id
int64 0
249
| text
stringlengths 31.9k
74k
| course
stringclasses 32
values | topic
stringclasses 2
values |
|---|---|---|---|
0
|
PROFESSOR: All right. So today's task is going to be to outline some of the basic experimental facts that we will both have to deal with and that our aim should be to understand and model through the rest of the course. Physics doesn't tell you some abstract truth about why the universe is the way it is. Physics gives you models to understand how things work and predict what will happen next. And what we will be aiming to do is develop models that give us an intuition for the phenomena and allow us to make predictions. And these are going to be the experimental facts I would like to both explain, develop an intuition for, and be able to predict consequences of. So we'll start off with-- so let me just outline them. So, first fact, atoms exist. I'll go over some of the arguments for that. Randomness, definitely present in the world. Atomic spectre are discrete and structured. We have a photoelectric effect, which I'll describe in some detail. Electrons do some funny things. In particular electron diffraction. And sixth and finally, Bell's Inequality. Something that we will come back to at the very end of the class, which I like to think of as a sort of a frame for the entirety of 8.04. So... we'll stick with this for the moment. So everyone in here knows that atoms are made of electrons and nuclei. In particular, you know that electrons exist because you've seen a cathode ray tube. I used to be able to say you've seen a TV, but you all have flat panel TVs, so this is useless. So a cathode ray tube is a gun that shoots electrons at a phosphorescent screen. And every time the electron hits the screen it induces a little phosphorescence, a little glow. And that's how you see on a CRT. And so as was pithily stated long ago by a very famous physicist, if you can spray them, they exist. Pretty good argument. There's a better argument for the existence of electrons, which is that we can actually see them individually. And this is one of the most famous images in high-energy physics. It's from an experiment called Gargamelle, which was a 30-cubic meter tank of liquid freon pulsing just at its vapor pressure 60 times a second. And what this image is is, apart from all the schmut, you're watching a trail of bubbles in this de-pressurizing freon that wants to create bubbles but you have to nucleate bubbles. What you're seeing there in that central line that goes up and then curls around is a single electron that was nailed by a neutrino incident from a beam at CERN where currently the LHC is running. And this experiment revealed two things. First, to us it will reveal that you can see individual electrons and by studying the images of them moving through fluids and leaving a disturbing wake of bubbles behind them. We can study their properties in some considerable detail. The second thing it taught us is something new-- we're not going to talk about it in detail-- is that it's possible for a neutrino to hit an electron. And that process is called a weak neutral current for sort of stupid historical reasons. It's actually a really good name. And that was awesome and surprising and so this picture is both a monument to the technology of the experiment, but also to the physics of weak neutral currents and electrons. They exist if you can discover neutrinos by watching them. OK. Secondly, nuclei. We know that nuclei exist because you can shoot alpha particles, which come from radioactive decay, at atoms. And you have your atom which is some sort of vague thing, and I'm gonna make the-- I'm gonna find the atom by making a sheet of atoms. Maybe a foil. A very thin foil of stuff. And then I'm gonna shoot very high-energy alpha particles incident of this. Probably everyone has heard of this experiment, it was done by Rutherford and Geiger and Marsden, in particular his students at the time or post-docs. I don't recall-- and you shoot these alpha particles in. And if you think of these guys as some sort of jelly-ish lump then maybe they'll deflect a little bit, but if you shoot a bullet through Jello it just sort of maybe gets deflected a little bit. But Jello, I mean, come on. And I think what was shocking is that you should these alpha particles in and every once in a while, they bounce back at, you know, 180, 160 degrees. Rutherford likened this to rolling a bowling ball against a piece of paper and having it bounce back. Kind of surprising. And the explanation here that people eventually came up upon is that atoms are mostly zero density. Except they have very, very high density cores, which are many times smaller than the size of the atom but where most of the mass is concentrated. And as a consequence, most of the inertia. And so we know that atoms have substructure, and the picture we have is that well if you scrape this pile of metal, you can pull off the electrons, leaving behind nuclei which have positive charge because you've scraped off the electrons that have negative charge. So we have a picture from these experiments that there are electrons and there are nuclei-- which, I'll just write N and plus-- which are the constituents of atoms. Now this leads to a very natural picture of what an atom is. If you're a 19th-century physicist, or even an early 20th-century physicist, it's very natural to say, aha, well if I know if I have a positive charge and I have a negative charge, then they attract each other with a 1 over r minus q1 q2-- sorry, q1 q2 over r potential. This is just like gravity, right. The earth and the sun are attracted with an inverse-r potential. This leads to Keplerian orbits. And so maybe an atom is just some sort of orbiting classical combination of an electron and a nucleus, positively charged nucleus. The problem with this picture, as you explore in detail in your first problem on the problem set, is that it doesn't work. What happens when you accelerate a charge? It radiates. Exactly. So if it's radiating, it's gotta lose energy. It's dumping energy into this-- out of the system. So it's gotta fall lower into the potential. Well it falls lower, it speeds up. It radiates more. Because it's accelerating more to stay in a circular orbit. All right, it radiates more, it has to fall further down. So on the problem set you're going to calculate how long that takes. And it's not very long. And so the fact that we persist for more than a few picoseconds tells you that it's not that-- this is not a correct picture of an atom. OK. So in classical mechanics, atoms could not exist. And yet, atoms exist. So we have to explain that. That's gonna be our first challenge. Now interestingly Geiger who is this collaborator of Rutherford, a young junior collaborator of Rutherford, went on to develop a really neat instrument. So suppose you want to see radiation. We do this all the time. I'm looking at you and I'm seeing radiation, seeing light. But I'm not seeing ultra high energy radiation, I'm seeing energy radiation in the electromagnetic waves in the optical spectrum. Meanwhile I'm also not seeing alpha particles. So what Geiger wanted was a way to detect without using your eyes radiation that's hard to see. So the way he did this is he took a capacitor and he filled the-- surrounded the capacitor with some noble gas. It doesn't interact. There's no-- it's very hard to ionize. And if you crank up the potential across this capacitor plate high enough, what do you get? A spark. You all know this, if you crank up a capacitor it eventually breaks down because the dielectric in between breaks down, you get a spontaneous sparking. So what do you figure it would look if I take a capacitor plate and I charge it up, but not quite to breakdown. Just a good potential. And another charged particle comes flying through, like an alpha particle, which carries a charge of plus 2, that positive charge will disturb things and will add extra field effectively. And lead to the nucleation of a spark. So the presence of a spark when this potential is not strong enough to induce a spark spontaneously indicates the passage of a charged particle. Geiger worked later with-- Marsden? Muller. Heck. I don't even remember. And developed this into a device now known as the Geiger counter. And so you've probably all seen or heard Geiger counters going off in movies, right. They go ping ping ping ping ping ping ping ping ping, right. They bounce off randomly. This is an extremely important lesson, which is tantamount to the lesson of our second experiment yesterday. The 50-50, when we didn't expect it. The white electrons into the harness box then into a color box again, would come out 50-50, not 100 percent. And they come out in a way that's unpredictable. We have no ability to our knowledge-- and more than our knowledge, we'll come back to that with Bell's Inequality-- but we have no ability to predict which electron will come out of that third box, white or black, right. Similarly with a Geiger counter you hear that atoms decay, but they decay randomly. The radiation comes out of a pile of radioactive material totally at random. We know the probabilistic description of that. We're going to develop that, but we don't know exactly when. And that's a really powerful example-- both of those experiments are powerful examples of randomness. And so we're going to have to incorporate that into our laws of physics into our model of quantum phenomena as well. Questions? I usually have a Geiger counter at this point, which is totally awesome, so I'll try to produce the Geiger counter demo later. But the person with the Geiger counter turns out to have left the continent, so made it a little challenging. OK. Just sort of since we're at MIT, an interesting side note. This strategy of so-called hard scattering, of taking some object and sending it at very high velocity at some other object and looking for the rare events when they bounce off at some large angle, so-called hard scattering. Which is used to detect dense cores of objects. It didn't stop with Rutherford. People didn't just give up at that point. Similar experience in the '60s and '70s which are conducted at Slack, were involved not alpha particles incident on atoms but individual electrons incident on protons. So not shooting into the nucleus, but shooting and looking for the effect of hitting individual protons inside the nucleus. And through this process it was discovered that in fact-- so this was done in the '60s and '70s, that in fact the proton itself is also not a fundamental particle. The proton is itself composite. And in particular, it's made out of-- eventually people understood that it's made out of, morally speaking, and I'm gonna put this in quotation marks-- ask me about it in office hours-- three quarks, which are some particles. And the reason we-- all this tells you is that it's some object and we've given it the name quark. But indeed there are three point-like particles that in some sense make up a proton. It's actually much more complicated than that, but these quarks, among other things, have very strange properties. Like they have fractional charge. And this was discovered by a large group of people, in particular led by Kendall and Friedman and also Richard Taylor. Kendall and Friedman were at MIT, Richard Taylor was at Stanford. And in 1990 they shared the Nobel Prize for the discovery of the partonic structure out of the nucleons. So these sorts of techniques that people have been using for a very long time continue to be useful and awesome. And in particular the experiment, the experimental version of this that's currently going on, that I particularly love is something called the relativistic heavy ion collider, which is going on at Brookhaven. So here what you're doing is you take two protons and you blow them into each other at ultra high energy. Two protons, collide them and see what happens. And that's what happens. You get massive shrapnel coming flying out. So instead of having a simple thing where one of the protons just bounces because there's some hard quark, instead what happens is just shrapnel everywhere, right. So you might think, well, how do we interpret that at all. How do you make sense out of 14,000 particles coming out of two protons bouncing into each other. How does that make any sense? And the answer turns out to be kind of awesome. And so this touches on my research. So I want to make a quick comment on it just for color. The answer turns out to be really interesting. First off, the interior constituents of protons interact very strongly with each other. But at the brief moment when protons collide with each other, what you actually form is not a point-like quirk and another point-like quark. In fact, protons aren't made out of point-like quarks at all. Protons are big bags with quarks and gluons and all sorts of particles fluctuating in and out of existence in a complicated fashion. And what you actually get is, amazingly, a liquid. For a brief, brief moment of time the parts of those protons that overlap-- think of them as two spheres and they overlap in some sort of almond-shaped region. The parts of those protons that overlap form a liquid at ultra high temperature and at ultra high density. It's called the RHIC fireball or the quark-gluon plasma, although it's not actually a plasma. But it's a liquid like water. And what I mean by saying it's a liquid like water, if you push it, it spreads in waves. And like water, it's dissipative. Those waves dissipate. But it's a really funny bit of liquid. Imagine you take your cup of coffee. You drink it, you're drinking your coffee as I am wont to do, and it cools down over time. This is very frustrating. So you pour in a little bit of hot coffee and when you pour in that hot coffee, the system is out of equilibrium. It hasn't thermalized. So what you want is you want to wait for all of the system to wait until it's come to equilibrium so you don't get a swig of hot or swig of cold. You want some sort of Goldilocks-ean in between. So you can ask how long does it take for this coffee to come to thermal equilibrium. Well it takes a while. You know, a few seconds, a few minutes, depending on exactly how you mess with it. But let me ask you a quick question. How does that time scale compare to the time it takes for light to cross your mug? Much, much, much slower, right? By orders of magnitude. For this liquid that's formed in the ultra high energy collision of two protons, the time it takes for the system-- which starts out crazy out of equilibrium with all sorts of quarks here and gluons there and stuff flying about-- the time it takes for it to come to thermal equilibrium is of order the time it takes for light to cross the little puddle of liquid. This is a crazy liquid, it's called a quantum liquid. And it has all sorts of wonderful properties. And the best thing about it to my mind is that it's very well modeled by black holes. Which is totally separate issue, but it's a fun example. So from these sorts of collisions, we know a great deal about the existence of atoms and randomness, as you can see. That's a fairly random sorting. OK so moving on to more 8.04 things. Back to atoms. So let's look at specifics of that. I'm not kidding, they really are related to black holes. I get paid for this. So here's a nice fact, so let's get to atomic spectra. So to study atomic spectra, here's the experiment I want to run. The experiment I want to run starts out with some sort of power plant. And out of the power plant come two wires. And I'm going to run these wires across a spark gap, you know, a piece of metal here, a piece of metal here, and put them inside a container, which has some gas. Like H2 or neon or whatever you want. But some simple gas inside here. So we've got an electric potential established across it. Again, we don't want so much potential that it sparks, but we do want to excite the H2. So we can even make it spark, it doesn't really matter too much. The important thing is that we're going to excite the hydrogen, and in exciting the hydrogen the excited hydrogen is going to send out light. And then I'm going to take this light-- we take the light, and I'm gonna shine this on a prism, something I was taught to do by Newton. And-- metaphorically speaking-- and look at the image of this light having passed through the prism. And what you find is you find a very distinct set of patterns. You do not get a continuous band. In fact what you get-- I'm going to have a hard time drawing this so let me draw down here. I'm now going to draw the intensity of the light incident on the screen on this piece of paper-- people really used to use pieces of paper for this, which is kind of awesome-- as a function of the wavelength, and I'll measure it in angstroms. And what you discover is-- here's around 1,000 angstroms-- you get a bunch of lines. Get these spikes. And they start to spread out, and then there aren't so many. And then at around 3,000, you get another set. And then at around 10,000, you get another set. This is around 10,000. And here's the interesting thing about these. So the discovery of these lines-- these are named after a guy named Lyman, these are-- these are named after a guy named-- Ballmer. Thank you. Steve Ballmer. And these are passion, like passion fruit. So. Everyone needs a mnemonic, OK. And so these people identified these lines and explained various things about them. But here's an interesting fact. If you replace this nuclear power plant with a coal plant, it makes no difference. If you replace this prism by a different prism, it makes no difference to where the lines are. If you change this mechanism of exciting the hydrogen, it makes no difference. As long as it's hydrogen-- as long as it's hydrogen in here you get the same lines, mainly with different intensities depending upon how exactly you do the experiment. But you get the same position of the lines. And that's a really striking thing. Now if you use a different chemical, a different gas in here, like neon, you get a very different set of lines. And a very different effective color now when you eyeball this thing. So Ballmer, incidentally-- and I think this is actually why he got blamed for that particular series, although I don't know the history-- Ballmer noticed by being-- depending on which biography you read-- very clever or very obsessed that these guys, this particular set, could be-- they're wavelengths. If you wrote their wavelengths and labeled them by an integer n, where n ran from 3 to any positive integer above 3, could be written as 36. So this is pure numerology. 