Good morning, everyone. So welcome to the quantum theory course. So my name is Fernando and I will be your lecturer over the next, hopefully if are of the electors over the next 16 letters. So this course will be about quantum theory, and I will start by telling you three things three important things about quantum theory or quantum mechanics, as they call it in. In other places. So the first thing about quantum mechanics is that this theory is the most. Fundamental. The audio of physics.
An all modern theories. Actually, all the theories over the last 50, 60, 70 years are actually build upon quantum mechanics. Any theory to be consistent? It has to agree with quantum mechanics. The second point, which is also very important, is that quantum mechanics, at least for us, will be very counterintuitive.
As we will see, the need for quantum mechanics arose because we have understood that all the theories, all the fit, all the laws that we knew from classical mechanics actually fail when we are talking about a very small scale. And because of that, people needed to develop a new theory. But because our intuition comes from classical mechanics, all of you know what happens if I throw this pen? I do this and the true false. This won't be true in quantum mechanics anymore.
So quantum mechanics is very counterintuitive because it doesn't agree with our intuition, which is based on classical physics. And this third point, which is so, so very important, especially for you as mathematicians, is that we will not give. A first principle celebration. And the reason for that is not because I am lazy. I am a bit lazy. But that's not the reason why I will not give up first principles derivation.
But the reason is that actually no one can. So quantum mechanics is just based on experiments. We do some experiments. We see that nature behaves an atomic scale in a crazy way, and we try to build a mathematical theory that agrees with these observations. Then we try to do predictions. These predictions over the last one billion experiments happen actually to be true. So we believe in quantum mechanics. But then in no way in which I am going to derive quantum mechanics for you.
OK. So instead, we will describe briefly the experiments that have led to quantum mechanics. And then we will try to present the formalism of quantum mechanics, and we will see what the implications of quantum mechanics are. Is that OK? Now as for material for this course? This course is based on three things. So for you, first, you have the lecture notes that you can see in in the Mathematical Institute web page. Then we also have the letters. But by Feynman.
So they are called the Feynman lectures. They had beautiful books, and the volume three is the one that deals with quantum mechanics. And then we have another classic book, which is a book by Lando. And, Richard, now this book is a book that you should not read unless you already know quantum mechanics. And even if you know quantum mechanics, it will confuse you. And this is still the best book ever written, but I have to tell you so this is the volume three and a disclaimer.
Read it. At your own risk. So your. Is that OK? And after my lectures, every time I will stay outside for 15 minutes, 10, 15 minutes, you can ask me as many questions as you one philosophy to stop me in the lecture, right? If something I am saying is not very clear for you, then you just stop me. Tell me, Fernando, why you start? Why is this? Is that OK? You can also drop me an email. So feel free to stop me at any time. Maybe a bit too early. But do you have any questions now? Fantastic.
So as I was just mentioning the motivation for quantum mechanics. It's basically the failure. Of classical physics. So basically, all the physics that you have learnt so far. At the atomic scale. By the way, it's people is the font large enough for people in the book? Yeah. Fantastic. Then what we will start to do today will be we will start by reviewing two of the basic concepts of classical mechanics, and I will just remind you of a few things that you have already learnt.
And then over the next few lectures, we will shatter all these concepts that we have learnt so far. But first, I want to just we are on the same page. We will learn. We will remind about blowing particles and waves and equations that go burn them. OK, so all these things are things that you have seen. But let me remind you a few things so that we we can build quantum mechanics and we can understand experiments why they are so crazy.
So first. We have the concept of point particle, a point particle is a particle the size of a point which at a given time is in a specific position. OK. So it's an idealised object. Of cirrhosis. Right. And basically, we say that at any time, the point particle is at some location. Ah, which depends on the time. OK, then we know. That the position. Is governed by the second law of Newton, so it satisfies.
The following a question for Constance Bass. So let's assume that the particle has constant mass. You know that the mass. Times the acceleration of the particle, which is the second derivative of AH, with respect to T. Is equal to the force acting on the particle. OK. In this course, we will consider what this call a conservative force. That is the force. As a function of air, our study will be the gradient of some V and these V, which is a function of our.
It's called the potential. And furthermore, in this in this course, we will consider the case of static potential. And this simply means that B of R is independent of the of the 20. OK, if we have this situation a well known result that you have derived in in dynamics or even in high school is that the total energy of the particle is actually conserved. And by definition, the total energy of the particle is the sum. Of the kinetic energy. Plus the potential energy or the potential.
So R is a better D, R D D is a better to buy this, we just mean the dot product of the R D D with itself, OK? Or the non square of the vector, the R D D on here. This piece is called the kinetic energy, and this one is called the potential energy. OK. You have seen all this right now, the the computation. That shows that the energy is actually conserve. It's very simple, but these are pretty instructive. So you can take the 80. And just from the definition, this is I'm.
