1 00:00:12,080 --> 00:00:18,700 Good morning, everyone. So welcome to the quantum theory course. 2 00:00:18,700 --> 00:00:26,890 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. 3 00:00:26,890 --> 00:00:30,970 So this course will be about quantum theory, 4 00:00:30,970 --> 00:00:40,300 and I will start by telling you three things three important things about quantum theory or quantum mechanics, as they call it in. 5 00:00:40,300 --> 00:00:52,550 In other places. So the first thing about quantum mechanics is that this theory is the most. 6 00:00:52,550 --> 00:01:09,000 Fundamental. The audio of physics. 7 00:01:09,000 --> 00:01:19,800 An all modern theories. Actually, all the theories over the last 50, 60, 70 years are actually build upon quantum mechanics. 8 00:01:19,800 --> 00:01:25,890 Any theory to be consistent? It has to agree with quantum mechanics. 9 00:01:25,890 --> 00:01:46,970 The second point, which is also very important, is that quantum mechanics, at least for us, will be very counterintuitive. 10 00:01:46,970 --> 00:01:55,220 As we will see, the need for quantum mechanics arose because we have understood that all the theories, all the fit, 11 00:01:55,220 --> 00:02:04,760 all the laws that we knew from classical mechanics actually fail when we are talking about a very small scale. 12 00:02:04,760 --> 00:02:08,420 And because of that, people needed to develop a new theory. 13 00:02:08,420 --> 00:02:15,800 But because our intuition comes from classical mechanics, all of you know what happens if I throw this pen? 14 00:02:15,800 --> 00:02:20,150 I do this and the true false. This won't be true in quantum mechanics anymore. 15 00:02:20,150 --> 00:02:28,100 So quantum mechanics is very counterintuitive because it doesn't agree with our intuition, which is based on classical physics. 16 00:02:28,100 --> 00:02:43,800 And this third point, which is so, so very important, especially for you as mathematicians, is that we will not give. 17 00:02:43,800 --> 00:02:57,930 A first principle celebration. And the reason for that is not because I am lazy. 18 00:02:57,930 --> 00:03:03,060 I am a bit lazy. But that's not the reason why I will not give up first principles derivation. 19 00:03:03,060 --> 00:03:11,310 But the reason is that actually no one can. So quantum mechanics is just based on experiments. 20 00:03:11,310 --> 00:03:13,440 We do some experiments. 21 00:03:13,440 --> 00:03:23,820 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. 22 00:03:23,820 --> 00:03:30,960 Then we try to do predictions. These predictions over the last one billion experiments happen actually to be true. 23 00:03:30,960 --> 00:03:37,620 So we believe in quantum mechanics. But then in no way in which I am going to derive quantum mechanics for you. 24 00:03:37,620 --> 00:03:45,810 OK. So instead, we will describe briefly the experiments that have led to quantum mechanics. 25 00:03:45,810 --> 00:03:53,700 And then we will try to present the formalism of quantum mechanics, and we will see what the implications of quantum mechanics are. 26 00:03:53,700 --> 00:04:00,760 Is that OK? Now as for material for this course? 27 00:04:00,760 --> 00:04:04,570 This course is based on three things. 28 00:04:04,570 --> 00:04:17,580 So for you, first, you have the lecture notes that you can see in in the Mathematical Institute web page. 29 00:04:17,580 --> 00:04:23,430 Then we also have the letters. But by Feynman. 30 00:04:23,430 --> 00:04:41,640 So they are called the Feynman lectures. They had beautiful books, and the volume three is the one that deals with quantum mechanics. 31 00:04:41,640 --> 00:04:50,500 And then we have another classic book, which is a book by Lando. 32 00:04:50,500 --> 00:05:04,450 And, Richard, now this book is a book that you should not read unless you already know quantum mechanics. 33 00:05:04,450 --> 00:05:07,540 And even if you know quantum mechanics, it will confuse you. 34 00:05:07,540 --> 00:05:17,260 And this is still the best book ever written, but I have to tell you so this is the volume three and a disclaimer. 