1 00:00:00,070 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,880 Your support will help MIT OpenCourseWare continue 4 00:00:06,880 --> 00:00:10,740 to offer high quality educational resources for free. 5 00:00:10,740 --> 00:00:13,350 To make a donation or view additional materials 6 00:00:13,350 --> 00:00:17,237 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,237 --> 00:00:17,862 at ocw.mit.edu. 8 00:00:23,620 --> 00:00:25,340 PROFESSOR: My first question is, do you 9 00:00:25,340 --> 00:00:29,240 have any questions about the last class, about 10 00:00:29,240 --> 00:00:34,930 the introduction, about the course overview, the syllabus, 11 00:00:34,930 --> 00:00:38,050 any requirements of the course, formalities? 12 00:00:42,340 --> 00:00:44,360 It should be pretty clear. 13 00:00:44,360 --> 00:00:50,055 So today, we start from first principles. 14 00:00:54,300 --> 00:01:03,370 I want to give you a rigorous derivation of what 15 00:01:03,370 --> 00:01:07,590 this course is based, namely, the Hamiltonian 16 00:01:07,590 --> 00:01:12,340 to describe the interaction between atoms and light. 17 00:01:12,340 --> 00:01:15,820 I mentioned to you last week that atomic physics can 18 00:01:15,820 --> 00:01:17,470 be defined about everything that is 19 00:01:17,470 --> 00:01:20,560 interesting about the building blocks of nature, atoms 20 00:01:20,560 --> 00:01:23,040 interacting with light electromagnetic fields, Coulomb 21 00:01:23,040 --> 00:01:26,190 fields, and today, we'll talk about the first principle, 22 00:01:26,190 --> 00:01:26,690 Hamiltonian. 23 00:01:38,010 --> 00:01:40,770 The reference for that, and I'm closely 24 00:01:40,770 --> 00:01:44,070 following that derivation, is the book Atom-Photon 25 00:01:44,070 --> 00:01:48,215 Interaction, the green book by Cohen-Tannoudji, Grynberg, 26 00:01:48,215 --> 00:01:52,100 and others, and it's taken from the appendix. 27 00:01:55,160 --> 00:01:57,360 And if you flip through the appendix, 28 00:01:57,360 --> 00:02:00,675 you will find out that there are about 100 equations. 29 00:02:04,170 --> 00:02:08,039 Today, we'll discuss every single of them. 30 00:02:08,039 --> 00:02:12,020 However, the good news is the result 31 00:02:12,020 --> 00:02:15,720 in the end is simple and intuitive. 32 00:02:22,210 --> 00:02:24,590 In the end, we have what we want, 33 00:02:24,590 --> 00:02:27,870 that the electromagnetic field couples different energy 34 00:02:27,870 --> 00:02:31,160 levels, and then we can play around with quantum case, 35 00:02:31,160 --> 00:02:32,030 with laser cooling. 36 00:02:32,030 --> 00:02:34,190 All we need is a coupling matrix element. 37 00:02:34,190 --> 00:02:37,850 And for most of the course, I will not even 38 00:02:37,850 --> 00:02:41,120 elaborate what this matrix element is about. 39 00:02:41,120 --> 00:02:43,480 But here, we derive it from first principles. 40 00:02:46,220 --> 00:02:49,110 The result we will use most often 41 00:02:49,110 --> 00:03:02,910 is actually the electric dipole approximation, 42 00:03:02,910 --> 00:03:09,310 which I'm sure you have already seen many, many times. 43 00:03:09,310 --> 00:03:15,010 So in that sense, I sometimes have second thoughts. 44 00:03:15,010 --> 00:03:16,950 Should I dedicate a whole lecture 45 00:03:16,950 --> 00:03:19,680 to derive something you know already? 46 00:03:19,680 --> 00:03:23,330 On the other hand, this is sort of the meat of atomic physics, 47 00:03:23,330 --> 00:03:26,440 and I want to go as deep as possible 48 00:03:26,440 --> 00:03:29,140 into it, that whenever you wonder 49 00:03:29,140 --> 00:03:31,750 what form of other interactions exist, 50 00:03:31,750 --> 00:03:33,780 you have a reference to look it up. 51 00:03:33,780 --> 00:03:36,780 And so to some extent, I want to also encourage 52 00:03:36,780 --> 00:03:41,080 you to read more about the fundamental nature of how light 53 00:03:41,080 --> 00:03:42,790 and atoms interact. 54 00:03:42,790 --> 00:03:44,890 I will give you another reference a little bit 55 00:03:44,890 --> 00:03:46,820 later on. 56 00:03:46,820 --> 00:03:50,460 But as a motivation, I want to tell you 57 00:03:50,460 --> 00:03:56,520 what I learned when I prepared this lecture, 58 00:03:56,520 --> 00:03:58,340 and I hope that there is something 59 00:03:58,340 --> 00:04:00,400 that you will learn from this treatment. 60 00:04:10,950 --> 00:04:14,785 One is a rigorous separation of local fields 61 00:04:14,785 --> 00:04:16,045 from radiation fields. 62 00:04:22,310 --> 00:04:26,230 Let me just write it down, rigorous separation 63 00:04:26,230 --> 00:04:45,115 of local Coulomb fields from radiation fields. 64 00:04:48,640 --> 00:04:50,820 So to clearly separate the terms in the Hamiltonian, 65 00:04:50,820 --> 00:04:51,653 let me explain that. 66 00:04:54,990 --> 00:04:57,630 When we introduce electromagnetic fields 67 00:04:57,630 --> 00:05:00,560 into our world, we would say we introduce them 68 00:05:00,560 --> 00:05:02,990 by saying particles have charge. 69 00:05:02,990 --> 00:05:06,770 But some parts of the charged particle physics 70 00:05:06,770 --> 00:05:12,190 is something we want to include in the description of atoms, 71 00:05:12,190 --> 00:05:14,310 and that's the Coulomb field of protons. 72 00:05:14,310 --> 00:05:16,860 The structure of atoms is electromagnetic, 73 00:05:16,860 --> 00:05:20,200 but when we now introduce electromagnetic field 74 00:05:20,200 --> 00:05:22,310 as a new degree of freedom-- photons 75 00:05:22,310 --> 00:05:24,200 can be admitted, absorbed, and such-- 76 00:05:24,200 --> 00:05:27,450 we only want to talk about the electric field which 77 00:05:27,450 --> 00:05:30,630 belongs to the photons and not the electric field which 78 00:05:30,630 --> 00:05:32,230 belongs to the atoms. 79 00:05:32,230 --> 00:05:35,820 And you may wonder, can you distinguish 80 00:05:35,820 --> 00:05:38,930 between two kinds of electric fields? 81 00:05:38,930 --> 00:05:42,420 The answer is yes, and I want to show you how mathematically you 82 00:05:42,420 --> 00:05:46,110 distinguish when you introduce electromagnetic fields 83 00:05:46,110 --> 00:05:50,700 between the fields which stay with the atom, which don't have 84 00:05:50,700 --> 00:05:53,960 an extra degree of freedom, and if you 85 00:05:53,960 --> 00:05:55,815 do canonical quantization, they don't enter 86 00:05:55,815 --> 00:05:59,700 as an extra variable, and the fields which are the photons, 87 00:05:59,700 --> 00:06:02,543 and these are the extra objects we have to consider. 88 00:06:06,170 --> 00:06:11,490 What is also related is when you do second quantization, 89 00:06:11,490 --> 00:06:15,350 if you wanted, I could quantize electromagnetic field for you 90 00:06:15,350 --> 00:06:16,320 in five minutes. 91 00:06:16,320 --> 00:06:18,180 Just say it's harmonic oscillator. 92 00:06:18,180 --> 00:06:19,888 You're familiar with harmonic oscillator, 93 00:06:19,888 --> 00:06:22,390 and let's just do it. 94 00:06:22,390 --> 00:06:25,780 Yes, this is correct, and I will remind you of that a little bit 95 00:06:25,780 --> 00:06:29,660 later, but there is one non-trivial aspect 96 00:06:29,660 --> 00:06:34,460 if you want to do quantization of a field theory, 97 00:06:34,460 --> 00:06:37,040 you have to make sure that before you quantize it, 98 00:06:37,040 --> 00:06:41,050 you know which degrees of freedom are really independent. 99 00:06:41,050 --> 00:06:45,440 So the question is, how many independent fields 100 00:06:45,440 --> 00:06:47,530 does the electromagnetic field have? 101 00:06:47,530 --> 00:06:50,920 The naive answer is three electric field components, 102 00:06:50,920 --> 00:06:54,030 three magnetic field components, that makes six. 103 00:06:54,030 --> 00:06:55,790 But of course they're not independent, 104 00:06:55,790 --> 00:06:57,015 and we will discuss that. 105 00:07:03,200 --> 00:07:07,750 So we'll spend some time in the identification 106 00:07:07,750 --> 00:07:20,390 of truly independent field components, or degrees 107 00:07:20,390 --> 00:07:24,530 of freedom of the electromagnetic field. 108 00:07:28,970 --> 00:07:38,600 So all that is an excursion into classical physics 109 00:07:38,600 --> 00:07:43,830 because it is just reformulating classical electrodynamics 110 00:07:43,830 --> 00:07:45,460 to be ready for quantization. 111 00:07:53,160 --> 00:08:00,860 Actually, all the work is done to discuss and derive 112 00:08:00,860 --> 00:08:02,550 the appropriate classical description. 113 00:08:13,820 --> 00:08:16,850 That means to eliminate all redundant variables, 114 00:08:16,850 --> 00:08:19,970 and then at the end, have variables which, yes, 115 00:08:19,970 --> 00:08:22,440 will look like a harmonic oscillator. 116 00:08:22,440 --> 00:08:26,440 And once we have reduced it to degrees of freedom which 117 00:08:26,440 --> 00:08:33,340 look like a harmonic oscillator, it is very straightforward 118 00:08:33,340 --> 00:08:38,030 to do the field quantization following 119 00:08:38,030 --> 00:08:39,640 the recipe of the harmonic oscillator. 