Fundamentals of Physics, II : 19. Quantum Mechanics I: The key experiments and wave-particle duality (Lecture 19 of 25)

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PHYS 201: Fundamentals of Physics, II

Lecture 19 - Quantum Mechanics I: Key experiments and wave-particle duality

Overview:

The double slit experiment, which implies the end of Newtonian Mechanics, is described. The de Broglie relation between wavelength and momentum is deduced from experiment for photons and electrons. The photoelectric effect and Compton scattering, which provided experimental support for Einstein's photon theory of light, are reviewed. The wave function is introduced along with the probability interpretation. The uncertainty principle is shown to arise from the fact that the particle's location is determined by a wave and that waves diffract when passing a narrow opening.

Resources:
Notes: Quantum Mechanics [PDF]
Problem Set 10 [PDF]
Problem Set 10 Solutions [PDF]

Fundamentals of Physics, II: Lecture 19 Transcript

March 31, 2010

Chapter 1: Recap of Young's Double Slit Experiment [00:00:00]

Professor Ramamurti Shankar: So this is a very exciting day for me, because today, we're going to start quantum mechanics and that's all we'll do till the end of the term. Now I've got bad news and good news. The bad news is that it's a subject that's kind of hard to follow intuitively, and the good news is that nobody can follow it intuitively. Richard Feynman, one of the big figures in physics, used to say, "No one understands quantum mechanics." So in some sense, the pressure is off for you guys, because I don't get it and you don't get it and Feynman doesn't get it. The point is, here is my goal. Right now, I'm the only one who doesn't understand quantum mechanics. In about seven days, all of you will be unable to understand quantum mechanics. Then you can go back and spread your ignorance everywhere else. That's the only legacy a teacher can want. All right, so that's the spirit in which we are doing this. I want you to think about this as a real adventure. Try to think beyond the exams and grades and everything. It's one of the biggest discoveries in physics, in science, and it's marvelous how people even figured out this is what's going on.
So I want to tell you in some fashion, but not strictly historical fashion. Purely historical fashion is pedagogically not the best way, because you go through all the wrong tracks and get confused, and there are a lot of battles going on. When the dust settles down, a certain picture emerges and that's the picture I wanted to give you. In a way, I will appeal to experiments that were perhaps not done in the sequence in which I describe them, but we know that if you did them, this is what the answer would be, and everyone agrees, and they are the simplest experiments.
All right, so today we're going to shoot down Newtonian mechanics and Maxwell's theory. So we are like the press. We build somebody up, only to destroy them. Built up Newton; shot down. Built up Maxwell; going to get shot down. So again, I have tried to drill into all of you the notion that people get shot down because somebody else does a new experiment that probes an entirely new regime which had not been seen before. So it's not that people were dumb; it's that given the information they had, they built the best theory that they could. And if you give me more additional information, more refined measurements, something to the tenth decimal place, I may have to change what I do.
That's how it's going to be. So there's always going to be — for example, in the big collider, people are expecting to see new stuff, hopefully stuff that hasn't been explained by any existing theory. And we all want that, because we want some excitement, we want to find out new things. The best way not to worry about your old theories is to not do any experiments. Then you can go home. But that's not how it goes. You probe more and more stuff. So here's what you do to find out what's wrong with electrodynamics, I mean, with Maxwell's theory. It all starts with a double slit experiment. You have this famous double slit and some waves are coming from here. You have some wavelength l. Then in the back, I'm going to put a photographic plate. A photographic plate, as you know, is made of these tiny little pixels which change color when light hits them and then you see your picture. And that's the way to detect light, a perfectly good way to detect light.
So first thing we do is, we block this hole or this slit. This is slit 1 and slit 2. We block this and we look at what happened to the photographic plate. What you will find is that the region in front of it got pretty dark, or let's say had an image, whereas if you go too far from the slit, you don't see anything. So that's called intensity, when one is open. Then you close that guy and you get similar pattern. Then you open both. Then I told you, you may expect that, but what you get instead — let's see, I've got to pick my graph properly — is something that looks like this. Now that is the phenomenon of interference, which we studied last time. So what's the part that's funny? What's the part that makes you wonder is if you go to some location like this, go to a location like this. This used to be a bright location when one slit was open. It was also a bright location, reasonably bright, when the other slit was open. But when both are open, it becomes dark. You can ask, "How can it be that I open two windows, room gets darker? Why doesn't it happen there, and why does it happen here?" The answer is that you're sending light of definite wavelength and the wave function, Y, whatever measures the oscillation, maybe electric field, magnetic field, obeys the superposition principle.
And when two slits are open, what you're supposed to add is the electric field, not the intensity. The intensity is proportional to the square of the electric field. You don't add E2; you add E. E is what obeys the wave equation. E1 is a solution, E2 is a solution. E1 + E2 is a solution. No one tells you that if you add the two sources, I1 + I2 is going to be the final answer. The correct answer is to find E1 + E2 and then square that. But when you do E1 + E2, since E1 and E2 are not necessarily positive definite, when you add them, sometimes they can add with the same sign, sometimes they can add with opposite signs, and sometimes in between, so you get this pattern. So we're not surprised. And I've told you many times why we don't see it when we open big windows, first of all, when you open a window, the slit sizes are all many, many, many million times bigger than the wavelength of light. Plus the light is not just one color and so on. So you don't pick up these oscillations. These oscillations are very fine. I draw them this way so you can see them. In a real life thing, if this were really windows and this was the back wall of your house, the oscillations would be so tightly spaced, I'll just draw some of them, that the human eye cannot detect these oscillations. It will only pick up the average value. The average value will in fact look like I1 + I2.
So this is all very nice. This is how Young discovered that light is a wave. By doing the interference, I told you, he could even find a wavelength, because it's a simple matter of geometry to see where you've got to go for two guys to cancel. And once you know that angle, you know where on that screen you will get a minimum or a maximum. So you've got the wavelength. He didn't know what was doing it. He didn't know it was electromagnetic, but you can get the wavelength. So interference is a hallmark of waves. Any wave will do interference. Water will do the same thing. For example, this is your beach house. You've got some ocean front property. This is a little lagoon and you have a wall to keep the ocean waves out of your mansion. And then suddenly one day, there is a break in the wall. Break in the wall, the waves start coming in and you're having a little boat here, trying to get some rest. The boat starts jumping up and down because of the waves. So you have two options.
