This program is brought to you by Stanford on iTunes U at Stanford University. Please visit us at itunes.stanford.edu. I just want to do one thing actually before we start the material of the class. A whole bunch of people, a lot of people, particularly people from Europe, who actually follow these things on the internet, seem very puzzled.
They send me email, even some of my friends in Europe, who are physicists who are following it, are very curious about... what's going on here they don't see you people they only see me and they know that i'm teaching a class in this or that special uh... special topic in classical mechanics or whatever and they can sense that there's some something a little bit different about it they can sense that it's not a standard undergraduate or graduate course in these subjects and they ask me what is this what is this about or who are these people you're teaching to they can tell it and most of them don't know what continuing education means they didn't know what it meant and so they asked me i thought i would tell them over the over the internet what this class is. It's continuing education, which means education for people not from Stanford, people not, oh they can be from Stanford, some of the people probably are either employed by Stanford, maybe were students at Stanford, but almost everybody in this class is a little bit too old to be an undergraduate student, even a little bit too old to be a graduate student.
They're people from the... community from Palo Alto from Mountain View from who lives further than Mountain View okay what's the furthest there you come for this where what's the city okay so that's 20 miles away so people from within a certain radius I don't know whatever the radius is are allowed to come and take courses at Stanford they pay a bit of money for it and a professor teaches them Now, this is not, it is most definitely not your standard freshman physics. There's another guy on the internet who teaches freshman physics and totally does it very well. That's not what this is. real McCoy, theoretical physics at the full scale level. We use equations.
We not only use equations, we sometimes use hard equations, but we tend to try to use the simplest equations that will do the job. Basically, we try to keep it minimal. Just out of curiosity, I more or less know the answer just by looking, but what the age distribution of people here is by comparing...
Is there anybody here under 40? Yeah, there's... maybe one handful of people under 40 incidentally for those who can't see the audience it's probably about a hundred people in the room I'm not sure it's a it's a small theater with a lot of seats it's a lecture theater more or less filled up and Maybe four or five or three or four people were under 40. Let's see, let's go to the other end.
How many people here are over 70? A good deal more. Maybe 10 or 12. Anybody over 80?
We got a couple of people over 80. Anybody over 90 this time? Oh, there have been people over 90. There have been people over 90 in this class. So you see the age distribution is such that basically people want to get to the basic ideas fast.
They don't have a hell of a lot of time. They want to get there fast. So I try to tell them in some minimal way what the basic things you really have to know in order to get on to the next thing. Sometimes the basic things that you need to know are a little more difficult, a little more elaborate. we do them anyway, we do them as efficiently and as straightforwardly and with the minimal amount of stuff to get them right, to get them really right.
Not metaphor, not the analogy, but equations when necessary. What else can I say about this class? I guess it has various names.
Quantum mechanics, physics 25. I'm inclined to call it quantum mechanics for old people. Including myself. Anyway, the first course in the series, let me just say what the outline of the whole course of courses is.
These series of courses will consist of about six ten lecture series. The first of which was in classical mechanics. The classical mechanics is the basic basis of all classical physics.
Classical means before quantum mechanics. Anything that doesn't involve quantum mechanics or which ignores quantum mechanics or which is in a range of parameters. where quantum mechanics can be replaced by basic classical logic. That's called classical physics.
So the first class was classical mechanics, which is in some sense the basis for all of physics, the motion of objects, the energy of objects, the momentum of objects, what the characteristic behavior of systems is as they evolve with time, and so forth. We discussed that. that last quarter, and anybody who is thinking of following these classes should begin with that. The next class, which is already on the internet, was called, I believe, Quantum Entanglement.
Now this is also a class on quantum mechanics, and it will be self-contained, but I would strongly advise anybody who's following to go to course number two. Was it called Quantum Entanglement? does anybody remember? I think it was called quantum entanglement, it is also on the internet, it was the first one that was on the internet, and to get that under your belt first, before attempting to do the full-scale quantum mechanics that we're going to do here, although what I'm going to do is pretty self-contained.
I'm going to start this evening with some basic thoughts about the deep differences in the logic of classical mechanics and quantum mechanics. Or classical, maybe even mechanics is too strong a word, classical physics and quantum physics. There are some very queer phenomena in quantum physics that don't exist in classical physics. Now, one of them is the fact that quantum mechanics is based on statistical thinking, randomness. a certain degree of, not imprecision, that's not the right word, a certain degree of non-deterministic or indeterministic behavior, unpredictability, that's the word I'm looking for, unpredictability.
but it's a special kind of unpredictability. Einstein famously said, God does not play dice. Niels Bohr told him, Einstein, don't tell God what to do. But in fact, in a sense, God, I don't know if there's a God, I use the word God in the same sense that physicists always use it, meaning the laws of physics or something like that. The laws of physics don't play dice, at least not in the sense that they play dice.
the standard sense. Let me imagine for you a theory that's based on a bit of statistics, which is not the way quantum mechanics works, and then I'll illustrate it for you by showing you some of the differences. Imagine that there was an element of randomness in Newton's law. Which law?
F equals mA together with the law of gravity. So F equals mA together with the law of gravity tells us how objects move, let us say, how the moon moves around the earth. And it predicts with infinite precision, if you could work out the equations with infinite precision, and if you could...
account for every detail of the earth and the moon and all the material that's in between and so forth, it would predict with enormous detail, deterministically, the motion of the moon, which means if you know where the moon starts and you know how the moon is moving in the beginning, then you can predict forever after exactly how the moon moves around the earth. That's the deterministic classical mechanics. Now let me imagine a modification of the Modification involves a little bit of randomness. For example...
Okay, so let's imagine God sitting on his throne throwing dice. Every tenth of a second, God throws the die, and if he gets snake eyes, what he does is he gives the moon a little extra push in one direction. If he gets a seven, he pushes the moon a little bit in the other direction. A degree of randomness based on the random throwing of dice, and really random. a really random number generator being used to put a little bit of fluctuation into the motion of the moon.
That sounds like it introduces the kind of uncertainty, the kind of thing that one talks about in quantum mechanics. Uncertainty, non-predictability, unpredictability. But it is not anything like the randomness and unpredictability of quantum mechanics. The randomness and unpredictability of quantum mechanics is exceedingly special, exceedingly special and quite different. And it's that that we want to get our head around and learn and understand the difference between these things by the time this class is finished.
But also we want to learn how to use quantum mechanics a little bit to calculate something. things. Let's notice one thing about this law which includes a little bit of randomness, the throwing of dice in order to either kick the moon a little bit or inhibit the moon's motion a little bit.
One of the things that would do Is to add a little bit of energy or subtract a little bit of energy from the motion of the moon randomly If you randomly give the moon and knock this way and then a kick that way and a bunk that way and Keep doing it over and over again Each time you do it this on the average it may not change the energy But each little increment is randomly going to either increase the energy or decrease the energy of the moon And if you do that randomly, eventually that will build up to a statistical randomness in the energy. In other words, energy would not be exactly conserved in a world in which the laws of motion included... little bit of classical randomness. I call it classical randomness to distinguish it from quantum randomness. In quantum mechanics, you prepare a system the same way that you might prepare the moon in an initial situation.
You let it go for a while and then you look at it. And indeed you discover at the end of it that what you measure is a little bit unpredictable, but you also find that energy is exactly conserved. No hint, no remnant at all of energy being knocked this way and knocked that way. If you start it with a given energy, with a given precise energy, and you let it evolve for a while, and you measure the energy later, the energy is exactly the same as you started with. So there's something funny about this randomness.
It seems to affect something. things and not other things, and doesn't work the way you might expect randomness in classical physics to work. Let me give you two, oh, I think three other examples of the oddness of this randomness.