120 00:08:43,260 --> 00:08:49,260 There's one more highlight, if I want to say so, 121 00:08:49,260 --> 00:08:58,830 and this is a truly rigorous derivation 122 00:08:58,830 --> 00:09:00,596 of the electric dipole approximation. 123 00:09:07,490 --> 00:09:17,980 In most quantum mechanics textbooks, 124 00:09:17,980 --> 00:09:21,240 you do a dipole approximation and you wind up 125 00:09:21,240 --> 00:09:24,590 with the electric dipole Hamiltonian, which 126 00:09:24,590 --> 00:09:29,550 I have already put up there, but you do actually 127 00:09:29,550 --> 00:09:32,460 unnecessary approximations in terms 128 00:09:32,460 --> 00:09:36,040 of neglecting an a squared term, and there are often 129 00:09:36,040 --> 00:09:38,140 confusing discussions what happens 130 00:09:38,140 --> 00:09:39,720 with the a squared term. 131 00:09:39,720 --> 00:09:42,010 I want to show you a more rigorous deviation 132 00:09:42,010 --> 00:09:44,600 of the electric dipole approximation, which 133 00:09:44,600 --> 00:09:50,160 is including the quadratic term. 134 00:09:56,130 --> 00:09:58,550 That actually means-- that's the take home 135 00:09:58,550 --> 00:10:02,940 message-- the electric dipole approximation is actually 136 00:10:02,940 --> 00:10:05,515 better than many of you have thought so far. 137 00:10:12,820 --> 00:10:16,530 Since I don't want to write out 100 equations, 138 00:10:16,530 --> 00:10:20,500 I copied them and summarized them out of the book. 139 00:10:20,500 --> 00:10:23,050 You'll find them all, as I mentioned before, 140 00:10:23,050 --> 00:10:25,850 in the appendix of Atom-Photon Interaction. 141 00:10:25,850 --> 00:10:29,580 But I sort of want to walk you through. 142 00:10:29,580 --> 00:10:31,280 What I'm aiming at is that you have 143 00:10:31,280 --> 00:10:36,040 sort of a complete overview how is everything coming together, 144 00:10:36,040 --> 00:10:39,060 and if it's just mathematical, well, mathematics 145 00:10:39,060 --> 00:10:40,320 is always exact. 146 00:10:40,320 --> 00:10:42,890 I can go faster, but I really want 147 00:10:42,890 --> 00:10:46,550 to highlight here every single physical concept 148 00:10:46,550 --> 00:10:48,342 and approximation we are doing. 149 00:10:51,660 --> 00:10:55,330 When we start out with Maxwell's equations, 150 00:10:55,330 --> 00:11:04,570 we have actually six components of the electromagnetic field, 151 00:11:04,570 --> 00:11:10,720 and we will see in a moment that this is redundant. 152 00:11:16,330 --> 00:11:22,310 What will turn out to be very important to separate 153 00:11:22,310 --> 00:11:25,190 the local fields from the radiation field 154 00:11:25,190 --> 00:11:38,930 is a Fourier transformation, so we introduce an expansion 155 00:11:38,930 --> 00:11:41,490 into plain waves. 156 00:11:41,490 --> 00:11:43,880 And what then happens, of course, 157 00:11:43,880 --> 00:11:49,400 is the derivative operator becomes the k vector. 158 00:11:49,400 --> 00:11:52,660 So therefore, when you have Maxwell's equations 159 00:11:52,660 --> 00:12:00,050 with the curl, del cross B, it turns into k cross 160 00:12:00,050 --> 00:12:08,700 B. This is now important because we have now separate equations 161 00:12:08,700 --> 00:12:11,830 for the component of the electric and magnetic field, 162 00:12:11,830 --> 00:12:16,140 which is parallel to k, projected onto k, 163 00:12:16,140 --> 00:12:18,910 and of course, the cross product takes out 164 00:12:18,910 --> 00:12:21,150 the component which is perpendicular to k. 165 00:12:24,510 --> 00:12:27,800 So that's something which is sort of nice, 166 00:12:27,800 --> 00:12:34,750 that the Fourier transform it allows us to separate 167 00:12:34,750 --> 00:12:41,910 the fields into longitudinal fields, where 168 00:12:41,910 --> 00:12:45,110 the vector of the Fourier component 169 00:12:45,110 --> 00:12:52,253 is parallel to the k vector, and into transverse components. 170 00:12:52,253 --> 00:12:53,169 AUDIENCE: [INAUDIBLE]? 171 00:13:01,812 --> 00:13:04,020 PROFESSOR: The difference between curly and non-curly 172 00:13:04,020 --> 00:13:05,590 vector is one is a Fourier transform, 173 00:13:05,590 --> 00:13:07,630 the other one is other spatial components. 174 00:13:07,630 --> 00:13:16,760 I have to actually say the book API 175 00:13:16,760 --> 00:13:21,350 tries to be super accurate in choosing [INAUDIBLE]. 176 00:13:21,350 --> 00:13:23,670 Claude Cohen-Tannoudji is really a master 177 00:13:23,670 --> 00:13:25,810 of elegance and perfection, so they 178 00:13:25,810 --> 00:13:28,270 don't want to use the same [INAUDIBLE] 179 00:13:28,270 --> 00:13:32,190 for a spatial component and for the Fourier component. 180 00:13:32,190 --> 00:13:34,910 On the other hand, I have to tell you one thing. 181 00:13:34,910 --> 00:13:37,790 For the next hour, use a little bit your intuition. 182 00:13:37,790 --> 00:13:40,790 If you see e, it means electric field. 183 00:13:40,790 --> 00:13:43,290 Whether it is curly, whether it's italic, 184 00:13:43,290 --> 00:13:47,450 whether it's bold face, don't let 185 00:13:47,450 --> 00:13:50,130 those differences clutter your view. 186 00:13:50,130 --> 00:13:52,890 So I would say don't ask me too often, 187 00:13:52,890 --> 00:13:55,320 what exactly does it mean now because you 188 00:13:55,320 --> 00:13:58,720 have gone from curly to non-curly? 189 00:13:58,720 --> 00:14:00,750 It is explained in the appendix. 190 00:14:00,750 --> 00:14:03,350 I will tell you everything which is necessary, 191 00:14:03,350 --> 00:14:08,210 but these are subtleties which are 192 00:14:08,210 --> 00:14:12,620 more to present the mathematics in a more rigorous way. 193 00:14:16,300 --> 00:14:19,420 We have the spatial Fourier components, 194 00:14:19,420 --> 00:14:27,300 and as I mentioned, what you can do only because of the Fourier 195 00:14:27,300 --> 00:14:38,756 transform, you have now the distinction, 196 00:14:38,756 --> 00:14:49,180 a rigorous separation between the parallel 197 00:14:49,180 --> 00:14:50,346 and the longitudinal fields. 198 00:15:08,560 --> 00:15:11,090 So of course, you can also now get back 199 00:15:11,090 --> 00:15:14,900 what is the longitudinal field, not in Fourier space 200 00:15:14,900 --> 00:15:18,460 but in position space, by just taking 201 00:15:18,460 --> 00:15:23,000 the longitudinal or transverse components of the Fourier 202 00:15:23,000 --> 00:15:25,290 transform and transforming them back. 203 00:15:25,290 --> 00:15:27,300 Why I'm elaborating on that is we 204 00:15:27,300 --> 00:15:30,510 will see in the next few minutes that the transverse field 205 00:15:30,510 --> 00:15:32,200 is the field which propagates. 206 00:15:32,200 --> 00:15:34,720 The longitudinal field is the Coulomb field. 207 00:15:34,720 --> 00:15:37,080 It belongs to the atoms and will not 208 00:15:37,080 --> 00:15:38,830 become a new degree of freedom. 209 00:15:42,710 --> 00:15:48,150 And we see that immediately in the next equations. 210 00:15:48,150 --> 00:15:56,270 When we look now at Maxwell's equations, 211 00:15:56,270 --> 00:16:00,890 we find that the transverse and longitudinal fields decouple. 212 00:16:00,890 --> 00:16:06,810 We have two kinds of equations which are completely decoupled. 213 00:16:06,810 --> 00:16:11,890 The longitudinal electric field is the Coulomb field associated 214 00:16:11,890 --> 00:16:17,210 with a charge density, whereas the transverse fields are 215 00:16:17,210 --> 00:16:20,710 creating themselves, and this is radiation, 216 00:16:20,710 --> 00:16:23,675 how the electric and magnetic fields create themselves 217 00:16:23,675 --> 00:16:24,425 as they propagate. 218 00:16:28,330 --> 00:16:46,200 Now, the fact that the parallel electric field 219 00:16:46,200 --> 00:16:52,790 can be expressed by the charge density, 220 00:16:52,790 --> 00:17:06,930 or it's expressed by the momentary position of charges, 221 00:17:06,930 --> 00:17:14,079 and this means immediately that the longitudinal field is not 222 00:17:14,079 --> 00:17:18,169 an independent variable or an extra degree of freedom. 223 00:17:23,619 --> 00:17:25,514 Canonical, of course, with how we describe 224 00:17:25,514 --> 00:17:29,680 particles by their position, and here you see explicitly 225 00:17:29,680 --> 00:17:33,350 that the longitudinal field which 226 00:17:33,350 --> 00:17:39,850 we defined the way we did it, that this part of the field 227 00:17:39,850 --> 00:17:44,124 only depends on the momentarily electric field, the Coulomb 228 00:17:44,124 --> 00:17:45,540 field associated with the charges. 229 00:18:03,810 --> 00:18:06,480 The next thing, of course, is to define the vector 230 00:18:06,480 --> 00:18:13,080 potential and the scalar potential. 231 00:18:18,270 --> 00:18:20,640 By using the standard substitution, 232 00:18:20,640 --> 00:18:23,670 we can now express electric and magnetic fields 233 00:18:23,670 --> 00:18:26,735 by a scalar potential and by a vector potential. 234 00:18:34,140 --> 00:18:37,050 We continue to have the separation 235 00:18:37,050 --> 00:18:40,570 between longitudinal and transverse fields, 236 00:18:40,570 --> 00:18:49,210 namely the transverse electromagnetic fields 237 00:18:49,210 --> 00:18:53,700 depend only on the transverse component 238 00:18:53,700 --> 00:18:54,700 of the vector potential. 