36, 46 angstroms times the function n squared over n squared minus 4, where N is equal to 3, 4, dot dot dot-- an integer. And it turns out if you just plug in these integers, you get a pretty good approximation to this series of lines. This is a hallowed tradition, a phenomenological fit to some data. Where did it come from? It came from his creative or obsessed mind. So this was Ballmer. And this is specifically for hydrogen gas, H2. So Rydberg and Ritz, R and R, said, well actually we can do one better. Now that they realized that this is true, they looked at the whole sequence. And they found a really neat little expression, which is that 1 over the wavelength is equal to a single constant parameter. Not just for all these, but for all of them. One single numerical coefficient times 1 over m squared minus 1 over n squared-- n is an integer greater than zero and greater in particular than m. And if you plug in any value of n and any value of m, for sufficiently reasonable-- I mean, if you put in 10 million integers you're not going to see it because it's way out there, but if you put in or-- rather, in here-- if you put any value of n and m, you will get one of these lines. So again, why? You know, as it's said, who ordered that. So this is experimental result three that we're going to have to deal with. When you look at atoms and you look at the specter of light coming off of them, their spectra are discrete. But they're not just stupidly discrete, they're discrete with real structure. Something that begs for an explanation. This is obviously more than numerology, because it explains with one tunable coefficient a tremendous number of spectral lines. And there's a difference-- and crucially, these both work specifically for hydrogen. For different atoms you need a totally different formula. But again, there's always some formula that nails those spectral lines. Why? Questions? OK. So speaking of atomic spectra-- whoops, I went one too far-- here's a different experiment. So people notice the following thing. People notice that if you take a piece of metal and you shine a light at it, by taking the sun or better yet, you know, these days we'd use a laser, but you shine light on this piece of metal. Something that is done all the time in condensed matter labs, it's a very useful technique. We really do use lasers not the sun, but still it continues to be useful in fact to this day. You shine light on a piece of metal and every once in a while what happens is electrons come flying off. And the more light and the stronger the light you shine, you see changes in the way that electrons bounce off. So we'd like to measure that. I'd like to make that precise. And this was done in a really lovely experiment. Here's the experiment. The basic idea of the experiment is I want to check to see, as I change the features of the light, the intensity, the frequency, whatever, I want to see how that changes the properties of the electrons that bounce off. Now one obvious way-- one obvious feature of an electron that flew off a piece of metal is how fast is it going, how much energy does it have. What's its kinetic energy. So I'd like to build an experiment that measures the kinetic energy of an electron that's been excited through this photoelectric effect. Through emission after shining light on a piece of metal. Cool? So I want to build that experiment. So here's how that experiment goes. Well if this electron comes flying off with some kinetic energy and I want to measure that kinetic energy, imagine the following circuit. OK first off imagine I just take a second piece of metal over here, and I'm going to put a little current meter here, an ammeter. And here's what this circuit does. When you shine light on this piece of metal-- we'll put a screen to protect the other piece of metal-- the electrons come flying off, they get over here. And now I've got a bunch of extra electrons over here and I'm missing electrons over here. So this is negative, this is positive. And the electrons will not flow along this wire back here to neutralize the system. The more light I shine, the more electrons will go through this circuit. And as a consequence, there will be a current running through this current meter. That cool with everyone? OK. So we haven't yet measured the kinetic energy, though. How do we measure the kinetic energy? I want to know how much energy, with how much energy, were these electrons ejected. Well I can do that by the following clever trick. I'm going to put now a voltage source here, which I can tune the voltage of, with the voltage V. And what that's going to do is set up a potential difference across these and the energy in that is the charge times the potential difference. So I know that the potential difference it takes, so the amount of energy it takes to overcome this potential difference, is q times V. That cool? So now imagine I send in an electron-- I send in light and it leads an electron to jump across, and it has kinetic energy, kE. Well if the kinetic energy is less than this, will it get across? Not so much. It'll just fall back. But if the kinetic energy is greater than the energy it takes to cross, it'll cross and induce a current. So the upshot is that, as a function of the voltage, what I should see is that there is some critical minimum voltage. And depending on how you set up the sign, the sign could be the other way, but there's some critical minimal voltage where, for less voltage, the electron doesn't get across. And for any greater voltage-- or, sorry, for any closer to zero voltage, the electron has enough kinetic energy to get across. And so the current should increase. So there's a critical voltage, V-critical, where the current running through the system runs to zero. You make it harder for the electrons by making the voltage in magnitude even larger. You make it harder for the electrons to get across. None will get across. Make it a little easier, more and more will get across. And the current will go up. So what you want to do to measure this kinetic energy is you want to measure the critical voltage at which the current goes to zero. So now the question is what do we expect to see. And remember that things we can tune in this experiment are the intensity of the light, which is like e squared plus b squared. And we can tune the frequency of the light. We can vary that. Now does the total energy, does that frequency show up in the total energy of a classical electromagnetic wave? No. If it's an electromagnetic wave, it cancels out. You just get the total intensity, which is a square of the fields. So this is just like a harmonic oscillator. The energy is in the amplitude. The frequency of the oscillator doesn't matter. You push the swing harder, it gets more kinetic energy. It's got more energy. OK. So what do we expect to see as we vary, for example, the intensity? So here's a natural gas. If you take-- so you can think about the light here as getting a person literally, like get the person next to you to take a bat and hit a piece of metal. If they hit it really lightly they're probably not going to excite electrons with a lot of energy. If they just whack the heck out of it, then it wouldn't be too surprising if you get much more energy in the particles that come flying off. Hit it hard enough, things are just gonna shrapnel and disintegrate. The expectation here is the following. That if you have a more intense beam, then you should get more-- the electrons coming off should be more energetic. Because you're hitting them harder. And remember that the potential, which I will call V0, the stopping voltage. So therefore V0 should be greater in magnitude. So this anticipates that the way this curve should look as we vary the current as a function of v, if we have a low voltage-- sorry, if we have a low-intensity beam-- it shouldn't take too much potential just to impede the motion. But if we have a-- so this is a low intensity. But if we have a high-intensity beam, it should take a really large voltage to impede the electric flow, the electric current, because high-intensity beam you're just whacking those electrons really hard and they're coming off with a lot of kinetic energy. So this is high intensity. Everyone down with that intuition? This is what you get from Maxwell's electrodynamics. This is what you'd expect. And in particular, as we vary-- so this is our predictions-- in particular as we vary-- so this is 1, 2, with greater intensity. And the second prediction is that V-naught should be independent of frequency. Because the energy density and electromagnetic wave is independent of the frequency. It just depends on the amplitude. And I will use nu to denote the frequency. So those are the predictions that come from 8.02 and 8.03. But this is 8.04. And here's what the experimental results actually look like. So here's the intensity, here's the potential. And if we look at high potential, it turns out that-- if we look, sorry, if we look at intermediate potentials, it's true that the high intensity leads to a larger current and the low intensity leads to a lower current. But here's the funny thing that happens. As you go down to the critical voltage, their critical voltages are the same. What that tells you is that the kinetic energy kicked out-- or the kinetic energy of an electron kicked out of this piece of metal by the light is independent of how intense that beam is. No matter how intense that beam is, no matter how strong the light you shine on the material, the electrons all come out with the same energy. This would be like taking a baseball and hitting it with a really powerful swing or a really weak swing and seeing that the electron dribbles away with the same amount of energy. This is very counter-intuitive. But more surprisingly, V-naught is actually independent of intensity. But here's the real shocker. V-naught varies linearly in the frequency. What does change V-naught is changing the frequency of the light in this incident. That means that if you take an incredibly diffuse light-- incredibly diffuse light, you can barely see it-- of a very high frequency, then it takes a lot of energy to impede the electrons that come popping off. The electrons that come popping off have a large energy. But if you take a low-frequency light with extremely high intensity, then those electrons are really easy to stop. Powerful beam but low frequency, it's easy to stop those electrons. Weak little tiny beam at high frequency, very hard to stop the electrons that do come off. So this is very counter-intuitive and it doesn't fit at all with the Maxwellian picture. Questions about that? So this led Einstein to make a prediction. This was his 1905 result. One of his many totally breathtaking papers of that year. And he didn't really propose a model or a detailed theoretical understanding of this, but he proposed a very simple idea. And he said, look, if you want to fit this-- if you want to fit this experiment with some simple equations, here's the way to explain it. I claim-- I here means Einstein, not me-- I claim that light comes in packets or chunks with definite energy. And the energy is linearly proportional to the frequency. And our energy is equal to something times nu, and we'll call the coefficient h. The intensity of light, or the amplitude squared, the intensity is like the number of packets. So if you have a more intense beam at the same frequency, the energy of each individual chunk of light is the same. There are just a lot more chunks flying around. And so to explain the photoelectric effect, Einstein observed the following. Look, he said, the electrons are stuck under the metal. And it takes some work to pull them off. So now what's the kinetic energy of an electron that comes flying off-- whoops, k3. Bart might have a laugh about that one. Kinetic, kE, not 3. So the kinetic energy of electron that comes flying off, well, it's the energy deposited by the photon, the chunk of light, h-nu well we have to subtract off the work it took. Minus the work to extract the electron from the material. And you can think of this as how much energy does it take to suck it off the surface. And the consequence of this is that the kinetic energy of an electron should be-- look, if h-nu is too small, if the frequency is too low, then the kinetic energy would be negative. But that doesn't make any sense. You can't have negative kinetic energy. It's a strictly positive quantity. So it just doesn't work until you have a critical value where the frequency times h-- this coefficient-- is equal to the work it takes to extract. And after that, the kinetic energy rises with the frequency with a slope equal to h. And that fits the data like a champ. So no matter-- let's think about what this is saying again. No matter what you do, if your light is very low-frequency and you pick some definite piece of metal that has a very definite work function, very definite amount of energy it takes to extract electrons from the surface. No matter how intense your beam, if the frequency is insufficiently high, no electrons come off. None. So it turns out none is maybe a little overstatement because what you can have is two photon processes, where two chunks hit one electron at the right, just at the same time. Roughly speaking the same time. And they have twice the energy, but you can imagine that the probability of two photon hitting one electron at the same time of pretty low. So the intensity has to be preposterously high. And you see those sorts of multi-photon effects. But as long as we're not talking about insanely high intensities, this is an absolutely fantastic probe of the physics. Now there's a whole long subsequent story in the development of quantum mechanics about this particular effect. And it turns out that the photoelectric effect is a little more complicated than this. But the story line is a very useful one for organizing your understanding of the photoelectric effect. And in particular, this relation that Einstein proposed out of the blue, with no other basis. No one else had ever seen this sort of statement that the electrons, or that the energy of a beam of light should be made up of some number of chunks, each of which has a definite minimum amount of energy. So you can take what you've learned from 8.02 and 8.03 and extract a little bit more information out of this. So here's something you learned from 8.02. In 8.02 you learned that the energy of an electromagnetic wave is equal to c times the momentum carried by that wave-- whoops, over two. And in 8.03 you should have learned that the wavelength of an electromagnetic wave times the frequency is equal to the speed of light, C. And we just had Einstein tell us-- or declare, without further evidence, just saying, look this fits-- that the energy of a chunk of light should be h times the frequency. So if you combine these together, you get another nice relation that's similar to this one, which says that the momentum of a chunk of light is equal to h over lambda. So these are two enormously influential expressions which come out of this argument from the photoelectric effect from Einstein. And they're going to be-- their legacy will be with us throughout the rest of the semester. Now this coefficient has a name, and it was named after Planck. It's called Planck's Constant. And the reason that it's called Planck's Constant has nothing to do with the photoelectric effect. It was first this idea that an electromagnetic wave, that light, has an energy which is linearly proportional not to its intensity squared, none of that, but just linearly proportional to the frequency. First came up an analysis of black body radiation by Planck. And you'll understand, you'll go through this in some detail in 8.044 later in the semester. So I'm not going to dwell on it now, but I do want to give you a little bit of perspective on it. So Planck ran across this idea that E is equal to h/nu. Through the process of trying to fit an experimental curve. There was a theory of how much energy should be emitted by an object that's hot and glowing as a function of frequency. And that theory turned out to be in total disagreement with experiment. Spectacular disagreement. The curve for the theory went up, the curve for the experiment went down. They were totally different. So Planck set about writing down a function that described the data. Literally curve-fitting, that's all he was doing. And this is the depths of desperation to which he was led, was curve-fitting. He's an adult. He shouldn't be doing this, but he was curve-fitting. And so he fits the curve, and in order to get it to fit the only thing that he can get to work even vaguely well is if he puts in this calculation of h/nu. He says, well, maybe when I sum over all the possible energies I should restrict the energies which were proportional to the frequency. And it was forced on him because it fit from the function. Just functional analysis. Hated it. Hated it, he completely hated it. He was really frustrated by this. It fit perfectly, he became very famous. He was already famous, but he became ridiculously famous. Just totally loathed this idea. OK. So it's now become a cornerstone of quantum mechanics. But he wasn't so happy about it. And to give you a sense for how bold and punchy this paper by Einstein was that said, look, seriously. Seriously guys. e equals h/nu. Here's what Planck had to say when he wrote a letter of recommendation to get Einstein into the Prussian Academy of Sciences in 1917, or 1913. So he said, there is hardly one among the great problems in physics to which Einstein