The first comes from an experiment which to my mind shows the weirdness of quantum mechanics in the easiest and most straightforward least difficult way much easier for my money than Bell's inequalities and all that sort of stuff it's called a two-slit experiment all of you know about it or most of you know about it if not you learn about it right now but it is extremely odd when compared with classical randomness Classical randomness just being this idea that every now and then you give the system a little knock. The two-slit experiment involves a source of particles. Those particles could be photons, they could be electrons, they could be neutrons, they could be...
bowling balls except the effect for bowling balls is so minute that you'd never be able to measure it so when we speak about particles we think about things which are very light and because they're light the quantum effects associated with them are significant significant and measurable. All right, so we have a source of some kind. It could be a laser shooting out photons, but imagine the photons are coming through one at a time, very small number of them, quanta of light, one at a time, one every five minutes, if you like. I don't know, just to go to some extreme situation. These are the photons coming out of a photon gun.
And the photons pass through an obstacle with a little hole in it. All right, this is a two-dimensional diagram of a three-dimensional situation. This blue object over here is intended to be a disk with a tiny hole in it. So the photons can go through and they come out the other side of the hole.
And when they come out the other side of the hole, they eventually get to a screen over here. That screen records the photon by a flash, a flash of light at the screen, not the flash of the original photons, but another flash of energy appears at a point on the screen and records, or it could be just the blackening of a photographic plate, something that records the position of the photon when it goes through here. Now first let's think classically.
But classically with a bit of randomness. The bit of randomness that we can imagine is that when the photon goes through here, a little random kick might influence the photon and either kick it upwards or downwards. In fact, by a random amount.
What would we expect then? Well, if there was no randomness, then the photons would go straight through and illuminate a single point. Be completely deterministic, the photons would always arrive at exactly the same point, if the hole was small enough, and if the beam of photons was narrow enough. But now we can imagine a random kick. What does the random kick do?
Well, it changes the direction of the photon. Not the whole photon beam, but one photon at a time. It might kick the first photon up a little bit, the second photon down, and... and so forth and so on, eventually what you will see is a blob of illumination on the screen over here.
On the average, the photon might go straight ahead. So the blob might be most intense at the center. It might be highly improbable to knock the photon through 60 or 70 degrees. So the signal would fade as you moved away from the center.
You would see a blob with a maximum intensity near the center and thinning out as you moved far away. And you might describe it by a probability function, a probability function being the probability of the photon. probability that the photon arrives in different places.
Okay. Now we go, oh, now we do the same thing in real quantum mechanics. In other words, in the real world. We see essentially the same thing.
The photons go through the hole. With no ability to control the situation, we find out that the photons again create a blob like this. But now we're going to do something a little different.
And I think everybody here, more or less, or more or less everybody, knows what I'm going to do. I'm going to open a second hole. Okay, but let's think about what classical randomness would do. Classical randomness would simply mean that, and we'll also imagine, that this beam of photons is at its its origin a little bit uncertain and a little bit random, so that some photons begin a little bit upward, some photons a little bit downward, some photons go through the upper hole, some photons go through the lower hole.
If a photon goes through the upper hole, it may or may not get a random kick and get knocked off course. If it goes through the lower hole, it also gets a random kick. Now we're imagining the photons come through one at a time.
We could even imagine a... Okay. I do that all the time. Where was I?
Yes, the photons get a kick. We're imagining that the photons come through one at a time, very sparsely. And so what one photon does doesn't influence a later photon. because the later photon comes through so much later that whatever gave the first photon a kick is already finished happening and it's waiting for the next throw of the dice in fact the next photon may come through a hundred throws of the dice later and so we expect the next photon to be random, statistically independent of the first photon Under those circumstances, what we would expect, well let's decide what we would expect. Supposing only one hole is open, if only one hole was open, then we would see a blob of illumination like that, with a profile that might look something like that.
Supposing we closed up the first blob, the first hole, and opened the second hole. Close the first one, open the second one. Alright, so first we begin with just one hole.
Then we close the first hole and open the second hole. What would you expect to see? What you would expect to see under those circumstances is a different blob slightly displaced from the first one. The blue blob wouldn't be there because we closed the first hole. The green blob would be there because we opened the second hole.
Now, what happens if we open both holes? What happens if you open both holes in classical mechanics is the probability for a photon to get to the screen at any given point is the sum of the probabilities for it to get there by either root. The photon can either go through the upper root or it can go through the lower root. If both holes are open, the probability to get to this point over here, let's call it the green point over here, the probability to get over there is the probability to go through the upper hole and arrive at the green point, plus the probability to get to go through the lower hole and arrive at the green point. So the result is, in classical physics, you would always see the signal over here, the profile over here just being the sum.
Of the two probability distributions, and it would look, I wish I had another color. They never leave me enough colors. All right. We would see something which would look like just a higher, I don't know, it would be much bigger than that.
It would look like that. And in particular, if there was any point over here such that the photon could arrive either from one hole or from the other hole or both, then we would find illumination at. that point for sure by opening both holes.
That's what classical logic, let's call it by its right name, classical logic, that's what classical logic, classical statistics, classical probability would dictate. for a series of particles coming through here, one at a time. When they come through one at a time, they make blip, blip, blip, blip, but the average probability of the distribution of blips would be a distribution which would be the sum of the two distributions.
What happens if you really do this experiment? You find what's called an interference pattern. The interference pattern looks like this. Well, let's see if we get it right.
In particular... Well, first of all, what is this figure? This figure is a probability distribution, and it tells you, the horizontal axis here, tells you what the probability of a photon getting to a particular point at a particular height here, but in particular, It says that there are no photons which arrive at that point. There are no photons which arrive at that point.
There are no photons which arrive at that point. This is odd. If you opened only one hole, then you would find...
probability distribution which wasn't zero. In other words, you would find illumination at that point. You open the other hole, you still find illumination at that point. You open both holes and all of a sudden no photon gets to that point.
Even though they're coming in, coming through once every 20 minutes or once every 20 years, and therefore how can they know about each other? Nevertheless, if you open open both holes, there will be places where no photons can get to, despite the fact that photons arrived at those points when only one hole was open. That is, you might be able to sit down and work up some interesting but rather elaborate mechanism to make this happen.
You could imagine elaborate, complicated mechanisms where. somehow the this This screen here, some degrees of freedom inside the screen, remember how many photons went through and they remember what they're supposed to do. But it would be a rather elaborate mechanism just for this one purpose.
phenomenon of interference, of destructive interference, this is called destructive interference that the probabilities cancel instead of adding at certain points that's a very generic property in quantum mechanics, and so it requires a kind of explanation which is not some detailed, mechanical, complicated explanation, it requires a broad, new idea about how statistics works and how the logic of quantum mechanics works. So that's the first really weird thing that happens in quantum mechanics. Now let me give you another example.
If you remember, in the last course, we talked a little bit about reversibility. We talked about the laws of physics. In particular, we talked about the laws of physics for discrete systems. For example, we discussed the possible laws of physics, deterministic laws of physics for a coin which can either be heads or tails. Now, of course, flipping the coin, that introduces a level of uncertainty, a level of.
statistics, probability into things, I want to think about the deterministic laws. The truly deterministic laws are the ones that whatever the coin is doing, it will tell you what the coin is doing next. So when we discussed this last quarter, we talked about two possible laws of physics. The first was that if you find heads, in the next instant when you look at it, after an instant of time, you'll find heads again.
If tails, then you'll find tails. then you find tails again then your laws of physics just permit two possible evolutions heads heads heads heads heads heads heads or tails tails tails tails tails and the other possible laws of physics or law of physics was that when you see a head in the next instant microsecond or whatever your unit of time is you will see the opposite, tails then the two possible laws of physics are, not laws of physics, the two possible evolutions one One of them begins with heads, it goes heads, tails, heads, tails, heads, tails. And the other one begins with tails, it goes tails, heads, tails, heads, tails, heads. Whether you know it or not, those two different things were different. They began one with heads, one with tails.