239 00:18:59,530 --> 00:19:04,860 And the transverse component is gauge invariant. 240 00:19:04,860 --> 00:19:11,910 In other words, you know there are different gauges we 241 00:19:11,910 --> 00:19:15,050 can consider when we introduce electromagnetic potentials, 242 00:19:15,050 --> 00:19:20,000 but the choice of gauge does not affect the transverse vector 243 00:19:20,000 --> 00:19:20,500 potential. 244 00:19:27,530 --> 00:19:31,190 Now we want to pick our gauge and repeat 245 00:19:31,190 --> 00:19:33,820 the gauge which is most convenient. 246 00:19:33,820 --> 00:19:38,200 This is typically, for low energy quantum physics, 247 00:19:38,200 --> 00:19:42,380 for describing atoms and radiation, the Coulomb gauge. 248 00:19:42,380 --> 00:19:45,700 So we have the freedom of gauge, and the Coulomb gauge 249 00:19:45,700 --> 00:19:53,075 is written by the divergence of the vector potential is 0. 250 00:19:55,680 --> 00:19:58,800 If you now think about Fourier transform, 251 00:19:58,800 --> 00:20:03,130 this means k dot the Fourier component is 0, 252 00:20:03,130 --> 00:20:07,230 and that means the vector potential does not 253 00:20:07,230 --> 00:20:09,020 have any longitudinal component. 254 00:20:21,070 --> 00:20:33,660 So with that, we have now reduced equations 255 00:20:33,660 --> 00:20:36,860 for six variables, three electric fields, three 256 00:20:36,860 --> 00:20:38,640 magnetic field components. 257 00:20:38,640 --> 00:20:43,200 By introducing vector and scalar potential, you go to four. 258 00:20:43,200 --> 00:20:46,230 We've eliminated one more with the Coulomb gauge, 259 00:20:46,230 --> 00:20:48,780 so we have three. 260 00:20:48,780 --> 00:20:51,400 The scalar potential is the Coulomb potential. 261 00:20:51,400 --> 00:20:53,170 It's not an extra degree of freedom. 262 00:20:53,170 --> 00:21:02,320 And what is left are the two field components 263 00:21:02,320 --> 00:21:04,050 of the transverse vector potential. 264 00:21:14,630 --> 00:21:29,112 And these are now, as we will see, the independent variables 265 00:21:29,112 --> 00:21:30,890 of the radiation field. 266 00:21:36,730 --> 00:21:39,440 So in other words, when we talk about the electromagnetic field 267 00:21:39,440 --> 00:21:42,550 in the remainder of the course, the part 268 00:21:42,550 --> 00:21:45,580 of the electromagnetic field we are interested in 269 00:21:45,580 --> 00:21:49,450 are the fields generated by the transverse vector potential. 270 00:22:01,640 --> 00:22:10,960 Now, what is next is to go into normal modes. 271 00:22:10,960 --> 00:22:15,310 The reason is that we want to identify 272 00:22:15,310 --> 00:22:18,680 each normal mode of the electromagnetic field 273 00:22:18,680 --> 00:22:21,760 as an independent harmonic oscillator. 274 00:22:21,760 --> 00:22:24,680 Now, I'm showing you those equations, 275 00:22:24,680 --> 00:22:30,960 but you sort of have to sit back because now there will be 276 00:22:30,960 --> 00:22:35,230 alphas, a's, lower a's, upper A's. 277 00:22:35,230 --> 00:22:37,020 They all mean pretty much the same. 278 00:22:37,020 --> 00:22:42,940 In one case, you are normalized, maybe something 279 00:22:42,940 --> 00:22:44,430 has to be factored out or not. 280 00:22:47,110 --> 00:22:50,440 The derivation in the appendix of API 281 00:22:50,440 --> 00:22:53,140 is exactly [INAUDIBLE] distinction, 282 00:22:53,140 --> 00:22:56,090 but let me just give you a flyover here. 283 00:22:56,090 --> 00:23:00,560 And that is we are now taking the transverse components 284 00:23:00,560 --> 00:23:06,220 from the electromagnetic field, and we define new parameters, 285 00:23:06,220 --> 00:23:09,190 which are the normal modes. 286 00:23:09,190 --> 00:23:10,990 What we actually want to see in a moment 287 00:23:10,990 --> 00:23:14,000 is that those normal mode variables 288 00:23:14,000 --> 00:23:17,100 are harmonic oscillators. 289 00:23:17,100 --> 00:23:20,650 Those normal modes can be defined by the original field, 290 00:23:20,650 --> 00:23:23,450 which are transverse, but since we 291 00:23:23,450 --> 00:23:26,090 have expressed the transverse field by the transverse vector 292 00:23:26,090 --> 00:23:28,972 potential, they also simply can be expressed 293 00:23:28,972 --> 00:23:30,430 by the transverse vector potential. 294 00:23:33,040 --> 00:23:35,875 So to remind you, we've done a Fourier transform 295 00:23:35,875 --> 00:23:38,090 in the spatial coordinate, but not 296 00:23:38,090 --> 00:23:39,940 yet in the temporal coordinate. 297 00:23:49,580 --> 00:23:51,620 What we obtain now is an equation 298 00:23:51,620 --> 00:23:53,565 of motion for the normal mode variables. 299 00:24:01,630 --> 00:24:05,440 And here, I have reminded you that the transverse vector 300 00:24:05,440 --> 00:24:10,790 potential, how it can be expressed by the normal mode 301 00:24:10,790 --> 00:24:11,990 variables. 302 00:24:11,990 --> 00:24:16,180 In other words, yes, it looks complicated and a little bit 303 00:24:16,180 --> 00:24:19,300 messy, but it's classical physics, and all we have done 304 00:24:19,300 --> 00:24:24,000 is introduce normal modes in a just more complicated notation, 305 00:24:24,000 --> 00:24:27,690 as you may have seen it in 803 or some other course 306 00:24:27,690 --> 00:24:29,380 where for the first time you talked 307 00:24:29,380 --> 00:24:32,735 about normal modes of a pendulum or a chain of springs. 308 00:24:36,310 --> 00:24:40,230 So we have identified normal modes by the equation above, 309 00:24:40,230 --> 00:24:43,452 and then it's a mathematical identity 310 00:24:43,452 --> 00:24:47,260 that our radiation field, our transverse vector potential, 311 00:24:47,260 --> 00:24:50,020 can simply be expended into normal modes. 312 00:25:21,800 --> 00:25:44,250 So the equation of motion for the transverse fields 313 00:25:44,250 --> 00:25:47,400 involves of course, the transverse vector potential. 314 00:25:57,370 --> 00:25:58,640 I should have scrolled back. 315 00:26:05,822 --> 00:26:10,930 Here we have our equations for the transverse fields. 316 00:26:10,930 --> 00:26:13,480 This is what we want to describe, 317 00:26:13,480 --> 00:26:18,020 and if we express everything in terms of the vector potential, 318 00:26:18,020 --> 00:26:27,280 then we have an equation which involves the transverse vector 319 00:26:27,280 --> 00:26:31,720 potential, but in addition, because it's a differential 320 00:26:31,720 --> 00:26:34,283 equation, the first and second derivative. 321 00:26:43,680 --> 00:26:49,660 That means that if you have a second order differential 322 00:26:49,660 --> 00:26:53,100 equation, which is the full description 323 00:26:53,100 --> 00:26:55,580 of the electromagnetic field, then 324 00:26:55,580 --> 00:27:00,600 you know that the solution of the second order equation 325 00:27:00,600 --> 00:27:04,210 needs as an initial condition the field itself 326 00:27:04,210 --> 00:27:06,950 and the first derivative. 327 00:27:06,950 --> 00:27:11,060 Therefore, our classical fields are 328 00:27:11,060 --> 00:27:22,410 determined by a perpendicular and its derivative 329 00:27:22,410 --> 00:27:23,450 at the initial time. 330 00:27:47,580 --> 00:27:53,420 So we need a and we need a dot, and they are coupled, 331 00:27:53,420 --> 00:27:56,420 but what we are doing right now is-- 332 00:27:56,420 --> 00:28:01,270 and this is the idea behind the normal modes-- the normal modes 333 00:28:01,270 --> 00:28:04,690 coordinates combine a and a dot. 334 00:28:04,690 --> 00:28:07,430 It's the same when you have a harmonic oscillator 335 00:28:07,430 --> 00:28:09,560 and you want to introduce coordinates 336 00:28:09,560 --> 00:28:12,670 for the normal mode, they are a combination 337 00:28:12,670 --> 00:28:16,150 of position and momentum, and then they are decoupled. 338 00:28:16,150 --> 00:28:19,390 But position and momentum always couple or oscillate 339 00:28:19,390 --> 00:28:21,800 back and forth, and the same happens here 340 00:28:21,800 --> 00:28:25,190 between a and a dot, and the normal mode coordinates 341 00:28:25,190 --> 00:28:26,340 are the linear combination. 342 00:28:35,130 --> 00:28:46,550 In other words, the equation of motion 343 00:28:46,550 --> 00:28:53,810 coupled source and normal mode means 344 00:28:53,810 --> 00:29:05,960 that we have introduced he decoupled normal modes. 345 00:29:10,430 --> 00:29:15,200 And whenever, in classical physics, we have normal modes, 346 00:29:15,200 --> 00:29:19,480 normal modes means that time dependence is e to the i omega 347 00:29:19,480 --> 00:29:24,660 t, then we have distilled the problem of coupled components 348 00:29:24,660 --> 00:29:26,500 to decoupled harmonic oscillators. 349 00:29:30,690 --> 00:29:42,940 So at this point, each mode acts as an independent harmonic 350 00:29:42,940 --> 00:29:43,440 oscillator. 351 00:29:55,120 --> 00:30:00,470 So what I wanted to show you here is clearly, as a flyover, 352 00:30:00,470 --> 00:30:03,370 and as an appetizer to read more in the book, 353 00:30:03,370 --> 00:30:05,836 that we can start with the electromagnetic field 354 00:30:05,836 --> 00:30:06,335 components. 