35 00:05:17,260 --> 00:05:27,270 Read it. At your own risk. 36 00:05:27,270 --> 00:05:31,570 So your. Is that OK? 37 00:05:31,570 --> 00:05:37,720 And after my lectures, every time I will stay outside for 15 minutes, 10, 15 minutes, 38 00:05:37,720 --> 00:05:42,940 you can ask me as many questions as you one philosophy to stop me in the lecture, right? 39 00:05:42,940 --> 00:05:47,320 If something I am saying is not very clear for you, then you just stop me. 40 00:05:47,320 --> 00:05:51,220 Tell me, Fernando, why you start? Why is this? Is that OK? 41 00:05:51,220 --> 00:05:55,660 You can also drop me an email. So feel free to stop me at any time. 42 00:05:55,660 --> 00:06:00,540 Maybe a bit too early. But do you have any questions now? Fantastic. 43 00:06:00,540 --> 00:06:12,810 So as I was just mentioning the motivation for quantum mechanics. 44 00:06:12,810 --> 00:06:23,870 It's basically the failure. Of classical physics. 45 00:06:23,870 --> 00:06:45,130 So basically, all the physics that you have learnt so far. At the atomic scale. 46 00:06:45,130 --> 00:06:49,630 By the way, it's people is the font large enough for people in the book? 47 00:06:49,630 --> 00:07:01,900 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, 48 00:07:01,900 --> 00:07:07,060 and I will just remind you of a few things that you have already learnt. 49 00:07:07,060 --> 00:07:14,800 And then over the next few lectures, we will shatter all these concepts that we have learnt so far. 50 00:07:14,800 --> 00:07:18,790 But first, I want to just we are on the same page. 51 00:07:18,790 --> 00:07:37,030 We will learn. We will remind about blowing particles and waves and equations that go burn them. 52 00:07:37,030 --> 00:07:40,960 OK, so all these things are things that you have seen. 53 00:07:40,960 --> 00:07:53,410 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. 54 00:07:53,410 --> 00:08:14,860 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. 55 00:08:14,860 --> 00:08:26,340 OK. So it's an idealised object. 56 00:08:26,340 --> 00:08:42,240 Of cirrhosis. Right. And basically, we say that at any time, the point particle is at some location. 57 00:08:42,240 --> 00:08:47,040 Ah, which depends on the time. 58 00:08:47,040 --> 00:08:54,230 OK, then we know. 59 00:08:54,230 --> 00:09:13,850 That the position. Is governed by the second law of Newton, so it satisfies. 60 00:09:13,850 --> 00:09:24,500 The following a question for Constance Bass. So let's assume that the particle has constant mass. 61 00:09:24,500 --> 00:09:37,800 You know that the mass. Times the acceleration of the particle, which is the second derivative of AH, with respect to T. 62 00:09:37,800 --> 00:09:43,720 Is equal to the force acting on the particle. 63 00:09:43,720 --> 00:10:29,030 OK. In this course, we will consider what this call a conservative force. 64 00:10:29,030 --> 00:10:49,240 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. 65 00:10:49,240 --> 00:11:21,830 It's called the potential. And furthermore, in this in this course, we will consider the case of static potential. 66 00:11:21,830 --> 00:11:32,660 And this simply means that B of R is independent of the of the 20. 67 00:11:32,660 --> 00:11:43,970 OK, if we have this situation a well known result that you have derived in in dynamics or 68 00:11:43,970 --> 00:11:50,930 even in high school is that the total energy of the particle is actually conserved. 69 00:11:50,930 --> 00:12:01,010 And by definition, the total energy of the particle is the sum. 70 00:12:01,010 --> 00:12:17,230 Of the kinetic energy. Plus the potential energy or the potential. 71 00:12:17,230 --> 00:12:26,620 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? 72 00:12:26,620 --> 00:12:32,680 Or the non square of the vector, the R D D on here. 73 00:12:32,680 --> 00:12:47,020 This piece is called the kinetic energy, and this one is called the potential energy. 74 00:12:47,020 --> 00:12:53,560 OK. You have seen all this right now, the the computation. 75 00:12:53,560 --> 00:13:01,990 That shows that the energy is actually conserve. It's very simple, but these are pretty instructive. 