That was the basic idea of a deterministic law of physics. Now, you can have, of course, more complicated systems than just two state systems. We imagine, for example, a six state system.
A six state system was a die, a die that There can be one, two, three, four, five, six. But let's just take a simpler thing. Let's take a coin with three sides.
If you can't imagine a coin with three sides, then you'll have to do this using abstract mathematics. A coin with three sides has heads, tails, and what else? Feet. Heads, tails, and feet. Heads, tails, and feet.
All right, and there are two... Let's consider a simple law of physics. A simple law of physics could tell you that whenever you have heads, it goes to tails. Whenever you have tails, it goes to feet. Whenever you have feet, it goes to heads again.
And then... Wherever you start, you just cycle around endlessly, forever and ever. Now, there's a certain sense in which this is, what does it mean to say it's deterministic? What does it mean to say information is not lost? in this process what it means is that no matter how long the system evolves let's suppose you started with heads and you let it go a million units of time it will just go around and around in a cycle if at the end of that you reverse the law of physics now reversing the law of physics just means having it go in the opposite direction if you If you could somehow press a button or turn a knob, which had the effect of reversing the law of physics, in other words, changing the direction of every arrow, and think about actually if you could really do this by pressing a button, change the law of physics, then if you allowed the system to evolve for any length of time, at the end of that time, reverse it.
And let it evolve for the same length of time again, guess what? It magically comes back to the same original configuration. That's what it means to say that physics is reversible.
Or that's what it means to say that information is never lost in physics. That no matter how long you keep going, if you reverse, if you could find a way to re- reverse the laws of physics and run them backward for the same exact length of time, you'll come back to the starting configuration. Now to do that, you might not even need to know the laws of physics.
You might need to know very little. the only thing you would need to know is how to reverse the law. You needn't even know what the law is.
Whatever the law is, if you can find a button to push that reverses it, then you can test the determinism of physics by simply starting someplace, letting it evolve for a long period of time, and then letting it evolve with the reverse law of physics. If you come back to the same place every time, then your law of physics is deterministic. Now, what about a little bit of classical probability, a little bit of deity playing dice?
So let's suppose with some very, very small probability, the deity does something different that's not prescribed by the laws of physics. For example, in some random way, with a probability of one in a million, the law might say, don't move. So instead of moving the way the the diagram says stay still if you allow the system to evolve for a short period of time and then run it backward it will go back to the same point but if you allow it to run long enough that there's a significant statistical probability for a fluctuation for something to happen which is not deterministic.
In other words, if it was one in a million that you stand still, but you let the system run for 10 million units of time, then guess what? No chance, or not no chance, you have a chance that you'll come back. back to the same point, but you also have an equal chance that you'll come back to any of the other points. In other words, this test will fail. It'll fail.
One third of the time you will come back to the same point, two thirds of the time you will come back to two different points. So a little bit of classical randomness destroys the What I called last quarter the conservation of probability, no, the conservation of information, information gets lost. What about quantum mechanics? Is information lost?
Well, there is an element of statistics. statistical things in quantum mechanics. When an electron goes through a hole like this, it has a probability for getting kicked up.
Let's just take the case, let's simplify now. This is the one-slit experiment. We don't even need the two-slit experiment.
Let's start with the one-slit experiment. The electron goes through the slit. It's aimed toward the slit. Sometimes it more or less goes straight through. Sometimes it gets kicked up a little bit.
Sometimes it gets kicked down a little bit. And you might think that this is more or less like throwing dice. In fact, if you look for the electron afterwards, if you look for the electron out here, after you've given it... time to pass through. At the moment now let's just send one electron through.
One electron. If we send one electron through, it may get kicked up, it may go straight through, or may get kicked down. If we do the same experiment repeatedly, which of course is the same thing as sending many electrons through, but if we do the same experiment repeatedly, we'll find some go up, some go down. Let's ask the following question.
Supposing after a period of time, we send an electron through, one electron, after a certain period of time, having given it enough time to get to the other side, but not, but let's remove the screen over here. Let's remove the screen, send the electron through, it goes through, comes out someplace else. But we don't look at it.
We don't bother detecting where the electron is. Instead, we just reverse the law of physics. We do what we did over here.
Reverse the direction of time, if you like. Can you really do this? Is there really a button that you can push? Is there really something that you can do to a system that will reverse the motion? Yes, in many systems there is.
In many systems we actually know how to manipulate the system, how to change magnetic fields, how to do things to a system of electrons so as to run it backward. So, let's take it as a given that after a certain amount of time somebody can press the button that reverses the law of physics, that reverses the direction of time, so to speak, and runs the system backward for the same length of time. What happens?
Does that electron, we're not going to look at it, but do we later after the end of the experiment? We allow it to evolve for a time t, and then we allow it to evolve backward for another time t, 2t altogether. Do we find the electron moving backward along the original trajectory? Or do we find the probability the fluctuation's compounding, and that after we turn it around, there's even worse fluctuation? Particle comes through, gets knocked up.
We run it backward. It either gets knocked down or knocked up. Which is it? Does it reverse precisely along the original trajectory every single time?
Or does the statistical fluctuation, the imprecision, the unpredictability in one direction add to the unpredictability in going backward and make it even more unpredictable afterwards? than it was to begin with. The answer is very curious.
The answer is that if we don't look at the system in the intermediate stage after it's gone through the hole over here, don't look at it means don't interfere with the electron in any way. Take that screen away that might have converted that electron into a little pulse of light. Just remove any influence on the electron over here completely.
Do not look at it, do not interfere with it, do not do anything to disturb it, but just run the law of physics backward. Then we will exactly every single time detect the electron running back along the reverse trajectory on the other side over here. In particular, we'll find the electron just going right back into the gun. What if somebody does look at the electron when it goes through to the side over here?
For example, supposing somebody sets up an electron detector which detects where the electron is and then lets it go, having reversed the law of physics. We look at it and then reverse the law of physics. Now, in this case over here, the case where we're talking about classical coins, looking at a thing doesn't disturb it very much.
If I have a coin, where's my coin? I've lost my coin. It's okay, it's only a Chilean peso.
I lost my Chilean peso. I think it was actually a hundred pesos. Okay, now here's my Chilean peso. I put it down and I put it down heads. Don't look at it.
Now I look at it. It would be rather amazing if just the process of looking at it looking at it was capable of flipping it from heads to tails. Now of course if I look at it with an intense enough high frequency light beam, a beam of very, very energetic photons, sure enough, an energetic photon Could could hit the coin knock it into the air and spin it over that's true But you could but in classical physics you can look at an object and determine its state determine the heads or tails nests of it with an arbitrarily gentle interaction and So by looking at a system in classical mechanics and looking at it with a very very gentle Apparatus or a very gentle photon or whatever you like It does not entail disturbing the system So if you look at it after you allowed it to go a million times and then reverse the law, it will have no effect, no detrimental effect on the experiment, and you will come back to the same point that you started with. So just looking at the system in between. Has no effect, no, does not necessarily have an effect on it.
Even though whoever looks at it can determine, after a million... Units of time where the system is here, looking at it has no effect on what happens after you run it backward. Exactly the opposite in quantum mechanics.
If you detect the object, if you do anything to detect the object, electron in this case, and then run the law of physics backward, what you'll find out is that the probabilistic character of it gets compounded. So the fluctuation that sent it up here, when you run it backward and you now know that it came out up here and you run it backward, it's likely to come out down. here or up here, and it will disturb the system in such a way that the test of reversibility will fail. So this is curious, that whatever quantum logic is... The questions of the kind we're asking are deeply dependent on a ridiculous question.