has not made an important contribution. That he may sometimes have missed the target in his speculations as in his hypothesis of photons cannot really be held too much against him. It's not possible to introduce new ideas without occasionally taking a risk. Einstein who subsequently went on to develop special relativity and general relativity and prove the existence of atoms and the best measurement of Avogadro's Constant, subsequently got the Nobel Prize. Not for Avogadro's Constant, not for proving the existence of atoms, not for relativity, but for photons. Because of guys like Planck, right. This is crazy. So this was a pretty bold idea. And here, to get a sense for why-- we're gonna leave that up because it's just sort of fun to see these guys scowling and smiling-- there is, incidentally there's a great book about Einstein's years in Berlin by Tom Levenson, who's a professor here. A great writer and a sort of historian of science. You should take a class from him, which is really great. But I encourage you to read this book. It talks about why Planck is not looking so pleased right there, among many other things. It's a great story. So let's step back for a second. Why was Planck so upset by this, and why was in fact everyone so flustered by this idea that it led to the best prize you can give a physicist. Apart from a happy home and, you know. I've got that one. That's the one that matters to me. So why is this so surprising? And the answer is really simple. We know that it's false. We know empirically, we've known for two hundred and some years that light is a wave. Empirically. This isn't like people are like, oh I think it'd be nice if it was a wave. It's a wave. So how do we know that? So this goes back to the double-slit experiment from Young. Young's performance of this was in 1803. Intimations of it come much earlier. But this is really where it hits nails to the wall. And here's the experiment. So how many people in here have not seen a double-slit experiment described? Yeah, exactly. OK. So I'm just going to quickly remind you of how this goes. So we have a source for waves. We let the waves get big until they're basically plane waves. And then we take a barrier. And we poke two slits in it. And these plane waves induce-- they act like sources at the slits and we get nu. And you get crests and troughs. And you look at some distant screen and you look at the pattern, and the pattern you get has a maximum. But then it falls off, and it has these wiggles, these interference fringes. These interference fringes are, of course, extremely important. And what's going on here is that the waves sometimes add in-- so the amplitude of the wave, the height of the wave, sometimes adds constructively and sometimes destructively. So that sometimes you get twice the height and sometimes you get nothing. So just because it's fun to see this, here's Young's actual diagram from his original note on the double-slit experiment. So a and b are the slits, and c, d and f are the [INAUDIBLE] on the screen, the distant screen. He drew it by hand. It's pretty good. So we've known for a very long time that light, because of the double-slit experiment, light is clearly wavy, it behaves like a wave. And what are the senses in which it behaves like a wave? There are two important senses here. The first is answered by the question, where did the wave hit the screen? So when we send in a wave, you know, I drop a stone, one big pulsive wave comes out. It splits into-- it leads to new waves being instigated here and over here. Where did that wave hit the screen? Anyone? AUDIENCE: Everywhere. PROFESSOR: Yeah, exactly. It didn't hit this wave-- the screen in any one spot. But some amplitude shows up everywhere. The wave is a distributed object, it does not exist at one spot, and it's by virtue of the fact that it is not a localized object-- it is not a point-like object-- that it can interfere with itself. The wave is a big large phenomena in a liquid, in some thing. So it's sort of essential that it's not a localized object. So not localized. The answer is not localized. And let's contrast this with what happens if you take this double-slit experiment and you do it with, you know, I don't know, take-- who. Hmm. Tim Wakefield. Let's give some love to that guy. So, baseball player. And have him throw baseballs at a screen with two slits in it. OK? Now he's got pretty good-- well, he's got terrible accuracy, actually. So every once in a while he'll make it through the slits. So let's imagine first blocking off-- what, he's a knuckle-baller, right-- so every once in a while it goes, the baseball will go through the slit. And let's think about what happens, so let's cover one slit. And what we expect to happen is, well, it'll go through more or less straight, but sometimes it'll scrape the edge, it'll go off to the side, and sometimes it'll come over here. But if you take a whole bunch of baseballs, and-- so any one baseball, where does it hit? Some spot. Right? One spot. Not distributed. One spot. And as a consequence, you know, one goes here, one goes there, one goes there. And now, there's nothing like interference effects, but what happens is as it sort of doesn't-- you get some distribution if you look at where they all hit. Yeah? Everyone cool with that? And if we had covered over this slot, or slit, and let the baseballs go through this one, same thing would have happened. Now if we leave them both open, what happens is sometimes it goes here, sometimes it goes here. So now it's pretty useful that we've got a knuckle-baller. And what you actually get is the total distribution looks like this. It's the sum of the two. But at any given time, any one baseball, you say, aha, the baseball either went through the top slit, and more or less goes up here. Or it went through the bottom slit and more or less goes down here. So for chunks-- so this is for waves-- for chunks or localized particles, they are localized. And as a consequence, we get no interference. So for waves, they are not localized, and we do get interference. Yes, interference. OK. So on your problem set, you're going to deal with some calculations involving these interference effects. And I'm going to brush over them. Anyway the point of the double-slit experiment is that whatever else you want to say about baseballs or anything else, light, as we've learned since 1803 in Young's double-slit experiment, light behaves like a wave. It is not localized, it hits the screen over its entire extent. And as a consequence, we get interference. The amplitudes add. The intensity is the square of the amplitude. If the intensities add-- so sorry, if the amplitudes add-- amplitude total is equal to a1 plus a2, the intensity, which is the square of a1 plus a2 squared, has interference terms, the cross terms, from this square. So light, from this point of view, is an electromagnetic wave. It interferes with itself. It's made of chunks. And I can't help but think about it this way, this is literally the metaphor I use in my head-- light is creamy and smooth like a wave. Chunks are very different. But here's the funny thing. Light is both smooth like a wave, it is also chunky. It is super chunky, as we have learned from the photoelectric effect. So light is both at once. So it's the best of both worlds. Everyone will be satisfied, unless you're not from the US, in which case this is deeply disturbing. So of course the original Superchunk is a band. So we've learned now from Young that light is a wave. We've learned from the photoelectric effect that light is a bunch of chunks. OK. Most experimental results are true. So how does that work? Well, we're gonna have to deal with that. But enough about light. If this is true of light, if light, depending on what experiment you do and how you do the experiment, sometimes it seems like it's a wave, sometimes it seems like it's a chunk or particle, which is true? Which is the better description? So it's actually worthwhile to not think about light all the time. Let's think about something more general. Let's stick to electrons. So as we saw from yesterday's lecture, you probably want to be a little bit wary when thinking about individual electrons. Things could be a little bit different than your classical intuition. But here's a crucial thing. Before doing anything else, we can just think, which one of these two is more likely to describe electrons well. Well electrons are localized. When you throw an electron at a CRT, it does not hit the whole CRT with a wavy distribution. When you take a single electron and you throw it at a CRT, it goes ping and there's a little glowing spot. Electrons are localized. And we know that localized things don't lead to interference. Some guys at Hitachi, really good scientists and engineers, developed some really awesome technology a couple of decades ago. They were trying to figure out a good way to demonstrate their technology. And they decided that you know what would be really awesome, this thought experiment that people have always talked about that's never been done really well, of sending an electron through a two-slitted experiment. In this case it was like ten slits effectively, it was a grading. Send an electron, a bunch of electrons, one at a time, throw the electron, wait. Throw the electron, wait. Like our French guy with the boat. So do this experiment with our technology and let's see what happens. And this really is one of my favorite-- let's see, how we close these screens-- aha. OK. This is going to take a little bit of-- and it's broken. No, no. Oh that's so sad. AUDIENCE: [LAUGHTER] PROFESSOR: Come on. I'm just gonna let-- let's see if we can, we'll get part of the way. I don't want to destroy it. So what they actually did is they said, look, let's-- we want to see what happens. We want to actually do this experiment because we're so awesome at Hitachi Research Labs, so let's do it. So here's what they did. And I'm going to turn off the light. And I set this to some music because I like it. OK here's what's happening. One at a time, individual photons. [MUSIC PLAYING] PROFESSOR: So they look pretty localized. There's not a whole lot of structure. Now they're going to start speeding it up. It's 100 times the actual speed. [MUSIC PLAYING] PROFESSOR: Eh? Yeah. AUDIENCE: [APPLAUSE] PROFESSOR: So those guys know what they're doing. Let's-- there were go. So I think I don't know of a more vivid example of electron interference than that one. It's totally obvious. You see individual electrons. They run through the apparatus. You wait, they run through the apparatus. You wait. One at a time, single electron, like a baseball being pitched through two slits, and what you see is an interference effect. But you don't see the interference effect like you do from light, from waves on the sea. You see the interference effect by looking at the cumulative stacking up of all the electrons as they hit. Look at where all the electrons hit one at a time. So is an electron behaving like a wave in a pond? No. Does a wave in a pond at a spot? No. It's a distributed beast. OK yes, it interferes, but it's not localized. Well is it behaving like a baseball? Well it's localized. But on-- when I look at a whole bunch of electrons, they do that. They seem to interfere, but there's only one electron going through at a time. So in some sense it's interfering with itself. How does that work? Is an electron a wave? AUDIENCE: Yes. PROFESSOR: Does an electron hit at many spots at once? AUDIENCE: No. PROFESSOR: No. So is an electron a wave. No. Is an electron a baseball? No. It's an electron. So this is something you're going to have to deal with, that every once in awhile we have these wonderful moments where it's useful to think about an electron as behaving in a wave-like sense. Sometimes it's useful to think about it as behaving in a particle-like sense. But it is not a particle like you normally conceive of a baseball. And it is not a wave like you normally conceive of a wave on the surface of a pond. It's an electron. I like to think about this like an elephant. If you're closing your eyes and you walk up to an elephant, you might think like I've got a snake and I've got a tree trunk and, you know, there's a fan over here. And you wouldn't know, like, maybe it's a wave, maybe it's a particle, I can't really tell. But if you could just see the thing the way it is, not through the preconceived sort of notions you have, you'd see it's an elephant. Yes, that is the Stata Center. So-- look, everything has to happen sometime, right? AUDIENCE: [LAUGHTER] PROFESSOR: So Heisenberg-- it's often, people often give the false impression in popular books on physics, so I want to subvert this, that in the early days of quantum mechanics, the early people like Born and Oppenheimer and Heisenberg who invented quantum mechanics, they were really tortured about, you know, is it an electron, is it a wave. It's a wave-particle duality. It's both. And this is one of the best subversions of that sort of silliness that I know of. And so what Heisenberg says, the two mental pictures which experiments lead us to form, the one of particles the other waves, are both incomplete and have the validity of analogies, which are accurate only in limited cases. The apparent duality rises in the limitation of our language. And then he goes on to say, look, you developed your intuition by throwing rocks and, you know, swimming. And, duh, that's not going to be very good for atoms. So this will be posted, it's really wonderful. His whole lecture is really-- the lectures are really quite lovely. And by the way, that's him in the middle there, Pauley all the way on the right. I guess they were pleased. OK so that's the Hitachi thing. So now let's pick up on this, though. Let's pick up on this and think about what happens. I want to think in a little more detail about this Hitachi experiment. And I want to think about it in the context of a simple two-slit experiment. So here's our source of electrons. It's literally a gun, an electron gun. And it's firing off electrons. And here's our barrier, and it has two slits in it. And we know that any individual electron hits its own spot. But when we take many of them, we get an interference effect. We get interference fringes. And so the number that hit a given spot fill up, construct this distribution. So then here's the question I want to ask. When I take a single electron, I shoot one electron at a time through this experiment, one electron. It could go through the top slit, it could go through the bottom slit. While it's inside the apparatus, which path does it take? AUDIENCE: Superposition. PROFESSOR: Good. So did it take the top path? AUDIENCE: No. PROFESSOR: How do you know? [INTERPOSING VOICES] PROFESSOR: Good, let's block the bottom, OK, to force it to go through the top slit. So we'll block the bottom slit. Now the only electrons that make it through go through the top slit. Half of them don't make it through. But those that do make it through give you this distribution. No interference. But I didn't tell you these are hundreds of thousands of kilometers apart, the person who threw in the electron didn't know whether there was a barrier here. The electron, how could it possibly know whether there was a barrier here if you went through the top. This is exactly like our boxes. It's exactly like our box. Did it go through-- an electron, when the slits are both open and we know that ensemble average it will give us an interference effect, did the electron inside the apparatus go through the top path? No. Did it go through the bottom path? Did it go through both? Because we only see one electron. Did it go through neither? It is in a-- AUDIENCE: Superposition. PROFESSOR: --of having gone through the top and the bottom. Of being along the top half and being along the bottom path. This is a classic example of the two-box experiment. OK. So you want to tie that together. So let's nuance this just a little bit, though, because it's going to have an interesting implication for gravity. So here's the nuance I want to pull on this one. Let's cheat. OK. Suppose I want to measure which slit the electron actually did go through. How might I do that? Well I could do the course thing I've been doing which is I could block it and just catch the-- catch electrons that go through in that spot. But that's a little heavy-handed. Probably I can do something a little more delicate. And so here's the more delicate thing I'm going to do. I want to build a detector that uses very, very, very weak light, extremely weak light, to detect whether the particle went through here or here. And the way I can do that is I can sort of shine light through and-- I'm gonna, you know, bounce-- so here's my source of light. And I'll be able to tell whether the electron went through this slit or it went through this slit. Cool? So imagine I did that. So obviously I don't want to use some giant, huge, ultra high-energy laser because it would just blast the thing out of the way. It would destroy the experiment. So I wanna something very diffuse, very low energy, very low intensity electromagnetic wave. And the idea here is that, OK, it's true that when I bounce this light off an electron, let's say it bounces off an electron here, it's true it's going impart some momentum and the electron's gonna change its course. But if it's really, really weak, low energy light, then it's-- it's gonna deflect only a little tiny bit. So it will change the pattern I get over here. But it will change it in some relatively minor way because I've just thrown in very, very low energy light. Yeah? That make sense? So this is the experiment I want to do. This experiment doesn't work. Why. AUDIENCE: You know which slit it went through. PROFESSOR: No. It's true that it turns out that those are correlated facts, but here's the problem. I can run this experiment without anyone actually knowing what happens until long afterwards. So knowing doesn't seem to play any role in it. It's very tempting often to say, no, but it turns out that it's really not about what you