76 00:13:01,990 --> 00:13:18,600 So you can take the 80. And just from the definition, this is I'm. 77 00:13:18,600 --> 00:13:27,920 They are the de. The second that you are you. 78 00:13:27,920 --> 00:13:37,070 They are the discu. And in the second piece, we have simply DVD now. 79 00:13:37,070 --> 00:13:43,970 We say that we thus independently what we really mean, so we depends on air. 80 00:13:43,970 --> 00:13:54,560 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. 81 00:13:54,560 --> 00:14:01,310 We can write this down us. M. 82 00:14:01,310 --> 00:14:08,710 The RTT. But these are. 83 00:14:08,710 --> 00:14:20,090 The quarter here we are using the general. Plus. 84 00:14:20,090 --> 00:14:25,450 The radiant of Be. They are dirty. 85 00:14:25,450 --> 00:14:34,540 But then by definition. So this is a vector, and we take the product of this vector with the oddity, 86 00:14:34,540 --> 00:14:41,140 but we remember that by definition of potential, the gradient of the potential is minus the force. 87 00:14:41,140 --> 00:14:51,000 So this is equal to the R the T dot. 88 00:14:51,000 --> 00:14:58,620 And the second derivative of our respect to despair, minus the force. 89 00:14:58,620 --> 00:15:05,130 But our friend, Isaac Newton, tells us that this is actually policy. 90 00:15:05,130 --> 00:15:15,480 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. 91 00:15:15,480 --> 00:15:27,150 So we have also learnt that the momentum of the particle. 92 00:15:27,150 --> 00:15:35,450 That is what we the note that what we usually the note by be. 93 00:15:35,450 --> 00:15:55,990 Sequel to M. They are the deep and in terms of this momentum, the kinetic energy of the particle. 94 00:15:55,990 --> 00:16:07,170 Can be written. As P-Square. Over 2m, right? 95 00:16:07,170 --> 00:16:12,540 And finally, let me tell you another name. 96 00:16:12,540 --> 00:16:23,540 Remember that if there is no force over a particle? 97 00:16:23,540 --> 00:16:35,600 Then we call this a free particle. 98 00:16:35,600 --> 00:16:40,700 Is that OK? Fantastic. Is everyone happy with this, right? 99 00:16:40,700 --> 00:16:47,720 Great, and people was happy with all this too. OK, so they were, you know, in the late eighteen hundreds. 100 00:16:47,720 --> 00:16:54,200 They were quite happy with all this. Now let me tell you something about this. 101 00:16:54,200 --> 00:17:06,500 So the first point? The first a small note is that as we will see, all this works only. 102 00:17:06,500 --> 00:17:33,410 For massive particles. Which do not move very fast. 103 00:17:33,410 --> 00:17:34,700 As we will see, 104 00:17:34,700 --> 00:17:44,090 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. 105 00:17:44,090 --> 00:17:48,670 And for these particles, these kind of formulas don't actually work. 106 00:17:48,670 --> 00:17:52,640 OK? You can already see a problem here of what do you do with this? 107 00:17:52,640 --> 00:17:57,200 If the particle is much less, we will come back to that later. 108 00:17:57,200 --> 00:18:08,960 The second note and this more important for what we are going to to say next not to. 109 00:18:08,960 --> 00:18:23,520 This is actually important, is that the point particle? 110 00:18:23,520 --> 00:18:41,060 Moves in other terminals stick way. 111 00:18:41,060 --> 00:18:47,660 In other words, if you call the particle, you know, the location and the velocity of the particle, 112 00:18:47,660 --> 00:18:52,400 at some point you could just take mathematical or whatever. 113 00:18:52,400 --> 00:18:57,740 If you know the force, you could compute what the trajectory of the particle is. 114 00:18:57,740 --> 00:19:05,690 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. 115 00:19:05,690 --> 00:19:13,100 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. 116 00:19:13,100 --> 00:19:23,750 OK. And and then we say that classical physics, classical mechanics is actually a completely deterministic, then the another. 