One options is to go and try to plug that thing, but let's say you've got no bricks, no mortar, no time, no nothing. You've just got a sledgehammer. What you can do, you can make another hole. If these water waves are a definite wavelength, only in that case, you can make another hole so these two add up to 0 where you are. Similarly, if you don't like the music your roommate's playing, if you can manufacture the same music with a phase shift of p, you can add them together, you get 0. But you've got to figure out what the roommate's about to do and be synchronous with the person, but get a wavelength of p — I mean, a phase shift of p. So you can cancel waves. That's the idea behind all kinds of noise cancelation, but you've got to know the exact phase of the one signal that you're trying to cancel.
Chapter 2: The Particulate Nature of Light [00:09:12]
All right, so everything looks good for Maxwell, till you start doing the following experiment. You make the source of light, whatever it is, dimmer and dimmer, okay? So you may not be able to turn down the brightness. Maybe you can, maybe you cannot, but you can imagine moving the source further and further back. You move it further and further back, you know the energy falls like 1/r2, so you can make it weak.
So here's what we try to do. We put a new photographic film. We take the light source way back, then we wait for something to happen. We come the next morning, we find there's a very faint pattern that's taken place over night, because the film got exposed all through the night. Now we can see a faint pattern. Then you go and turn it down even more. You come back the next day, you look at the film. You find no pattern, just two or three spots which have been exposed. If you look at the screen — so let me show you a view of the screen. Normally you will have bright and dark and bright and dark patterns on the back wall if you turn on a powerful light. But I'm telling you, if you have a really weak source, you just find that got exposed, that got exposed and that got exposed, that's it. Only three points on the film are exposed, and that is very strange. Because if light is a wave, no matter how weak it is, it should hit the entire screen. It cannot hit certain parts of it. Waves don't hit certain parts. In fact, how can it hit just one? For example, if you make it weak enough, you can have a situation where in the whole day, you just get one hit. So something is hitting that screen and it's not a wave, because a wave is spread out over its full transverse dimension, but this is hitting one point on the screen. So you make further observations and you find out that what happens here is there is a certain amount of momentum and energy are delivered during that hit.
If you could measure the recoil of that film, you will find it gets hit and the momentum you get per hit looks like ℏ x k. I'll tell you what it means. This is 2ph/l, where l is your wavelength, h is the new constant called the Planck's constant. And its value is 6.6 x 10-34 joule seconds. You also find, every time you get a hit here, there's a certain energy deposited here and the energy deposited here happens to be ℏw where w, as you know, is 2pf.
So here's what I'm telling you. If you send light of a known frequency and known wavelength and you make it extremely dim, and you put a photographic plate and you wait till something happens, what happens is not a thin blur over the whole screen. What happens is a hit at one location. And what comes to that location seems to be a bundle of energy and momentum, i.e. a particle, right? When something hits you in the face, it's got energy, it's got momentum. So this film is getting hit at one point by a particle, and what we can say about the particle is the following: it has a momentum. It has the same momentum every time. You get this hit, you get that hit, you get that hit. As long as you don't screw around with the wavelength of incoming light, the momentum and energy of each packet is identical. It's more than saying light seems to be made of particles. Made of particles, each one of them carries an energy and momentum that's absolutely correlated with a wavelength and frequency.
Now let me remind you that w = kc for light waves. We've done this many, many times. That means the energy, which is ℏw, and the momentum is ℏk, are related by the relation E = pc. So these particles have a momentum which is related to energy by the formula E = pc. When you go back to your relativity notes from the last semester, you'll find the following relation is true. Any particle, E2 = c2p2 + m2c4. That's the connection between energy and momentum. Therefore this looks like a particle whose m is 0. If m is 0, E = pc. So these particles are massless. They have no rest mass and you know, something with no rest mass, if it is to have a momentum, it must travel at the speed of light. Because normally, the momentum of anything with mass is mv, in the old days, divided by this, after Einstein. And if you don't want to have an m, and yet you want to have a p, the only way it can happen is that v = c. Then you have 0/0, there is some chance, and nature seems to take advantage of that 0/0. These are the massless particles. So these photons are massless particles. So what is the shock?
The shock is that light, which you thought was a continuous wave, is actually made up of discrete particles. In order to see them, your light source has to be extremely weak, because if you turn on a light source like this one, millions of these photons come and the pattern is formed instantaneously. The minute you turn on the light, the film is exposed, you see these dark and light and dark and light fringes, you think it's happening due to waves that come instantaneously. But if you look under the hood, every pattern is formed by tiny little dots which occur so fast that you don't see them. That's where you've got to turn down the intensity to actually see them. When you see that, you see the corpuscular nature of light.
But here is the problem: if somebody told you light is made of particles and it's not continuous, it's not so disturbing, because water, which you think is continuous, is actually made of water molecules. Everything that you think of as continuous is made up of little molecules and in a bigger scale, much bigger than the atomic or molecular size, it looks like it's continuous. That's not the bad news. The bad news really has to do with the fact that if you have these particles called photons, if it really were a particle, namely a standard, garden variety particle, what should you find?
If you emit a particle from here, and only one slit is open, it will take some path going through the slit and it will come there. So let us say on a given day, 10 photons will come here, or let's say 4 photons will come here, with this one closed. Then let's close this one and open this one. In that, case, maybe 3 photons will come here. Or let's say again, 4. Now I'm claiming that when both are open, I get no photons. How can it be that when you open a second hole, you get fewer particles coming there? Particles normally either take path number 1 or path number 2. Either this slit or this slit. To all the guys going this way, they don't care if this slit is open or closed. They don't even know about the other slit. They do their thing, and guys going through this slit should do their thing, therefore you should get a number equal to the sum, but you don't.