They are the de. The second that you are you. They are the discu. And in the second piece, we have simply DVD now. We say that we thus independently what we really mean, so we depends on air. But of course, the are for the particle depends on the right, so it depends on time through ah and by using the chain rule. We can write this down us. M. The RTT. But these are. The quarter here we are using the general. Plus. The radiant of Be. They are dirty.
But then by definition. So this is a vector, and we take the product of this vector with the oddity, but we remember that by definition of potential, the gradient of the potential is minus the force. So this is equal to the R the T dot. And the second derivative of our respect to despair, minus the force. But our friend, Isaac Newton, tells us that this is actually policy.
So the vector product with seed oil. So the energy's concern is that, OK, now then there is another thing, another two little things. So we have also learnt that the momentum of the particle. That is what we the note that what we usually the note by be. Sequel to M. They are the deep and in terms of this momentum, the kinetic energy of the particle. Can be written. As P-Square. Over 2m, right? And finally, let me tell you another name. Remember that if there is no force over a particle?
Then we call this a free particle. Is that OK? Fantastic. Is everyone happy with this, right? Great, and people was happy with all this too. OK, so they were, you know, in the late eighteen hundreds. They were quite happy with all this. Now let me tell you something about this. So the first point? The first a small note is that as we will see, all this works only. For massive particles. Which do not move very fast.
As we will see, it happens actually that the nature that are massless particles and there are also particles that move at the speed very close to the speed of light. And for these particles, these kind of formulas don't actually work. OK? You can already see a problem here of what do you do with this? If the particle is much less, we will come back to that later. The second note and this more important for what we are going to to say next not to.
This is actually important, is that the point particle? Moves in other terminals stick way. In other words, if you call the particle, you know, the location and the velocity of the particle, at some point you could just take mathematical or whatever. If you know the force, you could compute what the trajectory of the particle is. OK, so if I am here and someone pushed me from my left, you know immediately that they will move here and I am not.
I am far from being a point particle. But if I was a point particle, you could use the second law of Newton to see which acceleration I will get. OK. And and then we say that classical physics, classical mechanics is actually a completely deterministic, then the another. The other important point is that E can take. Any real value? OK, so if I need my car, my car is still the kinetic energy of my car is cedo.
I start accelerating my car. It requires kinetic energy, but the kinetic energy grows in a continuous way. OK. And they can tune the velocity, choose the velocity in such a way that their energy takes any real value. OK with the correct units so he can take any real value. And it's also something continuous. Is that OK? Is everybody happy with that? So this is all what I am going to say about the point particle. So basically, we have the concept of energy.
We have the fact that it follows a deterministic law that is the second law of Newton and also its energy can take any real value. And the kinetic energy, for instance, is always positive or negative. And and all these things are known things. Is that OK? The second concept, which I want to describe today, is the concept of waves. So waves. Describe, for instance, the propagation of light or sound. Waves are described, but we but by what we call the classical wave equation.
And this is just one over every square. The second derivative of fi with respect to Discu. Is equal to the Le Plus, Jeune. Acting on fire and in this accusation, you have seen this with bear, with the strings in full here. For instance, this we hear. It's called is the speed of the wave, OK? So you have seen this in Futura in partial differential equations that that course. Do you remember what the solution to these sort of equations are is? What are the solutions? Can you tell me?
You are being recorded, so they will think you don't know what the wave equation is, so the solution is like science and cosine. Right? And it is very nice to to. So science and science. And we can repackage them. Into the following suffi of our A. is equal to some amplitude. Thanks to the exponential of I will tell what all these things are, OK? Ah, minus omega t. Now this type of solutions is called a plane wave.
Notice that if you take real an imaginary part of this plane waves, you go back to the sign Sancho signs that you are used to this, a in front is called the amplitude. Unfettered. And in principle, maybe complex. Then this victor came here. It's called the weight vector. And basically, it describes the direction. In which the waste moves. So this wave moves propagates in the direction of this vector. Then we have this omega here. And this coming out here is called the frequency.
OK, now what we do. We take the sunset and we plug it into the wave equation. We just do that. So notice that when you take two videotapes with respect to time, you will bring down and omega square. And we have the eye as well. So on the left hand side, we get minus omega square over this square and on the right hand side, we take the Le Plus jeune each one. When we take the the gradient, we have K. And if we take it twice, we have.
OK. A squat. OK, so this give us a relation between the velocity, the speed, subtly the speed. The frequency and the norm of this. Wait, no. OK. OK. Not this in particular, that the frequency for fixed wave number is proportional to the to the velocity. Then we have two other things of notation. So sometimes it is important. People will talk about. Instead of sunny, so this one sometimes is called the angular frequency omega.