know. It's really just about the experiment you're doing. So what principle that we've already run into today makes it impossible to make this work? If I want to shine really low-energy, really diffuse light through, and have it scatter weakly. Yeah. AUDIENCE: Um, light is chunky. PROFESSOR: Yeah, exactly. That's exactly right. So when I say really low-energy light, I don't-- I really can't mean, because we've already done this experiment, I cannot possibly mean low intensity. Because intensity doesn't control the energy imparted by the light. The thing that controls the energy imparted by a collision of the light with the electron is the frequency. The energy in a chunk of light is proportional to the frequency. So now if I want to make the effect the energy or the momentum, similarly-- the momentum, where did it go-- remember the momentum goes like h over lambda. If I want to make the energy really low, I need to make the frequency really low. Or if I want to make the momentum really low, I need to make the wavelength what? Really big. Right? So in order to make the momentum imparted by this photon really low, I need to make the wavelength really long. But now here's the problem. If I make the wavelength really long, so if I use a really long-wavelengthed wave, like this long of a wavelength, are you ever going to be able to tell which slit it went through? No, because the particle could have been anywhere. It could have scattered this light if it was here, if it was here, if it was here, right? In order to measure where the electron is to some reasonable precision-- so, for example, to this sort of wavelength, I need to be able to send in light with a wavelength that's comparable to the scale that I want to measure. And it turns out that if you run through and just do the calculation, suppose I send in-- and this is done in the books, in I think all four, but this is done in the books on the reading list-- if you send in a wave with a short enough wavelength to be able to distinguish between these two slits, which slit did it go through, the momentum that it imparts precisely watches-- washes out is just enough to wash out the interference effect, and break up these fringes so you don't see interference effects. It's not about what you know. It's about the particulate nature of light and the fact that the momentum of a chunk of light goes like h over lambda. OK? But this tells you something really interesting. Did I have to use light to do this measurement? I could have sent in anything, right? I didn't have to bounce light off these things. I could have bounced off gravitational waves. So if I had a gravitational wave detector, so-- Matt works on gravitational wave detectors, and so, I didn't tell you this but Matt gave me a pretty killer gravitational wave detector. It's, you know, here it is. There's my awesome gravitational wave detector. And I'm now going to build supernova. OK. And they are creeping under this black hole, and it's going to create giant gravitational waves. And we're gonna use those gravitational waves and detect them with the super advanced LIGO. And I'm gonna detect which slit it went through. But gravitational waves, those aren't photons. So I really can make a low-intensity gravitational wave, and then I can tell which slit it went through without destroying the interference effect. That would be awesome. What does that tell you about gravitational waves? They must come in chunks. In order for this all to fit together logically, you need all the interactions that you could scatter off this to satisfy these quantization properties. But the energy is proportional to the frequency. The line I just gave you is a heuristic. And making it precise is one of the great challenges of modern contemporary high-energy physics, of dealing with the quantum mechanics and gravity together. But this gives you a strong picture of why we need to treat all forces in all interactions quantum-mechanically in order for the world to be consistent. OK. Good. OK, questions at this point? OK. So-- oh, I forgot about this one-- so there are actually two more. So I want to just quickly show you-- well, OK. So, this is a gorgeous experiment. So remember I told you the story of the guy with the boat and the opaque wall and it turns out that's a cheat. It turns out that this opaque screen doesn't actually give you quantum mechanically isolated photons. They're still, in a very important way, classical. So this experiment was done truly with a source that gives you quantum mechanically isolated single photons, one at a time. So this is the analogue of the Hitachi experiment. And it was done by this pretty awesome Japanese group some number of years ago. And I just want to emphasize that it gives you exactly the same effects. We see that photons-- this should look essentially identical to what we saw at the end of the Hitachi video. And that's because it's exactly the same physics. It's a grating with something like 10 slits and individual particles going through one at a time and hitting the screen and going, bing. So what you see is the light going, bing, on a CCD. It's a pretty spectacular experience. So let's get back to electrons. I want another probe of whether electrons are really waves or not. So this other experiment-- again, you're going to study this on your problem set-- this other experiment was done by a couple of characters named Davisson and Germer. And in this experiment, what they did is they took a crystal, and a crystal is just a lattice of regularly-located ions, like diamond or something. Yeah? AUDIENCE: Before you go on I guess, I wanted to ask if the probability of a photon or an electron going through the 10 slits is about the same? PROFESSOR: Is what, sorry? AUDIENCE: Is exactly the same. PROFESSOR: You mean for different electrons? AUDIENCE: Yeah. PROFESSOR: Well they can be different if the initial conditions are different. But they could be-- if the initial conditions are the same, then the probabilities are identical. So every electron behaves identically to every other electron in that sense. Is that what you were asking? AUDIENCE: It is actually like through any [INAUDIBLE] the probability of it going like [INAUDIBLE]? PROFESSOR: Sure, absolutely. So the issue there is just a technological one of trying to build a beam that's perfectly columnated. And that's just not doable. So there's always some dispersion in your beam. So in practice it's very hard to make them identical, but in principle they could be if you were infinitely powerful as an experimentalist, which-- again, I was banned from the lab, so not me. So here's our crystal. You could think of this as diamond or nickel or whatever. I think they actually use nickel but I don't remember exactly. And they sent in a beam of electrons. So they send in a beam of electrons, and what they discover is that if you send in these electrons and watch how they scatter at various different angles-- I'm going to call the angle here of scattering theta-- what they discover is that the intensity of the reflected beam, as a function of theta, shows interference effects. And in particular they gave a whole calculation for this, which I'm not going to go through right now because it's not terribly germane for us-- you're going to go through it on your problem set, so that'll be good and it's a perfect thing for your recitation instructors to go through. But the important thing is the upshot. So if the distance between these crystal planes is L-- or, sorry, d-- let me call it d. If the distance between the crystal planes is d, what they discover is that the interference effects that they observed, these maxima and minima, are consistent with the wavelength of light. Or, sorry, with the electrons behaving as if they were waves with a definite wavelength, with a wavelength lambda being equal to some integer, n, over 2d sine theta. So this is the data-- these are the data they actually saw, data are plural. And these are the data they actually saw. And they infer from this that the electrons are behaving as if they were wave-like with this wavelength. And what they actually see are individual electrons hitting one by one. Although in their experiment, they couldn't resolve individual electrons. But that is what they see. And so in particular, plugging all of this back into the experiment, you send in the electrons with some energy, which corresponds to some definite momentum. This leads us back to the same expression as before, that the momentum is equal to h over lambda, with this lambda associated. So it turns out that this is correct. So the electrons diffract off the crystal as if they have a momentum which comes with a definite wavelength corresponding to its momentum. So that's experimental result-- oh, I forgot to check off four-- that's experimental result five, that electrons diffract. We already saw the electron diffraction. So something to emphasize is that-- so these experiments as we've described them were done with photons and with electrons, but you can imagine doing the experiments with soccer balls. This is of course hard. Quantum effects for macroscopic objects are usually insignificantly small. However, this experiment was done with Buckyballs, which are the same shape as soccer balls in some sense. But they're huge, they're gigantic objects. So here's the experiment in which this was actually done. So these guys are just totally amazing. So this is Zellinger's lab. And it doesn't look like all-- I mean it looks kind of, you know. It's hideous, right? I mean to a theorist it's like, come on, you've got to be kidding that that's-- But here's what a theorist is happy about. You know, because it looks simple. We really love lying to ourselves about that. So here's an over. We're going to cook up some Buckyballs and emit them with some definite known thermal energy. Known to some accuracy. We're going to columnate them by sending them through a single slit, and then we're going to send them through a diffraction grating which, again, is just a whole bunch of slits. And then we're going to image them using photo ionization and see where they pop through. So here is the horizontal position of this wave along the grating, and this is the number that come through. This is literally one by one counts because they're going bing, bing, bing, as a c60 molecule goes through. So without the grating, you just get a peek. But with the grating, you get the side bands. You get interference fringes. So these guys, again, they're going through one by one. A single Buckyball, 60 carbons, going through one by one is interfering with itself. This is a gigantic object by any sort of comparison to single electrons. And we're seeing these interference fringes. So this is a pretty tour de force experiment, but I just want to emphasize that if you could do this with your neighbor, it would work. You'd just have to isolate the system well enough. And that's a technological challenge but not an in-principle one. OK. So we have one last experimental facts to deal with. And this is Bell's Inequality, and this is my favorite one. So Bell's Inequality for many years languished in obscurity until someone realized that it could actually be done beautifully in an experiment that led to a very concrete experiment that they could actually do and that they wanted to do. And we now think of it as an enormously influential idea which nails the coffin closed for classical mechanics. And it starts with a very simple question. I claim that the following inequality is true: the number of undergraduate-- of the number of people in the room who are undergraduates, which I'll denote as U-- and not blonde, which I will denote as bar B-- so undergraduates who are not blonde-- actually let me write this out in English. It's gonna be easier. Number who are undergrads and not blonde plus the number of people in the room who are blonde but not from Massachusetts is strictly greater than or equal to the number of people in the room who are undergraduates and not from Massachusetts. I claim that this is true. I haven't checked in this room. But I claim that this is true. So let's check. How many people are undergraduates who are not blonde? OK this is going to-- jeez. OK that's-- so, lots. OK. How many people are blonde but not from Massachusetts? OK. A smattering. Oh God, this is actually going to be terrible. AUDIENCE: [LAUGHTER] PROFESSOR: Shoot. This is a really large class. OK. Small. And how many people are undergraduates who are not from Massachusetts? Yeah, this-- oh God. This counting is going to be-- so let's-- I'm going to do this just so I can do the counting with the first two rows here. OK. My life is going to be easier this way. So how many people in the first two rows, in the center section, are undergraduates but not blonde? One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen. We could dispute some of those, but we'll take it for the moment. So, fourteen. You're probably all undergraduates. So blonde and not from Massachusetts. One. Awesome. Undergraduates not from Massachusetts. One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen. Equality. AUDIENCE: [LAUGHTER] PROFESSOR: OK. So that-- you might say well, look, you should have been nervous there. You know, and admittedly sometimes there's experimental error. But I want to convince you that I should never, ever ever be nervous about this moment in 8.04. And the reason is the following. I want to prove this for you. And the way I'm gonna prove it is slightly more general, in more generality. And I want to prove to you that the number-- if I have a set, or, sorry, if the number of people who are undergraduates and not blonde which, all right, is b bar plus the number who are blonde but not from Massachusetts is greater than or equal to the number that are undergraduates and not from Massachusetts. So how do I prove this? Well if you're an undergraduate and not blonde, you may or you may not be from Massachusetts. So this is equal to the number of undergraduates who are not blonde and are from Massachusetts plus the number of undergraduates who are not blonde and are not from Massachusetts. It could hardly be otherwise. You either are or you are not from Massachusetts. Not the sort of thing that you normally see in physics. So this is the number of people who are blonde and not from Massachusetts, number of people who are blonde, who are-- so if you're blonde and not from Massachusetts, you may or may not be an undergraduate. So this is the number of people who are undergraduates, blonde, and not from Massachusetts plus the number of people who are not undergraduates, are blonde and are not from Massachusetts. And on the right hand side-- so, adding these two together gives us plus and plus. On the right hand side, the number of people that are undergraduates and not from Massachusetts, well each one could be either blonde or not blonde. So this is equal to the number that are undergraduates, blonde, and not from Massachusetts, plus-- remember that our undergraduates not blonde and not from Massachusetts. Agreed? I am now going to use the awesome power of-- and so this is what we want to prove, and I'm going to use the awesome power of subtraction. And note that U, B, M bar, these guys cancel. And U, B bar, M bar, these guys cancel. And we're left with the following proposition: the number of undergraduates who are not blonde but are from Massachusetts plus the number of undergrad-- of non-undergraduates who are blonde but not from Massachusetts must be greater than or equal to zero. Can you have a number of people in a room satisfying some condition be less than zero? Can minus 3 of you be blonde undergraduates not from Massachusetts? Not so much. This is a strictly positive number, because it's a numerative. It's a counting problem. How many are undergraduates not blonde and from Massachusetts. Yeah? Everyone cool with that? So it could hardly have been otherwise. It had to work out like this. And here's the more general statement. The more general statement is that the number of people, or the number of elements of any set where each element in that set has binary properties a b and c-- a or not a, b or not b, c or not c. Satisfies the following inequality. The number who are a but not b plus the number who are b but not c is greater than or equal to the number who are a but not c. And this is exactly the same argument. And this inequality which is a tautology, really, is called Bell's Inequality. And it's obviously true. What did I use to derive this? Logic and integers, right? I mean, that's bedrock stuff. Here's the problem. I didn't mention this last time, but in fact electrons have a third property in addition to-- electrons have a third property in addition to hardness and color. The third property is called whimsy, and you can either be whimsical or not whimsical. And every electron, when measured, is either whimsical or not whimsical. You never have a boring electron. You never have an ambiguous electron. Always whimsical or not whimsical. So we have hardness, we have color, we have whimsy. OK. And I can perform the following experiment. From a set of electrons, I can measure the number that are hard and not black, plus the number that are black but not whimsical. And I can measure the number that are hard and not whimsical. OK? And I want to just open up the case a little bit and tell you that the hardness here really is the angular momentum of the electron along the x-axis. Color is the angular momentum of the electron along the y-axis. And whimsy is the angular momentum of the electron along the z-axis. These are things I can measure because I can measure angular momentum. So I can perform this experiment with electrons and it needn't be satisfied. In particular, we will show that the number of electrons, just to be very precise, the number of electrons in a given set, which have positive angular momentum along the x-axis and down along the y-axis, plus up along the y-axis and down along the z-axis, is less than the number that are up. Actually let me do this in a very particular way. Up... zero down at theta. Up at theta, down at-- two theta is greater than the number that are up at zero and down at theta. Now here's the thing-- two theta. You can't at the moment understand what this equation means. But if I just tell you that these are three binary properties of the electron, OK, and that it violates this inequality, there is something deeply troubling about this result. Bell's Inequality, which we proved-- trivially, using integers, using logic-- is false in quantum mechanics. And it's not just false in quantum mechanics. We will at the end of the course derive the quantum mechanical prediction for this result and show that at least to a predicted violation of Bell's Inequality. This experiment has been done, and the real world violates Bell's Inequality. Logic and integers and adding probabilities, as we have done, is misguided. And our job, which we will begin with the next lecture, is to find a better way to add probabilities than classically. And that will be quantum mechanics See you on Tuesday. AUDIENCE: [APPLAUSE]