In other words, for particles, which have definite trajectories, opening a second slit should not affect the number going through the first slit. Do you understand that? Particles are local. They're moving along and they feel the local forces acting on them and they bend or twist or turn. They don't really care what's happening far away, whether a second slit may be open or closed. Therefore logically, the number coming here must be the sum of the number that would come with 1 open and 2 open. How can you cancel a positive number of particles coming somewhere with more positive number of particles coming from somewhere else? How do you get a 0? That is where the wave comes in. The wave has no trouble knowing how many slits are open, because the wave is not localized. The wave comes like this. It can hit both the slits and certainly cares about how many slits there are. Because there's only one, that wave will go. You'll have some amplitude here, which is kind of featureless. If that's open, it will be featureless. If both are open, there'll be interference. So we need that wave to understand what the photon will do, because when you send millions of photons and if you get the pattern like this — let's say you sent lots and lots of photons and you got a pattern like this.
Now I'm going to send million + 1 photon. Where will it go? We do not know where it will go. We only know that if you repeat the experiment a million times, you get this pattern. But on the million + 1th attempt, where it will go, we don't know. We just know that the odds are high when the function is high, or the intensity, and the odds are low when the function is small and the odds are 0 when the function is 0. So the role of the wave is to determine the probability that the photon will arrive at some point on the screen. And the probability is computed by adding one wave function to another wave function and then squaring.
So you've got to be very clear. If someone says to you, "Is the photon a particle or a wave? Make up your mind, what is it?" Well, the answer is, it's not going to be a yes or no question. People always ask you, "Is matter made of particles or waves, electron particles or waves?" Well, sometimes the vocabulary we have is not big enough to describe what's really happening. It is what it is. It is the following. It is a particle in the sense that the entire energy is carried in these localized places, unlike a wave. When the wave hits the beach, the energy's over the entire wave front.
This wave here is not a physical wave. It does not carry any energy and it's not even a property of a beam of photons. It's a property of one photon. Here's what I want you to understand: you send one photon at a time, many, many times, and you get this pattern. Each time you throw the die and ask where will the photon land, this function is waiting to tell you the probability it will land somewhere. So we have to play this game in two ways. It is particles, but its future is determined by a wave. The wave is purely mathematical. You cannot put an instrument that measures the energy due to that wave. It's a construct we use to determine what will happen in this experiment. So we have no trouble predicting this experiment, but we only make statistical predictions. So if someone tells you, "I got light from some mercury vapor or something, it's got a certain wavelength, therefore a certain frequency. I'm going to take two slits and I'm going to send the light from the left so weak that at a given time, only one photon leaves the source and hits the screen. What will happen?" We will say we don't know what it will do. We don't know where it will land. But we tell you if you do it enough times, millions of times, soon a pattern will develop. Namely, if you plot your histogram on where everybody landed, you'll get a graph. It's the graph that I can predict. And how do I predict that graph. I say, "What was the energy momentum of your photon?" If it was p, I will introduce a wave whose momentum is 2pℏ/p. Oh, I'm sorry, I forgot to tell you guys one thing. I apologize. I've been writing ℏ and h. I should have mentioned it long back, ℏ is h/2p. Since the combination occurs so often, people write ℏ. So you can write l = 2pℏ/p, or h/p. It doesn't matter. So I've stopped using h. Most people now in the business use ℏ, because the energy is ℏw, the momentum is ℏk. If you want, you can write this 2phk and k is 2p/l. Then you find p is h/l.
That's how some people used to write it in the old days, but now we write it in terms of ℏ and k. Anyway, I can make these predictions, if I knew the momentum of the photons. The photons were of a definite momentum, therefore there's a definite wavelength. I can predict the interference pattern.
Chapter 3: The Photoelectric Effect [00:23:18]
So where is the photon when it goes from start to finish? We don't know. I'll come back to that question now. But I want to mention to you a historical fact, which is, photons were not really found this way, by looking at the recoil of an emulsion plate. Just for completeness, I'm going to make a five minute digression to tell you how photons were found. So they were actually predicted by Einstein. He got the Nobel Prize for predicting the photon, rather than for the Theory of Relativity, which was still controversial at that time. So he predicted the photons, based on actually fairly complicated thermodynamic statistical mechanics arguments.
But one way to understand it is in terms of what's called the photoelectric effect. If you take a metal and you say "Where are the electrons in the metal?" As you know most electrons are orbiting the parent nucleus. But in a metal, some electrons are communal. Each atom donates one or two electrons to the whole metal. They can run all over the metal. They don't have to be near their parent nucleus. They cannot leave the metal. So in a way, they are like this. There's a little tank whose depth is h, and let's say mgh I want to call W. So these guys are somewhere in the bottom. They can run around; they cannot get out. So if you want to yank an electron out of the metal, you have to give an energy equal to W, which is called the work function. So how are you going to get an electron to acquire some energy? We all know. Electron is an electric charge. I have to apply an electric field and I know electromagnetic waves are nothing but electric and magnetic fields, so I shine a light, a source of light, towards this. The electric field comes and grabs the electron and shakes it loose. Hopefully it will shake it loose from the metal, giving it enough energy to escape. And once it escapes, it can take off. So they took some light source and they aimed it at the metal, to see if electrons come out. They didn't. So what do you think you will do to get some action? Yes?
Student: [inaudible]
Professor Ramamurti Shankar: So you make it brighter. You say, "Okay, let me crank up — " that's what anybody would do. They cranked up the intensity of light, make it brighter and brighter and brighter. Nothing happened. Then by accident they found out that instead of cranking up the brightness of the light, if you cranked up the frequency of light, slowly, suddenly beyond some frequency, you start getting electrons escaping the metal. So here's the graph you get. Let me just plot it if you like, ℏ times the w. In those days, they didn't know too much about ℏ. You can even plot w. It doesn't matter. And you plot here the kinetic energy of the emitted electron. And what you find is that below some minimum value, no electrons come out. There's nothing to plot. And once you cross a magical w, and anything higher than that, you get a kinetic energy that's linear in w. Now the kinetic energy is the energy you gave to the electron minus W. Energy given to the electron - W, because you paid W to get it out of the well, and whatever is left is the kinetic energy.