117 00:19:23,750 --> 00:19:35,950 The other important point is that E can take. 118 00:19:35,950 --> 00:19:50,380 Any real value? OK, so if I need my car, my car is still the kinetic energy of my car is cedo. 119 00:19:50,380 --> 00:19:58,120 I start accelerating my car. It requires kinetic energy, but the kinetic energy grows in a continuous way. 120 00:19:58,120 --> 00:20:05,320 OK. And they can tune the velocity, choose the velocity in such a way that their energy takes any real value. 121 00:20:05,320 --> 00:20:11,560 OK with the correct units so he can take any real value. 122 00:20:11,560 --> 00:20:18,100 And it's also something continuous. Is that OK? Is everybody happy with that? 123 00:20:18,100 --> 00:20:23,680 So this is all what I am going to say about the point particle. 124 00:20:23,680 --> 00:20:26,740 So basically, we have the concept of energy. 125 00:20:26,740 --> 00:20:37,390 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. 126 00:20:37,390 --> 00:20:44,380 And the kinetic energy, for instance, is always positive or negative. And and all these things are known things. 127 00:20:44,380 --> 00:20:56,180 Is that OK? The second concept, which I want to describe today, is the concept of waves. 128 00:20:56,180 --> 00:21:14,900 So waves. Describe, for instance, the propagation of light or sound. 129 00:21:14,900 --> 00:21:32,670 Waves are described, but we but by what we call the classical wave equation. 130 00:21:32,670 --> 00:21:44,500 And this is just one over every square. The second derivative of fi with respect to Discu. 131 00:21:44,500 --> 00:21:55,910 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. 132 00:21:55,910 --> 00:22:08,730 For instance, this we hear. It's called is the speed of the wave, OK? 133 00:22:08,730 --> 00:22:14,070 So you have seen this in Futura in partial differential equations that that course. 134 00:22:14,070 --> 00:22:19,920 Do you remember what the solution to these sort of equations are is? 135 00:22:19,920 --> 00:22:25,640 What are the solutions? Can you tell me? 136 00:22:25,640 --> 00:22:33,350 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. 137 00:22:33,350 --> 00:22:40,970 Right? And it is very nice to to. 138 00:22:40,970 --> 00:22:56,770 So science and science. And we can repackage them. 139 00:22:56,770 --> 00:23:09,120 Into the following suffi of our A. is equal to some amplitude. 140 00:23:09,120 --> 00:23:17,230 Thanks to the exponential of I will tell what all these things are, OK? 141 00:23:17,230 --> 00:23:31,410 Ah, minus omega t. Now this type of solutions is called a plane wave. 142 00:23:31,410 --> 00:23:36,420 Notice that if you take real an imaginary part of this plane waves, 143 00:23:36,420 --> 00:23:50,730 you go back to the sign Sancho signs that you are used to this, a in front is called the amplitude. 144 00:23:50,730 --> 00:24:03,890 Unfettered. And in principle, maybe complex. 145 00:24:03,890 --> 00:24:14,780 Then this victor came here. It's called the weight vector. 146 00:24:14,780 --> 00:24:21,910 And basically, it describes the direction. 147 00:24:21,910 --> 00:24:30,450 In which the waste moves. So this wave moves propagates in the direction of this vector. 148 00:24:30,450 --> 00:24:41,280 Then we have this omega here. And this coming out here is called the frequency. 149 00:24:41,280 --> 00:24:49,320 OK, now what we do. We take the sunset and we plug it into the wave equation. 150 00:24:49,320 --> 00:24:59,130 We just do that. So notice that when you take two videotapes with respect to time, you will bring down and omega square. 151 00:24:59,130 --> 00:25:01,590 And we have the eye as well. 152 00:25:01,590 --> 00:25:14,910 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. 153 00:25:14,910 --> 00:25:23,520 When we take the the gradient, we have K. And if we take it twice, we have. 154 00:25:23,520 --> 00:25:33,650 OK. A squat. OK, so this give us a relation between the velocity, the speed, subtly the speed. 155 00:25:33,650 --> 00:25:38,100 The frequency and the norm of this. 156 00:25:38,100 --> 00:25:49,380 Wait, no. OK. OK. Not this in particular, that the frequency for fixed wave number is proportional to the to the velocity. 157 00:25:49,380 --> 00:25:55,110 Then we have two other things of notation. 158 00:25:55,110 --> 00:26:02,260 So sometimes it is important. People will talk about. 