|
8-04-quantum-physics-i-spring-2013
|
physics_bio
|
1
|
JACK HARE: Right, so today, we are going to start a series of several lectures on refractive index diagnostics. So these are diagnostics which use the fact that the refractive index of a plasma is not 1 in order to make measurements about the plasma. So first of all, what we're going to do is a very quick recap of electromagnetic waves in a plasma. And you have doubtless seen this before in some courses that you've done, like it's in Chen. There's also a very good explanation of it in Hutchinson. And I'm going to give a little taster of how this derivation goes, just to remind folks of how this result actually comes about. And I'm going to do it in a rather restrictive way here so that we can make some rapid progress. And we'll go back and add in more bits of theory later on as we need it. So for our electromagnetic waves in a plasma, we have two fields. We have E and B, like this. And we have Maxwell's equations. So we have that the curl of E is equal to minus B dot, and we have C squared times the curl of B equals E dot plus J over epsilon 0. You might be more used to seeing this equation with a mu 0. I've just moved the C squared over the other side because it makes the math a little bit easier on the next step here. And so these equations are just true. They're true in any medium. And so what we want to do is try and reduce them down a little bit and then put some plasma physics in. So the first thing we normally do when we're deriving electromagnetic waves is we say, what if all of our vectors had some sort of time and space variation that looked like exponential of i, some wave vector k dot, some position vector x minus omega the frequency times time, like that. And so this is a little bit like Fourier transforming our equations here. And we end up with k cross E equals i omega B, and we have C squared k cross B equals minus i omega E plus J upon epsilon 0. And the astute amongst you have noticed I made a sign error in the first equation here. So this is plus i omega B, not minus i omega B. Very good. And so we could call these equations 1 and 2. And we can note that if we do k cross with equation 1, that is equal to i omega upon C squared of equation 2. So what we're doing here is we're just looking at this term and this term and being like, hey, they both got B in them. We can probably make them look the same. So then we can equate. We can do this calculation and we can equate the two sides of the equation. And with a little bit of vector magic, calculus magic, we would end up with something that looks like k k dot E plus k squared E is equal to i omega J over epsilon 0 C squared plus omega squared upon C squared E, like that. Now, we are searching for transverse waves here, electromagnetic waves that tend to be transverse, and that means that k dot E is 0. So we're just going to look for transverse wave solutions here. We can drop that. And that means that our equation can be rewritten as omega squared minus C squared k squared times E equals minus i omega J upon epsilon 0, like that. And I just want to point out, there's absolutely no plasma physics in this at the moment. All we've done is manipulate Maxwell's equations. We haven't said anything about the plasma. So if I take a standard limit here, say I let the current equal 0, which is what it would be in the vacuum of space where there's no particles to carry any current, then we would simply end up with an equation for light in free space and you would have a dispersion relationship omega squared equals C squared k squared, like that. So those are light waves. So life is good. Any questions on that before we put some plasma physics in? Hopefully, you're dredging up your memories of Griffiths and Jackson and all sorts of wonderful things like this, and this is all making sense. So let's keep going. The next thing we want to do then is add in some plasma physics. And we're going to add in some plasma physics with some serious assumptions which let us make significant progress quickly. And these assumptions can all be justified for most plasmas. And if you can't justify it for your plasma, you may want to revisit these a little bit. So the first assumption we're going to make is that we're using high-frequency waves here. And high frequency, this is kind of a wishy-washy term. We want to make that more precise by having a dimensionless parameter. And so we're going to say that omega is much, much larger than omega pi here. So the frequency of our waves is much higher than the ion plasma frequency. What does that condition physically respond to? What are we saying about the ions and their interaction with this wave by putting this condition in? STUDENT: You're saying they essentially don't interact at that point. JACK HARE: Yeah. So the ions are frozen in place. And they are not going to participate in any of the physics that we're interested in here. And again, you can work out the ion plasma frequency for some density that you're interested in, and you'll find out it's pretty low. So this is pretty reasonable, but if you start using very low-frequency waves, it won't be reasonable. So another thing that we're going to do is make the cold plasma approximation. And again, we can't just use the word cold. We have to say what we mean by cold. And we're going to say the thermal of the electrons is much, much less than the speed of light here. And this condition is equivalent to us not worrying about the Maxwellian distribution of the electrons. So all the electrons are just going to be moving with no-- they will be moving, unlike the ions, which are frozen, but they will be moving all at the same speeds. There's no spread of velocities here. So we have a delta function of velocities, and we do not have our Maxwellian distribution that we might normally think about. And the final condition we're going to write down is unmagnetized. Now, this is the one which I think is most complicated. Because, in fact, there are a dozen different ways you can write down a dimensionless parameter for unmagnetized, and they all mean slightly different things. We could think about collisionality. We could think about pressure balance, all sorts of things like that. For the purposes of this derivation, unmagnetized means omega is much, much larger than capital omega. I know it's confusing. And I'll put a little subscript e here because we've frozen the ions. But this is the gyro motion of the electrons. So the electrons may be gyrating around field lines. There may be some magnetic field. It doesn't have to be 0. But on the time scale that the wave goes by and does its stuff, the electrons do not move appreciably around their gyro orbit. So we don't have to care about their gyro motion. This is one of the places we will definitely have to relax later on when we want to do Faraday rotation imaging which relies on that gyro motion to give us the effect. So this is the final condition. Any questions on those three assumptions? OK, so then, we can write down that the current inside our plasma is simply going to be equal to the charge on the electrons, the number of electrons, and how fast they're moving. And so we've simply transferred our lack of knowledge about J into our lack of knowledge about V. So we better do something about that. And we're going to do that using the electron equation of motion. And that electron equation of motion looks like m dVe dt is equal to minus e the charge times capital E, the vector electric field, like that. So we could rearrange that so that we have Ve is equal to e vector E over im omega. We've done the same Fourier transform trick that we did with all the other quantities. We assume it's going to be oscillating in some way. Substitute that back in to our equation for E, and we'll get omega squared minus C squared k squared E, as we had before, is equal to nee squared over epsilon 0 and E, capital E. Sorry, was there a question? I heard someone speak. STUDENT: No. That was an accident. Sorry. JACK HARE: No worries. All good. And just to be clear here, this equation of motion doesn't have the V cross B term in it, which would be like the magnetic field term, because we've dropped it because we're making this unmagnetized approximation. But that's where you put it back in. You put in a little term that was like, cross V cross B in here and put little brackets around. But for now, we're just setting that equal to 0. OK. Good. And the solution to this equation that we've got now is omega squared equals omega p squared plus C squared K squared. So this looks an awful lot like what we had before, which was just these two terms. But now, we've got this additional term, which includes some plasma physics. And I could write that this is omega pe squared and make it clear that it's the electrons, but remember, we've dropped the arm motion already. So there's only one plasma frequency that's very interesting, the electron plasma frequency for this derivation here. And then, we can go ahead and do all the standard things we do with one of these functions, which is to try and write down the phase velocity. And the phase velocity squared is just equal to omega squared upon k squared. And so that means that our phase velocity is C upon 1 minus omega p squared upon omega squared. It's a half. And then, we can write down the refractive index, because our refractive index, capital N, is just equal to C over the phase velocity. And that is going to be equal to 1 minus omega p squared over omega squared. And we often write this not in terms of frequencies, but in terms of densities, because the plasma frequency has inside it an electron density. We write 1 minus ne over n critical, like this, where n critical is some critical density. And we'll get on to why it's so critical in a moment. But if you want to approximate it, then n critical in centimeters to the minus 3, so particles per cubic centimeter, is roughly 10 to the 21 over lambda when lambda is in microns here. So if you're using a laser beam at 1 micron, that would be a very standard laser wavelength from a neodymium YAG or a neodymium glass laser, then you have a critical density of about 10 to the 21. Which depending on which field you're working in is either hilariously high and unreachable or crazy low and happens all the time. So again, this is one of the exciting things about doing plasma diagnostics course where we span 16 orders of magnitude in density. Any questions on that? STUDENT: Yeah, what's the critical density point again? JACK HARE: As in, what is its physical significance? STUDENT: Yeah. Like, why did you choose that number? JACK HARE: We're going to look into that in a moment. It's a good question. It's a solid question. But just to be clear, the reason I've got there is I've taken this omega p squared and this omega squared and I've noticed that omega p squared has inside it the electron density, and I've rewritten all the other terms. So this n critical now has inside it things like omega squared. I made a critical mistake in my equation up here. This is lambda squared here for approximating this in terms of simple quantities. So the n critical now depends on the laser wavelength or the frequency of your probing electromagnetic radiation. So it's different for every frequency. But that's the only thing it depends on. The rest of it is all like, fundamental constants, like E and the electron mass that don't change. So this is a parameter there's a critical density for every single laser wavelength or electromagnetic wave frequency. I'm going to talk about lasers a lot. Of course, this also applies to microwaves and things like that as well. But some sort of source of radiation. So I'm going to get on to what n critical is in just a moment. But any other questions on this before we keep going? OK, so we just said that n is equal to the square root of 1 minus ne over n critical. If we work in a regime where ne is much, much less than n critical, so we work far from the critical density, we can do a Taylor expansion, which is what we often end up doing, and we write 1 minus ne over 2 n critical, like this. So my questions for you now-- and we will try and work out what n critical is doing together-- for n, for a density which is greater than the critical density, what happens to n, the refractive index? STUDENT: The wave becomes evanescent, right? JACK HARE: Yeah. Well, that's the result. So what happens to n itself? STUDENT: Big N? It's imaginary. JACK HARE: Right. Exactly. Yeah. Yeah. You've got the right answer. I just wanted to take in a few more steps. So n is imaginary because it's going to be the square root of a negative number, which we can see here. And so that means our wave becomes evanescent. So it's going to have properties which decay in time. So it's going to look like e to the minus alpha x e to the minus gamma t, like this. So it dies off. So that means that we can't propagate a wave at densities greater than the critical density. What happens to all the energy, then? Because this wave is carrying energy. STUDENT: It's absorbed by the plasma. JACK HARE: No absorption mechanism in our equations, actually. STUDENT: Yeah, so that-- I mean, from the math we have, the only option's reflection, right? JACK HARE: Yeah. So reflection is indeed the answer. So we will get reflection of this wave. It will bounce off the critical surface and go somewhere else. We'd only be able to have absorption if we put, for example, collisions into the equation of motion, all the way back there. And then, we could have what's called inverse Bremsstrahlung, which is, effectively, the electrons get oscillated by the wave, and then they collide with some ions and transfer the energy to the ions. So that's a damping mechanism. There could be other damping mechanisms like lambda, but in the equations we've got so far, we don't actually have those. So reflection is the only thing that can happen. Now, we also had this equation for the phase velocity, which was that Vp is equal to c over 1 minus omega p squared upon omega squared to the 1/2. Does anyone want to comment on what happens to the phase velocity as we go above the critical density? STUDENT: Does it go to infinity? JACK HARE: I mean, we'll do that eventually, yes. Yes. Right there, at that point-- STUDENT: It diverges? JACK HARE: Extremely large. Yes. Yeah. So this is going to start doing very silly things here. And those things don't seem very physical, of course, because we don't want things traveling faster than the speed of light. Fortunately, this isn't a big problem because the phase velocity doesn't carry any information, and there are limitations on things going faster than the speed of light have to do with information. And the information is encoded in a quantity called the group velocity, and the group velocity just looks like this, c times the square root of 1 minus omega p squared upon omega squared. And so as we get close to the critical density here, all that happens is the group velocity goes to 0, and effectively, we transmit no information through that evanescent region. So we don't have to worry about the fact that phase velocity. It's superluminal because our group velocity is still subluminal, as we'd like. Sean? STUDENT: So if the wave's being reflected here, and the group velocity is going to 0, to me, that seems like the wave information is sort of stagnating at the reflection point. How do we see from these equations that the wave information is actually being reflected back out. JACK HARE: Very good question. STUDENT: I think-- excuse me. I think you need to impose boundary conditions to do that. Because there would be some sort of discontinuity, right? JACK HARE: Yeah. I suspect the model I'm presenting is a little bit too simplistic to handle this stuff. STUDENT: So we would need to bake in some more information. JACK HARE: I think so. There's certainly-- as you get very, very slow group velocities, you're going to start-- we've been making some assumptions about the homogeneity here, and so, reflectively, there's going to be some length scale in here, which is going to be like, k, the size of the wave vector of the light at a given point, and we're going to be comparing that to the length scale associated with how quickly the electron density changes, this gradient length scale here. And as the group velocity gets very, very low that k is going to get very, very long, and we're going to start violating our assumption that k is-- let's see. Maybe I can write this as lambda. That lambda is going to be much, much less than this change in the gradient. So for any realistic system, our density has to ramp up. It can't just immediately get up to the critical density. This could be the critical density here. And we're going to find out that there's some region where some of the approximations we've implicitly been making break down. And then, you need to start doing wbk, and all that sort of stuff and doing everything properly. So I think, effectively, this simple model breaks down. But if you do it properly-- and