Quantum physics concerns the behavior of the smallest things we know. These smallest things are very small indeed. Although the world of the very small is remote from our senses, it shapes everyday experience. When this classic text was first published in , it fulfilled the goal of its authors "to produce a textbook of practical quantum mechanics for the chemist, the experimental physicist, and the beginning student of theoretical physics.
The book focuses on the evolution of the subject, the underlying ideas, the concepts, the laws and the mathematical apparatus for the formulation of the subject in a systematic and comprehensible manner.
Provides comprehensive coverage of all the fundamentals of quantum physics. Author : A. Full mathematical treatments are given. Uses examples from different areas of physics to demonstrate how theories work in practice.
Text derived from lectures delivered at Massachusetts Institute of Technology. Many worked examples serve to illustrate the material while biographical and historical footnotes round off the content. Author : David J. Author : F. Author : Michael E. The authors make these subjects accessible through carefully worked examples illustrating the technical aspects of the subject, and intuitive explanations of what is going on behind the mathematics. After presenting the basics of quantum electrodynamics, the authors discuss the theory of renormalization and its relation to statistical mechanics, and introduce the renormalization group.
This discussion sets the stage for a discussion of the physical principles that underlie the fundamental interactions of elementary particle physics and their description by gauge field theories. The main features of the book are: Emphasis on the key principles with minimal mathematical formalism Demystifying discussions of the basic features of quantum systems, using dimensional analysis and order-of-magnitude estimates to develop intuition Comprehensive overview of the key concepts of quantum chemistry and the electronic structure of solids Extensive discussion of the basic processes and applications of light-matter interactions Online supplement with advanced theory, multiple-choice quizzes, etc.
It has three apertures, an in port and two out ports, one which sends out black electrons and one which sends out white electrons. And the utility of this box is that the color can be inferred from the position. If you find the particle, the electron over here, it is a white electron. If you find the electron here, it is a black electron.
Similarly, we can build a hardness box, which again has three apertures, an in port. And hard electrons come out this port, and soft electrons come out this port. Now, if you want, you're free to imagine that these boxes are built by putting a monkey inside. And you send in an electron, and the monkey, you know, with the ears, looks at the electron, and says it's a hard electron, it sends it out one way, or it's a soft electron, it sends it out the other. The workings inside do not matter.
And in particular, later in the semester I will describe in considerable detail the workings inside this apparatus. And here's something I want to emphasize to you. It can be built in principle using monkeys, hyper intelligent monkeys that can see electrons. It could also be built using magnets and silver atoms. It could be done with neutrons.
It could be done with all sorts of different technologies. And they all give precisely the same results as I'm about to describe. They all give precisely the same results. So it does not matter what's inside. But if you want a little idea, you could imagine putting a monkey inside, a hyper intelligent monkey. I know, it sounds good. So a key property of these hardness boxes and color boxes is that they are repeatable. And here's what I mean by that.
If I send in an electron, and I find that it comes out of a color box black, and then I send it in again, then if I send it into another color box, it comes out black again. So in diagrams, if I send in some random electron to a color box, and I discover that it comes out, let's say, the white aperture. And so here's dot dot dot, and I take the ones that come out the white aperture, and I send them into a color box again.
So this is a persistent property. You notice that it's white. You measure it again, it's still white.
Do a little bit later, it's still white. It's a persistent property. Ditto the hardness. If I send in a bunch of electrons in to a hardness box, here is an important thing. Well, send them into a hardness box, and I take out the ones that come out soft. And I send them again into a hardness box, and they come out soft. Never do they come out the hard aperture. Any questions at this point? So here's a natural question.
Might the color and the hardness of an electron be related? And more precisely, might they be correlated? Might knowing the color infer something about the hardness? So for example, so being male and being a bachelor are correlated properties, because if you're male, you don't know if you're a bachelor or not, but if you're a bachelor, you're male. That's the definition of the word.