355 00:30:09,220 --> 00:30:11,810 We haven't just assumed we have now empty space 356 00:30:11,810 --> 00:30:13,490 and only [INAUDIBLE] radiation. 357 00:30:13,490 --> 00:30:16,460 We have rigorously separated the electromagnetic field 358 00:30:16,460 --> 00:30:19,330 in what belongs to the atoms and what belongs to radiation, 359 00:30:19,330 --> 00:30:22,730 and the trick was a spatial Fourier component. 360 00:30:22,730 --> 00:30:26,020 At that moment, we had a description 361 00:30:26,020 --> 00:30:28,380 in terms of the transverse vector potential, 362 00:30:28,380 --> 00:30:31,630 and by using purely classical physics, 363 00:30:31,630 --> 00:30:35,310 by combining the vector potential and its derivative, 364 00:30:35,310 --> 00:30:38,567 we found normal modes, and they are now 365 00:30:38,567 --> 00:30:40,400 completely independent harmonic oscillators. 366 00:30:43,820 --> 00:30:46,740 This approach, and I wanted to show that to you, 367 00:30:46,740 --> 00:30:49,380 involved a lot of notation-- a's, little alphas, 368 00:30:49,380 --> 00:30:51,670 normal modes, and such. 369 00:30:51,670 --> 00:30:55,240 Let me just simply show you another much, much shorter 370 00:30:55,240 --> 00:30:58,630 pathway how you see that everything looks and smells 371 00:30:58,630 --> 00:31:01,110 like a harmonic oscillator, and then we'll 372 00:31:01,110 --> 00:31:03,950 do the last step, which is rather straightforward. 373 00:31:03,950 --> 00:31:05,990 We quantize the electromagnetic field. 374 00:31:05,990 --> 00:31:08,015 I could quantize it right now, but we 375 00:31:08,015 --> 00:31:11,430 should take a short break from the appendix in Atom-Photon 376 00:31:11,430 --> 00:31:14,700 Interaction and do a more intuitive shortcut 377 00:31:14,700 --> 00:31:16,550 to the same physics. 378 00:31:16,550 --> 00:31:18,010 Any questions at this point? 379 00:31:21,850 --> 00:31:24,870 If you feel it's confusing, remind yourself 380 00:31:24,870 --> 00:31:26,980 this is just classical physics. 381 00:31:26,980 --> 00:31:31,810 We are just rewriting Maxwell's equations in new variables. 382 00:31:41,850 --> 00:31:45,420 So what we do now is, as I said, let's repeat 383 00:31:45,420 --> 00:31:59,610 some of this deviation by focusing on energy. 384 00:32:18,220 --> 00:32:26,150 If you write down energy for our systems of atoms, photons, 385 00:32:26,150 --> 00:32:33,020 and Coulomb fields, we have particles alpha 386 00:32:33,020 --> 00:32:36,270 with mass m alpha and velocity v alpha. 387 00:32:36,270 --> 00:32:39,340 This is our kinetic energy. 388 00:32:39,340 --> 00:32:42,950 And then we use the well known expression 389 00:32:42,950 --> 00:32:46,040 for the electromagnetic field, which 390 00:32:46,040 --> 00:32:52,710 is the spatial integral over E squared plus B squared. 391 00:33:06,280 --> 00:33:13,210 This integral over the electromagnetic field energy 392 00:33:13,210 --> 00:33:18,060 density can actually be nicely separated 393 00:33:18,060 --> 00:33:25,190 into integral over the longitudinal field which 394 00:33:25,190 --> 00:33:30,790 we have introduced and a second integral, which 395 00:33:30,790 --> 00:33:33,830 involves the transverse field. 396 00:33:36,980 --> 00:33:39,880 Of course, the magnetic field is only transverse 397 00:33:39,880 --> 00:33:42,460 because the divergence of the magnetic field is 0, 398 00:33:42,460 --> 00:33:44,847 and that means the magnetic field does not 399 00:33:44,847 --> 00:33:46,180 have any longitudinal component. 400 00:33:56,410 --> 00:34:05,560 So this part here, as we actually have shown, 401 00:34:05,560 --> 00:34:22,320 is given by simply the Coulomb energy where 402 00:34:22,320 --> 00:34:26,630 we have charge density at position r 403 00:34:26,630 --> 00:34:30,580 and r prime interacting with the Coulomb interaction. 404 00:34:40,480 --> 00:34:43,429 Eventually, if you want to treat that further, 405 00:34:43,429 --> 00:34:51,199 this Coulomb energy should be split into some divergent self 406 00:34:51,199 --> 00:34:54,600 energy, which is the energy of the electron interacting 407 00:34:54,600 --> 00:34:56,829 with itself, and people know how to deal 408 00:34:56,829 --> 00:35:02,090 with that, and the interaction energy, 409 00:35:02,090 --> 00:35:06,360 for example, between the proton and the electron, 410 00:35:06,360 --> 00:35:08,400 which is responsible for the atomic structure. 411 00:35:11,450 --> 00:35:13,750 So in other words, we know how to deal with that. 412 00:35:13,750 --> 00:35:17,880 This actually becomes atomic structure. 413 00:35:17,880 --> 00:35:21,870 This is in contrast to this part here, 414 00:35:21,870 --> 00:35:36,020 which is the classical energy of the radiation field, which 415 00:35:36,020 --> 00:35:39,670 is sort of the transverse part of the energy. 416 00:35:49,530 --> 00:35:55,790 To describe now radiation, we have 417 00:35:55,790 --> 00:36:02,750 to focus on this transverse part, 418 00:36:02,750 --> 00:36:10,240 and we are now expressing the transverse part 419 00:36:10,240 --> 00:36:16,340 by the vector potential and its derivatives. 420 00:36:16,340 --> 00:36:19,260 So we introduce the vector potential as before, 421 00:36:19,260 --> 00:36:21,600 and I don't want to go through the re-derivation. 422 00:36:21,600 --> 00:36:24,740 I just want to show you how the total energy now 423 00:36:24,740 --> 00:36:28,790 appears in terms of the vector potential. 424 00:36:28,790 --> 00:36:33,840 And what we need for that is, of course, the vector 425 00:36:33,840 --> 00:36:38,880 potential and its derivative. 426 00:36:38,880 --> 00:36:40,790 The vector potential depends on polarization, 427 00:36:40,790 --> 00:36:44,630 depends on the Fourier component, 428 00:36:44,630 --> 00:36:47,780 and we call this derivative of the vector 429 00:36:47,780 --> 00:36:51,560 potential the conjugate momentum. 430 00:36:56,760 --> 00:37:00,440 So now focusing on energy, we, of course, 431 00:37:00,440 --> 00:37:04,874 reduce everything as before to the vector potential 432 00:37:04,874 --> 00:37:05,665 and its derivative. 433 00:37:09,840 --> 00:37:14,280 So we sum over polarization. 434 00:37:14,280 --> 00:37:16,420 We integrate over all Fourier components. 435 00:37:26,770 --> 00:37:46,160 and the integral involves now the vector potential, 436 00:37:46,160 --> 00:37:49,060 the square of it, or in complex notation, 437 00:37:49,060 --> 00:37:53,650 the complex conjugate, depends on polarization. 438 00:37:53,650 --> 00:37:58,010 We have c squared k square. 439 00:37:58,010 --> 00:38:00,840 This part, of course, comes from the magnetic field, 440 00:38:00,840 --> 00:38:02,760 which depends on the vector potential. 441 00:38:02,760 --> 00:38:04,730 It's a spatial derivative, the curl, 442 00:38:04,730 --> 00:38:07,830 and this gives the k squared, whereas the part 443 00:38:07,830 --> 00:38:10,430 of the electromagnetic energy, which 444 00:38:10,430 --> 00:38:14,400 is related to the electric field, the electric field 445 00:38:14,400 --> 00:38:17,440 is a temporal derivative of the vector potential, 446 00:38:17,440 --> 00:38:21,540 and therefore, it involves the temporal derivatives which 447 00:38:21,540 --> 00:38:23,440 are now the canonical momenta. 448 00:38:28,990 --> 00:38:33,830 So this equation should really remind you now-- 449 00:38:33,830 --> 00:38:40,920 it's an energy equation-- of the energy of a harmonic oscillator 450 00:38:40,920 --> 00:38:46,160 because the energy is now a sum over all oscillators, 451 00:38:46,160 --> 00:38:50,010 but this is sort of x squared, the amplitude of the oscillator 452 00:38:50,010 --> 00:38:52,100 squared, this is the potential energy. 453 00:38:52,100 --> 00:38:56,160 And here we have the derivative of x, the velocity or momentum, 454 00:38:56,160 --> 00:38:58,160 and this should remind you of the kinetic energy 455 00:38:58,160 --> 00:38:59,201 of a harmonic oscillator. 456 00:39:02,760 --> 00:39:05,900 In other words, this should tell you 457 00:39:05,900 --> 00:39:08,470 that by focusing on the transverse component 458 00:39:08,470 --> 00:39:13,640 of the vector potential, it's about lots of summation-- k 459 00:39:13,640 --> 00:39:16,050 summation, polarization summation. 460 00:39:16,050 --> 00:39:18,799 We in the end find that each such mode 461 00:39:18,799 --> 00:39:19,840 is a harmonic oscillator. 462 00:39:27,892 --> 00:39:28,475 Any questions? 463 00:39:34,160 --> 00:39:36,480 Let me just show you now, and it may 464 00:39:36,480 --> 00:39:40,520 help you to go through the previous derivation, 465 00:39:40,520 --> 00:39:46,530 let me know introduce what I did before, the normal modes, 466 00:39:46,530 --> 00:39:49,210 and then show you how the energy looks 467 00:39:49,210 --> 00:39:53,300 like defined in normal modes. 468 00:39:53,300 --> 00:39:57,370 So we have these normal mode variables, 469 00:39:57,370 --> 00:40:06,790 which are defined as a superposition of A and A dot. 470 00:40:13,370 --> 00:40:15,850 And I'm not telling you here with all the indices 471 00:40:15,850 --> 00:40:19,500 whether it's the polarization component, the k component. 472 00:40:19,500 --> 00:40:21,450 I just want to tell you the structure. 