I think Hutchinson does this in the reflectometry section. So we may end up doing it. You can get the answer about the reflection there. It's a good question. This is very hand-wavy at this point. I agree. STUDENT: Thanks. JACK HARE: Cool. Any other questions on this? Because this equation here, we are now going to use an awful lot. So I'd like you to agree that it's valid within the assumptions that we've made. And if you don't agree, we should have a chat about it. OK, good. So let's keep going. So there is a series of different measurements that we can make. And these are the refractive index diagnostics. So I'm going to just call these n measurements or n diagnostics. Because they rely on the change in refractive index here. So one type is when the refractive index is not equal to 1. There's a refractive index inside our plasma is not equal to 1, which is true anytime there's any density inside the plasma. This sort of diagnostic causes a phase shift. So the plasma ends up-- the laser beam going through or the electromagnetic radiation going through the plasma ends up with a different phase than it would have done in the absence of the plasma. And we can measure that phase shift using a technique called interferometry. And with interferometry, we can therefore say something about the density inside the plasma. Another technique is when the gradient of the refractive index is not equal to 0. So this is when there is any change in refractive index. And in a plasma, that corresponds very clearly to just changes in the electron density, so gradients in the electron density. But of course, in general, this technique can be used for any medium where the refractive index changes. So air, if you heat it up, the refractive index changes, and so you could use these techniques. These are not specific to plasma physics. And these diagnostics tend to be called refraction diagnostics, because the light refracts and it bends. And we end up doing techniques such as schlieren and sonography. And then, the final type that I'm going to talk about are ones where, actually, the polarization, the medium is birefringent. It treats different polarizations differently. And so we can have polarizations of light which are circular. We can have the left-handed and the right-handed polarizations, which we sometimes refer to as plus and minus. And here, we would say that the refractive index for the plus wave is not equal to the refractive index to the minus wave here. And so here, we measure the polarization. And this is using a technique called Faraday or Faraday rotation. Which we briefly discussed in the context of magnetic field measurements using Verdet glass. And in fact, it turns out that you need to have magnetic fields that are non-zero, and we also need to relax our assumption that the plasma is unmagnetized in the sense that the light frequency is much larger than the electron cyclotron frequency. So those are three different types of refractive index diagnostic, and we're going to start with what I think is conceptually the simplest, but still often causes us lots of problems, which are the refraction diagnostics here. I see some people writing, so I'm just going to pause on this slide for a moment. Okey doke. Now, I just want to have a little aside. And this is on conceptual models for electromagnetic propagation. Because I'm going to be switching quite a lot between different ways of thinking about electromagnetic radiation and how it moves through a plasma. Because sometimes, some models are easier to work with than others. Sometimes, models are simplifications and they throw away physics, but they make the intuition much simpler. So I just want to show you two different models that we're going to be using in these next few lectures so that you have an idea of what's going on. One model would be a model of wavefronts. So this is based on the idea that, as we said, our electric and magnetic fields can be written as just a single Fourier component. So there's some strength and polarization of the electric field here, and this is multiplied by the exponential of what we call the phase factor. So i k dot x minus omega t, like this, which we could write as E0 exponential of i times some scalar quantity, which is the phase here. And so if we think of our electromagnetic wave as having a phase, and the electromagnetic wave still exists in all places, all points in time, blah, blah, blah, blah, blah, but that's a very difficult thing for me to sketch on my iPad here or on the board in front of you. So what I'll probably end up sketching are what we call isophase contours. So these are controls along which the phase is constant. And so, for example, it could be at some integer multiple of pi, right? Yes. And belonging to z. So this might look like some waves like this. This would be an electromagnetic wave which is diverging here. Actually, it could be an electromagnetic wave which is converging to the left. But at the moment, it looks like it's diverging to the right. So that's one way I could draw a wave here. Another way I could do it is with a ray model. And this gets into a topic which is called geometric optics. And it turns out what you can do, if you have some isophase contours like the ones I just drew, say, these contours here, they're doing something slightly strange, but perhaps there's a plasma there, which is like moving the phase contours around. If there's a change of refractive index, will affect the phase. The rays that we draw here, I can just take these phase contours and I can draw rays such that they are everywhere normal to the isophase contours. So this ray would look like this. This one would look like this. And this one would look like that. So they are perpendicular to the wavefronts. Conveniently, they are also parallel to the Poynting vector. At least, I'm pretty convinced they are. If someone knows more about geometric optics and thinks I'm wrong, please, shout out, because this was a very hard fact to check in like, 10 minutes before the lecture. But I'm pretty certain they represent the direction of the energy flux in electromagnetic waves. So they're quite conceptually useful as well. They tell us where the power is flowing. Now, when we think about these rays here, we can start thinking a little bit like it's a particle trajectory. And I put particle here in speech marks. I don't think you really need to think about these as photons, but you can think about them as little point particles that move around inside a plasma. And we'll find out some rules for how they move inside the plasma in a moment. And if you track their trajectory, that's where the ray's gone on. And then, you also know some places where you have lines which are normal to the wavefronts. So maybe you could reconstruct the wavefronts later on. But it's important that when we're doing this, we ignore the wave effects. So we no longer track the phase of each particle. It's now just a little billiard ball. And billiard balls don't have phase. And so we're going to get rid of effects like interference and diffraction, and we're going to keep effects only like refraction here. So no interference. No diffraction. Just refraction. So this is our ray model. So does anyone have any questions on these models before we start trying to use them? STUDENT: I was just wondering, so if the Poynting vector right is E cross B, Does that mean that if we have any parallel electric field to k-- I'm just wondering, your point about the Poynting vector, would that break if there was like a parallel e to the main background magnetic field, or is that just the oscillating B there? If that question makes sense. JACK HARE: It does, actually. And I know the answer to it. That's good. So if you had some background magnetic field, like in a tokamak. And then E was parallel to it. Well, let's put it this way. The Poynting flux oscillates. And so, when you're averaging it, time averaging it, that's what gives you the actual power that's moving. If you've got a static magnetic field, your average power will go to 0. STUDENT: OK. Yeah. Thank you. JACK HARE: So you'll only get power flow from oscillating components here because that's what's transporting the electromagnetic energy. But it's a really good question. STUDENT: Thank you. JACK HARE: Like I said, I'm not completely 100% sure that rays follow the trajectory of the Poynting vector, but I'm pretty certain, after thinking about it for about 10 minutes, that they do. So if someone finds out that's wrong, please let me know and I'll take it out. OK, so everyone is going to be pretty happy if I start drawing ray diagrams. And they'll understand that these ray diagrams represent the trajectory of little beamlets of light, and you can also reconstruct the wavefronts from them, and therefore, you could reconstruct visually what the entire electromagnetic field looks like. And we're implicitly assuming everywhere here that our magnetic field is perpendicular to our electric field, which is a pretty good approximation to the assumptions we've made so far. OK, so now, let's try putting an electromagnetic wave through a plasma. And it's not going to be any old plasma here. I'm going to choose a slab of plasma like this. And this slab is going to be much denser at the top than it is at the bottom, which I've tried to really clumsily do with some shading here. So it's going to have a gradient of electron density going up, like that. And of course, you remember our formula, 1 minus ne over 2n critical for our refractive index. We're going to work in this regime where the density is much less than the critical density, so we don't have to worry about what happens if we get close to the critical density. And so you can see, then, that if the gradient in the electron density is in this direction, then the gradient in the refractive index is in the opposite direction, like that. And we're going to start by putting through some phase fronts. And we're going to start with a plane wave. So this is a wave in which the phase fronts are flat and parallel and uniformly spaced. So those are my wavefronts. I'll put a little coordinate system in here. I'm going to tend to put the z-coordinate in the direction waves are going. And so there'll be two transverse coordinates, y and x. And I'll probably just write y on most of these. I'll try and do things in a one-dimensional sense. But everything I say you can imagine could be applied to a three-dimensional picture here. I'm going to say that this plasma slab has some length L, and it's homogeneous within that length apart from the gradient in the density here. Does anyone know what happens to the phase fronts as they emerge out from this plasma? The wavefronts or the rays. I don't mind. STUDENT: They bend in the up or down, right? JACK HARE: Sorry? Yeah, Daniel? STUDENT: Oh. Yeah. They're bent downward, right? Because you've got a lower refractive index in the upper half. JACK HARE: Yeah. You're absolutely right. So these rays will emerge or these wavefronts will emerge bent, like this. And so the rays-- which I didn't draw on before, but I meant to. So here are some rays for you here. You see how they're all normal to the wavefronts. Here are some rays for you here. And they're going to be bent by some angle, which we'll call theta here. And it turns out, if you go and look how to do this, theta is going to be equal to d phase dy times lambda over 2pi. And so we can actually put that all together and we can say it's going to be equal to d dy times the integral of capital N dz, like that. Which, for our plasma, is minus 1 over 2 n critical integral of gradient of the electron density dz. And this dz here is going to be running from 0 to L. Now, I don't know how clear this is to everyone that the rays should bend or that they bend downwards or why they bend. There's lots of different ways of thinking about it. You can go and just solve a load of equations, if you want to. I like to think of it-- and you may laugh at me for this-- as a bunch of soldiers marching arm in arm through some mixed terrain here. So here's my soldiers. I'm looking at them from above. You can see how I'm lining them up nicely with the wavefronts very suggestively. And maybe some of the soldiers over here have got some sort of marsh that they've got to walk through, and these soldiers are going to fall behind. And because they've all linked arms, they've still got to stay in a straight line with each other. And so, as they go, they turn more and more round like this, and this is what leads to our bending here. And you can make this a bit more rigorous if you start thinking about the rays as particles, and you think about their velocity, and you think about the speed that they're going at inside the plasma, and you realize that they're actually going slower in the denser regions, and that's going to start giving you a twist. So they're going faster in the denser regions. They're going slower in the regions with high refractive index. And that's what gives you the bending here. So this is just like a little mental model to think about when you're trying to work out why it is that the rays of light are turning. But there's many, many different ways to get this. STUDENT: Is there a great density over all you've got use this? Hi, is there a gradient and density along the z-axis? The way it's drawn it looks like it's only within the y-axis. JACK HARE: It isn't only in the y-axis. Yes. STUDENT: So an integral of the gradient of density along z from that constant, then, was it-- is there no-- is there a density change in z, si what I'm asking. JACK HARE: Not in this really simple model I'm proposing. Of course, in general, there can be a density change. Really, this should read gradient of density dot dl, where L is an infinitesimal. No. Sorry. Ignore that. Yeah, there is no gradient in density in z. And we don't need one to get any bending. And in fact, if there was a gradient of density in z, it wouldn't have any effect on the light. It would just go forwards at a different speed, but it wouldn't get bent. STUDENT: All right. So then that integral is just gradient of ne times L. JACK HARE: Yes. For this very simple model, you're absolutely right. I'm just introducing the generality because we may have something different. But you're quite right. We could write this as minus 2 ncr times L. And maybe I'll put a subscript z so that I know that it's my length scale z and times by the gradient in ne. And if this is some simple density ramp, so I would have any 0 times 1 minus x upon Ly, or something like that, I can simply put this in and say that the entire beam is now twisted by a nice linear angle, which has an Lz inside it, an ne0, and minus 1 upon Ly, like this. So this, is if I give you some analytical result, you can then go and work out what the angle would be. And that's a super useful thing to be able to do. As I segue perfectly into my next remark, which is to do with the first problem that this causes. So this is issues with deflection. The first issue is if you've got some plasma and you're trying to put some electromagnetic radiation in it, you want to collect that radiation. You want to put it onto a detector. Maybe that detector is a camera, or it might be a waveguide that you're collecting microwaves with. And so that camera has some physical size. And so maybe the camera is represented by this lens here. It's got some physical size D. And if your rays get deflected-- that was a terrible straight line. OK. If your rays get deflected by an angle greater than theta max, where tangent of theta max is equal to d over 2 times L-- I forgot to put in this L here. There we go. Then your ray is going to be lost. So for theta greater than theta max, you lose your rays. So that means you can't collect them. You can't detect them anymore. So this causes big problems because it means that we're going to start losing light here. And for most situations, we can use the small angle paraxial approximation and just replace the tan theta with theta here. So that means you want to keep your deflection angle theta, which is equal to, as we said, ddy of the integral of Ndl. That wants to be less than theta max. So there's a few things that you can do to try and do this. You can have a nice big lens. You can have a close lens. You can put it nice and close in. Or you can use a shorter wavelength. Because if you go to a shorter wavelength, you get a smaller deflection angle, which you can see if you go back to maybe this formula here. A shorter wavelength corresponds to a larger n critical, and so you'll get a smaller angle. Now, not all of these things are possible in a standard experiment. If you've got a tokamak, you may have a limit on how big your detector can be, because it's got to fit in a gap between some magnets. You'll certainly have a limit on how close you can put it to the plasma because you don't want to stick it right inside. And you may not be able to choose whatever wavelength you want. Perhaps you're looking at electron cyclotron emission and you've got no choice but to use the wavelength that's emitted at. So this can cause big problems. And so if you're doing some electromagnetic probing of your plasma, one of the first things you should probably do is check whether the density gradients are going to make it hard to actually measure anything. Any questions on this? All right, so we're now going to plunge in to our first diagnostic. And the point I want to make here is although deflection can be frustrating, it can also be useful. Because we can use it to measure something about the plasma. The first thing we're going to talk about is schlieren imaging. This word, schlieren, people often assume refers to a person. It does not. So it doesn't have a capital, despite what Overleaf will tell you. And it's actually after a German word, schlierer, which is like streaks, because this was first used for looking at small imperfections in optics. And so looking at these little streaks here. So it's a way of imaging things which would otherwise be impossible to see because they cause small gradients in