1 plus sex and then affects is small the next square is very small so we can throw it away and that means we can write the 2 B H new upon Katie is 1 plus additional upon Katie and then if we make that substitution we find that as long as the frequency is not high compared with the temperature Katie that we get exactly the same formula that really jeans go but if the temperature is the frequency is hot 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 now the physical meaning of Katie is the K T is really a measure of the random thermal energy that's available at temperature T 1 something's hot things are moving their colliding there banging around and there's lots of energy available to excite things and to create photon 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 go well but things that are heated up finally do start to amid a global like an electric element ,comma still so at high frequency the problem is we just don't have enough energy to make even a single photon we just keep run out before we get there and so the distribution has to fall off sharply and the analogy I can give you is that supposed the smallest amount of currency the smallest coins were a thousand dollars 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 life in daytoday observation we don't notice that it looks like Sanders something like that is because of the relative size of these 2 constants K and age bowls miscast in this 1 times 10 minus 23 jewels Patel's and plants constant is 6 times demise 34 and that means there is a factor of about 10 of the 11 or 100 billion between them and that means that usually at low frequency there are plenty of photons around and highfrequency we start to notice that age knew the quantum of light has a minimum value the other experiment that really sealed the deal with respect to the core postulated or quantized nature of life 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 evacuated chamber and you have a clean metal surface and you shine a light on the surface and what was observed is that electrons would come off the surface and this is how
19:48
you make a cathode ray tube in fact but electric electrons would come off the surface of the metal and would come into the vacuum and they would be ejected and ideally flight were than what should happen is if you have a wave coming in and it should sort of excited 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 rejected you can measure that by chopping the lights 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 3 things 1st of all the photo electrons when they came out were ejected essentially instantaneously so there was no delay the 2nd point is that below a threshold frequency there were no photo electrons at all In the 3rd point is that turning up the intensity of a lowfrequency lighting made no difference is 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 1 particle hits 1 electron and it's that 1 particle that hits the electron doesn't have enough votes to kick the electron out then the electron doesn't come out and having a lot of particles none of which can weigh on a single here hit the electron out doesn't help you you need 1 particle that has not loans instigated in 1 Our goal and he found that the particle an energy exactly in accordance with plants formula so now we have another experiment which is indicating that light depending on its frequency has a quantized energy and that that we call that a photon and we think of it as a particle so as I said if 1 photon hit electron the energy is good enough kicks it otherwise the electron stays in the metal but here's an idealized view of the experiment I apologize for the kind of gray of the potassium is hard to see that this is a potassium metal used because it's very easy to kick electrons out and their 3 wavelengths and the energy source quoted to an electron volts and electron volt is the energy to that 1 electron gets by being dropped through 1 bolt of potential difference and its 1 . 6 times 10 months 19 jewels if we have 700 enemy light which is red we get no electrons if we shine in green light at 550 we get electrons to come out and they come out with a maximum speed that we can measure by timing when they get a detector of about 3 times to the 5 meters per 2nd and if we use more energetic light toward the pilot and of the visible spectrum then we also get full of electrons out instantaneously but now the owner at the speed of the electrons is higher so the electrons have more so here's a diagram that shows the energy balance it takes a certain amount of energy In this case for potassium to electron volts 2 get the electrons that the part ways from the potassium atoms and then whatever energies
24:04
left over since and we believe energy is conserved even in this crazy you realm of quantum mechanics that energy must be done 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 Bark then there is no way that the electron is going to be the objective yeah and the energy fires called the work functions of the metal it's too bolts for potassium but not for other metals and the kinetic energy as I showed us just the difference so we can express that equations 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 onehalf squared is equal to the difference and so Arkansas for V as the square root of 2 Asian minus5 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 faced is not up to the Red bars it's less than 5 that 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 it in this case when we interpreted as a velocity it we'd have to figure out what an imaginary velocity actually meant in terms of what we would see no it turns out that we could 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 rejects the electron the potassium the silvery surface almost like a mirror and so we could get us and we'll be right the light doesn't really penetrate through like a window and come out the other side is so if we ejected electron from an atom further down the electron might come up and yet another Adam and