473 00:40:21,450 --> 00:40:26,390 The normal mode is a superposition of A dot and A. 474 00:40:26,390 --> 00:40:32,890 And with that, the energy of the electromagnetic field 475 00:40:32,890 --> 00:40:36,290 can be written-- well, we always have 476 00:40:36,290 --> 00:40:38,817 to sum over Fourier components. 477 00:40:38,817 --> 00:40:40,650 We always have to sum over the polarization. 478 00:40:40,650 --> 00:40:52,410 But then we have something which is very, very simple 479 00:40:52,410 --> 00:40:55,360 and intuitive. 480 00:40:55,360 --> 00:41:04,750 It's just the square of alpha with the correct polarization, 481 00:41:04,750 --> 00:41:09,920 the pre-factor is h bar omega over 2. 482 00:41:09,920 --> 00:41:16,110 It looks like quantum mechanics, h bar omega times A dagger 483 00:41:16,110 --> 00:41:23,730 A plus 1/2, but this is purely classical. 484 00:41:23,730 --> 00:41:26,240 There is no quantization, there is no operator. 485 00:41:26,240 --> 00:41:31,180 We've simply defined something new, namely alpha 486 00:41:31,180 --> 00:41:33,322 in terms of the vector potential. 487 00:41:33,322 --> 00:41:36,060 In that sense, if you're going to go back to the derivation, 488 00:41:36,060 --> 00:41:39,960 the alphas are nothing else than some elaborate combination 489 00:41:39,960 --> 00:41:42,271 of the transverse electric and magnetic field. 490 00:41:42,271 --> 00:41:42,770 Colin? 491 00:41:42,770 --> 00:41:44,150 AUDIENCE: Where does the h bar come in? 492 00:41:44,150 --> 00:41:45,530 Is that the coefficient of alpha? 493 00:41:48,750 --> 00:41:50,670 PROFESSOR: h bar only enters through 494 00:41:50,670 --> 00:41:53,920 the constant in the definition of the normal mode parameter. 495 00:41:53,920 --> 00:41:56,640 So actually, I have introduced h bar 496 00:41:56,640 --> 00:42:00,070 by choosing this parameter wisely in such a way 497 00:42:00,070 --> 00:42:02,310 that it connects with quantum mechanics. 498 00:42:02,310 --> 00:42:08,270 But it's a completely arbitrary introduction here. 499 00:42:08,270 --> 00:42:09,930 I could have said h bar to 1. 500 00:42:17,460 --> 00:42:21,030 I hope you enjoyed, or at least did not 501 00:42:21,030 --> 00:42:24,520 dislike this excursion into classical physics. 502 00:42:24,520 --> 00:42:30,730 We have now two equations for describing 503 00:42:30,730 --> 00:42:32,780 the energy of a harmonic oscillator. 504 00:42:38,740 --> 00:42:44,550 So both of these equations look like a harmonic oscillator, 505 00:42:44,550 --> 00:42:47,100 but if something looks like a harmonic oscillator, 506 00:42:47,100 --> 00:42:48,410 it is a harmonic oscillator. 507 00:42:53,520 --> 00:42:58,230 And I think I should always remind you 508 00:42:58,230 --> 00:43:05,100 that so far, everything has been purely classical. 509 00:43:07,870 --> 00:43:22,846 And also, let me write down that h bar enters solely 510 00:43:22,846 --> 00:43:30,052 through the constant in the definition of the normal mode 511 00:43:30,052 --> 00:43:30,760 parameter, alpha. 512 00:43:54,680 --> 00:43:56,410 If you want to know more about it, 513 00:43:56,410 --> 00:43:59,555 there is a second reference. 514 00:43:59,555 --> 00:44:03,560 I will actually show you the cover page in a few moments. 515 00:44:03,560 --> 00:44:07,040 But there is a second book by Claude Cohen-Tannoudji 516 00:44:07,040 --> 00:44:10,440 and collaborators, not Atom-Photon Interaction, 517 00:44:10,440 --> 00:44:12,560 but called Photons and Atoms. 518 00:44:12,560 --> 00:44:17,130 It's a whole book on rigorously defining QED. 519 00:44:17,130 --> 00:44:20,430 So a whole book has been written about the subject 520 00:44:20,430 --> 00:44:25,690 of this lecture, and on page 27, you can read more about that. 521 00:44:42,790 --> 00:44:45,870 So enough of classical physics. 522 00:44:45,870 --> 00:44:49,470 All we have done so far-- I'm sorry for repeating it, 523 00:44:49,470 --> 00:44:53,400 but I think I can't repeat it too often with so many 524 00:44:53,400 --> 00:44:58,300 complicated equations on the screen-- all we have done so 525 00:44:58,300 --> 00:45:01,020 far is we have rewritten Maxwell's equations. 526 00:45:01,020 --> 00:45:04,830 We have rewritten Maxwell's equations in Fourier space, 527 00:45:04,830 --> 00:45:07,970 with the vector potential, eventually with normal modes. 528 00:45:07,970 --> 00:45:10,820 That's all we have done. 529 00:45:10,820 --> 00:45:13,010 But now what we are doing is we do 530 00:45:13,010 --> 00:45:15,210 the step in quantum mechanics which you have already 531 00:45:15,210 --> 00:45:19,650 seen a few times, and this is you have written equations 532 00:45:19,650 --> 00:45:23,080 in such a way that they look like harmonic oscillators, 533 00:45:23,080 --> 00:45:28,480 and then you postulate that the classical quantities become 534 00:45:28,480 --> 00:45:30,840 operators. 535 00:45:30,840 --> 00:45:34,220 The transverse vector potential and the conjugate canonical 536 00:45:34,220 --> 00:45:37,400 momentum fulfill now commutators, 537 00:45:37,400 --> 00:45:43,735 and we use those commutators to define them as operators. 538 00:45:46,810 --> 00:45:50,560 Now, it looks particularly easy when we use normal mode 539 00:45:50,560 --> 00:45:53,110 operators because the normal mode operators 540 00:45:53,110 --> 00:45:56,190 after quantization become our a's and a daggers. 541 00:45:59,620 --> 00:46:02,020 That was one of the things I wanted to show you, 542 00:46:02,020 --> 00:46:05,500 that you can go through everything introducing 543 00:46:05,500 --> 00:46:07,990 classical normal modes, and then the normal modes 544 00:46:07,990 --> 00:46:11,920 turns into operators, and now we have all your creation 545 00:46:11,920 --> 00:46:13,130 and annihilation operators. 546 00:46:16,000 --> 00:46:21,520 In other words, quantization cannot be rigorously proven. 547 00:46:21,520 --> 00:46:24,594 I mean, you cannot prove quantum theory from first principles. 548 00:46:24,594 --> 00:46:26,135 You can have a mathematical framework 549 00:46:26,135 --> 00:46:29,470 and check it against nature. 550 00:46:29,470 --> 00:46:33,400 What we have done here is to formulate the quantum theory. 551 00:46:33,400 --> 00:46:39,140 At this point, we've made a postulate 552 00:46:39,140 --> 00:46:47,440 that we have operators which fulfill a commutation relation. 553 00:46:47,440 --> 00:46:50,230 And it is now your choice if you want 554 00:46:50,230 --> 00:46:55,080 to formulate postulate the commutator 555 00:46:55,080 --> 00:46:58,710 for the transverse vector potential 556 00:46:58,710 --> 00:47:03,770 and its conjugate momentum, or if you immediately 557 00:47:03,770 --> 00:47:05,930 want to jump at the normal modes, 558 00:47:05,930 --> 00:47:09,470 and that would mean you have the commutator for a and a dagger. 559 00:47:20,382 --> 00:47:21,590 We are almost done with that. 560 00:47:24,850 --> 00:47:27,960 We started with the electromagnetic fields. 561 00:47:27,960 --> 00:47:30,150 We went through transverse fields, vector 562 00:47:30,150 --> 00:47:33,850 potential, normal modes, now we quantize. 563 00:47:33,850 --> 00:47:37,760 But now all the equations, all the substitutions we have made, 564 00:47:37,760 --> 00:47:39,570 we can go backward. 565 00:47:39,570 --> 00:47:43,870 Therefore, we can now express the transverse vector 566 00:47:43,870 --> 00:47:46,710 potential, the electric and the magnetic field, 567 00:47:46,710 --> 00:47:49,230 by the normal modes, or that would 568 00:47:49,230 --> 00:47:52,520 mean in quantum mechanics, after quantization, we 569 00:47:52,520 --> 00:47:56,240 can express them by the a's and a daggers, 570 00:47:56,240 --> 00:47:59,221 and this is how we define the operator of the electric, 571 00:47:59,221 --> 00:48:01,096 the magnetic field, and the vector potential. 572 00:48:07,060 --> 00:48:11,040 So now, our fully quantized theory 573 00:48:11,040 --> 00:48:17,840 has operators, a of a dagger, or if you want now, 574 00:48:17,840 --> 00:48:20,870 the first defined operators of vector potential, electric 575 00:48:20,870 --> 00:48:23,140 and magnetic field. 576 00:48:23,140 --> 00:48:27,190 And just to remind you, since atomic physics is 577 00:48:27,190 --> 00:48:29,890 between fields and particles, we have 578 00:48:29,890 --> 00:48:35,130 exactly the standard definition of the particle operators. 579 00:48:35,130 --> 00:48:38,620 Each particle is described by its momentum and its position. 580 00:48:47,910 --> 00:48:55,680 So with that, we have our Hamiltonian, 581 00:48:55,680 --> 00:48:58,790 and a lot of what we discuss in this course 582 00:48:58,790 --> 00:49:01,620 are understanding this Hamiltonian, 583 00:49:01,620 --> 00:49:04,440 understanding its solution, understanding 584 00:49:04,440 --> 00:49:06,210 what is the physics described by that. 585 00:49:19,520 --> 00:49:23,340 So this Hamiltonian, this is the kinetic energy. 586 00:49:23,340 --> 00:49:25,630 The canonical momentum minus the vector potential 587 00:49:25,630 --> 00:49:27,500 is the mechanical velocity. 588 00:49:27,500 --> 00:49:29,880 This is kinetic energy. 