refractive index. So let's have a little example, building up towards schlieren imaging. This first thing I'm going to show you is not schlieren imaging. This is just imaging. But my impression is that some folks need a refresher with some optics. So we're going to start with a solid object. It's going to be this nice little chalice here. And we're going to put in some rays of light. Like this. Now, this object is solid, and so it blocks any rays of light which hit it, these two centers ones, and allows through rays of light going past it. Allows through rays of light going past it. Very good. And what we would probably do here if we're doing a standard imaging system is we would have a lens. So this is how you form an image. So we'll put our lens here. It's going to have a focal length F. And we're going to place it at a distance, which is 2F away from the object we're trying to image. I'll just put that F up there. Now, behind this lens, if we're doing a standard 2F imaging system, we're going to have a focal point, and that's going to be at F away. And then, we're going to have an object plane, which is also at F away. Sorry, an image plane. This is the object plane. And this is the lens with focal length F. So hopefully, some of you have seen this sort of thing before. You know that the rays will pass through the focal point here. He says, drawing them carefully. And what we'll end up with-- can't do this on a chalkboard-- is a copy of our image, of our object here. But it's going to be inverted. And you can tell that because you can see the rays have changed place. So this is a nice 1 to 1 image. It's at magnification 1, and it is inverted. So this is the simplest-- I think, the simplest possible imaging system you could possibly develop. It simply takes whatever is at the object plane and puts it at the image plane some distance away. This could be a microscope. This could be a camera. All sorts of things like that. Yes, Vincent? STUDENT: I think I missed it. What was F again? JACK HARE: I beg your pardon. STUDENT: What was F, like, in the diagram? JACK HARE: The focal length of the lens. STUDENT: Oh. Thank you. JACK HARE: Cool. Any other questions? OK, let's make this more interesting. Let's put a plasma here instead. And we're still going to have our lens. There still can be a focal point. And there's still going to be an image plane here. But the plasma doesn't block the rays of light, as long as we've got, for example, ne much, much less than the critical density here so that the rays can pass through easily. Instead, what we're going to have is rays that come in. And then, they're going to be deflected slightly inside this plasma. So I haven't drawn the density gradient. We can imagine, we've just got a whole range of exciting density gradients that cause some deflections. So they deflect this ray slightly downwards. They deflect this ray slightly upwards. They deflect this bottom ray-- what have I done to this one? Let's have this one go straight. For some reason, there's no density gradient exactly there, so the ray just goes through. And this bottom one gets deflected downwards as well. And let's say it just about makes it onto the lens. And I'll move the lens downwards to make that true. Can't do that on a chalkboard either. OK, good. So what will the lens do to these rays now? Well, it's still going to reflect the rays, and it's going to reflect them-- let's start with this one that actually didn't get deflected at all. So its angle hasn't changed. It's going to go straight through the focal point, as you'd expect. This one that was deflected upwards is going to be deflected down, but it's going to slightly miss the focal point. It's going to be slightly above it, like that. This one that was deflected downwards is going to be the opposite way. It's going to be slightly below the focal point. And this one was deflected downwards. It's also going to be slightly below the focal point. And that was a mistake. There we go. There we go. It's not a very good image compared to the one I was hoping to draw, but there we go. Nothing quite works out. So we should have an image of our plasma here. This image of the plasma should still be 1 to 1 mag 1 and inverted. The fact that I haven't quite managed to get it to work is probably just a flaw with how I managed to draw the rays this time round. Not quite sure what went wrong there. Looks good on my notes, anyway. This stuff gets a little bit tricky. The point is, although the rays here look like they've all gone upwards slightly, they should actually still end up in the same places that they did before. And the fact that I can't get it to work right now just means that I've made a mistake while drawing it live. OK, good. STUDENT: Jack, this is a very basic question, but what's the point of having all the rays go through the trouble of going through the lens when we could just have them go straight through and hit our image plane? Guess I missed that. You know what I mean? JACK HARE: Yeah, absolutely. So in the top case, the solid object, the rays could go-- if the rays went straight through and hit the image plane, they will be deflected slightly at the edges. And so you'll end up with something fuzzy, so it'll be out of focus. So you need a lens to bring it to focus, which effectively is mapping the rays from where they came from back to the same place on the object plane. If in the case of the plasma, if you don't have the lens and you just put-- if you don't have a lens and you just let the rays propagate to a screen, that's a technique called shadowgraphy, which we'll talk about next lecture, which I actually think is more difficult even though it's simpler to draw. And so I want to talk about it after this one. So we haven't done anything here at the moment. And in fact, if you do this with a plasma, you won't see anything at all. Because all of the rays are mapped back to where they started from. And that means that you are going to end up with just the same laser beam that you originally started with or the same microwaves you originally started with. So this will be invisible. So the only way we can make this visible is to notice that the rays do not all pass through the focal point. Now, you saw in the case where we did the imaging that all the rays did, indeed, still pass through a focal point. But here, some of them have gone above and some of them gone below. And in fact, the distance they've gone above and the distance they've gone below is directly proportional to the angle with which they exited the plasma here. And so we can learn something about the angle they can select exited the plasma by placing a filter at this focal plane. And this filter maybe looks a little bit like this. This filter, for example, here is like a little aperture. And it lets through these two rays. Let me color code them. This one and this one. And it blocks off these two rays. And so light is coming from that bit of the plasma, where the density gradients were large will be blocked and it will no longer appear on our final image. And this is what Schlieren imaging is. So we place a stop at the focal plane and we filter by angle. I'm going to do some exhaustive examples of this to try and build some intuition for what's going on if you're a little bit confused right now. Any questions on this before we keep going? STUDENT: I have sort of an overall conceptual question. I feel like the highest gradients, a lot of the time, or-- well, maybe this isn't quite true. But I can imagine, in a lot of cases, this is going to be affected most by the edges where it's entering and leaving the plasma because you're-- yeah, depending on how uniform things are. So just curious what you actually get an image of. I mean, if you-- especially if you don't have a great sense of where the gradients are or if you have gradients inside you don't know about or something. JACK HARE: Yeah, absolutely. These images are difficult to interpret. So this is not a generic diagnostic technique that will immediately tell you what's going on. You need to know something about your plasma. Maybe you have a simulation and you do a synthetic diagnostic on it. Or maybe you've set up your experiment such that it's particularly simple. We'll talk a little bit about some simple distributions and what patterns they make and that will give us an idea for what sorts of things we might be able to measure with this. Turbulence in the edge of a tokamak or something like that, this is maybe not the ideal diagnostic for it. STUDENT: Fair enough. JACK HARE: Yeah. I want to make very clear because I don't know if it came across when we were talking about it before. But we only get deflections from density gradients, which are perpendicular to the direction here. So if our ray is going in this direction, in the z direction, we sense ddy of ne and d dx sub ne, but we do not sense ddz of any. So if there's density gradients in the direction the ray is propagating, the ray will slow down or speed up, but it won't actually deflect from that. And so that helps you a little bit. You're only sensitive to gradients perpendicular to the probing direction. That might also tell you if you've got a plasma, which you think has some geometry. There may be a good direction to send the probing beam through there maybe a bad direction. So you want to think about that a little bit. So let's have a talk about some of these stops, right? So I've said that we can place these stops here. Let's have a chat about what sort of different stops are available to us. So types of stop. The first type is to decide whether our stop is going to be dark field or light field. So I'm putting darken in brackets because it will save me writing in a moment. The difference between dark field and light field is that the dark field blocks undeflected rays. So ones that do pass through the focal point. And the light field blocks deflected rays, ones which do not pass through the focal point. So you can either look for regions where there are density gradients or where there aren't density gradients. We can also choose the shape of our stop because our stop is a two dimensional plane at the focal plane here. So we can have a circular stop. And that doesn't care what direction the ray is deflected in, it only cares on the size of the angle. So the size of theta. So that is basically, are there any large density gradients? Or we can have what's called a knife edge, which is linear like this. And that is sensitive to density gradients in only one direction and it still cares about the size of the density gradient. That is like x hat here. So we could, for example, have a stop at the focal plane. No, it's not going to do it. OK, fine. I can't get a nice, round circle. We can have a stop which is an opening inside an opaque sheet of material here. And this opening could be positioned such that the focal spot in the absence of any plasma sits inside it. What sort of stop would this be? STUDENT: Light field? JACK HARE: So this is a light field stop. And what shape is it? [INTERPOSING VOICES] Yes, OK, circle. Thank you. We could also have a stop that looks like this. And we can position it such that the focal spot is actually within the opaque region. And what sort of stop would this be? STUDENT: Dark field. STUDENT: Dark field. JACK HARE: OK, dark field. And what shape is it? STUDENT: Linear. JACK HARE: So the knife edge here. Yeah. We call it a knife edge because actually using a razor blade is a pretty good thing to have because you get a very nice, sharp, uniform edge to it. OK, and so depending on these stops you can think of as filters in angle space, right? So they allow through certain angles. You can think of arbitrarily complicated versions of this. There's a technique called angular-- not fringe. Angular filter refractometry, which has a set of nested annuli, which let through light which has been deflected by certain specific angles. So the world is your oyster. You can come up with all sorts of exciting different stops if you want to. One thing I will note is that the dynamic range of your diagnostic, which we'll talk about more later, depends a great deal on your focal spot size. So I've shown these focal spots to be relatively small here. But that focal spot size, at least in a diffraction limited sense, it covers an angle, which is equal to the wavelength of your light over the size of your lens, the diameter of your lens here. And so you might end up having focal spots which are not small, but actually could be rather large. And then, part of the focal spot could be obscured and part of the focal spot could be clear for some given deflection angle. And in general, when we've got a plasma here, we have, say, our small focal spot before we put any plasma in the way. When the plasma is gone in the way, different rays of light have been deflected by different amounts. So this thing may take on some complicated shape here. And this is the shape that you're filtering. You might be filtering it with your knife edge like this, or you might be filtering it with your circular stop like this. So we're basically filtering the rays based on how far they've been deflected. At the focal plane, there's no information about where the rays came from inside their plasma. So their spatial information has been lost. The only thing we know about them is their angular position. So rays, which are deflected by an angle theta 1 from at the top of the plasma, are rays which are deflected by the same angle from the bottom of plasma. It ends up at the same place in the focal plane, even though they came from different parts of the plasma. This is the magic of geometric optics. So any questions on this or should we do a little example? OK, let's do our example. So let us consider a very simple plasma. This plasma-- we'll have the coordinate system y vertically and we'll have z in the direction of propagation as we discussed before. We'll have rays. Well, I'll draw the plasma first. So the plasma is going to have a density distribution that sort of looks Gaussian ish, some sort of nice peaked function. So this is density, ne. So you can think about this, for example, as like a cylinder. So you've got a cylinder of plasma, like a z pinch, and you're probing down the axis of this z pinch and it's got a Gaussian distribution of density to it. Anything like that. And then, we'll have the rays of light coming through. So we'll have rays of light, which are sampling very small density gradients at the edges here. And these rays e will just go straight through. There's also be a ray that goes through the center, which also sees a very small density gradient here, right? At the center of this distribution, the density gradient is zero. But the edges here where the density gradient is large will have some deflection. And if we place our lens, as we did before, some distance away, it's going to focus those rays onto our focal plane. And in our focal plane, we're going to put some sort of stop here. So the undeflected rays are just going to go straight through the focal point. But the deflected rays are not going to go through the focal point. The deflected rays are going to go above and below. So now we can put a series of stops inside here. So we can have a stop on that blue dashed line that looks like a light field knife edge. We can have a stop that looks like dark field knife edge. We can have a stop that looks like a light field circle. And we could have a stop that looks like a dark field circle. And what we're going to do is sketch out what we expect the intensity to look like from each of these different knife edges here. So if I plot this one-- I'll just draw the density distribution again like that. So that's our density. Now we want to know what our intensity distribution looks like. So this is now intensity. We have our initial intensity 1 and we have 0 intensity corresponding to all the rays being blocked. So where for the light field knife edge do we see no intensity? STUDENT: In the upper side where there is the largest density gradient. JACK HARE: I'm sorry, I didn't hear that properly. Can you say it again? STUDENT: I think the upper portion where there is the largest density gradient. JACK HARE: So I think what you said was in the upper side where there's the largest density gradient, right? So this is where the density gradient is very large. So we expect outside of that region our intensity would be pretty much constant. But inside that region, we'd expect the intensity would drop to 0. Because we're blocking the rays which have a large deflection angle in one direction upwards and the rays which have a large deflection angle in one direction upwards corresponds to that specific density gradient there. OK, anyone want to have a go at telling me what happens with this one? What's happening at the edges? What's happening out in these regions here where the density gradients are small? STUDENT: They're cropped out because they mostly go through the focal spot. JACK HARE: Right. Yeah, so these are going to be 0, right? So 0, 0, is there any region where it's not zero? STUDENT: I think it's not 0 for the high gradient region that's lower in y. Because it's deflected. JACK HARE: So you think it's not 0 for this region here? STUDENT: Yes. JACK HARE: Does anyone agree with Sara or does anyone disagree? STUDENT: I think that looks right. JACK HARE: OK, so we're talking about this ray here. Yeah. So this ray is, indeed, passing low down compared to the knife edge. So we'd expect this to show some intensity here. But not for the upper one, which passes higher up. OK, good. We're getting there. What about for the circular light field? Anyone want to tell me what the intensity looks like here? STUDENT: You'd probably get three peaks. So on the edges where the density isn't really-- where the density is just low and then right in the middle where the density is high, but not really changing too much. JACK HARE: Yeah. So you say three peaks, I'm going to think about them as two notches, but I agree with what you're saying. So these notches here correspond to the points where