And you can also talk about the way frequency and the way frequency eater is omega over two pi. And we have also the wavelength and the wavelength lambda is two pi over the norm. OK. So for instance, on these two things, you can see that the velocity you can write it as the wave frequency times the wavelength. OK? Because the two pies cancel each other, the two two pies cancel each other.
Is that OK? Any questions about this? So this is like a very brief review of of things that you have seen before, but the but still they are very important and very much like before we have that the the dynamics that controls waves is a still deterministic. So it so happens that actually every solution to the wave equation can be written as a linear combination of this playing waves.
So these playing waves are a basis of solutions. And once you have done that, you know the time dependence of your wave. And in addition, something that I have not explained. But again, the energy is continuous in this model. Is that OK? Great. So this was the view, so I need it, sorry, I need to worry you with this. But this was the situation of physics. About 100 years ago, and people with these two laws, they have either waves, they have particles and waves satisfy the wave equation.
Particles satisfy the second law of Newton, and everything was fine. But then there were three experiments, and these three experiments changed utterly completely our vision of physics at the atomic level. So now we will describe three experiments. That changed the world. First experiment in the first experiment is something that you can try at home. Well, you can do the classical version at home. So imagine that you have a bowl of cereal, right?
This is milk in the milk. You have sun cereal, but you shouldn't use vita weeks. You should use these hoops, right? Discapacidad. And then what you do, maybe with the straw or something you spit milk on the bowl of cereal. OK, that's one thing. Your mum will love this, but you can do this so you start spitting milk. And if you spit milk really, really fast and strong. Some of the cereals would just be emitted from this right.
And it's actually something that is true that if you do it really fast and super strongly admit that cereal will go really far. OK, fantastic. So now we are going to do a slightly more fancy version of that experiment, but basically it's the same. So we will take a metal. Plate, so this is metal, and this metal has some electrons on it. In minus a minus a minus your minus in minus. And then we get the laser. OK. And with our laser, we shoot this metal plate. So here you have your laser.
So this is light and this is light of. Frequency Omega. OK. So you buy your laser. You adjust to have frequency omega. And then you just shoot your plate, your metal plate that has a lot of electrons on it. And what will happen is that exactly as the serial over there, some electrons will be emitted. This experiment is called the photoelectric effect. And the question we want to answer in this in this experiment is what is the kinetic energy?
Off the. Emitted electrons. OK. So basically, if we go to to our to the experiments, we can do with our laser. Basically, you want to spit milk and then you are asking how far the krill reach us. OK, so here the question to ask is we have our metal with electrons. We shoot it with a laser with light of angular frequency onaga. And we ask, what is the kinetic energy? Some electrons will be emitted? And we ask, what is the kinetic energy of the emitted electrons?
OK, great. So far, so good it happens what people discover. What's actually something quite surprising and what they discover was that the energy, the kinetic energy of the electrons was equal to Sun Energy minus some energy is acito, plus some constant times omega the frequency of the light that you are shooting. So these. It's a constant. So this depends on the metal. And this one here. It's another constant you have to read this constant US page bar.
It's also divided to pay, but no one more, no one any more uses. So we always use a bar. And this constant is actually very fundamental in quantum mechanics. In quantum mechanics, we will see it over and over again, and it has some value. It's not very important, but it's important that these quite the small cell is one point zero five 10 to the minus 34 joules times second and the the surprising thing that that people have found. It was the following, so first. If the frequency of the laser.
It's a smaller than CEDAW, divided by a bar, then no electrons. Admit it does, OK, I mean, it's quite cool. What we are saying is that if that formula formulates through the kinetic energy of electrons cannot be negative in the quantum theory, crazy, but not that crazy. So it's good that that at least we got this, but this is the result of an experiment and people were happy with this. But more importantly. The kinetic energy.
S independent. Of the intensity of the laser, and this is completely crazy. We are saying that if we have a laser that is 10 times more powerful, the kinetic energy of the electrons will still be the same. The only thing that happens is that more electrons are emitted, but each electron has actually the same kinetic energy minus e C2 plus omega. Just to see on the left hand side, it means that if you spit milk, no matter how strong it is, this serious.
The hoops always go the same distance. Faraway, OK, so either they are not emitted or they are always a metre to the same distance. OK. And actually, if you if you have a look at the classical theory of Maxwell, that was the accepted theory of of light and electromagnetism. So. Maxwell cannot explain these observations. Can I see these observations then came along? Einstein, it's a nice thing, said. What if light? Of angular frequency. Omega. It's made up. Of Tiny Quanta.