26:14
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 photo electron spectroscopy is a very good technique to interrogate a solid surface because you will only attract 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 area maker material but you find out is that the surface tends to be enriched in 1 kind of element and the ball In the Senate tends to be enriched in another kind and that could be very important if you're designing parts that are gonna fit together and you think the surface has a certain kind of composition and in fact because some Adams preferred beyond the surface because of the way they bombed the composition is much much different in that case you can use photo electron spectroscopy and you can do this ancient experiment and since now all 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 but insists the wavelength times the frequency is the velocity which relied is given the special symbols C we can also write the the energy in terms of the wavelength and here's the equation the energy of the photons is age time see over finally if you take very energetic photons like Cameron then the speed of the electron could approach see the speed of light and in that case we had to use a different formula but which I won't arrive but we have to use what's called the toll relativistic energy which is given by this formula he squared equals peace PSquare C squared plus and squared seated before and I think you can see that it if P vanishes then the rest mass if you have no other kinetic energy Is he squared equals and square feet at the 4th and that's where equals mc squared came from I am not it's just call the rest mass of the electron cements when it's not moving and the relativistic momentum P is still just sometimes the but and this 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 we note that a photon has 0 rest mass and therefore the energy is just the squared equals peace Corps he squared and we can take the square root of that and find that the momentum this year upon and since lights quantized that's H overseas and since Sears the frequency times a wavelength we just end up with the With sorry with P equals aged over land which is the momentum and that means that short wavelength photons with small wavelength I have high momentum and there's an application the proposal this phenomenon here a spaceship that set sail a couple years ago and it's a giant solar sail across you don't have to worry about their resistance if you're out in the middle of space and you can weather 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 you could just use the sun itself to push you around and you can turn things this way and that kind of an interesting application of the photon momentum and I'll let you speculate about how fast you think that the spaceship could go in open space now there is a saying to take note of the matter is even if you're going pretty fast you don't have to worry about relativity unless year about 10 per cent Of the speed of light if you're 10 per cent of the speed of light or something like that then you may have to start wearing a little bit about relativity but normally found in chemistry we don't worry about relativistic corrections so the only point of introducing this formula was to show that the photon although it has no mass has a moment the Leicester practice problems solicitors confirmed that the speeds that we quoted on the potassium photoelectric effect diagrams are in fact the right speeds well we could use either the wavelength of the light or we could use the energy in electron and as I told you 1 is 1 . 6 times and my mine minus 19 jewels but since we have the work function for potassium in electron volts I think that's probably going to be easier course so for the 2 . 2 TV photons the green light we can set up our energy balance equation that the kinetic energy is the difference between the photon energy when the work function and then we can solve for the velocity of the speed of the electron comes out and you notice that when I assaulted 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 1 that you want to get me and so taking 2 times . 2 5 Evie dividing it by that the mass of the electron and kg and converting the lead the jewels and then I'm remembering that jewel is a kilogram meters squared per 2nd squared I can remember that because I know that forces mass times acceleration acceleration I can remember is meters per 2nd squared so forces kg meter per 2nd squared and Julissa Newton needed and I can remember that too and so I have another meeting and I see the kg go away these go away the jewels go away and I have the square root of meters squared per 2nd squared which is meters per 2nd 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 a very likely going to have some funny units left over like the square root of the view over jeweler something else and without the units there to let you know that they did not fold up like a hat trick you just get the wrong numerical answer and if you're right the wrong numerical answer on an exam or submitted in a report or build a bridge with it and falls down nobody is interested in why that happened there are only interested in the state so we went through this and you can see that the way I quoted as I keep