Did somebody look at the system during the course of its evolution? And as I said, of course, looking at a system can disturb it, but in classical physics we can look at a system without disturbing it. We can look at a system, detect the system, measure the system as gently as we like, arbitrarily gently, and have arbitrarily small effect on it. it so that running it backwards will be exactly as if we didn't look at it.
Not so in quantum mechanics. Determining the state of a system is never a small thing to do to the system. And this is an example that completely destroys the experiment.
The two-slit experiment has also a similar story that goes with it. The story that I told you a moment ago about the interference pattern and the destructive interference is only true if nobody records which way the electron went through. Now when I say nobody I don't mean a human being necessarily.
I mean that nothing in the environment of the experiment records and remembers which way the electron went through. In other words, after the experiment, there was nothing in the environment of the this experiment which has recorded which way the electron went through, nothing in here, only, no, this also hasn't recorded where the electron went through. If nothing records where the electron goes through, then there's an interference pattern. But if you were to put a little demon over there, a real physical demon, now a real physical demon could be another electron.
It could be some gas in the apparatus, some gas molecules in the apparatus. this, whatever it is, in such a way that something records which way the electron went through. Something remembers it afterwards. Perhaps a molecule over here gets disturbed, gets excited if the electron goes through the upper hole. Gets excited and changes the character of that molecule.
If it goes through the lower hole, it changes the character of a different molecule. molecule. So looking afterwards, after the electron went through, you can tell which hole the electron went through, then the interference pattern is destroyed, and the answer is exactly the same as the classical answer, namely the probabilities add, just exactly as in classical physics. So again, there's no way to record whether the electron went through the upper hole or the lower hole. or the lower hole without seriously disturbing the experiment without so seriously disturbing the experiment that you drastically change the conclusion That's a quantum...
That's roughly speaking the general character of quantum mechanics that you cannot... do measurements on systems without disturbing them, and disturbing them can change completely the character of a, yeah, question. Yeah.
Yeah. That is, yeah, that's a very, very good question. That's an excellent, outstanding question.
Wish I knew the answer. No, I know the answer. Yeah, I will. The answer has to do with the probability distribution for the position of the, let's call this the detector. Let's call this the detector.
Okay, now, if the detector. is very well localized in space, as it would be if it were a heavy, massive classical detector, then by the uncertainty principle, which we haven't talked about yet, if we're going to... Let me skip ahead and imagine that we have talked about the uncertainty principle.
That's a sophisticated question, so I suspect you've thought about it a little bit. If the position of this detector, the up-down position of this detector, delta x, is very, very small, that means that the uncertainty in the momentum of the detector is large. Okay?
That means if I were to plot the probability distribution of the momentum of the detector, Let's plot the probability distribution of the momentum of the detector. It's rather broad because for a heavy detector, its location is so well defined. Now the electron comes through and kicks this thing a little bit.
It gives it a small kick. And what does it do? It shifts this probability distribution a small amount.
But unless the probability distribution has been shifted by something approximately equal to its width, then you can't tell afterwards whether it got a kick or not. You have to be, in order to be able to be certain, that it got a kick, you have to kick it by an amount large by comparison with the uncertainty. So it's the uncertainty principle that comes in and rescues you, or rescues me, and makes sure that the interference pattern is not destroyed.
Okay? It's the uncertainty principle itself. So let's go to the uncertainty principle since it's been raised, since I raised it. The uncertainty principle is another factor in quantum mechanics which is extremely different than classical physics, conceptually very different.
I know we haven't talked about what the uncertainty principle is yet, but let's discuss it anyway. Let's jump ahead. My main motivation now is to explain the strangeness of, not to explain the strangeness of quantum mechanics, but to point some fingers at the strangeness of quantum mechanics so that you see that it really is fundamental.
Fundamentally logically different than classical mechanics logically different whole logic of quantum mechanics is different okay, it's easy to imagine a Bit of uncertainty in classical physics as I said the coin throwing or the dice throwing deity who sticks his finger into the system and gives it a push this way, a push that way and so forth and that way things after a short period of time become uncertain and it's easy to imagine that that uncertainty can affect both position and momentum and if you wait a little while both the position and the momentum may be uncertain. In fact if you wait a little bit while more than a tiny fraction of a second for a particle, you might discover that inevitably both the position and the momentum get jostled about, and so that there's a good deal or a certain amount of uncertainty in position and uncertainty in momentum. But you would be unlikely to say that that uncertainty and position and momentum is, what's the word I'm looking for, well first of all that's fundamental in any sense, you would say it was a result of hitting the system randomly but I think most of us would agree that under those circumstances it was a bit of laziness that didn't allow us to watch the particle carefully enough to see what exactly the momentum was in the position was, we always imagine in that kind of classical context, an example of that classical context would be the random walk of a Brownian moving particle, right? So if we watch a whole bunch of particles, a whole bunch of particles might form a cloud and that cloud might spread. And perhaps even the velocity of the cloud, or the cloud describing the velocities might also spread.
But every one of those particles. if we cared to, we could look at with better precision, better and better precision. We could look at it gently, as gently as we liked in classical physics, and determine both its velocity and its position simultaneously. Certainly that would be true of a classical Brownian motion particle that was being knocked around by ordinary collisions with some gas or something like that.
In principle, we could just get ourselves a better microscope. Better accuracy, better precision, better resolving power, and do it very gently so as not to disturb the system. When we measure the position we don't want to jostle the momentum. When we measure the momentum...
we don't want to do something funny to the position, and we just measure it gently enough so that we measure both the position and the momentum. That would be expected to be true for a Brownian particle. So the uncertainty in position and momentum quantum is something which in a certain sense is due to our own laziness and not spending enough money and buying a good enough detector and so forth. To be able to detect both the position and the velocity at the same time.
On the other hand, in quantum mechanics, there's a really fundamental obstruction, a logical obstruction, a deep obstruction to knowing, to ever being able to measure both the position and. and the velocity of a particle. I'm going to work it out for you and show you how it works. I'm going to show you how Heisenberg first thought about it. Well, he first thought about it entirely through abstract mathematics.
For us, that's going to come later. But then, when questioned by Bohr, what are you talking about that you can't measure the position and the velocity simultaneously? Your mathematics? is a crock of baloney.
Don't tell me that x times p is not equal to p times x. Come on, it's like saying 3 times 5 is not 5 times 3. Did I write it right? Yeah.
Not equal to. Don't tell me such nonsense stories. give me some physics.
And so Heisenberg cooked up his experiment to show, to illustrate the fact, not show, but illustrate the fact that there was a good, deep, consistent reason why it's impossible to do that. possible to ever, ever under any circumstances, simultaneously determine the position and the momentum of an object. So let's go through that a little bit.
It has a similar character to some of the other illustrations that I've given here. What Heisenberg and Bohr and Einstein and all those people knew around 1926 was the property of photons. In fact, whenever they thought about measuring the properties of a particle, they were always thinking roughly speaking, of looking at it under a microscope, detecting it by bombarding it with photons.
Now it doesn't matter whether they were photons or not, it's just that they were thinking about microscopes and they were thinking in a language where you optically look at things. It doesn't have to be optical, but this is what they were thinking. So Heisenberg imagined putting his particle under a microscope and detecting its position and its velocity and seeing what he could learn.
Okay, the measurement involved interacting with a particle with a photon. A photon would be the thing which would be used to determine the position of momentum. In other words, looking at it really meant hitting it with some light, letting the light scatter off, and then focusing the light waves, focusing the light waves in order to see exactly where the particle was, just as you would do it for a billiard ball or for anything else. Focus the light on your retina, focus the light on some screen, and reconstruct the position of the particle.