So is it possible that color and hardness are similarly correlated? So, I don't know, there are lots of good examples, like wearing a red shirt and beaming down to the surface and making it back to the Enterprise later after the away team returns.
Correlated, right? Negatively, but correlated. So the question is, suppose, e. Does that determine the hardness? So we can answer this question by using our boxes. So here's what I'm going to do. I'm going to take some random set of electrons. That's not random. And I'm going to send them in to a color box.
And I'm going to take the electrons that come out the white aperture. And here's a useful fact. When I say random, here's operationally what I mean. I take some piece of material, I scrape it, I pull off some electrons, and they're totally randomly chosen from the material. And I send them in. If I send a random pile of electrons into a color box, useful thing to know, they come out about half and half.
It's just some random assortment. Some of them are white, some of them come out black. Suppose I send some random collection of electrons into a color box. And I take those which come out the white aperture.
And I want to know, does white determine hardness. So I can do that, check, by then sending these white electrons into a hardness box and seeing what comes out. Hard, soft. And ditto if we reverse this. If we take hardness, and take, for example, a soft electron and send it into a color box, we again get So if you take a white electron, you send it into a hardness box, you're at even odds, you're at chance as to whether it's going to come out hard or soft.
And similarly, if you send a soft electron into a color box, even odds it's going to come out black or white. So knowing the hardness does not give you any information about the color, and knowing the color does not give you any information about the hardness.
These are independent facts, independent properties. They're not correlated in this sense, in precisely this operational sense. So measuring the color give zero predictive power for the hardness, and measuring the hardness gives zero predictive power for the color.
And from that, I will say that these properties are correlated. So H, hardness, and color are in this sense uncorrelated. So using these properties of the color and hardness boxes, I want to run a few more experiment's. I want to probe these properties of color and hardness a little more. And in particular, knowing these results allows us to make predictions, to predict the results for set a very simple experiments.
Now, what we're going to do for the next bit is we're going to run some simple experiments. And we're going to make predictions. And then those simple experiments are going to lead us to more complicated experiments. But let's make sure we understand the simple ones first. So for example, let's take this last experiment, color and hardness, and let's add a color box. One more monkey. So color in, and we take those that come out the white aperture.
And we send them into a hardness box. And we take those electrons which come out the soft aperture. And now let's send these again into a color box. So it's easy to see what to predict. Black, white. So you can imagine a monkey inside this, going, aha. You look at it, you inspect, it comes out white. Here you look at it and inspect, it comes out soft. And you send it into the color box, and what do you expect to happen? Well, let's think about the logic here.
Anything reaching the hardness box must have been measured to be white. And then we send them into a color box, and what happens? Well, since colors are repeatable, the natural expectation is that, of course, it comes out white. That seem like a reasonable-- let's just make sure that we're all agreeing. So let's vote. How many people think this is probably correct?
OK, good. How many people think this probably wrong? That's reassuring. Except you're all wrong. So let's think about what's going on here. This is really kind of troubling. We've said already that knowing the color doesn't predict the hardness. So that's surprising.
What that tells you is you can't think of the electron as a little ball that has black and soft written on it, right? You can't, because apparently that black and soft isn't a persistent thing, although it's persistent in the sense that once it's black, it stays black. So what's going on here? Now, I should emphasize that the same thing happens if I had changed this to taking the black electrons and throwing in a hardness and picking soft and then measuring the color, or if I had used the hard electrons.
Any of those combinations, any of these ports would have given the same results, Is not persistent in this sense. Apparently the presence of the hardness box tampers with the color somehow. So it's not quite as trivial is that hyper intelligent monkey. Something else is going on here. So this is suspicious. So here's the first natural move. The first natural move is, oh, look, surely there's some additional property of the electron that we just haven't measured yet that determines whether it comes out the second color box black or white.
There's got be some property that determines this. And so people have spent a tremendous amount of time and energy looking at these initial electrons and looking with great care to see whether there's any sort of feature of these incident electrons which determines which port they come out of.