159 00:26:02,260 --> 00:26:10,210 Instead of sunny, so this one sometimes is called the angular frequency omega. 160 00:26:10,210 --> 00:26:24,580 And you can also talk about the way frequency and the way frequency eater is omega over two pi. 161 00:26:24,580 --> 00:26:39,940 And we have also the wavelength and the wavelength lambda is two pi over the norm. 162 00:26:39,940 --> 00:26:52,090 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. 163 00:26:52,090 --> 00:26:59,380 OK? Because the two pies cancel each other, the two two pies cancel each other. 164 00:26:59,380 --> 00:27:10,610 Is that OK? Any questions about this? So this is like a very brief review of of things that you have seen before, 165 00:27:10,610 --> 00:27:31,310 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. 166 00:27:31,310 --> 00:27:41,750 So it so happens that actually every solution to the wave equation can be written as a linear combination of this playing waves. 167 00:27:41,750 --> 00:27:49,880 So these playing waves are a basis of solutions. And once you have done that, you know the time dependence of your wave. 168 00:27:49,880 --> 00:28:03,890 And in addition, something that I have not explained. But again, the energy is continuous in this model. 169 00:28:03,890 --> 00:28:13,780 Is that OK? Great. So this was the view, so I need it, sorry, I need to worry you with this. 170 00:28:13,780 --> 00:28:18,010 But this was the situation of physics. 171 00:28:18,010 --> 00:28:29,800 About 100 years ago, and people with these two laws, they have either waves, they have particles and waves satisfy the wave equation. 172 00:28:29,800 --> 00:28:35,650 Particles satisfy the second law of Newton, and everything was fine. 173 00:28:35,650 --> 00:28:46,780 But then there were three experiments, and these three experiments changed utterly completely our vision of physics at the atomic level. 174 00:28:46,780 --> 00:29:12,860 So now we will describe three experiments. That changed the world. 175 00:29:12,860 --> 00:29:30,500 First experiment in the first experiment is something that you can try at home. 176 00:29:30,500 --> 00:29:39,530 Well, you can do the classical version at home. So imagine that you have a bowl of cereal, right? 177 00:29:39,530 --> 00:29:46,010 This is milk in the milk. You have sun cereal, but you shouldn't use vita weeks. 178 00:29:46,010 --> 00:29:58,460 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. 179 00:29:58,460 --> 00:30:07,010 OK, that's one thing. Your mum will love this, but you can do this so you start spitting milk. 180 00:30:07,010 --> 00:30:11,570 And if you spit milk really, really fast and strong. 181 00:30:11,570 --> 00:30:17,810 Some of the cereals would just be emitted from this right. 182 00:30:17,810 --> 00:30:28,070 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. 183 00:30:28,070 --> 00:30:38,450 OK, fantastic. So now we are going to do a slightly more fancy version of that experiment, but basically it's the same. 184 00:30:38,450 --> 00:30:53,150 So we will take a metal. Plate, so this is metal, and this metal has some electrons on it. 185 00:30:53,150 --> 00:31:00,440 In minus a minus a minus your minus in minus. 186 00:31:00,440 --> 00:31:07,880 And then we get the laser. OK. And with our laser, we shoot this metal plate. 187 00:31:07,880 --> 00:31:12,230 So here you have your laser. 188 00:31:12,230 --> 00:31:23,550 So this is light and this is light of. 189 00:31:23,550 --> 00:31:29,310 Frequency Omega. 190 00:31:29,310 --> 00:31:36,060 OK. So you buy your laser. You adjust to have frequency omega. 191 00:31:36,060 --> 00:31:43,800 And then you just shoot your plate, your metal plate that has a lot of electrons on it. 192 00:31:43,800 --> 00:32:04,890 And what will happen is that exactly as the serial over there, some electrons will be emitted. 193 00:32:04,890 --> 00:32:23,170 This experiment is called the photoelectric effect. 194 00:32:23,170 --> 00:32:47,400 And the question we want to answer in this in this experiment is what is the kinetic energy? 195 00:32:47,400 --> 00:32:57,730 Off the. Emitted electrons. 