589 00:49:29,880 --> 00:49:33,970 And we have separated the energy of the electromagnetic field 590 00:49:33,970 --> 00:49:38,790 into the Coulomb energy, which is written again here, 591 00:49:38,790 --> 00:49:41,070 and the radiation field. 592 00:49:41,070 --> 00:49:44,590 There is now one more term which we 593 00:49:44,590 --> 00:49:50,580 need later on, which is this one here. 594 00:49:50,580 --> 00:50:00,350 And at this point, I would say-- I'm just heuristically adding 595 00:50:00,350 --> 00:50:10,370 it by hand-- classical we don't have spin, 596 00:50:10,370 --> 00:50:14,070 but our particles have spin, and now we need [INAUDIBLE] 597 00:50:14,070 --> 00:50:15,530 coupling, how does the spin couple 598 00:50:15,530 --> 00:50:17,620 to the rest of the world? 599 00:50:17,620 --> 00:50:21,600 Well, the spin, if you multiply it with the g factor, 600 00:50:21,600 --> 00:50:23,970 the Bohr magneton has a magnetic moment, 601 00:50:23,970 --> 00:50:27,650 and what we simply add here is mu dot p, 602 00:50:27,650 --> 00:50:30,430 be the interaction of the magnetic moment 603 00:50:30,430 --> 00:50:32,342 with the magnetic field. 604 00:50:32,342 --> 00:50:34,550 So in that sense, we have been classical all the way. 605 00:50:34,550 --> 00:50:36,730 Now at the end we said, we need the spin. 606 00:50:36,730 --> 00:50:41,020 Let's put it on by simply taking an interaction, which 607 00:50:41,020 --> 00:50:43,000 is mu dot p. 608 00:50:47,160 --> 00:50:49,340 If you don't like that because I try 609 00:50:49,340 --> 00:50:53,170 to be very fundamental today and go from first principles, 610 00:50:53,170 --> 00:50:57,890 for electrons, you can get that by taking 611 00:50:57,890 --> 00:51:02,330 the non-relativistic unit of the Dirac equation. 612 00:51:02,330 --> 00:51:07,140 So if you have electrons, you can 613 00:51:07,140 --> 00:51:13,050 start with the Dirac equation and do the Pauli approximation, 614 00:51:13,050 --> 00:51:28,950 which is the non-relativistic limit of the Dirac equation. 615 00:51:32,357 --> 00:51:32,940 Any questions? 616 00:51:42,010 --> 00:51:45,650 So these are all the terms in the Hamiltonian. 617 00:51:49,060 --> 00:51:51,160 I mentioned, but I don't want to dwell on, 618 00:51:51,160 --> 00:51:56,740 that the Coulomb energy has to be separated into a Coulomb 619 00:51:56,740 --> 00:51:59,250 self energy and an interaction energy 620 00:51:59,250 --> 00:52:02,140 between the charged particles, but all of that 621 00:52:02,140 --> 00:52:05,760 becomes just one term in our Hamiltonian. 622 00:52:05,760 --> 00:52:08,010 This is the atomic structure. 623 00:52:08,010 --> 00:52:10,825 When we assume that atoms have energy levels, 624 00:52:10,825 --> 00:52:13,320 all the energy is included in that, 625 00:52:13,320 --> 00:52:16,535 so we will not discuss any further the Coulomb energy. 626 00:52:16,535 --> 00:52:20,690 We will simply assume we have an atom which has certain energy 627 00:52:20,690 --> 00:52:28,480 levels, and that includes all the Coulomb terms. 628 00:52:28,480 --> 00:52:32,150 But we will talk a lot about the Hamiltonian 629 00:52:32,150 --> 00:52:33,960 for the radiation field. 630 00:52:33,960 --> 00:52:36,380 The Hamiltonian for the radiation field 631 00:52:36,380 --> 00:52:40,000 can be conveniently written in a and a dagger, 632 00:52:40,000 --> 00:52:44,110 but I have a few equations up there I rigorously 633 00:52:44,110 --> 00:52:49,950 defined for you operators E and B in terms of a and a daggers, 634 00:52:49,950 --> 00:52:53,210 and those two equations are identical. 635 00:52:53,210 --> 00:52:57,240 So this looks very quantum, this looks very classical, 636 00:52:57,240 --> 00:52:59,880 but if you interpret the electric and magnetic field 637 00:52:59,880 --> 00:53:02,600 as operators, we have identical equations 638 00:53:02,600 --> 00:53:05,520 for the Hamiltonian of the radiation field. 639 00:53:28,730 --> 00:53:31,020 What we want to study are interactions 640 00:53:31,020 --> 00:53:34,140 between light and atoms, and of course, this 641 00:53:34,140 --> 00:53:36,960 comes from the vector potential. 642 00:53:36,960 --> 00:53:40,740 And in particular, when we square it out, 643 00:53:40,740 --> 00:53:46,390 the canonical momentum of the atoms times the vector 644 00:53:46,390 --> 00:53:50,500 potential has this cross term, p dot a. 645 00:53:50,500 --> 00:53:53,990 So let me know just take this Hamiltonian 646 00:53:53,990 --> 00:53:58,546 and write it in the way how we will need it for this course. 647 00:54:05,070 --> 00:54:09,190 We want to take the Hamiltonian, and I will often refer to that. 648 00:54:09,190 --> 00:54:12,110 We want to take the Hamiltonian and split it 649 00:54:12,110 --> 00:54:17,550 into three parts, the atoms, the radiation 650 00:54:17,550 --> 00:54:22,200 field, and the interaction between the two. 651 00:54:22,200 --> 00:54:27,900 The Hamiltonian for the particles, H subscript P, 652 00:54:27,900 --> 00:54:31,870 has the momentum squared, and it has 653 00:54:31,870 --> 00:54:37,320 the part of the electromagnetic fields which are longitudinal, 654 00:54:37,320 --> 00:54:40,152 the energy of which can be described by a Coulomb 655 00:54:40,152 --> 00:54:40,652 integral. 656 00:54:43,960 --> 00:54:47,130 We have already discussed the radiation field. 657 00:54:47,130 --> 00:54:50,740 The radiation field was nothing else 658 00:54:50,740 --> 00:54:57,360 than h bar omega, a dagger a, but the new part 659 00:54:57,360 --> 00:55:01,790 which we need now is the interaction term. 660 00:55:01,790 --> 00:55:03,990 I want to show you now, or remind you 661 00:55:03,990 --> 00:55:10,369 by just summarizing the term, that the interaction part has 662 00:55:10,369 --> 00:55:11,660 actually three different terms. 663 00:55:18,030 --> 00:55:22,990 The first one is the cross term between p and A. 664 00:55:22,990 --> 00:55:25,925 The second one, when we had p minus A and squared it, 665 00:55:25,925 --> 00:55:28,750 is the A squared term. 666 00:55:28,750 --> 00:55:31,290 And the third one is the interaction 667 00:55:31,290 --> 00:55:34,790 of the spin with the magnetic component of the radiation 668 00:55:34,790 --> 00:55:35,480 field. 669 00:55:35,480 --> 00:55:37,820 This is the mu dot B interaction. 670 00:55:41,540 --> 00:55:44,460 I don't think it's an exaggeration when I tell you 671 00:55:44,460 --> 00:55:48,110 that with those three equations, you 672 00:55:48,110 --> 00:55:49,970 can understand all of atomic physics. 673 00:55:58,800 --> 00:56:07,340 This term here that's important is second order 674 00:56:07,340 --> 00:56:11,400 in A, which of course also means it 675 00:56:11,400 --> 00:56:16,070 will be very important for very strong laser palaces, 676 00:56:16,070 --> 00:56:18,360 but there's more to be said about it. 677 00:56:40,560 --> 00:56:43,710 If you're interested in this subject, 678 00:56:43,710 --> 00:56:46,550 and I couldn't do full justice to it, 679 00:56:46,550 --> 00:56:48,750 you may want to look at this book from Claude 680 00:56:48,750 --> 00:56:51,170 Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert 681 00:56:51,170 --> 00:56:54,000 Grynberg about photons and atoms. 682 00:56:54,000 --> 00:56:57,625 The whole book is dedicated how to describe 683 00:56:57,625 --> 00:57:00,180 the interaction of atoms with electromagnetic fields. 684 00:57:03,980 --> 00:57:07,950 In this book, they go through different ways 685 00:57:07,950 --> 00:57:10,170 of formulating the electromagnetic field 686 00:57:10,170 --> 00:57:10,990 classically. 687 00:57:10,990 --> 00:57:14,160 They use Lagrangian formalism for the electromagnetic field 688 00:57:14,160 --> 00:57:18,370 and show the Euler-Lagrange equation or Maxwell's equation. 689 00:57:18,370 --> 00:57:21,690 There are lots of different approaches, 690 00:57:21,690 --> 00:57:27,780 and also, one can say what we did 691 00:57:27,780 --> 00:57:30,630 here, which is the simplest way of describing 692 00:57:30,630 --> 00:57:41,080 light-atom interaction, will work in the Coulomb 693 00:57:41,080 --> 00:57:44,240 gauge, which is not the gauge you would choose when you want 694 00:57:44,240 --> 00:57:46,040 to describe relativistic physics. 695 00:57:46,040 --> 00:57:48,440 There is the other gauge, the Lawrence gauge. 696 00:57:48,440 --> 00:57:52,230 It makes the field quantization much more difficult, 697 00:57:52,230 --> 00:57:57,680 but when you ever wondered, why is the quantization done 698 00:57:57,680 --> 00:58:01,890 in the way I did, and you think something I did was arbitrary-- 699 00:58:01,890 --> 00:58:03,550 why did I pick the Coulomb gauge? 700 00:58:03,550 --> 00:58:04,305 Read this book. 701 00:58:04,305 --> 00:58:07,350 There are hundreds of pages which explain it to you. 702 00:58:07,350 --> 00:58:11,540 And I'm not joking here, it's wonderful reading. 