And the shocker is no one's ever found such a property. No one has ever found a property which determines which port it comes out of. As far as we can tell, it is completely random. Those that flip and those that don't are indistinguishable at beginning. And let me just emphasize, if anyone found such a-- it's not like we're not looking, right?
If anyone found such a property, fame, notoriety, subverting quantum mechanics, Nobel Prize. People have looked. And there is none that anyone's been able to find. And as we'll see later on, using Bell's inequality, we can more or less nail that such things don't exist, such a fact doesn't exist.
But this tells us something really disturbing. This tells us, and this is the first real shocker, that there is something intrinsically unpredictable, non-deterministic, and random about physical processes that we observe in a laboratory.
There's no way to determine a priori whether it will come out black or white from the second box. Probability in this experiment, it's forced upon us by observations. OK, well, there's another way to come at this. You could say, look, you ran this experiment, that's fine. But look, I've met the guy who built these boxes, and look, he's just some guy, right? And he just didn't do a very good job. The boxes are just badly built. So here's the way to defeat that argument. No, we've built these things out of different materials, using different technologies, using electrons, using neutrons, using bucky-balls, C60, seriously, it's been done.
We've done this experiment, and this property does not change. It is persistent. And the thing that's most upsetting to me is that not only do we get the same results independent of what objects we use to run the experiment, we cannot change the probability away from at all. Within experimental tolerances, we cannot change, no matter how we build the boxes, we cannot change the probability by part in And to anyone who grew up with determinism from Newton, this should hurt.
This should feel wrong. But it's a property of the real world. And our job is going to be to deal with it. Rather, your job is going to be to deal with it, because I went through this already. So here's a curious consequence-- oh, any questions before I cruise? So here's a curious consequence of this series of experiments. Here's something you can't do. Are you guys old enough for you can't do this on television?
This is so sad. OK, so here's something you can't do. We cannot build, it is impossible to build, a reliable color and hardness box. We've built a box that tells you what color it is. We've built a box that tells you what hardness it is. But you cannot build a meaningful box that tells you what color and hardness an electron is. So in particular, what would this magical box be?
It would have four ports. And its ports would say, well, one is white and hard, and one is white and soft, one is black and hard, and one is black and soft. So you can imagine how you might try to build a color and hardness box. So for example, here's something you might imagine. Take your incident electrons, and first send them into a color box. And take those white electrons, and send them into a hardness box. And take those electrons, and this is going to be white and hard, and this is going to be white and soft.
And similarly, send these black electrons into the hardness box, and here's hard and black, and here's soft and back. Everybody cool with that? So this seems to do the thing I wanted. It measures both the hardness and the color. What's the problem with it? So the color is not persistent. So you tell me this is a soft and black electron, right?
That's what you told me. Here's the box. But if I put a color box here, that's the experiment we just ran. And what happens? Does this come out black? No, this is a crappy source of black electrons. So this box can't be built. And the reason, and I want to emphasize this, the reason we cannot build this box is not because our experiments are crude. And it's not because I can't build things, although that's true.
I was banned from a lab one day after joining it, actually. So I really can't build, but other people can. And that's not why. We can't because of something much more fundamental, something deeper, something in principle, which is encoded in this awesome experiment. This can be done. It does not mean anything, as a consequence. It does not mean anything to say this electron is white and hard, because if you tell me it's white and hard, and I measure the white, well, I know if it's hard, it's going to come out It does not mean anything.
So this is an important idea. This is an idea which is enshrined in physics with a term which comes with capital letters, the Uncertainty Principle. And the Uncertainty Principle says basically that, look, there's some observable, measurable properties of a system which are incompatible with each other in precisely this way, incompatible with each other in the sense not that you can't know, because you can't know whether it's hard and soft simultaneously, deeper.
It is not hard and white simultaneously. It cannot be. It does not mean anything to say it is hard and white simultaneously. That is uncertainty. And again, uncertainty is an idea we're going to come back to over and over in the class. But every time you think about it, this should be the first place you start for the next few weeks. No questions? So at this point, it's really tempting to think yeah, OK, this is just about the hardness and the color of electrons.