196 00:32:57,730 --> 00:33:07,160 OK. So basically, if we go to to our to the experiments, we can do with our laser. 197 00:33:07,160 --> 00:33:13,810 Basically, you want to spit milk and then you are asking how far the krill reach us. 198 00:33:13,810 --> 00:33:20,020 OK, so here the question to ask is we have our metal with electrons. 199 00:33:20,020 --> 00:33:24,400 We shoot it with a laser with light of angular frequency onaga. 200 00:33:24,400 --> 00:33:28,570 And we ask, what is the kinetic energy? Some electrons will be emitted? 201 00:33:28,570 --> 00:33:32,890 And we ask, what is the kinetic energy of the emitted electrons? 202 00:33:32,890 --> 00:33:45,750 OK, great. So far, so good it happens what people discover. 203 00:33:45,750 --> 00:33:56,010 What's actually something quite surprising and what they discover was that the energy, 204 00:33:56,010 --> 00:34:10,620 the kinetic energy of the electrons was equal to Sun Energy minus some energy is acito, 205 00:34:10,620 --> 00:34:20,340 plus some constant times omega the frequency of the light that you are shooting. 206 00:34:20,340 --> 00:34:28,900 So these. It's a constant. 207 00:34:28,900 --> 00:34:45,870 So this depends on the metal. And this one here. 208 00:34:45,870 --> 00:34:54,960 It's another constant you have to read this constant US page bar. 209 00:34:54,960 --> 00:35:01,920 It's also divided to pay, but no one more, no one any more uses. 210 00:35:01,920 --> 00:35:11,500 So we always use a bar. And this constant is actually very fundamental in quantum mechanics. 211 00:35:11,500 --> 00:35:18,370 In quantum mechanics, we will see it over and over again, and it has some value. 212 00:35:18,370 --> 00:35:25,000 It's not very important, but it's important that these quite the small cell is one point zero five 10 to the 213 00:35:25,000 --> 00:35:37,840 minus 34 joules times second and the the surprising thing that that people have found. 214 00:35:37,840 --> 00:35:45,320 It was the following, so first. If the frequency of the laser. 215 00:35:45,320 --> 00:36:00,570 It's a smaller than CEDAW, divided by a bar, then no electrons. 216 00:36:00,570 --> 00:36:06,750 Admit it does, OK, I mean, it's quite cool. 217 00:36:06,750 --> 00:36:14,370 What we are saying is that if that formula formulates through the kinetic energy of electrons cannot be negative in the quantum theory, 218 00:36:14,370 --> 00:36:23,580 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. 219 00:36:23,580 --> 00:36:30,430 But more importantly. The kinetic energy. 220 00:36:30,430 --> 00:36:49,240 S independent. Of the intensity of the laser, and this is completely crazy. 221 00:36:49,240 --> 00:36:58,870 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. 222 00:36:58,870 --> 00:37:10,390 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. 223 00:37:10,390 --> 00:37:20,680 Just to see on the left hand side, it means that if you spit milk, no matter how strong it is, this serious. 224 00:37:20,680 --> 00:37:31,330 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. 225 00:37:31,330 --> 00:37:50,630 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. 226 00:37:50,630 --> 00:38:15,560 So. Maxwell cannot explain these observations. 227 00:38:15,560 --> 00:38:22,270 Can I see these observations then came along? 228 00:38:22,270 --> 00:38:29,570 Einstein, it's a nice thing, said. 229 00:38:29,570 --> 00:38:41,310 What if light? Of angular frequency. 230 00:38:41,310 --> 00:38:53,460 Omega. It's made up. 231 00:38:53,460 --> 00:39:02,390 Of Tiny Quanta. 232 00:39:02,390 --> 00:39:13,700 Of energy. E. H. 233 00:39:13,700 --> 00:39:20,390 Homemaker. And if this is true. 234 00:39:20,390 --> 00:39:27,620 Right. We will see in a second, how dare we explain the results of that experiment? 235 00:39:27,620 --> 00:39:36,840 But it turns out. So this would be. 236 00:39:36,840 --> 00:39:45,540 What we call photons. And photons. 237 00:39:45,540 --> 00:40:02,970 Are massless particles. That move at the speed of light. 238 00:40:02,970 --> 00:40:33,270 Speak of such. I just said, Ace, one of the most important things that move at the speed of light. 