That was what they were thinking about. Okay, now, here's what Einstein and de Broglie and others had taught them about photons. First of all, Einstein had told them that the energy of a photon is equal to Planck's constant times the frequency of the light describing the photon. This is equivalent to Planck's other constant, h bar, times the angular frequency where f stands for the frequency of a wave, number of cycles per second, measured in hertz. Hertz.
Usually a frequency of a light wave is a lot of Hertz. How many Hertz? How many? Ten to the fifteenth.
Ten to the fifteenth for ordinary light. That's just a number, ten to the fifteenth Hertz. Omega is the same as the frequency except multiplied by two pi.
It's the angular frequency instead of the number of cycles per second. And h bar is just h divided by 2 pi. So these are the same expressions. I'll use this one over here for the moment.
Energy is equal to h times frequency. That's something that was known by Einstein. Now, here's another fact about a beam of light. Beams of light have not only energy, but they also have momentum. momentum You can take a beam of light and shine it on something and it will warm it it will heat it that tells you it Has energy, but you can also take a beam of light and you can shine it at that door And if the beam of light has sufficient intensity, it'll just push the door open in other words the beam of light has momentum So that when the door absorbs the beam of light the door gets a kick Just as if Just as if you threw a ball at the door and the door or collide with the wall, it has momentum.
Okay, now what is the relationship between the energy of a beam of light and its momentum? I'll just tell you what it is. This is classical Maxwell.
theory of light. If a beam of light moving in a particular direction, assume it's moving in a particular direction, has energy E, then it also has momentum, and the relationship is that the energy is the speed of light times the momentum. Incidentally, let's just check the units of that equation in another way.
Let's compare that with Newton's theory of momentum and energy. For an ordinary particle, the energy is p squared over 2m. That's non-relativistic.
The momentum, p squared over 2m, yeah, let's... Which is equal to p times p over m and an extra 1 over 2. One half momentum times momentum divided by velocity. Now what's momentum divided by velocity?
Sorry, momentum divided by mass. Velocity. So this is equal to one half momentum times velocity. Well, it's almost the same for a photon.
It's not the same because we have to use the theory of relativity for a photon. But the correction from the theory of relativity is not so enormous. This is one half the velocity of the particle times its momentum.
This is just... equal to the velocity of the particle times the momentum. So for highly relativistic particles, the formula is very similar, except a half goes away.
Among other things, this argument tells you that the units are right. Energy is velocity times momentum. Let me just solve that.
Momentum is energy divided by the speed of light. We can now plug in and find that the momentum of a photon, or momentum of any little piece of light, let's do a single photon now, the energy of the photon is h times f. And so the momentum is h times the frequency divided by the speed of light. Now if you have a wave that's moving with the speed of light, The other way of moving with the speed of light, it's moving down and it has a certain frequency and a velocity.
There's a connection between the frequency and the wavelength. Think about it for a moment. Here's a wavelength.
Let's call the wavelength lambda. Lambda is the standard notation for the wavelength of light, the wavelength of anything. How far does the wave move in one cycle?
The answer is lambda. That's what lambda is. It's the distance that the wave moves in one cycle.
If you stand there with your nose watching that light ray and it's moving past you, it moves past you one wavelength per cycle. One wavelength per unit, per cycle. How long does a cycle take?
How long does one cycle take? What's the period that goes with one cycle? The inverse of the... of the frequency. So the time that it takes, the time that it takes to move distance lambda is one over the frequency.
So let's just write that down. Time to go one cycle is equal to one over the frequency. The distance that it goes in that same time is just lambda.
So what's the velocity of the wave? The velocity of the wave is the distance that it goes divided by the time that it travels. Distance over time is the velocity of the wave. So what's times velocity, c is equal to lambda divided by t, which means lambda times the frequency.
So this is a general formula relating velocity, lambda, and frequency. And let's see, let's plug it into here now. Let's get rid of the frequency. Frequency is c divided by lambda. So frequency is C divided by lambda, and then there's another C down here.
And we get de Broglie's equation that the momentum is Planck's constant divided by the wavelength. The smaller the wavelength, the larger the momentum. If you want a beam of particles with very high momentum, give it a small wavelength. Or, say it the opposite way, if you want a very short wavelength particle, it's at the cost of having that particle have a large momentum.
Momentum and wavelength are inverse to each other. So now let's go back to... He wants to measure, or roughly speaking, he wants to get a photograph of the electron where the electron is not fuzzy on scales larger than delta x.
This is once they take a photograph of the electron. Boy, it doesn't have to be an electron. It could be a golf ball. It doesn't matter what it is.
Once they get a photograph of it and wants the photograph to be non-fuzzy on a certain scale delta x. Well, every photographer knows that, or anybody who understands anything about waves. and images and so forth, knows that to form an image which is precise or non-fuzzy the size delta x, you have to use wavelengths that are shorter than delta x.
If you try to make an image of a golf ball with a radio wave, a radio wave of let's say 10 meters, the golf ball will look fuzzy on the scale of 10 meters. If you try to do it with a wavelength of a tenth of a golf ball, ball, a tenth of a centimeter, the golf ball will look pretty good. So the rule is that lambda, the wavelength of the light, must be less than delta x if you want to get an image with precision delta x.
Now I erased an equation. The equation that I erased is that the momentum is equal to Planck's constant divided by lambda. So now Heisenberg was caught in a bind.
If he wants to measure the position to a high accuracy delta x, he's got to use a high momentum. electrons, he's a short wavelength electron. If he wants delta x to be smaller than a centimeter, then he's got to be using a wavelength smaller than a centimeter. But if he's using a wavelength smaller than a centimeter, that means the molecular weight of the electron is going the momentum of the photon has to be larger than h over one centimeter. So the smaller delta x, the larger the momentum of the photon that he has to use to make the image, the shorter the wavelength of the photon that he has to use to make the image.
If lambda is small, then p is going to be large. Well now, what does that mean? That means that we're going to wind up bombarding this object. with a high-momentum photon.
The high-momentum photon may make a very good image of the position, but then it's going to collide with this and knock it off in some random direction. Knock it off in a random direction with a momentum, uncertain amount, uncertain momentum of order of magnitude of this momentum here. This particle will come in, and just like the photon hitting the slit, it will get knocked in some random direction and so the conclusion will be that immediately after you try to measure the position Immediately afterwards, the momentum has become very uncertain. It has been kicked hard. Having been kicked hard, if you measure its momentum afterwards, it will have nothing to do with the momentum beforehand.
So, you cannot determine both the position and the momentum at the same time. Measuring the position necessarily imparts a random momentum, a kick, a random momentum kick to the particle. The result is, whatever the momentum was beforehand, it won't be that afterwards.
And that's another example of the fact that there's no such thing as a gentle determination in quantum mechanics. Well, you can do a gentle determination, but it will be a very imprecise determination. In classical physics, incidentally, oh, keep in mind, what about classical physics? Why is it different in classical physics? The reason is because light doesn't come in discrete packets in classical physics.
We've used the fact that light comes in discrete, indivisible quanta, discrete, indivisible photons. If we have a wavelength... there's a minimum amount of energy that can go with that, minimum amount of momentum that can go with that wavelength, namely one photon.
You can't have less than a single photon. In classical physics, energy does not come in discrete multiples of some basic unit, and so you can do the same experiment with as small an energy as you like. It's as though in classical physics you could supplement...
subdivide that photon into arbitrarily small units of value, take just one of them with the same wavelength, and form an image with it, in quantum mechanics you're always stuck by the fact that the energy of a light wave comes in these discrete packets. And that discrete packet, if it has wavelength lambda, will have a momentum, h over lambda, and therefore give this an inevitable kick. of a mount H over lambda.