It's just a weird thing about electrons. It's not a weird thing about the rest of the world. The rest of the world's completely reasonable. And no, that's absolutely wrong. Every object in the world has the same properties. If you take bucky-balls, and you send them through the analogous experiment-- and I will show you the data, I think tomorrow, but soon, I will show you the data. When you take bucky-balls and run it through a similar experiment, you get the same effect.
Now, bucky-balls are huge, right, 60 carbon atoms. But, OK, OK, at that point, you're saying, dude, come on, huge, 60 carbon atoms. So there is a pendulum, depending on how you define building, in this building, a pendulum which is used, in principle which is used to improve detectors to detect gravitational waves. There's a pendulum with a, I think it's 20 kilo mirror. And that pendulum exhibits the same sort of effects here. We can see these quantum mechanical effects in those mirrors.
And this is in breathtakingly awesome experiments done by Nergis Malvalvala, whose name I can never pronounce, but who is totally awesome.
She's an amazing physicist. And she can get these kind of quantum effects out of a 20 kilo mirror. So before you say something silly, like, oh, it's just electrons, it's 20 kilo mirrors.
And if I could put you on a pendulum that accurate, it would be you. These are properties of everything around you. The miracle is not that electrons behave oddly. The miracle is that when you take 10 to the 27 electrons, they behave like cheese. That's the miracle. This is the underlying correct thing. OK, so this is so far so good.
But let's go deeper. Let's push it. And to push it, I want to design for you a slightly more elaborate apparatus, a slightly more elaborate experimental apparatus. And for this, I want you to consider the following device. I'm going to need to introduce a couple of new features for you. Here's a hardness box. And it has an in port. And the hardness box has a hard aperture, and it has a soft aperture. And now, in addition to this hardness box, I'm going to introduce two elements.
First, mirrors. And what these mirrors do is they take the incident electrons and, nothing else, they change the direction of motion, change the direction of motion. And here's what I mean by doing nothing else. If I take one of these mirrors, and I take, for example, a color box. It does not change the observable color.
All it does is change the direction. Similarly, with the hardness box, it doesn't change the hardness. It just changes the direction of motion.
And every experiment we've ever done on these, guys, changes in no way whatsoever the color or the hardness by subsequent measurement. Just changes the direction of motion. And then I'm going to add another mirror. It's actually a slightly fancy set of mirrors.
All they do is they join these beams together into a single beam. And again, this doesn't change the color. You send in a white electron, you get out, and you measure the color on the other side, you get a white electron.
You send in a black electron from here, and you measure the color, you get a black electron again out. So here's my apparatus. And I'm going to put this inside a big box. And I want to run some experiments with this apparatus. Everyone cool with the basic design? Any questions before I cruise on? This part's fun. So what I want to do now is I want to run some simple experiments before we get to fancy stuff.
And the simple experiments are just going to warm you up. They're going to prepare you to make some predictions and some calculations. And eventually we'd like to lead back to this guy. So the first experiment, I'm going to send in white electrons. I'm going to send in white electrons. And I'm going to measure at the end, and in particular at the output, the hardness. So I'm going to send in white electrons.
And I'm going to measure the hardness. So this is my apparatus. I'm going to measure the hardness at the output. And what I mean by measure the hardness is I throw these electrons into a hardness box and see what comes out.
So this is experiment 1. And let me draw this, let me biggen the diagram. So you send white into-- so the mechanism is a hardness box. Mirror, mirror, mirrors, and now we're measuring the hardness out. And the question I want to ask is how many electrons come out the hard aperture, and how many electrons come out the soft aperture of this final hardness box. So I'd like to know what fraction come out hard, and what fraction come out soft. I send an initial white electron, for example I took a color box and took the white output, send them into the hardness box, mirror, mirror, hard, hard, soft.
And what fraction come out hard, and what fraction come out soft. So just think about it for a minute. And when you have a prediction in your head, raise your hand.