239 00:40:33,270 --> 00:40:50,410 Then the explanation so how this explains our results is because the electrons, so the electrons, the minus absorbed. 240 00:40:50,410 --> 00:40:56,720 One for them. Right. 241 00:40:56,720 --> 00:41:10,040 Then they use a CDO to overcome. 242 00:41:10,040 --> 00:41:29,390 The binding energy. Of the metal. 243 00:41:29,390 --> 00:41:33,740 So a metal is like a country that requires a visa to get out. 244 00:41:33,740 --> 00:41:39,530 So you have to pay a little bit if you want to get out of a metal. And this is the binding energy you. 245 00:41:39,530 --> 00:41:44,090 And then they started with this energy, you are omega. 246 00:41:44,090 --> 00:41:49,430 Then they use Israel to overcome the binding energy of the metal. 247 00:41:49,430 --> 00:41:58,970 And finally, they remain. They get away. 248 00:41:58,970 --> 00:42:16,270 With the remaining energy. And Einstein got the Nobel prise for this explanation. 249 00:42:16,270 --> 00:42:20,030 And there is a lot of crazy things about this explanation. 250 00:42:20,030 --> 00:42:25,960 Notice that first, the first was the crazy idea that actually light, 251 00:42:25,960 --> 00:42:33,090 which was always believed to be a wave and that it is actually made up of tiny particles. 252 00:42:33,090 --> 00:42:40,810 OK, but also these particles are very special because they are particles that they travel at the speed of light. 253 00:42:40,810 --> 00:42:47,470 As you will learn next year in general relativity, if you take that option, 254 00:42:47,470 --> 00:42:52,210 you will see that the only particles that can travel at the speed of light are mass less particles, 255 00:42:52,210 --> 00:43:00,070 and also notice that the energy of these particles is proportional to the frequency or to the velocity. 256 00:43:00,070 --> 00:43:06,580 Well, usually for massive particles, the kinetic energy is proportional to the square of the velocity. 257 00:43:06,580 --> 00:43:10,780 That's not true for for these photons. 258 00:43:10,780 --> 00:43:26,320 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. 259 00:43:26,320 --> 00:43:33,550 And the example was the example of the photoelectric, photoelectric effect. 260 00:43:33,550 --> 00:44:02,000 Let me describe now the second experiment. So the second experiment experiment to. 261 00:44:02,000 --> 00:44:16,690 It's about. Mission. And absorption. 262 00:44:16,690 --> 00:44:41,860 Spectra of atoms. Atoms have these very curious feature, and it's the fact that they emit. 263 00:44:41,860 --> 00:44:50,300 Observed. Light. 264 00:44:50,300 --> 00:45:05,330 With very particular frequencies. 265 00:45:05,330 --> 00:45:16,100 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. 266 00:45:16,100 --> 00:45:26,090 But so specific frequencies will be absorbed by the atom, and these are the same frequencies that the gas will emit as well. 267 00:45:26,090 --> 00:45:37,280 OK. And of course, people was very surprised by this fact, but it is a bit harder to make this something quantitative. 268 00:45:37,280 --> 00:45:51,740 But then they realise that for the hydrogen atom, which is the simplest atom that one can consider the frequencies. 269 00:45:51,740 --> 00:46:01,370 Satisfied. A beautiful law, and they depend on two natural numbers and one and two. 270 00:46:01,370 --> 00:46:08,450 And this was equal to a constant two pi r zero times the speed of light. 271 00:46:08,450 --> 00:46:17,150 One over and one square minus one over and two square. 272 00:46:17,150 --> 00:46:26,750 So here. And one is a smaller than and two the size, so the frequencies are always positive. 273 00:46:26,750 --> 00:46:32,040 This is the speed of light. See, and this is what it is. 274 00:46:32,040 --> 00:46:38,030 So no, I don't care too much about numbers a metre to the minus one. 275 00:46:38,030 --> 00:46:45,070 And this is called. The Wrightbus. 276 00:46:45,070 --> 00:46:51,700 Constant, is someone in the street ask you what the Wrightbus constant is, you know? 277 00:46:51,700 --> 00:46:58,420 So it's this constant here. Imagine how hard it is to come up with such a formula, right? 