Now, that's... This is several examples of the same kind of thing that doing experiments in quantum mechanics is different than doing experiments in classical mechanics. You can always imagine classical mechanics doing a very gentle experiment that doesn't disturb a system. and then just go on from there and do a later experiment, and do a later experiment, the earlier experiments, not having influenced the outcome of the later experiments.
So, for example, you can measure the position, and that not affect the outcome of a later detection of the velocity, which is not true in quantum mechanics. These are a lot of examples, but the examples add up to a notion that the basic logic of classical mechanics is incorrect. The basic underlying logic is not sufficient to understand measurement process in quantum mechanics.
The whole setup is wrong. The whole setup. Not just the thinking, not just each experiment. You can go and analyze it and try to figure out what's wrong with it and try to correct for it. No.
The whole underlying structure of classical physics is inadequate to discuss quantum mechanical phenomena. Let's take a break for five minutes and then start quantum mechanics proper. OK. Somebody asked me about how do you measure the velocity of a particle.
The answer is gently, first of all. Here's a simple conceptual way to do it. Velocity or momentum. Momentum. Let's suppose we know the mass of the particle, that by measuring its velocity, we also measure its momentum.
A simple conceptual way to measure velocity is to measure location at two different times and then take the difference of location, that's how far it travels, and divide by the time between measurements and that's the velocity. That's the way you would measure velocity. Now you have to be careful.
You want to measure the position twice in succession. But you don't want to measure the position with such a good determination that it gives a random kick to the velocity, which is what you're trying to measure. You wouldn't want to measure the velocity by beginning the experiment with a random kick, which randomizes the velocity and sends it off in some direction very different than it was moving with to begin with, direction and velocity.
So your two measurements of velocity, sorry, of position, Should be very gentle measurements that don't change the velocity very much, or that don't change the momentum very much. That means that they must be done with photons of very long wavelength. If you don't want your first initial detection of the location of this particle to... To give it a good whack, then you want to do it with a very long wavelength photon.
A long wavelength photon will then tell you only that the electron is in some region of size lambda. So what you know then is the position of the electron x, let's say plus or minus lambda, meaning to say that there's an uncertainty of magnitude lambda. Then you wait a really, really long time until the electron...
has moved a long, long ways. It has some velocity. It started out with some velocity.
You've changed the velocity only by a very small amount by using a very long wavelength photon. So we take the wavelength of the photon. photon to be so long that there's been an unappreciable change in the momentum.
And then at a much later time, we discover that the particle is at x plus or minus lambda, maybe even plus or minus 2 lambda, plus vt. It's velocity times the time between the two measurements. So true, there's a little bit of sloppiness in the measurement of the position.
a large sloppiness, excuse me, in the measurements of the positions, and that's going to lead to a sloppiness in the distance that the particle moves. What's the sloppiness? The sloppiness, the distance that the particle will move, will be Vt plus a sloppiness of order lambda. And we take lambda to be very big so that we don't... disturb the velocity very much.
Then how do you find this is how far the particle moves? Let's call it d, distance that it moves. In order to find the velocity we have to divide the distance that it moves by the time.
So let's divide it by the time. This is now a measurement in the sloppiness of the velocity. The sloppiness of the velocity measurement is lambda divided by t. No matter how big lambda is, if we wait long enough, if we let t be very, very long, large, then the sloppiness in the velocity can be made small.
So the way we measure the velocity is by taking a long time to do it, do very very gentle measurements of the position so that we don't whack the velocity, and in that way we can measure the velocity with great precision. However, it's been at the cost of knowing where the particle is. The particle has not been determined to a precision better than lambda.
The velocity has been determined to a precision of something like lambda over t. So whatever you do, you'll never be able to determine both the position and the velocity in the same experiment. You do one and not the other. One or the other. Kind of quantum mechanics...
Yeah? Yes, well, um, say it again? Yeah. Uh, let's see. You wait a very long time so that...
Uh... That's because you're taking this to be the uncertainty and the velocity, but there's also an uncertainty from here that has to do with the original setup. Let me think about it.
I understand your question. Let me think about it and come back to you with an answer. Yeah, it's a good...
Let me come back to it. Yeah. Mm-hmm.
Mm-hmm. Yes, I see, I see, yes, you can determine the position of this wall by averaging over a long period of time with photons, many, many photons banging off it. Yeah, yeah.
Trying to remember what I was going to talk about. We'll come back to the uncertainty principle for sure, but I've lost the thread of my thought. All right, so I forgot what I was going to say, but let's just move on.
So the fundamental logic of quantum mechanics is not the same as the fundamental logic of classical mechanics. And that shows up at the earliest possible stage, namely, what do you mean by the state of a particle? What do you mean by the configuration? What do you mean by knowing everything that there is to know about a system, or everything that can be known about a system?
In classical mechanics, the space of states, what we call the phase space. We had some various versions of it. In one version, it was just a set of points.
I used the word set, and I used the word set on purpose, a set of points, the states of a system. Heads, tails, or one, two, three, four, five, six for a die. In particle mechanics, or mechanics that's a little bit less primitive than this very simple system here.
It's phase space. The states of the system are phase space, p's and x's. p's and x's.
Points in a phase space are the states of a system, but again, the collection of possible states of a system form a set. They form a set, they form a set of points, and the basic logic of classical physics, of classical phase space, is set theory. Sets of points describe the possible states of a system, and transitions or motions from one point in that set to another point in that set describe the mechanics or the dynamics of a system.
In phase space, it's the flow through phase space. What exactly is a state? A state is a point in that set, a member of the set.
An x and a p for a particle, a point h or t for heads and tails. Or a bunch of p's and x's for a general system of many particles, but a point or a member of a set. Now, you could think of something a little more general and possibly call it a state. in which you introduce a bit of statistics, a bit of a probability or a bit of uncertainty from the beginning You might say, look, my apparatuses are not sufficient to determine with infinite precision the position and velocity of a particle So instead of doing that, I will say the particle is somewhere in some little region here Or I might assign a probability distribution.
A probability distribution is a function of x and p, which might be peaked at the center here and fall off as you move away from it, and so forth. A probability distribution on phase space might even be a more general version of what you mean by the state of a system. But if I told you that the notion of a state of a system... Is a probability distribution I think most of you will come back and say yeah But you know if I look at more carefully at the system if I look more carefully I can always reduce that probability distribution and I can always get as close as I like By doing delicate experiments and again in classical physics doing experiments which don't disturb the system terribly much Doing experiments on the system which will get it to be closer and closer to a point.
So when you speak about a statistical distribution as being a state, you're not talking about the maximum amount of knowledge that you could have about a system. You're talking about a practical limitation because of the coarseness of your apparatuses and so forth might force you to use a probability distribution. And only talk about things to within a precision that you can measure them, but you wouldn't think that that was very fundamental and eventually you would say with sufficient accuracy in your apparatuses that That the maximum you can know about a particle Corresponds to a point in the phase space and in that sense States in classical physics are points in a set points in a set And you can know everything that's implied by knowing a point in the set.
In particular, in this case, knowing a P and an X. In quantum mechanics, states are not sets, do not form sets. The natural way of manipulating states and asking questions about them is not set theory. States are vectors in a vector space in quantum mechanics.
Quite a different mathematical object than a set. A vector space is mathematically extremely different than a set. And in order to understand quantum mechanics, this is not some abstraction that is really sort of unnecessary to understanding the subject.
This is so central that to not talk about it, we would completely miss the basics of quantum mechanics. What, in order to understand the logic of quantum mechanics, we have to understand the mathematics of vector spaces. We're just forced to it. There's no way around it. If I tried to do things without it, I'd be faking.
So let's talk about vector spaces. Now, the use of the term vector, oh, these are vector spaces, linear vector spaces. Every vector space is linear.