So Einstein predicted photons from independent arguments, and according to him, light and frequency w is made up of particles, each of which contains energy ℏw. So you can see what's happening. If you've got low frequency light, you're sending millions of photons, each carries an energy ℏw somewhere here. None of them has the energy to lift the electron out of the metal. It's like sending a million little kids to lift something and they cannot do it. They cannot do it, but if you send 10 tall, powerful people, they will lift it out. So what's happening with light is that as you crank up the w, even if it's not very bright, the individual packets are carrying more and more energy and more and more momentum, and that's why they succeed in knocking the electron out. And in fact, if you set the energy of each photon, it's ℏw, then the kinetic energy of the electron is the energy you gave with 1 photon, take away the W, that's the price you pay to leave the metal. The rest of it is kinetic energy. So when plotted as a function of w, K should look like a straight line with intercept W. And that's what you find. In fact, this is one way to measure the work function. How much energy do we need to rip an electron out of a metal depends on the metal. And you shine light and you crank up the frequency, till something happens. And just to be sure, you go a little beyond that and you find that the kinetic energy grows linearly in w. Anyway, this is how one confirmed the existence, indirect existence, of photons. There's another experiment that also confirmed the existence of photons. Look, that's the beauty. Once you've got the right answer, everything is going to be on your side.
Before I forget, I should mention to you, you've probably heard that Einstein is very unhappy with quantum mechanics. And yet if you look at the history, he made enormous contributions to quantum mechanics. Even Planck didn't have the courage to stand behind the photons that were implied in his own formula. Einstein took it to be very real and pursued it. So when you say he doesn't like quantum mechanics, it's not that he couldn't do the problem sets. It's that he had problems with the problems. He did not like the probabilistic nature of quantum mechanics, but he had no trouble divining what was going on. So it's quite different. It's like saying, "I don't like that joke." There are two reasons. Some guys don't get it and they don't like it. Some guys get it and don't think it's funny. So this was like Einstein certainly understood all the complexities of quantum mechanics. He said he had spent more time on quantum, much more on either the special or the general theory of relativity, because he said that was a real problem. That's a problem I couldn't track. Now it turns out that even till the end, he didn't find an answer that was satisfactory to him. The answer I'm giving you certainly works, makes all the predictions, never said anything wrong. Until something better comes to replace it, we will keep using it.
Anyway, going back, the second experiment that confirmed the reality of photons. See, if you say light is made of particles and each one has an energy and momentum, do you understand why the photoelectric effect is a good test. It agrees with that picture. Individual particles come. Some have the energy to liberate the electron and some don't. And if individually, they cannot do it, it doesn't matter how many you send. Now you may have thought of one scenario in which all of these tiny little kids can get something lifted out of the well. How will they do that? Maybe 10 kids together, like ants, can lift the thing out. So if you had 10 photons which can collectively excite the electron, it can happen, but in those days, they didn't have a light whose intensity was enough to send enough of these photons. But nowadays, it turns out that if you really, really crank up the intensity, you can make electrons come out, even below the frequency. That's because more than one photon is involved in ejecting the electron.
Chapter 4: Compton's Scattering [00:31:19]
So luckily, we didn't have that intensity then, so we go the picture of the photons right. Anyway, Compton said the following thing: it turns out that if you have an electron here and you send a beam of light, it scatters off the electron and comes off in some direction at an angle q to the original direction. The wavelength here changes by an amount Dl, and Dl happens to be 2pℏ/mc x 1 - cosine q. Are you with me? You send light in at a known wavelength. It scatters off the electron and comes at an angle q, no longer preserving its wavelength, having a different wavelength. And the shift in the wavelength is connected to the angle of scattering. For example, if q is 0, Dl is 0 in the forward direction. If it bounces right back, that cos q is -1. That number is 2 and you get a huge Dl. And you can find the l of it by putting a diffraction grating.
Now, what one could show is that if you took this to be made of particles, and each particle has an energy, ℏw, and each particle has a momentum, ℏk, and that that collides with an electron, then you just balance energy conservation and momentum conservation. In any collision, energy and momentum before = energy and momentum after. You set them equal and you fiddle around, you can find the new momentum after scattering. From the new momentum, you can extract the new wavelength and you will find this formula actually works. So I did that in Physics 200, I think, so if you want, you can go look at that, or maybe it was done for you. I don't know. But Compton's scattering, the scattering due to Compton, can be completely understood if you think of the incoming beam of light as made up of particles with that momentum and that energy. In other words, you're always going to go back and forth. Light will be characterized by a wavelength and by a momentum. It will be characterized by a frequency and by an energy. When you think about the particles, you'll think of the energy and momentum. When you think about the waves, you'll think of frequency and wave number. So this is what really nailed it. After this, you could not doubt the reality of the photons.
Okay, now I go back to my old story. Let's remember what it is. The shock is that light, which we were willing to believe was waves, because Young had done the interference experiment, is actually made up of particles. That's the first thing. So who needs the wave? If you send a single photon into a double slit, we don't know what it will do. We can only give the odds. To find the odds, we take the photon's wavelength and we form this wave, and then we form the interference pattern. And we find out that whenever it is high, it is very likely to come. Wherever it's low, it's very unlikely, but at 0, it won't come. So to test this theory, it's not enough to send 1 photon. 1 photon may come here; that doesn't show you anything. You've got to send millions of photons, because if a prediction is probabilistic, to test it, you've got to do many times. If I give you a coin, and I tell you it's a fair coin, I toss it a couple of times and I get 1 head and 1 tail, it doesn't mean anything. You want to toss it 500,000 times and see if roughly half the time it's heads and half the time it's tails. That's when a probabilistic theory is verified. It's not verified by individuals.
Insurance companies are always drawing pictures of when I'm going to die. They've got some plot, and that's my average chance. I don't know when I will be part of that statistic, because in fact — sorry, it usually looks like this. Life expectancy of people looks like that, but doesn't mean everybody dies at one day. People are dying left and right, so there's probability on either side. So to verify this table that companies have got, you have to watch a huge population. Then you can do the histogram and then you get the profile. So whenever you do statistical theories, you've got to run it many times. I'll tell you more about statistics and quantum mechanics. It's different from statistics and classical mechanics and we'll come to that later. But for now, you must understand the peculiar behavior of photons. They are not particles entirely, they are not waves entirely. They are particles in the sense they're localized energy and momentum, but they don't travel like Newtonian particles. If they were Newtonian particles, you'll never understand why opening a second slit reduced the amount of light coming somewhere. All right, so this is the story.