278 00:46:58,420 --> 00:47:02,710 I mean, you are an experimentalist and you have numbers, right? 279 00:47:02,710 --> 00:47:11,560 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. 280 00:47:11,560 --> 00:47:17,710 And at the end of the day, you find out that all the numbers you found follow this pattern. 281 00:47:17,710 --> 00:47:29,440 OK. It's quite cool, actually. And then so this was actually discover long ago saw in 1888. 282 00:47:29,440 --> 00:47:34,180 But then this is not OK. It's a result. 283 00:47:34,180 --> 00:47:44,090 But now we can think, what is the meaning of Experiment one together with the experiment to write? 284 00:47:44,090 --> 00:47:49,280 We know that light is made. Apple photons. 285 00:47:49,280 --> 00:47:56,600 Right. So we can imagine that the atom of hydrogen absorbs these photons. 286 00:47:56,600 --> 00:48:01,280 OK, but these photons don't have random energies. 287 00:48:01,280 --> 00:48:07,940 They have some energy that we we know. 288 00:48:07,940 --> 00:48:12,570 And what that means. So. 289 00:48:12,570 --> 00:48:23,890 Experiment one. Plus the explanation by Einstein. 290 00:48:23,890 --> 00:48:32,200 It implies that the photons. 291 00:48:32,200 --> 00:48:46,060 Admitted that absorbed by this hydrogen atom have energy. 292 00:48:46,060 --> 00:48:50,290 H Bar Omega and one and two, right? 293 00:48:50,290 --> 00:49:03,960 Because these are the frequencies that we are observing. And this is consistent with the result of the experiment provided. 294 00:49:03,960 --> 00:49:29,350 If the possible energies. Of the hydrogen atom. 295 00:49:29,350 --> 00:49:38,380 Are. In. 296 00:49:38,380 --> 00:49:52,070 Equals minus two PI. R zero, c h bar over and square. 297 00:49:52,070 --> 00:50:01,460 And then what happens if we assume that the energy levels of the hydrogen atom can only be these? 298 00:50:01,460 --> 00:50:12,710 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? 299 00:50:12,710 --> 00:50:17,600 Or if it absorbs an electron of the rival photon study a photon. 300 00:50:17,600 --> 00:50:29,900 If it absorbs a photon of the right frequency, then it will emit a photon of that of that frequency over there. 301 00:50:29,900 --> 00:50:35,490 Is that OK? This was something absolutely crazy. 302 00:50:35,490 --> 00:50:45,210 So people couldn't believe that, like all the examples in physics that we have known so far, the energy is something continuous, right? 303 00:50:45,210 --> 00:50:49,920 You can have a seed of 0.02 0.00 two to two. 304 00:50:49,920 --> 00:50:57,050 They whatever real number, you can have that energy, the hydrogen atom cannot. 305 00:50:57,050 --> 00:51:05,900 And the energies of the hydrogen atom are actually given by the embers of perfect squares. 306 00:51:05,900 --> 00:51:13,220 How cool is that? So this was absolutely crazy. 307 00:51:13,220 --> 00:51:25,620 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. 308 00:51:25,620 --> 00:51:35,550 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. 309 00:51:35,550 --> 00:51:44,880 I will be super happy. But so this was also one of the biggest successes of quantum mechanics to derive this formula. 310 00:51:44,880 --> 00:51:52,020 But basically, this formula tells us that, OK, things are very different to how we thought they were. 311 00:51:52,020 --> 00:51:55,470 Now that is Experiment three that will. 312 00:51:55,470 --> 00:52:01,170 I will explain at the beginning of next lecture, which is so, so very crazy. 313 00:52:01,170 --> 00:52:07,920 So we have seen in Experiment one that sometimes light behaves as it was. 314 00:52:07,920 --> 00:52:12,060 If it was particle a particle, the photon. In this case, 315 00:52:12,060 --> 00:52:18,060 we will actually see an experiment in which it is clear that some particles also 316 00:52:18,060 --> 00:52:25,830 behave like waves and things that you thought were particles are actually waves. 317 00:52:25,830 --> 00:52:31,980 And that tells us that actually, we need to reconsider all the things that we have learnt in classical mechanics. 318 00:52:31,980 --> 00:52:53,007 Thank you.