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a lot of digits and then I put the ones that I don't think a significant into parentheses and then I can rounded together 3 times 10 to the 5 meters per 2nd and the same thing with the violet light you just put in a slightly different numbers and again on leaders 6 . 2 times 10 to the 5 meters per 2nd always retained in significant digits in case you want to continue the calculation further on and never ever round in the middle of a calculations your calculator will hold took 12 digits 12 digits 5th whole 15 digits everything's 15 digits never around and you can see in my examples I take the time to write 1 . 6 0 to etc. 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 fell years of work and to just say Well I can't even be bothered to punch and my calculator is really almost a crime so in many experiments like behaves like a and the question is if it behaves like a particle in these experiments but it behaves like a wave was too slow and other things like that which is it like a particle or it away and the answer is it apparently depends on the nature of the experiments lend itself seems to have both qualities at once and even tho we we think of them as completely different kinds of things apparently these 2 qualities are not mutually exclusive but if light is a particle then the thought is that if I shoot 1 particle at a time With the 2 sled that the interference phenomena would have to go away because the reason we're getting interference phenomena as we're getting all these waves going through together and then they would add subtract but if they are aren't there simultaneously if there's only 1 particle going through a time pick pick pick pick then we would see just 2 piles did you go through this let you get a pile here go through this let pile there but interestingly enough this fails and so on this to such experiment where you get an interference pattern you get exactly the same pattern even if you can verify the issuing photons 1 at a time so you shoot 1 photon another another you detect where they end up and at the end of the day but you get an interference pattern and the only way you can really try to explain that is it seems like the photon Witcher claiming it a particle when it suits you it is now the particle is somehow slipping through both slipped 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 it once led we know it went through that slept and it doesn't somehow a breakup and then reconstitute itself on the other side and this was 1 of the most discomfiting things about this new theory of quantum mechanics because it seemed to suggest something that was very foreign to our physical intuition and art even foreign to our commonsense notion about what a particle hits so if we fire up 1 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 1 slip we get the pattern on the left and if we have 2 to slits we get the patent on the right and we get the pattern on the right with 2 slits whether we fire the photons 1 at a time and take forever to do it or whether we fire a bunch of them at once and and they all go through and that means that whatever the wave nature of light it's like it doesn't seem to have much to do with water even stranger an electron as a certain amount of charge it has a certain Mass and I've never seen half an electron but if we fire electrons no From an electron guns like an electron microscope and we fire them 1 at a time and then we lost quick to what happens we would expect to get again to piles of shot of the electron went through here it would go here and at a certain angle we get a big pile and then we get a pile over here for the ones that apparently went through this list because we can't control exactly like a marksman where the electrons go so they may go through but which are ones go through the slats we should get too what if you do this and you fire them 1 at a time again but what you find is that you do not get to lumps so what you find in it 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 there may be a glitch because the 11 if you look closely but anyway you get 10 spots and you notice that when the electrons hit the screen you get us a spot as if it were a very tiny article In this slits are much farther apart then those of any kind of work vs . and that when he do 200 electrons is 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 10 roof the appearance of the pattern of intensity which even earlier shot the electrons through 1 at a time seems to be indicating that each electrons 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 came up but 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
42:02
which slid did the charge go through which led to the mass through why is it that I whenever I look at electron I see exactly the same manner in exactly the same charge and then when I fire them through these 2 sleds I do not and the short answer to that is that if you look at which slipped the electron goes through you get to lumps of shops so if you tried to intercept the electron and you try to see it looking damn you get to lumps of shot and the electron says OK you're gonna look at what I'm doing I'm going to go through with the last letter the rights slip that's it but if you don't look which a cost they were not looking ahead to set up by electrons if you look normally we think well but I see the ride I wanna see the monitor I wanna see whatever I simply look at it but in fact what's happening is we've got
43:11
lined up we've got light on and because I'm so heavy in the light 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 slip I can just see it it's too tiny I need to shine some light on and when I shine a light on it I know the photon can even kickin electron out of a metal and so the photon interacts with the electron and changes and so by trying to observe where it is I actually change the nature of the experiment and a very frustrating because when I don't look it does something incredible but I can hardly believe it when I do look it behaves exactly the way I would have thought that it would be behaved and what will do then and is will close out their effort this lecture and in the next lecture what I want to talk about is the connection that a man by the name of the 2 broadly made between the wavelength of these particles and the wavelength of light which seemed to be a major advance in on this kind of interesting that that's that's the 1 thing he did and he did that and that was great men who never did much else after that OK thanks very much