Chapter 5: Particle-Wave Duality of Matter [00:36:10]
So now comes the French physicist, de Broglie, and he argued as follows: you'll find his argument quite persuasive, and this is what he did for his PhD. He said, "If light, which I thought was a particle — I'm sorry, which I thought was a wave, is actually made up of particles, perhaps things which I always thought of as particles, like electrons, have a wave associated with them." And he said, "Let me postulate that electrons also have a wave associated with them and that the wavelength associated with an electron of momentum p will be 2pℏ/p; and that this wave will produce the same interference pattern when you do it with electrons, as you did with light." So what does that mean? It means if you did a double slit experiment, and you sent electrons of momentum p, one at a time, and you sit here with an electron detector, or you have an array of electron detectors, he claims that the pattern will look like this, where this pattern is obtained by using a certain wavelength that corresponds to the momentum of the incoming beam of electrons. Now there the shock is not that the electron hits one point on the screen. It supposed to; it's a particle. What is shocking is that when two slits are open, you don't get any electrons in the location where you used to get electrons.
That is the surprising thing, because if an electron is a Newtonian particle and you used to go like that through hole 1, and you used to go like that through hole 2, if you open the two holes and two slits, you've got to get the sum of the two numbers. You cannot escape that, because in Newtonian mechanics, an electron either goes through slit 1 or through slit 2. And therefore, the number coming here is simply the sum of the ones that went here, + the ones that went here. Now sometimes people think, "Well, if you have a lot of electrons coming here, maybe these guys bumped into these guys and collided and therefore didn't hit the screen at that point." That's a fake. You know you don't have much of a chance with that explanation, because if there are random collisions, what are the odds they'll form this beautiful, repeatable pattern? Not very big.
Furthermore, you can silence that criticism by making the electron gun that emits electrons so feeble that at a given time, there's only one electron. There's only one electron in the lab. It left here, then it arrived there. And it cannot collide with itself. And yet it knows two slits are open. A Newtonian particle cannot know that two slits are open. So it has an associated wave, and if you do this calculation and you find the interference pattern, that's what electrons do. Originally, it was not done with a double slit. It was done with a crystal. I have given you one homework problem where you can see how a crystal of atoms regularly arranged can also help you find the wavelength of anything. And you shine a beam of electrons on a crystal, you find out that they come out in only one particular angle, and using the angle, you can find the wavelength, and the wavelength agrees with the momentum. The momentum of the electron is known, because if you accelerate them between two plates with a certain voltage, V, and the electron drops down the voltage, it gains an energy eV, which is ½ mv2, which you can also write as p2/2m. So you can find the momentum of an electron before you send it in.
Okay, so this is the peculiarity of particles now. Electron also behaves like a particle or a wave. So now you can ask yourself the following question. Why is it that microscopic bodies — first of all, I hope you understand how surprising this is. Suppose it was not electrons. Suppose this was not an electron gun, but a machine gun, okay? And these are some concrete barriers. The barrier has a hole in it and that's you. They've tied you to the back wall and they're firing bullets at you, and you're of course very anxious when a friend of yours comes along and says, "I want to help you." So let me do that. So you know that that's not a friend, and if you do it with bullets, it won't help. You cannot reduce the number of bullets. And why is it with electrons — if instead of the big scenario, we scale the whole thing down to atomic dimensions, and you're talking about electrons and slits which are a few micrometers away, why is it that with electrons, you can do that? Why is it with bullets you don't do that? The answer has to do with this wavelength p. If you put for p, m x v and you put for m the mass of a cannonball or a bullet, say 1 kilogram, you will find this wavelength is 10-27 something. That means these oscillations will have maybe 1020 oscillations per centimeter and you cannot detect that. So oscillation, the human eye cannot detect that, and everything else looks like you're just adding the intensities, not adding the wave function. It looks like the probabilities are additive, and you don't see the interference pattern.
Now there's another very interesting twist on this experiment, which is as follows. You go back to that experiment, and you say, "Look, I do not buy this notion that an electron does not go through one slit. I mean, come on. How can it not go through one particular slit?" So here's what I'm going to do. I'm going to put a light bulb here. I'm going to have the light bulb look at the slit, and when this guy goes past, I will see whether the guy went through this slit or through that slit. Then there's no talk about going through both slits or not going through a definite slit or not having the trajectory. All that's wrong, because I'm going to actually catch the electron in the act of going through one or the other by putting a light source. So you put a light source, and whenever it hits an electron, you will see a flash and you will know whether it was near this hole or that hole. You make a tally.
So you find that a certain number went through hole 1, a certain number went through hole 2. You add them up, you get the number, you cannot avoid getting the number. Let's imagine that of our 1,000 electrons, about 20 got by without your seeing them. It can happen. When you turn the light, you don't see it; it misses. Then you will find a pattern that looks like this. There'll be a 2 percent wiggle on top of this featureless curve. In other words, the electrons that you caught and identified as going through slit 1 or slit 2, their numbers add up the way they do in Newtonian mechanics, but the electrons you did not catch, who slipped by, pretend as if they went through both the slits, or at least they showed the interference pattern. That's a very novel thing, that whether you see the electron or not, makes such a difference. That's all I did. In one case, I caught the electron. In the other case, I slipped by. And whenever it's not observed, it seems to be able to somehow be aware of two slits. And this was a big surprise, because normally when we study anything in Newtonian mechanics, you say here's a collision, ball 1 collides and goes there, you do all the calculations. Meanwhile, we are watching it. Maybe we are not watching it. Who cares? The answer doesn't depend on whether we are watching or not? For example, if you have a football game, and somebody throws the pass, and you close your eyes, which sometimes my kids do, because they don't know what's happening, that doesn't change the outcome of the experiment. It follows its own trajectory. So what does seeing do to anything? And you can say maybe he didn't see it, but maybe people in the stadium were looking at the football. So turn off all the lights. Then does the football have a definite trajectory from start to finish? It does, because it's colliding with all these air molecules. To remove all the air molecules, of course, first you remove all the spectators, then you remove all the air molecules. Then does it have a definite trajectory? You might say, "Of course it does. What difference does it make?" But then you would be wrong. You would be wrong to think it had a trajectory, because the minute you said it had a trajectory, you will never understand interference, which even a football can show. But the condition is, for a football to show this kind of quantum effects, it should not be disturbed by anything. It should not be seen. Nothing can collide with it. The minute you interact with a quantum system, it stops doing this wishy-washy business of "Where am I?" Till you see it, it's not anywhere. Once you see it, it's in a different location. Till you see it, it's not taking any particular path. To assume it took this or that path is simply wrong. But the act of observation nails it.