All right, good. Walk me through your prediction. How come? So let me say that again. So we've done the experiment, you send a white electron into the hardness box, and we know that it's non-predictive, Now if you take the one that comes out the hard aperture, then you send it up here or send it up here, we know that these mirrors do nothing to the hardness of the electron except change the direction of motion.
We've already done that experiment. So you measure the hardness at the output, what do you get? Hard, because it came out hard, mirror, mirror, hardness, hard. And with soft, mirror, mirror, hardness, you know it comes out soft. Was this the logic? How many people agree with this? How many people disagree?
No abstention. So here's a prediction. Oh, yep. Could you justify that prediction without talking about oh, well, half the electrons were initially measured to be hard, and half were initially measured to be soft, by just saying, well, we have a hardness box, and then we joined these electrons together again, so we don't know anything about it.
So it's just like sending white electrons into one hardness box instead of two. So let's see. We're going to see in a few minutes whether that kind of an argument is reliable or not. But so far we've been given two different arguments that lead to the same prediction, Like when you get them to where This is a very good question. So here's a question look you're sending a bunch of electrons into this apparatus. But if I take-- look, I took You take two electrons and you put them close to each other, what do they do?
They interact with each other through a potential, right? So yeah, we're being a little bold here, throwing a bunch of electrons in and saying, oh, they're independent. So I'm going to do one better. I will send them in one at a time. One electron through the apparatus. And then I will wait for six weeks. See, you guys laugh, you think that's funny. But there's a famous story about a guy who did a similar experiment with photons, French guy. And, I mean, the French, they know what they're doing.
So he wanted to do the same experiment with photons. But the problem is if you take a laser and you shined it into your apparatus, there there are like, 10 to the 18 photons in there at any given moment. And the photons, who knows what they're doing with each other, right? So I want to send in one photon, but the problem is, it's very hard to get a single photon, very hard.
So what he did, I kid you not, he took an opaque barrier, I don't remember what it was, it was some sort of film on top of glass, I think it was some sort of oil-tar film. Barton, do you remember what he used?
So he takes a film, and it has this opaque property, such that the photons that are incident upon it get absorbed. Once in a blue moon a photon manages to make its way through. Literally, like once every couple of days, or a couple of hours, I think. So it's going to take a long time to get any sort of statistics.
But he this advantage, that once every couple of hours or whatever a photon makes its way through. That means inside the apparatus, if it takes a pico-second to cross, triumph, right? That's the week I was talking about. So he does this experiment. But as you can tell, you start the experiment, you press go, and then you wait for six months.
Side note on this guy, liked boats, really liked yachts. So he had six months to wait before doing a beautiful experiment and having the results.
So what did he do? Went on a world tour in his yacht. Comes back, collects the data, and declares victory, because indeed, he saw the effect he wanted. So I was not kidding. We really do wait. So I will take your challenge.
And single electron, throw it in, let it go through the apparatus, takes mere moments. Wait for a week, send in another electron. No electrons are interacting with each other. Just a single electron at a time going through this apparatus. Other complaints? I have a hard time resisting. So here's a prediction, We now have two arguments for this. So again, let's vote after the second argument. You sure? How many people don't think so? Very small dust. It's correct. So, good.
I like messing with you guys. So remember, we're going to go through a few experiments first where it's going to be very easy to predict the results. We've got four experiments like this to do. And then we'll go on to the interesting examples. But we need to go through them so we know what happens, so we can make an empirical argument rather than an in principle argument. So there's the first experiment.
Now, I want to run the second experiment. And the second experiment, same as the first, a little bit louder, a little bit worse. The second experiment, we're going to send in hard electrons, and we're going to measure color at out. So again, let's look at the apparatus. We send in hard electrons. And our apparatus is hardness box with a hard and a soft aperture.
And now we're going to measure the color at the output. Color, what have I been doing? And now I want to know what fraction come out black, and what fraction come out white.