So linear is a redundant word. But they're vector spaces over the complex numbers. That may mean nothing to you right now, but it will as we go along.
Vector space, the space of states, the state space, space of states, is not a set. But it is a vector space. Now the use of the word vector can be a vector space over the complex numbers. Over C. C stands for the complex numbers, and I'll tell you exactly what this means. If you don't know what it means, don't worry, because you will know.
Okay. First of all, most of us are familiar with the notion of vector. from the classical notion of a pointing, of a thing pointing in a direction in ordinary three-dimensional space with a certain length.
What I'm talking about is not vectors in space in that sense. These are are abstract vectors in abstract vector spaces, which have nothing to do with your naive concept of a vector in space as an arrow pointing in some direction in space with a given length. When I teach this, I usually make a linguistic distinction between vector spaces. And pointers in space, which are the things you usually think about as vectors. I can never quite figure out how to do this without getting everybody confused, including myself.
I will try as much as possible when I'm talking about a three-dimensional vector in space, the kind of vector that could correspond to velocity or position or momentum and that sort of thing. I'm going to use the term pointer, whenever there is any ambiguity about what I'm talking about, a pointer meaning a direction in ordinary three-dimensional space together with a length. I'll use the term pointer in order to not get confused with a completely different mathematical concept, or not a different mathematical, to a much more general mathematical concept called a vector space.
A vector space is a space of vectors, but of course these vectors are not pointers in ordinary directions, ordinary space. Pointers in ordinary space are a special case of vector spaces, but not the special case we're going to be interested in. Okay, so let me tell you what a vector space is. First of all, it's a collection of mathematical objects called vectors. Again, I'll emphasize over and over, not vectors, just abstract objects called vectors.
Maybe I should change their name and not the name of the vectors in three-dimensional space. We could call them wectors or schmectors, but one way or another, we've got to make some distinctions. All right. It's a collection of objects which we can label. Let's label them.
We can also label sets by points and give the points labels like a, b, and c, and so forth, but now I'm labeling vectors and to indicate that they are vectors of the right kind, namely the kind that come into quantum mechanics, we're going to draw a symbol like that. This is called a ket vector. Called a ket vector by Dirac.
There's another kind of vector called a bra vector. And if you put them next to each other, they make a bra ket or a bracket. That's where the name came from.
But we'll come to brackets in a little while. It's half a bracket, a bracket being what happens when you put a bra next to a ket. Is that clear? Yeah, OK.
OK, so there's a collection of objects, collection of objects called vectors. Now, what does it mean to say it's a vector space over the complex numbers? What it means is that, first of all, you can multiply vectors by complex numbers.
Now, the ordinary vectors in ordinary space are a vector space over the real numbers. If I have a vector, for example, three units of length pointing in the direction north by northwest that are 45 degrees to the horizontal, that would define a pointer, if you like. Alright, so ordinary three-dimensional pointers have a length and a direction. You can multiply them by numbers. If you multiply a certain vector, this one here, sorry, a certain pointer, by 2, it just gets twice as long in the same direction.
If you multiply it by minus 1, it's the same length but pointing in the opposite direction. If you multiply it by 7 it becomes 7 times as long. So you can multiply a vector, a pointer, a pointer by an ordinary real number, positive or negative.
Mathematicians would say that pointers form a vector space over the real numbers, vectors over the real numbers, meaning to say that you can multiply them by real numbers, and when you multiply them by a real number you get another pointer back. We're going to make a generalization of that idea, where vectors can be multiplied by complex numbers. Now eventually we'll see some examples.
There's nothing like an example to make it clear. But what that means is that given any vector, you can multiply it, given A, you can multiply it by any complex number. Let's use for complex numbers, let's use the Greek alphabet. So if Alpha is any complex number, I assume everybody here is familiar with of complex numbers. I'm going to assume that.
I'm not going to teach you complex numbers here. You can multiply any vector by a complex number alpha. And it's a new vector.
Not the same vector, it's another vector. So the operation of multiplying by a complex number is well defined in a complex vector space. We can call this B, for example. Multiplying by alpha maps every vector into another vector.
Alpha being any complex number. That's the first property of a vector space over the complex numbers. Sometimes we can call it a Hilbert space.
I mean, sometimes it's just called a Hilbert space. Vector spaces over the complex numbers. number, a Hilbert space. Hilbert was a mathematician.
So that's the first property. Given a vector, you can multiply it by any complex number and it's a new vector. The second property is that if you have two vectors, any two vectors, let's call them A and B, you can add them and you get another vector. So adding vectors give new vectors.
Now first of all, here's something which is really, both of these things are really new when you go from classical mechanics to quantum mechanics. In classical mechanics, the notion of a state is a point in a space of points, in other words, in a set. It doesn't make any sense to multiply points in a set by a number. For example, let's take heads and tails. What does that mean?
What does it mean to multiply heads by 3? Or heads by a complex number? It doesn't mean anything. You just have two points, heads and tails, and that's all you have. You don't have a thing that you would call 3 times heads, or minus 3 times heads, or 2 plus 4 times i times heads.
So in saying that state vector, that states of a system are a vector space, this is something new and radical and weird. You should not understand that. this at this point, unless of course you've done quantum mechanics before.
This should not make any sense at all, but for the moment we're doing some abstract mathematics to get some definitions, and then I'll show you how those definitions apply. So first of all you can multiply a vector, an abstract vector, by any complex number, and you get a new vector. That's an operation that you can do.
And the next operation is that you can add any two vectors. Now of course if you can add any two vectors and you can also multiply multiply them by complex numbers, then you can also multiply the first vector by a complex number, the second vector by a complex number, and that gives you some nu. I won't call it c, I'll call it c prime. So what you can do with a vector space is you can take any pair of vectors, multiply both of them by arbitrary complex numbers, add them, and you get a new vector.
So the notion of addition is well defined. Of course, for ordinary pointers, the notion of addition is also defined, and it's defined by the operation of drawing the parallelogram. Ordinary vectors satisfy both of these rules, that you can multiply a vector by any real number.
I'm sorry, not complex numbers. Ordinary pointers, as I said, are vector spaces over the real numbers. And that means that you can multiply any of them by a real number, and you can add any pair of pointers to get a new pointer.
So ordinary pointers in space are a special case of a vector space over a vector space. over the real numbers. Let me give you a couple of other examples, in particular vector spaces over the complex numbers. This was the reason Hilbert originally invented the notion of Hilbert space to describe functions. Functions of x, let's say, functions of one variable.
For simplicity, that's the simplest example. Take the class of functions of one variable, x. Let's take x to be the variable.
But complex functions of one variable. Functions psi of x, which are the sum of two terms. Let's call it a real part, psi real of x, where psi real of x always takes on real values, plus i.
i being the square root of minus 1 times the imaginary part of psi. Every complex number is the sum of a real part plus an imaginary part. And every complex function, now this is a function. of only one variable.
It's not a function of a complex variable. It's a function of an ordinary variable, but the function itself can be complex. All right? So this means that for every point x, there is a complex number psi of x, or equivalently two real functions.
Now, let's take any complex function. Let's take any complex function like this. We can multiply it by a complex number. Any complex number defines a new function.
You can multiply the complex number psi plus psi real plus psi imaginary by a complex number alpha. In other words, complex numbers can be multiplied, and therefore, you can take any complex number and multiply it by any function. So first of all, functions of x satisfy the first rule.
The collection of functions, the collection of functions of x, the collection of complex functions of x form a vector space. That's the assertion. We're going to check that. First of all, you can multiply any function by a complex number, and you get a new function. So it satisfies rule number one.