So why is observation so important? You have to ask how we observe things. We shine light. You've already seen, the light is made of quanta, and each quantum carries a certain momentum and certain energy. If I want to locate the electron with some waves, with some light, I want the momentum of the light to be weak, because I don't want to slam the electron too hard in the act of finding it. So I want p to be very small. If p is very small, l, which is 2pℏ/p becomes large, and once l's bigger than the spacing between the slits, the picture you get will be so fuzzy, you cannot tell which slit it went through. In other words, to make a fine observation in optics, you need a wavelength smaller than the distances you're trying to resolve. So you've got to use a wavelength smaller than these two slits.
So this p should be such that this l is comparable to this slit, or even smaller. But then you will find the act of observing the electron imparts to it an unknown amount of momentum. Once you change the momentum, you change the interference pattern. So the act of observation, which is pretty innocuous for you and me — right now, I'm getting slammed by millions of photons, but I'm taking it like a man. But for the electron, it is not that simple. One collision with a photon is like getting hit by a truck. The momentum of the photon is enormous in the scale of the electron. So it matters a lot to the electron. For example, when I observe you, I see you because photons bounce back and forth. Suppose it's a dark room and I was swinging one of those things you see in Gladiator. What's that thing called? Trying to locate you. So the act of location, you realize it will be memorable for you, because it's a destructive process. But in Newtonian mechanics, we can imagine finding gentler ways to observe somebody and there's no limit to how gentle it is. You just say make the light dimmer and dimmer and dimmer till the person doesn't care. But in quantum theory, it's not how dim the light is. If the light is too dim, there are too few photons and nobody catches the electron. In order to see the electron, you've got to send enough photons. But the point is, each one carries a punch which is minimum. It cannot be smaller than this number, because if the wavelength is bigger than this, you cannot tell which hole it went through. That's why in quantum theory, the act of observation is very important, and it can change the outcome. Okay, so what can we figure out from this. Well, it looks like the act of observing somehow affects the momentum of the electron.
Chapter 6: The Uncertainty Principle [00:48:34]
So people often say that's why, when you try to measure the position of the electron, you do something bad to the momentum of the electron. We change it, because you need a large momentum to see it very accurately. But that statement is partly correct but partly incomplete and I'll tell you what it is. The trouble is not that you use a high momentum photon to see an electron precisely. That's not a problem. The problem is that when it bounces off the electron and comes back to you, it would have changed, the momentum by an amount that you cannot predict, and I'll tell you why that is the case. So I told you long back that if you have a hole and light comes in through it, it doesn't go straight, it fans out, that the profile of light looks like this. It spreads out and the angle by which it spreads out obeys the condition dsinq = l. Remember that part from wave theory of light. Now here is the person trying to catch an electron, which is somewhere around this line. And he or she brings a microscope that looks like this. Here's the opening of the microscope, and you send some light. This opening of the microscope has some extent d.
Let's say it's got a sharp opening here of width d. The light comes, hits an electron, if it is there, and goes right back to the microscope. If I see a flicker of reflected light, I know the electron had to be somewhere here, because if it's here, it's not going to collide with the light. So you agree, this is a way to locate the electron's position with an uncertainty, which is roughly d, right? The electron had to be in front of the opening of the microscope for me to actually see that flash. So I make an electron microscope with a very tiny hole, and I'm scanning back and forth, hoping one day I will hit an electron and one day I hit the electron, it sends the light right back. This has momentum p. It also sends back with momentum p, but there's one problem. You know that light entering an aperture will spread out. It won't go straight through. This is this process. So if you think of this light entering your microscope, it spreads out. If it spreads out, it means the photon that bounced back can have a momentum anywhere in this cone. And we don't know where it is.
All we know is it re-entered the microscope, entered this cone, but anywhere in this cone is possible, because there's a sizeable chance the light will come anywhere into this diffracted region. That means the final photon's momentum magnitude may be p0, but its direction is indefinite by an amount q. Therefore the photon's momentum has a horizontal part, p0sine q, which is an uncertainty in the momentum of the photon in the x direction. This is my x direction. So now you can see that Dpx = p0sine q. Sine q is l over the width of the slit. And l was 2pℏ/p0 over d. You can see that these p0's cancel, then you get d x Dpx = 2pℏ. By the way, another good news is I'm going to give you very detailed notes on quantum mechanics. I'm not following the textbook, and I know you have to choose between listening to me and writing down everything. So everything I'm saying here, you will find in those notes, so don't worry if you didn't get everything. You will have a second chance to look at it. But what you find here is that d x Dpx is 2pℏ, but d is the uncertainty in the location of the electron, so you get Dx, Dpx, I'm not going to say =, roughly of order, ℏ. Forget the 2p's and everything.
This is a very tiny number, 10-34, so we don't care if there are 2p's. But what this tells you is that in the act of locating the electron — so let's understand why. It's a constant going back and forth between waves and particles, okay? That's why this happens. I want to see an electron and I want to know exactly where I saw it. So I take a microscope with a very small opening, so that if I see that guy, I know it has to be somewhere in front of that hole. But the photon that came down and bounced off it, if you now use wave theory, the wave will spread out when it re-enters the cone by minimal angle q, given by dsine q = l. That means the photon will also come at a range of angles, spread out, but if it comes at a tilted angle, it certainly has horizontal momentum. That extra horizontal momentum should be imparted to the particle, because initially, the momentum of this thing was strictly vertical.