the density gradient is largest here. And so those rays got a big deflection angle there being blocked out. What about for the final one here, dark field circular aperture? STUDENT: Well, there you're basically notching out all the minimally deflected rays, so you lose the ones on the edges and center. JACK HARE: Yeah, so we'd have something that looks like this, right? So we would just see light where there was a significant deflection angle. OK, this may still seem a little bit abstract, so we're going to try one more thing. And I hope that you don't hate me for spending so much time on Schlieren, but I absolutely love it, so that's your loss. Which is going to be a two dimensional example here. I'll just draw on this distribution function. There we go, density function now on that last one just so you've got it. So this actually goes back to a sketch that I did during the b dot lecture where we stuck a little b dot inside the plasma. And we have plasma flow coming from left to right. And because it collides with this, we get some sort of bow shock like this. And as we all know from our shop physics, at the bow shock, we have strong density gradients like this where the density jumps across the shock here. So if we look at this system and we have our probing laser coming towards us, so our laser is looking towards us like this, we are the camera. What would we see about this bow shock? What could we tell about it? So maybe the first thing we could do is ask for a light field circular aperture, what do we think we would see here? And if you're not quite sure, at each of these places where I've drawn this little arrow, you could say that the density maybe as a simple model looks like a sort of hyperbolic tangent type thing like that. It's got some region where the density is ne0 some region where the density is ne1, and then some region where the density changes rapidly. That's not true for a shock, but it's a model just to get us thinking about what this looks like. So what would I see on my image? I've got this expanded laser beam going through this shock and I've decided to use a circular light field stop. STUDENT: I might be doing this backwards, but from-- it looks like what we drew above. That would mean that you're not getting your steep gradient sections, so it should be dark where the bow shock is. JACK HARE: So you'd actually have an image. Yeah, you'd have an image, which if I was on a chalkboard, I could do as an eraser, but it's actually quite hard here. If you imagine this is filled, a green laser beam beautiful nice laser beam image. Then you would have a region where it-- no, the arrays is not very good. If I can make the eraser smaller that would work much better you would have a region where there was no light whatsoever and there was just darkness, right? So you just have this dark region here corresponding to the bow shock. And if you did this with a dark field circular aperture, then you'd have the opposite. You'd have complete darkness and you would just have a region where the bow shock shows up very nicely like this. And so this is actually typically what we use for shock measurements is dark field circular aperture. If you do happen to do something like dark field with a knife edge and you put your knife edge like this, so you were measuring density gradients that were in this direction, you would end up seeing something a little bit like just one half of the bow shock like that. So if you set-- your probe was sitting here, you would just see a little bit of the bow shock. You wouldn't see this section because the density gradients wouldn't be in the correct direction. So you wouldn't be able to observe those. But the knife edge might be a better choice for some shock geometries if you're only interested in gradients in one direction. Aidan, I see your hand. STUDENT: Yeah, I'm curious how distinct the actual image gradients would be given that this is like a cylindrical phenomenon, right? The shock. So I assume there's some density gradients that's they slowly become parallel to your propagation direction as you rotate to the-- JACK HARE: Yeah, so you won't see those density gradients, but you will see the ones on the edges of the shock. Yeah, it depends on the exact shock morphology, whether it's some extended like a bow shock that's extended like this or whether it's a bow shock that's sort of rotated like the tip of a nose cone on an aircraft or something like that. So it will make a difference. But this is just to try and get a feel for what this looks like in terms of imaging here. I can send around some papers later, which have some very nice images of Schlieren of shock structures that show what sort of quality of data you can get out from this. So any other questions on Schlieren? I will then just summarize exactly how to use it and when not to use it and things like that. But if anyone has any questions on these sort of worked examples we've done, please go ahead and shout out. So Schlieren is good for visualizing density gradients, right? And in particular, it's good for visualizing strong density gradients. So in particular, it's good at visualizing shocks. So if you're looking at shocks in plasmas, this is a nice diagnostic. And it's particularly nice because it's very simple to set up. As I showed you, it's just got at minimum a single optic. You need to form a focal point, so you need to have an optic. You need a focal point and you need a stop. So this is a very simple thing to set up. You can do it very, very quickly. The trouble is, although it's simple, it's very difficult to be quantitative. We can say there is some sort of density gradient there. We think the density gradient has to be larger than some certain limiting value. But other than that, we can't really say much more than that. So it's useful for seeing where shocks are and their morphology, but is not useful for measuring gradient of any. So we can't measure gradient Ne. We can measure shape and location of the density gradients. Now, there are some ways where you can make this more quantitative. So imagine you've got a little beam of light coming in like this. This is going in this direction. And you have a knife edge that looks like this. That knife edge is going to be entirely blocking what's coming through, we'll have 0 light coming through. But if we have a beam that comes through and it's lined up like this, you can see that now about half the light is going to get through. And if we have a beam coming through that's lined up above the knife edge, we'll have all of the light coming through. And it turns out that if you have a large enough spot that you can actually see this partial obscuration of the focal point, you end up in a regime where the intensity that you see on your detector eye is, in fact, proportional directly to the gradient of the electron density. I'm putting it in brackets because you need a large uniform focal spot. And I'll talk in a moment about why that's actually extremely difficult using a laser, which is what we normally use here. OK, if you can't guarantee that you have a nice large uniform focal spot, you could also use a graded neutral density filter. So a neutral density filter is sort of smoky glass that blocks light. And you can change the amount of impurities in it to block more light. So we could carefully fabricate for ourselves a neutral density filter that has a changing absorption. So we can have a gradient in, say, we'll call the absorption alpha here. And then, the idea is that different rays of light at different points of your stop will either come through not at all or very attenuated. Or they will come through partially attenuated or they will come through lazy not attenuated at all. And so if you have a really good graded neutral density filter, you can, again, end up in this regime where you actually get some sensitivity. To the position of the beam. And that can get you back into this nice regime up here where the intensity of your signal is actually proportional to the gradient of the electron density. So that'd be a very nice place to be, but this is hard to make. And you still need a uniform beam to start with, which is also hard to make. So these techniques, a lot of Schlieren was actually developed using light sources, which are very different from the light sources that we have to end up using in our experiments with plasmas. And so you end up, yeah, so just looking at the beautiful pictures that Matthew put in the chat here. These pictures here are absolutely glorious. And you can see there's a huge amount of detail on them. And this detail is due to the fact that actually I think for these ones they're using the sun as the back lighter. I may have forgotten this exactly, but light source is the sun. And the sun is actually quite large. It's an extended object, you may have noticed, in the sky. And this gives you really nice Schlieren imaging. But we don't tend to use nice large objects for our experiments with plasmas. We tend to use things like lasers. And lasers are not a good Schlieren source. And if you want to read in more detail why lasers are actually a really bad idea, you should go read the book that's listed in the bibliography by Settles, which is an absolutely cracking book. Has lots of lovely pictures, like the ones which were just posted in the chat. And also, a very detailed description of this stuff. But can anyone tell me why-- if lasers aren't very good, why do we end up using lasers as the light source for Schlieren imaging in plasma physics? What property of a laser is it that we particularly want? STUDENT: Monochromatic light. STUDENT: Monochromatic. JACK HARE: So monochromatic is somewhat useful. Actually, it turns out you can do really cool Schlieren techniques with a broadband light source as well and filters because the different wavelengths will be reflected by different amounts. And so if you have multiple cameras with different filters, you can really carefully reconstruct the deflection angles. So monochromatic is a good guess. But actually, we'll be OK with broadband. What else-- when you think laser, what do you think? STUDENT: Coherence. JACK HARE: Coherence. Actually, coherence isn't good for this. And we've been using a picture and the coherence screws up our nice simple picture. It's going to cause problems. So we really want coherence for interferometry and we absolutely hate it for Schlieren. And that's another thing that Settles says in his book, so coherence is a bad thing. So I'm going to put coherence. No, we actually don't want it. What else do we think about lasers? STUDENT: Well, they tend to produce nice round spots. So it's also easy to deflect them without changing them that. JACK HARE: Nice beam. Yeah, you can do that with other light sources. But fine, it is a nice beam. Anyone ever shone a laser pointer in their eye? If not, why not? STUDENT: High energy density. JACK HARE: They're very bright. Should we put it that way? OK, so lasers are extremely bright. Why do we want a bright light source when we're dealing with plasmas? STUDENT: The plasmas glow. JACK HARE: Yes, so we need to overcome the plasma glow, should we call it, or should I say emission here. So when you're working with jet planes flying around and you're taking Schlieren images of them using the sun as your background, you don't actually have to worry about the plasma or the shocked air around the plane glowing and ruining your measurements. Whereas, a plasma, you really do. It makes an awful lot of light. So you need an incredibly bright light source. And so despite the fact lasers are terrible for Schlieren it's the brightest light source that we have available so we end up using that. But the trouble with lasers is that they have a very small focal spot. So there are, in fact, incredibly easy to focus down to a point. And we don't want that. We want a nice large focal spot. And it's actually very difficult to get a laser to decohere enough to get a nice big focal spot. But maybe there are some techniques we can do. And because we've got a small focal spot, laser Schlieren is effectively binary. And I'll explain what I mean by that when I remember how to spell Schlieren. What I mean by that is the spot is so small, the laser Schlieren, it is either completely blocked by the knife edge or it is completely visible, unblocked, by the knife edge. And so when you do a laser Schlieren image, you get an image which is either dark or light, 0 or 1, no intensity or full intensity. And so you don't get those nice images that we were just looking at that were linked. And we also don't have the ability to use the change in intensity to measure the gradients in the electron density because the intensity is either 0 or 1. So we have very, very limited dynamic range. You can think about it that way. We have no dynamic range. We even know the density gradient is larger than the number or it's smaller than a number and that's it. So it makes very good shock pictures, but you can't do all the beautiful techniques that people use Schlieren for in standard fluid dynamics where they have access to different sources. So someone came up with a nice, bright, incoherent large area laser that we could use. That would be absolutely great. I have a few ideas along this direction, but haven't had a chance to try them out yet. So yeah, they have some serious limitations. So Schlieren is a lovely technique for plasmas. It's very limited. It's obviously useful when you have shocks, so you need to be having fast moving plasmas. And so this is not really a technique that we would normally use inside say a tokamak or a magnetically confined device, but it is very good when you have larger densities. All right, so any questions on this? Nigel, yeah. STUDENT: So did we ever say how exactly the critical density was determined? Is that for a later lecture? JACK HARE: Do you mean like, what is a numeric-- what is an analytical result for the critical density? STUDENT: Yeah, when we had like 1 minus n over n critical. Where that was ever determined? Or is it just kind of guess and check. JACK HARE: No, absolutely not. You can write it down. It's a function. And I neglected to write it down here. And I am not going to try and remember what it is. But it is a function which has inside it the frequency of the laser light, epsilon naught, e and me and the-- not the density. Definitely doesn't have the density inside it. So this effectively comes from rearranging the plasma frequency here. And you can find this in Hutchinson's book, and I really should have written in my notes, but I just didn't, OK? And I'm not going to try and bullshit you and look it up now. So yeah, the critical density is a number that we can find. It is uniquely defined for every electromagnetic wavelength and that is the density, which you cannot go above. And if you're sensible, you'll work well away from the critical density. Because if you get anywhere close to it, it really screws up most of these calculations. So most of the time, we're relying on this approximation that the density we're using is much less than the critical density. And we'll come across that approximation very strongly when we deal with interferometry in a little bit. But we did also use it already when we were deriving the Schlieren angle. When we got to this point here, we used this linear approximation where the density is much less than the critical density. Any other questions? Yeah? No? STUDENT: So can you hear me? JACK HARE: I can hear you. Yeah. STUDENT: OK, cool. So would it not be possible to use the quote unquote glow of the plasma itself to use in some other part of the plasma that isn't glowing itself or that it's glowing at some other frequency? Say, once we're putting-- JACK HARE: I thought about this. I think it's a really cool idea. You definitely have to arrange your plasma so there's a bit that's glowing and a bit that isn't. It's a bit you want to measure as also glowing that's going to make things very, very difficult. So yeah, you could potentially have something like that, like a ICF hotspot backlighting the whole plasma. That sort of thing could work. Yeah. But I don't know of anyone who's done it. STUDENT: OK, Thank you. JACK HARE: All right, we're past the hour now. So I think we'll leave it here. I will be back in the classroom next Tuesday. I look forward to seeing all of you there and all of the Columbia folks online. And yeah, enjoy your weekend and bye for now.
|
22-67j-principles-of-plasma-diagnostics-fall-2023
|
physics_bio
|
2
| "The following content is provided under a Creative Commons license. Your support will help MIT Open(...TRUNCATED)
|
8-04-quantum-physics-i-spring-2013
|
physics_bio
|
3
| " JACK HARE: OK, so you'll remember, in the last lecture, we discussed the technique called schliere(...TRUNCATED)
|
22-67j-principles-of-plasma-diagnostics-fall-2023
|
physics_bio
|
4
| "PROFESSOR: I'm Ernst Frankel. I'll be teaching next two lectures. I'd like to encourage you to cont(...TRUNCATED)
|
7-91j-foundations-of-computational-and-systems-biology-spring-2014
|
physics_bio
|
5
| "PROFESSOR: Welcome back, everyone. I hope you had a good break. Hopefully you also remember a littl(...TRUNCATED)
|
7-91j-foundations-of-computational-and-systems-biology-spring-2014
|
physics_bio
|
6
| "ANNOUNCER: The following content is provided under a Creative Commons license. Your support will he(...TRUNCATED)
|
7-91j-foundations-of-computational-and-systems-biology-spring-2014
|
physics_bio
|
7
| "The following content is provided under a Creative Commons license. Your support will help MIT Open(...TRUNCATED)
|
8-591j-systems-biology-fall-2014
|
physics_bio
|
8
| "The following content is provided under a Creative Commons license. Your support will help MIT Open(...TRUNCATED)
|
8-286-the-early-universe-fall-2013
|
physics_bio
|
9
| "The following content is provided under a Creative Commons license. Your support will help MIT Open(...TRUNCATED)
|
8-286-the-early-universe-fall-2013
|
physics_bio
|
End of preview. Expand
in Data Studio
README.md exists but content is empty.
- Downloads last month
- 12