What about rule number two? You can take any pair of functions and add them together, and you get a new function. So if you have two functions, psi and phi, let's call them, That's another function, a perfectly good one. So if psi and phi are complex functions, their sum is a complex function.
That's enough to tell you that complex functions are a vector space over the complex numbers. Real functions are a vector space over the real numbers, but they're not a vector space over the complex numbers. Why not?
Because you can't multiply a real function by a complex number and get back a real function. You'll get back a complex function. You won't get back a real function.
So complex functions form a complex vector space. And that actually is the vector space that Hilbert was first interested in. That's an example of a Hilbert space. It's an important example in quantum mechanics, maybe the most important example.
But for now, I'm just illustrating the abstract mathematical definitions. Complex functions of x form a vector space over the complex numbers. Let me give you another one. It's very similar.
Let's. Let's invent a thing called a column vector. A column vector is just a collection of numbers, and we can decide how many. We fix the number of numbers.
One, two, three, any number we like. Let's fix it. And we simply arrange them in a column.
Let's call those numbers, let's give them, let's call them A1, A2, A3, and A4. This would be a vector space of dimension 4. We could also think of vector spaces of dimension 2, A1, and A2. There's pairs of numbers, pairs of complex numbers. Here's quadruples of complex numbers.
We could make quintuples of complex numbers, sextuples of complex numbers, and we could even imagine infinite columns of complex numbers, or just... Just one complex number, just one complex number, one unit long, a1 of x. All of these, each one separately, not all together, but each one separately, four dimensions or two dimensions or three dimensions. Each of these form the collection of such objects, form a complex vector space, vector space over the complex numbers.
So let's take this particular complex vector space here. a1, a2, a3, a4, where the a's are arbitrary complex numbers. If these are arbitrary complex numbers and we display them in a column like this, then I assert that there are rules which will allow this to be a vector space.
So what are the rules? I'm going to invent now a rule of addition. Notice so far there's been no rule of multiplication of vectors. Only addition of vectors.
So the rule for adding two vectors, here's two vectors. One I'll call A, the other one I'll call B. B1, B2, B3, B4. Just add the entries. This is the definition of adding of.
column vectors, a1 plus b1, a2 plus b2, a3 plus b3, a4 plus b4. Obviously with this definition, if you take any two vectors and you add them, you get another vector. The a's and b's are arbitrary complex numbers, but each specification of four complex numbers defines a vector. There are two vectors, one labeled a, one labeled b. Here's a third one that you can make out of it.
All right, so I started, for some reason, I started with the addition rule here. But it's also true that we can invent the idea of multiplication by a complex number. If we want to take a vector, let's call it a1, a2, a3, and a4.
and multiply it by the complex number alpha, we just do that by multiplying every entry by alpha. Alpha A1, alpha A2, alpha A3, and alpha A4. We multiply every entry by the same number. That defines multiplication by a complex number. And all I've done is multiply complex numbers here.
Obviously, given any vector and any complex number, I can make another vector by multiplication. Not multiplication of vectors, but multiplication of numbers by vectors. And given any two vectors, you can add them. So the collection of objects arranged in a column this way, also form a vector space.
Now, if I were to take the collection that consists of vectors of length 2, length 3, length 4, and combine them all together, that's not a vector space, because I haven't given you... any rule for adding a vector with length 2 to a vector of length 3. I haven't told you how to do that. And I don't see any obvious rule for that, so it's, with this set of rules...
The collection of two-dimensional vectors, three, four, the whole shebang does not form a vector space. But for a given length, fix the length, it forms a vector space. And the terminology is that it forms a vector space in this. case of dimension four.
Now of course this has an analog for ordinary pointers. For pointers we can just think of these as the components of the point. along the different axes. Pick some axes, x, y, and z, and pointers have components. So we can also specify an ordinary pointer by specifying a collection of components, a collection of components along different axes, and that also defines a pointer.
So it's natural to call these numbers the components. Thanks for watching! of a vector, and we'll come back to the uniqueness of components.
You can, if you like, think of such a column as a function of the index variable here. Well, that's too abstract. Let's drop that for the moment.
We'll come back to it. It can be thought of as a function, but we don't need to. Now, of course, I've told you nothing about in what possible sense, what possible sense can it make to identify vectors with states of systems.
We're not going to do that tonight. Tonight we're just going to talk about the abstract notion of a vector. Tonight and a bit of next week, we'll talk about the abstract notion of a vector. And then we'll talk about how vector spaces define the states of a system. And then go on and work out some examples of quantum mechanical systems described by such vector spaces.
At the moment, this should be sort of mumbo jumbo, right? Hmm? What success? In making mumbo-jumbo. No, I think I've given you rather accurate definitions that you can understand of what vectors are, and how to manipulate them, how to how to manipulate them, how to do the abstract operation.
with them. These abstract operations are in many ways analogous to the abstract operations that you do with set theoretic logic, with Boolean logic. These abstract, these abstractions...
for how you manipulate vector spaces are the generalizations, if you like, of how Boolean set theory allows you to combine concepts together. And, or, all these ideas have some meaning in set theory. What's and?
And is you take two sets and you combine them and make the union of them. Or you take the intersection. Or is it the other way? It's the other way, isn't it?
It's the other way, sorry. And is the intersection. intersection or is the union of two sets.
The analogous logic in quantum mechanics doesn't operate on sets, it acts on vector spaces. So it's best that we get rid, that we go through vector spaces once and for all. It's the basic underlying mathematics and once we get familiar with it, it won't seem so much mumbo-jumbo, but at the moment we're doing abstract mathematics.
Any other questions? Okay, so let's continue with the abstractions. Now...
There's a notion for every such vector space. Well, let's leave this here. There's a notion of a dual vector space.
The dual vector space is in one-to-one correspondence with the original vector space. Well, let's go back a step. Let's just talk about complex numbers.
For complex numbers, Incidentally, complex numbers are themselves a vector space. You can multiply complex numbers by a complex number. You can add complex numbers. They're just the case of one dimension. One-dimensional vector spaces...
over the complex numbers are just the complex numbers. Now, there's an operation that you can do on complex numbers which is called complex conjugation. It's a very important operation. And all it is, if you think about the Cartesian plane, if you think about the complex plane...
the real part of a number being on the horizontal axis and the imaginary part being on the vertical axis, then the number then is described By a real part, the real part is the horizontal component and the imaginary part is the vertical component. We would just write that this number, usually called z, is equal to x plus iy. The complex conjugate number is just a number reflected About the horizontal.
If this is a number called z, x plus iy, then x minus iy is called z star. x minus i y is the complex conjugate of x plus i y, or z star is the complex conjugate of z. Let's manipulate something.
Let's consider the product of z times z star. z times z star, what is it? It's equal to x plus iy times x minus iy. That's x squared. Now we get ixy and then we get minus ixy from the cross terms.
They cancel. And then plus y squared. Why did I write plus y squared?
Because i times minus i is plus 1. i times... times i is minus 1. So i times minus i is plus 1. Z star z is x squared plus y squared. What is x squared plus y squared? X squared plus y squared is the square of the length of this hypotenuse here.
So z star z is the square of the length, or the square of the magnitude of the number, or its distance from the origin. So complex conjugation is a convenient operation. Among other things, the complex conjugate times a number is the square of the magnitude of the number.
And if you're not, most of you of course are quite familiar with complex conjugation, I want you to think of it as a very very basic operation, but more than that I want you to think of it as a mapping of the numbers back into the numbers. Given any number, any complex number, you can map it to its complex conjugate. What about vector spaces?
I think we should probably quit now. I think it's time to quit now. Next time, we'll go through a little more about vector spaces, vector spaces, operators, Hermitian operators, and eigenvalues. And then we'll begin applying it to quantum mechanical problems. The preceding program was brought to you by Stanford on iTunes U.
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