So the photon has given a certain horizontal momentum to the electron and you don't know how much it has given. And smaller your opening, so the better you try to locate the electron, bigger will be the spreading out, and bigger will be the uncertainty in the reflected photon and therefore uncertainty in the electron after collision. So before the collision, you could have had an electron with perfectly well known momentum in the x direction. But after you saw it, you don't know its x momentum very well, because the photon's x momentum is not known. I want you to appreciate, it's not the fact that the photon came in its large momentum that's the problem; it is that it went back into the microscope with a slight uncertainty in its angle, that comes from diffraction of light. It's the uncertainty of the angle that turns into uncertainty in the x component of the momentum. So basically, collision of light with electrons leaves the electron with an extra momentum whose value we don't know precisely, because the act of seeing the photon with the microscope necessarily means it accepts photons with a range of angles.
Okay, so now I want to tell you a little more about the uncertainty principle in another language. The language is this: here is a slit. Okay, here's one way to state the uncertainty principle. I challenge you to produce for me an electron whose location is known to arbitrary accuracy and whose momentum, in the same dimension, same direction, is also known to arbitrary accuracy. I dare you to make it. In Newtonian mechanics, that's not a big deal. So let's say this is the y direction, and you say, "I'll give you an electron with precisely known y coordinate, and no uncertainty in y momentum by the following trick. I'll send a beam of electrons going in this direction, in the x direction, with some momentum p0 and I put a hole in the middle. The only guys escaping have to come out like this. So right outside, what do I have? I have an electron whose vertical momentum is exactly 0, because the beam had no vertical momentum, whose vertical position = the width of the slit. It's uncertain by the width of the slit, and I can make the width as narrow as I like. I can make my filter finer and finer and finer, till I'm able to give the electrons a perfectly well defined position and perfectly well defined momentum, namely 0.
That's true in Newtonian mechanics, but it's not true in the quantum theory, because as you know, this incoming beam of electrons is associated with a wave, the wave is going to fan out when it comes out. And we sort of know how much it's going to fan out. That's why I did that diffraction for you. It fans out by an angle q, so that dsine q = l. That means light can come anywhere in this cone to your screen. That means the electrons leaving could have had a momentum in any of these directions. So the initial photon at a momentum p0, the final one has a momentum of magnitude p0, but whose direction is uncertain. The uncertainty in the y momentum, simply p0sine q. You understand? Take a vector p0. If that angle is q, this is p0sine q. And we don't know. Look, it's not that we know exactly where it's going to land. It can land anywhere inside this bell shaped curve, so it can have any momentum in this region. So the electrons you produce, even though the position was well known to the width of the slit, right after leaving the slit, are capable of coming all over here. That means they have momenta which can point in any of these allowed directions.
So let's find the uncertainty in y momentum as this. The uncertainty in y position is just the width of the slit. So take the product now of D py. Let me call it Dy. That happens then to be p0sine q times d but dsine q is l and l is 2pℏ/p0. Cancel the p0, you get some number. Forget the 2p's that look like ℏ. So this is the uncertainty principle. So the origin of the uncertainty principle is that the fate of the electrons is determined by a wave. And when you try to localize the wave in one direction, it fans out. And when it fans out, the probability of finding the electron is not 0 in the non-forward direction. It's got a good chance of being in the range of non-forward directions. That means momentum has a good chance of lying all the way from there to here. That means the y momentum has an uncertainty. And more you make the purchase smaller to nail its position, broader this will be, keeping the product constant. So it's not hard mathematically to understand. What is hard to understand is the notion that somehow you need this wave, but it was forced upon us. The wave is forced upon us, because there's no way to understand interference, except through waves.
So when people saw the interference pattern of the electrons, they said there's got to be a wave. They said, "What is the role of that wave?" That's what I want you to understand. With every electron now — so let's summarize what we have learned. When I say electron, I mean any other particle you like, photon, neutron, doesn't matter. They all do this. Quantum mechanics applies to everything. Therefore, with every electron, I'm going to associate a function, Y(x) — or Y(x,y,z), so that if you find its absolute value, that gives — or absolute value squared, that gives the odds of finding it at the point x, y, z. Let me say it's proportional. This function is stuck. We are stuck with this function. And what else do we know about the function? We know that if the electron has momentum p, then the function Y has wavelength l, which is 2pℏ/p. This is all we know from experiment. So experiment has forced us to write this function Y. And the theory will make predictions. Later on we'll find out how to calculate the Y in every situation.
But the question is, what is the kinematics of quantum mechanics compared to kinematics of classical mechanics? In classical mechanics, a particle has a definite position, it has a definite momentum. That describes the state of the particle now. Then you want to predict the future, so you want to know the coordinate and momentum of a future time. How are you going to find that? Anybody know? How do you find the future of x and p? In Newtonian mechanics; I'm not talking about quantum mechanics.
Student: [inaudible]
Professor Ramamurti Shankar: Which one? Just use Newton's laws. That's what Newton's law does for you. It tells you what the acceleration is in a given context. Then you find the acceleration to find the new velocity. Find the old velocity to find the new position a little later and keep on doing it, or you solve an equation. So the cycle of Newtonian mechanics is give me the x and p, and I know what they mean, and I'll tell you x and p later if you tell me the forces acting on it. Or if you want to write the force as a gradient of a potential, you will have to be given the potential. In quantum mechanics, you are given a function Y. Suppose the particle lives in only one dimension, then for one particle, not for a swarm of particles, for one particle, for every particle there can be a function Y associated with it at any instant. That tells you the full story. Remember, we've gone from two numbers, x and p, to a whole function. What does the function do? If you squared the function at this point — square will look roughly the same thing — that height is proportional to the odds of finding it here, and that means it's a very high chance of being found here, maybe no chance of being found here and so on.
That's called the wave function. The name for this guy is the wave function. So far we know only one wave function. In a double slit experiment, if you send electrons of momentum p, that wave function seems to have a wavelength l connected to p by this formula, this. This is all we know.
So let's ask the following question: take a particle of momentum p. What do you think the corresponding wave function is in the double slit experiment? Can you cook up the function in the double slit experiment at any given time? So I want to write a function that can describe the electron in that double slit experiment, and I'll tell you the momentum is p. So what can you tell from wave theory?