We're using lots of monkeys in this process. OK, so this is not rocket science. Rocket science isn't that complicated. Neuroscience is much harder. This is not neuroscience. So let's figure out what this is. So again, think about your prediction your head, come to a conclusion, raise your hand when you have an idea.
And just because you don't raise your hand doesn't mean I won't call on you. I like it. Tell me why. So the statement, I'm going to say that slightly more slowly.
That was an excellent argument. We have a hard electron. We know that hardness boxes are persistent. If you send a hard electron in, it comes out hard. So every electron incident upon our apparatus will transit across the hard trajectory. It will bounce, it will bounce, but it is still hard, because we've already done that experiment. The mirrors do nothing to the hardness. So we send a hard electron into the color box, and what comes out?
Well, we've done that experiment, too. Hard into color, So the prediction is This is your prediction. Is that correct? OK, let us vote. How many people think this is correct? Gusto, I like it. How many people think it's not? All right. Yay, this is correct. Third experiment, slightly more complicated.
But we have to go through these to get to the good stuff, so humor me for a moment. Third, let's send in white electrons, and then measure the color at the output port. So now we send in white electrons, same beast. And our apparatus is a hardness box with a hard path and a soft path. Do-do-do, mirror, do-do-do, mirror, box, join together into our out. And now we send those out electrons into a color box.
And our color box, black and white. And now the question is how many come out black, and how many come out white. Again, think through the logic, follow the electrons, come up with a prediction. Raise your hand when you have a prediction. And then it'll go back into the color box. But earlier when we did the same thing without the weird path-changing, it came out still. So I would say still So let me say that again, out loud. And tell me if this is an accurate extension of what you said.
I'm just going to use more words. But it's, I think, the same logic. We have a white electron, initially white electron. We send it into a hardness box. When we send a white electron into a hardness box, we know what happens. Consider those electrons that came out the hard aperture. Those electrons that came out the hard aperture will then transit across the system, preserving their hardness by virtue of the fact that these mirrors preserve hardness, and end up at a color box.
We've done that experiment, too. The other half of the time, the single electron in the system will come out the soft aperture. It will then proceed along the soft trajectory, bounce, bounce, not changing its hardness, and is then a soft electron incident on the color box. But we've also done that experiment, and we get out, black and white.
And the logic then leads to , twice, Was that an accurate statement? It's a pretty reasonable extension. OK, let's vote. How many people agree with this one? OK, and how many people disagree? Yeah, OK. So vast majority agree. And the answer is no, this is wrong. Never ever does an electron come out the black aperture. I would like to quote what a student just said, because it's actually the next line in my notes, which is what the hell is going on?
So let's the series of follow up experiments to tease out what's going on here. So something very strange, let's just all agree, something very strange just happened. We sent a single electron in. And that single electron comes out the hardness box, well, it either came out the hard aperture or the soft aperture. And if it came out the hard, we know what happens, if it came out the soft, we know what happens. And it's not So we need to improve the situation.
Hold on a sec. Hold on one sec. Well, OK, go ahead. So with the second hardness box, are we collecting both the soft and hard outputs? This guy? Oh, that's a mirror, not a hardness box.
Oh, thanks for asking. Yeah, sorry. I wish I had a better notation for this, but I don't. There's a classic-- well, I'm not going to go into it. Remember that thing where I can't stop myself from telling stories? So all this does, it's just a set of mirrors.
It's a set of fancy mirrors. And all it does is it takes an electron coming this way or an electron coming this way, and both of them get sent out in the same direction.
It's like a beam joiner, right? It's like a y junction. That's all it is. So if you will, imagine the box is a box, and you take, I don't know, Professor Zwiebach, and you put him inside. And every time an electron comes up this way, he throws it out that way, and every time it comes in this way, he throws it out that way.
And he'd be really ticked at you for putting him in a box, but he'd do the job well. It works reliably. So the question was, what's the difference between this experiment and the last one. Yeah, good question. So we're going to have to answer that. So it's like as you weren't measuring it at all, right?
Because we send in the white electron, and at the end we get out that it's still white.
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