Of energy. E. H. Homemaker. And if this is true. Right. We will see in a second, how dare we explain the results of that experiment? But it turns out. So this would be. What we call photons. And photons. Are massless particles. That move at the speed of light. Speak of such. I just said, Ace, one of the most important things that move at the speed of light. Then the explanation so how this explains our results is because the electrons, so the electrons, the minus absorbed. One for them. Right.
Then they use a CDO to overcome. The binding energy. Of the metal. So a metal is like a country that requires a visa to get out. So you have to pay a little bit if you want to get out of a metal. And this is the binding energy you. And then they started with this energy, you are omega. Then they use Israel to overcome the binding energy of the metal. And finally, they remain. They get away. With the remaining energy. And Einstein got the Nobel prise for this explanation.
And there is a lot of crazy things about this explanation. Notice that first, the first was the crazy idea that actually light, which was always believed to be a wave and that it is actually made up of tiny particles. OK, but also these particles are very special because they are particles that they travel at the speed of light.
As you will learn next year in general relativity, if you take that option, you will see that the only particles that can travel at the speed of light are mass less particles, and also notice that the energy of these particles is proportional to the frequency or to the velocity. Well, usually for massive particles, the kinetic energy is proportional to the square of the velocity. That's not true for for these photons.
OK, so this was the first experiment, and it was important because it is showing us that actually the sometimes light can behave as a particle. And the example was the example of the photoelectric, photoelectric effect. Let me describe now the second experiment. So the second experiment experiment to. It's about. Mission. And absorption. Spectra of atoms. Atoms have these very curious feature, and it's the fact that they emit. Observed. Light. With very particular frequencies.
In other words, if you have a gas of a given atom hydrogen, helium and you throw light to it, most of the frequencies will just go through. But so specific frequencies will be absorbed by the atom, and these are the same frequencies that the gas will emit as well. OK. And of course, people was very surprised by this fact, but it is a bit harder to make this something quantitative. But then they realise that for the hydrogen atom, which is the simplest atom that one can consider the frequencies.
Satisfied. A beautiful law, and they depend on two natural numbers and one and two. And this was equal to a constant two pi r zero times the speed of light. One over and one square minus one over and two square. So here. And one is a smaller than and two the size, so the frequencies are always positive. This is the speed of light. See, and this is what it is. So no, I don't care too much about numbers a metre to the minus one. And this is called. The Wrightbus.
Constant, is someone in the street ask you what the Wrightbus constant is, you know? So it's this constant here. Imagine how hard it is to come up with such a formula, right? I mean, you are an experimentalist and you have numbers, right? You have a lot of numbers and then you just try a lot of formulas because there is a lot of trial and error and error. And at the end of the day, you find out that all the numbers you found follow this pattern.
OK. It's quite cool, actually. And then so this was actually discover long ago saw in 1888. But then this is not OK. It's a result. But now we can think, what is the meaning of Experiment one together with the experiment to write? We know that light is made. Apple photons. Right. So we can imagine that the atom of hydrogen absorbs these photons. OK, but these photons don't have random energies. They have some energy that we we know. And what that means. So.
Experiment one. Plus the explanation by Einstein. It implies that the photons. Admitted that absorbed by this hydrogen atom have energy. H Bar Omega and one and two, right? Because these are the frequencies that we are observing. And this is consistent with the result of the experiment provided. If the possible energies. Of the hydrogen atom. Are. In. Equals minus two PI. R zero, c h bar over and square.
And then what happens if we assume that the energy levels of the hydrogen atom can only be these? What happens is that when the hydrogen atoms jumps from one of these levels to another level, it emits an electron with the difference OK? Or if it absorbs an electron of the rival photon study a photon. If it absorbs a photon of the right frequency, then it will emit a photon of that of that frequency over there. Is that OK? This was something absolutely crazy.
So people couldn't believe that, like all the examples in physics that we have known so far, the energy is something continuous, right? You can have a seed of 0.02 0.00 two to two. They whatever real number, you can have that energy, the hydrogen atom cannot. And the energies of the hydrogen atom are actually given by the embers of perfect squares. How cool is that? So this was absolutely crazy.
Uncle, at the same time and one of the biggest successes of what will be our aim in this course is to derive that formula. We will write down the axioms, the principles of quantum mechanics, and we will derive the formula and we will be all super happy about it. I will be super happy. But so this was also one of the biggest successes of quantum mechanics to derive this formula. But basically, this formula tells us that, OK, things are very different to how we thought they were.
Now that is Experiment three that will. I will explain at the beginning of next lecture, which is so, so very crazy. So we have seen in Experiment one that sometimes light behaves as it was. If it was particle a particle, the photon. In this case, we will actually see an experiment in which it is clear that some particles also behave like waves and things that you thought were particles are actually waves.
And that tells us that actually, we need to reconsider all the things that we have learnt in classical mechanics. Thank you.