703 00:58:11,540 --> 00:58:13,840 I got the book just looked up a few things 704 00:58:13,840 --> 00:58:16,300 and I almost got hooked on it and read more and more. 705 00:58:16,300 --> 00:58:21,250 It's a fascinating story how deeply people have thought 706 00:58:21,250 --> 00:58:26,830 about how to describe this aspect of the course 707 00:58:26,830 --> 00:58:28,590 and how profound the thoughts are. 708 00:58:31,710 --> 00:58:34,080 What this book will emphasize-- and this 709 00:58:34,080 --> 00:58:36,060 is what I want to tell you in closing 710 00:58:36,060 --> 00:58:41,842 about the quantization of the electromagnetic field-- 711 00:58:41,842 --> 00:58:42,550 is the following. 712 00:58:47,230 --> 00:58:53,230 We were really working hard with classical field equations 713 00:58:53,230 --> 00:58:58,090 to describe the electromagnetic field classically by completely 714 00:58:58,090 --> 00:59:00,360 eliminating redundant variables. 715 00:59:00,360 --> 00:59:03,260 We started with six components of the electric and magnetic 716 00:59:03,260 --> 00:59:05,530 field and reduced it to two components, 717 00:59:05,530 --> 00:59:09,800 the two components of the transverse vector potentials. 718 00:59:09,800 --> 00:59:13,450 And as those masters say, as a result, 719 00:59:13,450 --> 00:59:16,890 the field can be quantized with a great economy 720 00:59:16,890 --> 00:59:19,250 in the formalism. 721 00:59:19,250 --> 00:59:25,430 If you want to have more symmetric formulations, where 722 00:59:25,430 --> 00:59:29,050 you don't eliminate the variables using the Coulomb 723 00:59:29,050 --> 00:59:32,760 gauge, the problem is you have variables, 724 00:59:32,760 --> 00:59:35,310 which are not independent. 725 00:59:35,310 --> 00:59:37,370 And now I think if you quantize, you 726 00:59:37,370 --> 00:59:40,700 have to formulate auxiliary conditions 727 00:59:40,700 --> 00:59:44,380 between the not independent operators. 728 00:59:44,380 --> 00:59:46,560 Aesthetically, it may be pleasing 729 00:59:46,560 --> 00:59:50,310 because the approach is more symmetric, 730 00:59:50,310 --> 00:59:53,290 but mathematically, it's much more involved. 731 00:59:53,290 --> 00:59:54,940 Anyway, if you're interested, this book 732 00:59:54,940 --> 00:59:58,940 has a wonderful discussion on the different ways how 733 00:59:58,940 --> 01:00:01,114 you can approach interactions of atoms 734 01:00:01,114 --> 01:00:02,280 with electromagnetic fields. 735 01:00:06,400 --> 01:00:09,250 I hope nobody has a question about that, 736 01:00:09,250 --> 01:00:11,880 because I'm not able to explain to you much more than that. 737 01:00:14,430 --> 01:00:15,340 Anyway, questions? 738 01:00:21,870 --> 01:00:26,970 We have derived from first principles the Hamiltonian, 739 01:00:26,970 --> 01:00:30,760 and the result is something many of you 740 01:00:30,760 --> 01:00:34,208 are already familiar with. 741 01:00:34,208 --> 01:00:40,250 Let me now spend the next 10 or 15 minutes 742 01:00:40,250 --> 01:00:43,275 on what is called the dipole approximation 743 01:00:43,275 --> 01:00:44,400 and the dipole Hamiltonian. 744 01:00:52,060 --> 01:00:58,210 We are talking know about one further expansion 745 01:00:58,210 --> 01:01:01,380 of our expressions, and this is the multipole expansion. 746 01:01:07,170 --> 01:01:10,670 Let me first give you the simple derivation, which 747 01:01:10,670 --> 01:01:12,442 you may have seen, but then you can 748 01:01:12,442 --> 01:01:13,900 appreciate the rigorous derivation. 749 01:01:19,510 --> 01:01:23,230 Either directly or through the vector potential, 750 01:01:23,230 --> 01:01:26,360 we have formulated our theory in terms 751 01:01:26,360 --> 01:01:29,610 of the electric and magnetic field 752 01:01:29,610 --> 01:01:31,890 as a function of position. 753 01:01:31,890 --> 01:01:37,500 But what we now want to exploit is that atoms are tiny, 754 01:01:37,500 --> 01:01:40,400 and usually, the electric or magnetic field 755 01:01:40,400 --> 01:01:45,420 is not changing over the extent of an atom or molecule. 756 01:01:45,420 --> 01:01:46,980 In other words, what I'm telling you 757 01:01:46,980 --> 01:01:51,450 is the relevant frequencies, the relevant components 758 01:01:51,450 --> 01:01:53,790 of the electric and magnetic fields 759 01:01:53,790 --> 01:01:55,480 will, of course, be the components 760 01:01:55,480 --> 01:01:58,790 which are in resonance with the atoms and molecules. 761 01:01:58,790 --> 01:02:02,020 What I'm telling you is that in many, many situations, 762 01:02:02,020 --> 01:02:04,560 the wavelengths of the relevant modes 763 01:02:04,560 --> 01:02:07,320 of the electromagnetic field are much, much longer 764 01:02:07,320 --> 01:02:09,750 than the size of an atom. 765 01:02:09,750 --> 01:02:15,990 So therefore, if we pinpoint our atom at the origin, 766 01:02:15,990 --> 01:02:20,940 we may be able to neglect the spatial dependence 767 01:02:20,940 --> 01:02:23,840 of the electromagnetic fields and replace it 768 01:02:23,840 --> 01:02:27,520 by electric and magnetic fields at the origin. 769 01:02:27,520 --> 01:02:30,930 Or, if you want to go higher in a Taylor expansion, 770 01:02:30,930 --> 01:02:32,700 we use gradients of it. 771 01:02:36,920 --> 01:02:39,220 We've talked more about that, actually, 772 01:02:39,220 --> 01:02:44,260 in 8.421, when we talked about multipole transitions, 773 01:02:44,260 --> 01:02:52,290 but here I want to only focus on the lowest order, which 774 01:02:52,290 --> 01:03:03,560 is called the electric dipole approximation, 775 01:03:03,560 --> 01:03:10,510 because this is the most important case which 776 01:03:10,510 --> 01:03:13,410 is used in atomic physics. 777 01:03:13,410 --> 01:03:21,740 So for the electric dipole approximation, 778 01:03:21,740 --> 01:03:26,370 we make actually several assumptions. 779 01:03:26,370 --> 01:03:32,720 One is we neglect the quadratic term, the a squared term. 780 01:03:32,720 --> 01:03:38,490 And now if you look at the A dot p term, 781 01:03:38,490 --> 01:03:41,820 and we are looking for matrix elements between two 782 01:03:41,820 --> 01:03:48,650 atomic levels, we expand the vector potential 783 01:03:48,650 --> 01:03:49,555 into plain waves. 784 01:03:53,440 --> 01:03:56,220 Well, it's called the electric dipole approximation, 785 01:03:56,220 --> 01:03:57,670 so maybe we want to express things 786 01:03:57,670 --> 01:04:03,090 by the electric field, which is the derivative of the vector 787 01:04:03,090 --> 01:04:05,320 potential. 788 01:04:05,320 --> 01:04:13,990 So with that, we have now described things 789 01:04:13,990 --> 01:04:16,850 by the electric field. 790 01:04:16,850 --> 01:04:24,810 And what is needed now is if we assume that kr equals 0, 791 01:04:24,810 --> 01:04:28,490 or r over lambda is very, very small, 792 01:04:28,490 --> 01:04:33,060 the exponential, the plain wave vector, is approximated by 1, 793 01:04:33,060 --> 01:04:37,330 and we are simply left in the electric dipole approximation 794 01:04:37,330 --> 01:04:45,425 with the a element of the momentum operator. 795 01:04:50,850 --> 01:04:59,460 We can simplify it further by writing the momentum operator 796 01:04:59,460 --> 01:05:02,430 as the commutator between the position 797 01:05:02,430 --> 01:05:04,620 operator and the Hamiltonian. 798 01:05:04,620 --> 01:05:10,660 If you take the commutator between r or x and p squared, 799 01:05:10,660 --> 01:05:13,050 you'd simply get p and p factors, 800 01:05:13,050 --> 01:05:14,950 so that's what we are using. 801 01:05:14,950 --> 01:05:24,850 And now the matrix element becomes-- so what 802 01:05:24,850 --> 01:05:31,510 we have here now is rH minus Hr, but H acting on 1, 803 01:05:31,510 --> 01:05:35,160 because 1 is an eigenstate, gives the energy, E1. 804 01:05:35,160 --> 01:05:37,570 And for the flipped part, we have Hr, 805 01:05:37,570 --> 01:05:42,670 and now H acting on state two gives us the energy, E2. 806 01:05:42,670 --> 01:05:47,940 So therefore, by taking care of the Hamiltonian, which 807 01:05:47,940 --> 01:05:53,500 gives us, depending on which side H appears, E1 or E2, 808 01:05:53,500 --> 01:06:00,180 we have reduced it to now a position matrix element. 809 01:06:00,180 --> 01:06:03,430 And what we have here is, of course, 810 01:06:03,430 --> 01:06:08,782 the transition frequency between the levels one and two. 811 01:06:08,782 --> 01:06:14,120 I know you have seen it, but if I can derive it in two minutes, 812 01:06:14,120 --> 01:06:15,880 it's maybe worth the exercise. 813 01:06:15,880 --> 01:06:20,380 What I've shown you is that the momentum matrix element 814 01:06:20,380 --> 01:06:25,920 can be replaced by the position matrix element multiplied 815 01:06:25,920 --> 01:06:35,140 by the resonance frequency. 816 01:06:35,140 --> 01:06:37,280 But now we want to put things together, 817 01:06:37,280 --> 01:06:39,040 and what I want to point out for you 818 01:06:39,040 --> 01:06:41,730 is that we have an omega here, which 819 01:06:41,730 --> 01:06:44,360 is the frequency of the electromagnetic field, 820 01:06:44,360 --> 01:06:47,950 and we have an omega 12 here, which is the transition energy. 821 01:06:50,640 --> 01:06:55,020 So therefore, if I put it together, 822 01:06:55,020 --> 01:06:59,220 I started with a p dot a Hamiltonian, 823 01:06:59,220 --> 01:07:02,965 but the p matrix element became an r matrix element. 824 01:07:02,965 --> 01:07:05,815 The a was expressed by the electric field. 