Let's say the wave is traveling, this is the x direction. What can you say about Y(x) at some given time? It's got some amplitude and it's oscillating, so it's cosine 2px/l. Forget the time dependence. At one instant of time, it's going to look like this. This is the wavelength l for anything. But now I know that l is connected to momentum as follows: l is 2pℏ/p, so let's put that in. So 2p/l = Acosine p over ℏx. This has the right wavelength for the given momentum. In other words, if you send electrons of momentum p, and you put that p into this function exactly where it's supposed to go, it determines a wavelength in just the right way, that if you did interference, you'll get a pattern you observe. But this is not the right answer.
This is not the right answer, because if you took the square of the Y, it's real. I don't care whether it's absolute square or square, you get cosine squared px over ℏ, and if you plot that function, it's going to look like this, the incoming wave. I'm talking not about interference but the incoming wave, if I write it this way. But incoming wave, if it looks like this, I have a problem, because the uncertainty principle says Dx Dpx is of order ℏ. It cannot be smaller, so the correct statement is, it's bigger than ℏ over some number. Take this function here. Its momentum is exactly know, do you agree? The uncertainty principle says if you know the position well, you don't know the momentum too well. If you know the momentum exactly, so Dpx is 0, Dx is infinity, that means you don't know where it is. A particle of perfectly known momentum has perfectly unknown position. That means the probability of finding it everywhere should be flat. This is not flat. It says I'm likely to be here, not likely to be here, likely to be there, so this function is ruled out. Because I want for Y2, for a situation where it has a well defined momentum, I want the answer to look like this. The odds of finding it should be independent of where you are, because we don't know where it is. Every place is equally likely. And yet this function has no wavelength. So how do I sneak in a wavelength, but not affect this flatness of Y2? Is there a way to write a function that will have a magnitude which is constant but has a wavelength hidden in it somewhere, so that it can take part in interference? Pardon me? Any guess? Yes?
Student: [inaudible]
Professor Ramamurti Shankar: A complex function. So I'm going to tell you what the answer is. We are driven to that answer. Here's a function I can write down, which has all the good properties I want: Y(x) looks like some number Aeipx/ℏ. This is just cosine(px + i) sine(px/ℏ). It's got a wavelength, but the absolute value of Y is just A2, because the absolute value of this guy is 1. Ae to the thing looks like this. This is the number A, this is px/ℏ is the angle. That complex number Y at a given point x has got a magnitude which is just A2. So we are driven to the conclusion that the correct way to describe an electron with wave function, with a momentum p, is some number in front times eipx/ℏ, because it's got a wavelength associated with it, and it also has an absolute value that is flat.
Do you understand why it had to be flat? The uncertainty principle says if you know its momentum precisely, and you seem to know it, because you put a definite p here, you cannot know where it is. That means the probability for finding it cannot be dependent on position. Any trigonometric function you take with some wavelength will necessarily oscillate, preferring some points over other points. The exponential function, it will oscillate and yet its magnitude is independent. That's a remarkable function. It's fair to say that if you did not know complex exponentials, you wouldn't have got beyond this point in the development of quantum mechanics. The wave function of an electron of definite momentum is a complex exponential. This is the sense in which complex functions enter quantum mechanics in an inevitable way. It's not that the function is really cosine px/ℏ and I'm trying to write it as a real part of something. You need this complex beast.
So the wave functions of quantum mechanics. There are electrons which could be doing many things, each one has a function Y. Electron of definite momentum we know is a reality. It happens all the time. In CERN they're producing protons of a definite momentum, 4 point whatever, 3 point p tev. So you know the momentum. You can ask what function describes it in quantum theory; this is the answer. This is not derived. In a way, this is a postulate. I'm only trying to motivate it. You cannot derive any of quantum mechanics, except looking at experiments and trying to see if there is some theoretical structure that will fit the data. So I'm going to conclude with what we have found today, and it's probably a little weird. I try to pay attention to that and I will repeat it every time, maybe adding a little extra stuff. So what have you found so far? It looks like electrons and photons are all particles and waves, except it's more natural to think of light in terms of waves with the wavelength and frequency. What's surprising is that it's made up of particles whose energy is ℏw, and whose momentum is ℏk. Conversely, particles like electrons, which have a definite momentum, have a wavelength associated with them. And when does the wavelength come into play?
Whenever you do an experiment in which that wavelength is comparable to the geometric dimensions, like a double slit experiment at a single slit diffraction, it's the wave that decides where the electron will go. The height squared of the wave function is proportional to the probability the electron will end up somewhere. And also, in a double slit experiment, it is no longer possible to think that the electron went through one slit or another. You make that assumption, you cannot avoid the fact that when both slits are open, the numbers should be additive. The fact they are not means an electron knows how many slits are open, and only a wave knows how many slits are open because it's going everywhere. A particle can only look at one slit at a time. In fact, it doesn't know anything, how many slits there are. It usually bangs itself into the wall most of the time, but sometimes when it goes through the hole, it comes up. And so what do you think one should do to complete the picture? What do we need to know? We need to know many things. Y(x)2 is the probability that if you look for it, you will find it somewhere. Instead of saying the particle is at this x in Newtonian mechanics, we're saying it can be at any x where Y doesn't vanish and the odds are proportional to the square of Y at that point. Then you can say, what does the wave function look like for a particle of definite momentum? Either you postulate it or try to follow the arguments I gave, but this simply is the answer.
This is the state of definite momentum. And the uncertainty principle tells you this is an agreement to the uncertainty principle that any attempt to localize an electron in space by an amount Dx leads to a spread in momentum in an amount Dpx. That's because it's given by a wave. If you're trying to squeeze the wave this way, it blows up in the other direction. And the odds for finding in other directions are non 0, that means the momentum can point in many directions coming out of the slit. That's the origin of the uncertainty principle. So I'm going to post whatever I told you today online. You should definitely read it and it's something you should talk about, not only with your analyst, because this can really disturb you, talk about it with your friends, your neighbors, talk about it with senior students. The best thing in quantum is discussing it with people and getting over the weirdness.

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