825 01:07:10,980 --> 01:07:15,530 So therefore, we have now the electric field 826 01:07:15,530 --> 01:07:19,210 times the position matrix element, 827 01:07:19,210 --> 01:07:21,060 or the dipole matrix element, if you 828 01:07:21,060 --> 01:07:24,207 multiply the position of the electron with the charge two. 829 01:07:24,207 --> 01:07:35,190 But in addition, we have this prefactor. 830 01:07:35,190 --> 01:07:41,530 So this becomes the dipole Hamiltonian 831 01:07:41,530 --> 01:07:52,950 E dot d when we replace that by 1, assuming 832 01:07:52,950 --> 01:07:58,560 that we are interested anyway only in interactions with atoms 833 01:07:58,560 --> 01:08:00,950 where the radiation field is near resonance. 834 01:08:06,120 --> 01:08:14,870 So it seems that we have actually made three assumptions 835 01:08:14,870 --> 01:08:18,520 to derive the electric dipole approximation. 836 01:08:18,520 --> 01:08:26,439 One was the long wavelengths approximation. 837 01:08:26,439 --> 01:08:31,720 The second one was that we are near resonance, 838 01:08:31,720 --> 01:08:38,640 and the third one was that we neglected the quadratic term 839 01:08:38,640 --> 01:08:40,080 in the light-atom interaction. 840 01:08:59,460 --> 01:09:02,580 There are often people who are wondering about it. 841 01:09:02,580 --> 01:09:04,460 When I said we are only interested when 842 01:09:04,460 --> 01:09:08,970 the atoms interact with electromagnetic radiation 843 01:09:08,970 --> 01:09:12,649 near resonance, that's OK, but atomic physics 844 01:09:12,649 --> 01:09:16,460 is the area of physics with the highest precision. 845 01:09:16,460 --> 01:09:19,689 We can make measurements with 10 to the minus 16 and 10 846 01:09:19,689 --> 01:09:21,750 to the minus 17 precision. 847 01:09:21,750 --> 01:09:25,229 So therefore, if you have an electromagnetic field which 848 01:09:25,229 --> 01:09:27,939 resonates with atoms-- the typical frequency 849 01:09:27,939 --> 01:09:30,899 of visible light is 10 to the 14 Hertz-- 850 01:09:30,899 --> 01:09:33,740 and you're just one megahertz away from the resonance, 851 01:09:33,740 --> 01:09:35,620 it's 10 to the minus 8. 852 01:09:35,620 --> 01:09:39,399 So the question is, would we observe a correction 853 01:09:39,399 --> 01:09:45,950 to the dipole approximation if omega 12 is not exactly omega? 854 01:09:45,950 --> 01:09:55,520 Well, the answer is no because, as I want to show you now 855 01:09:55,520 --> 01:09:58,650 in the next few minutes, and this is also 856 01:09:58,650 --> 01:10:01,090 the end of our flyover over the appendix 857 01:10:01,090 --> 01:10:04,399 of atom-photon interaction, the last two assumptions 858 01:10:04,399 --> 01:10:05,440 are not really necessary. 859 01:10:10,710 --> 01:10:15,370 In other words, we go back to atom-photon interaction, 860 01:10:15,370 --> 01:10:26,510 to this long appendix on the derivation of the QED 861 01:10:26,510 --> 01:10:27,010 Hamiltonian. 862 01:10:30,730 --> 01:10:34,410 We go back to our fundamental Hamiltonian, 863 01:10:34,410 --> 01:10:37,130 and we want to do one approximation 864 01:10:37,130 --> 01:10:42,060 and this is the only approximation which is needed. 865 01:10:42,060 --> 01:10:48,770 It is the dipole approximation, and that 866 01:10:48,770 --> 01:10:54,770 means that the vector potential-- of course, we 867 01:10:54,770 --> 01:11:00,235 only need the transverse part of it-- is replaced by its value 868 01:11:00,235 --> 01:11:01,470 at the origin. 869 01:11:06,590 --> 01:11:09,610 That's the only approximation. 870 01:11:09,610 --> 01:11:13,120 So here is our Hamiltonian in its full beauty, 871 01:11:13,120 --> 01:11:16,890 and the only thing we have done is we've put in the origin 872 01:11:16,890 --> 01:11:19,080 here. 873 01:11:19,080 --> 01:11:25,720 For later convenience, we introduce the dipole operator, 874 01:11:25,720 --> 01:11:28,300 which is the position operator of particle 875 01:11:28,300 --> 01:11:31,630 alpha times the charge of particle alpha. 876 01:11:31,630 --> 01:11:40,110 And then we do simply a transformation. 877 01:11:40,110 --> 01:11:50,200 So this is a unitary transformation, 878 01:11:50,200 --> 01:12:00,910 and we transform from the original Hamiltonian, which 879 01:12:00,910 --> 01:12:05,295 includes the p dot a term and the a squared term 880 01:12:05,295 --> 01:12:09,860 to a transformed Hamiltonian, H prime. 881 01:12:09,860 --> 01:12:19,120 And this transformed Hamiltonian has now the electric dipole 882 01:12:19,120 --> 01:12:20,565 approximation. 883 01:12:20,565 --> 01:12:23,560 It's sort of written in polarization Fourier space, 884 01:12:23,560 --> 01:12:29,992 but this is the electric field, and a, a dagger gives position. 885 01:12:29,992 --> 01:12:30,632 I'm sorry. 886 01:12:30,632 --> 01:12:31,590 These are normal modes. 887 01:12:31,590 --> 01:12:35,890 It's a dipole operator times the electric field, 888 01:12:35,890 --> 01:12:39,700 and the A squared term has disappeared. 889 01:12:39,700 --> 01:12:55,430 So after the transformation, we have no quadratic term, 890 01:12:55,430 --> 01:13:04,010 and all we are left is this is nothing else 891 01:13:04,010 --> 01:13:08,480 than d dot e, the dipole interaction. 892 01:13:08,480 --> 01:13:14,270 And there is no prefactor omega 12 over omega. 893 01:13:14,270 --> 01:13:20,970 This is a rigorous unitary transformation 894 01:13:20,970 --> 01:13:21,745 to another basis. 895 01:13:25,754 --> 01:13:34,170 In full disclosure, it involves a dipolar self energy, 896 01:13:34,170 --> 01:13:36,890 which is sort of a constant energy. 897 01:13:36,890 --> 01:13:43,560 And if you look at the transformed velocities, 898 01:13:43,560 --> 01:13:46,480 the transformed velocities v prime are now 899 01:13:46,480 --> 01:13:52,050 identical to the original canonical momentum. 900 01:13:52,050 --> 01:13:57,330 So after that, what was the canonical momentum 901 01:13:57,330 --> 01:14:00,860 in the original Hamiltonian, H, becomes 902 01:14:00,860 --> 01:14:05,730 now velocity times mass. 903 01:14:05,730 --> 01:14:08,445 So p becomes now the mechanical momentum. 904 01:14:12,090 --> 01:14:14,692 And therefore, p squared is the kinetic energy. 905 01:14:19,510 --> 01:14:21,610 Anyway, I think this is important 906 01:14:21,610 --> 01:14:24,960 that we have this much more rigorous deviation 907 01:14:24,960 --> 01:14:26,423 of the dipole approximation. 908 01:14:38,057 --> 01:14:39,390 I don't want to go through that. 909 01:14:39,390 --> 01:14:41,760 I just want to make sure you have seen it. 910 01:14:44,620 --> 01:14:47,190 If you want to do it really rigorously, 911 01:14:47,190 --> 01:14:51,700 then you have to distinguish between the electric field 912 01:14:51,700 --> 01:14:59,180 d and E. So you have an electric field here, 913 01:14:59,180 --> 01:15:04,480 and here you have the polarization. 914 01:15:04,480 --> 01:15:07,675 It's pretty much the same as in classical e and m. 915 01:15:14,150 --> 01:15:27,010 At the end of the day, we have now, for the prime operator, 916 01:15:27,010 --> 01:15:29,720 for the Hamiltonian after the canonical transformation, 917 01:15:29,720 --> 01:15:33,770 after the unitary transformation, 918 01:15:33,770 --> 01:15:36,790 the only interaction term which remains 919 01:15:36,790 --> 01:15:41,070 is the electric dipole interaction between the dipole 920 01:15:41,070 --> 01:15:46,830 and this field d, but this field d prime corresponds 921 01:15:46,830 --> 01:15:50,510 to the original a transverse electric field. 922 01:15:50,510 --> 01:15:52,855 I'm not going to explain the difference between E, 923 01:15:52,855 --> 01:15:54,660 E prime, d, and d prime. 924 01:15:54,660 --> 01:15:56,310 It can be done rigorously. 925 01:15:56,310 --> 01:15:58,680 But the take home message is-- and with that, I 926 01:15:58,680 --> 01:16:03,657 want to conclude-- that for most of this course, 927 01:16:03,657 --> 01:16:05,490 the interaction of the electromagnetic field 928 01:16:05,490 --> 01:16:08,690 with our atoms is described by this term. 929 01:16:08,690 --> 01:16:11,290 We take the transverse electric field. 930 01:16:11,290 --> 01:16:13,360 This is our radiation field, and it 931 01:16:13,360 --> 01:16:17,078 couples to the operator of the atomic dipole moment. 932 01:16:22,950 --> 01:16:24,690 I hope you got the take home message 933 01:16:24,690 --> 01:16:29,540 that whenever you have any doubts about any part 934 01:16:29,540 --> 01:16:32,600 of this deviation, or if you have any doubts whether it's 935 01:16:32,600 --> 01:16:35,380 rigorous or not, you know now where 936 01:16:35,380 --> 01:16:39,294 to look it up and learn everything about it. 937 01:16:39,294 --> 01:16:40,268 Any questions? 938 01:16:53,930 --> 01:16:56,460 OK, then that's it. 939 01:16:56,460 --> 01:17:00,570 We'll probably post the first homework assignment tomorrow-- 940 01:17:00,570 --> 01:17:04,240 I will meet with the TAs-- and it will be due in about a week, 941 01:17:04,240 --> 01:17:06,820 but you will see all of that on the website 942 01:17:06,820 --> 01:17:08,070 and hear from me on Wednesday. 943 01:17:08,070 --> 01:17:10,186 Have a good afternoon.