1 00:00:00,070 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,030 under a Creative Commons license. 3 00:00:04,030 --> 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 00:00:27,360 --> 00:00:29,650 PROFESSOR: Before I continue with the material, 9 00:00:29,650 --> 00:00:34,830 I want to show you at least the title of a recent paper 10 00:00:34,830 --> 00:00:39,260 in Nature, because it's related to material 11 00:00:39,260 --> 00:00:41,450 we have covered in this course. 12 00:00:41,450 --> 00:00:44,630 It's about the Kerr effect, the effect 13 00:00:44,630 --> 00:00:49,520 that one photon can create a phase shift for another photon. 14 00:00:49,520 --> 00:00:52,540 And one goal, of course, for quantum computation 15 00:00:52,540 --> 00:00:55,310 where things are about single photons 16 00:00:55,310 --> 00:00:58,760 is to have a single-photon Kerr effect 17 00:00:58,760 --> 00:01:01,800 that one photon can change the phase of the other photon 18 00:01:01,800 --> 00:01:03,390 in a strong, noticeable way. 19 00:01:03,390 --> 00:01:05,750 So maybe one photon should create a phase shift 20 00:01:05,750 --> 00:01:07,620 on the order of pi. 21 00:01:07,620 --> 00:01:11,760 And this was reported here in this paper. 22 00:01:11,760 --> 00:01:17,960 Of course, the non-linearity created by nonlinear crystals 23 00:01:17,960 --> 00:01:21,850 is much too weak for that. 24 00:01:21,850 --> 00:01:28,176 But what those authors did is they used microwave photos, 25 00:01:28,176 --> 00:01:30,330 had microwave photons into cavities. 26 00:01:30,330 --> 00:01:34,320 And they were coupled through a sapphire substrate 27 00:01:34,320 --> 00:01:36,330 with a Josephson junction. 28 00:01:36,330 --> 00:01:38,840 So the non-linearity here is the non-linearity 29 00:01:38,840 --> 00:01:42,390 of a Josephson junction, which is actually 30 00:01:42,390 --> 00:01:45,260 realized with a superconducting qubit. 31 00:01:45,260 --> 00:01:47,394 I can't explain you many more details, 32 00:01:47,394 --> 00:01:48,810 but I just thought it sort of cool 33 00:01:48,810 --> 00:01:53,122 to see how the Kerr effect, which we discussed 34 00:01:53,122 --> 00:01:54,830 and which we discussed for single-photon, 35 00:01:54,830 --> 00:01:59,770 is realized, at least in the microwave domain. 36 00:01:59,770 --> 00:02:03,290 And also, just sort of to illustrate 37 00:02:03,290 --> 00:02:09,220 that I hope this course enables you to read recent research 38 00:02:09,220 --> 00:02:11,630 papers, what those people measured 39 00:02:11,630 --> 00:02:14,680 is a [? key ?] representation. 40 00:02:14,680 --> 00:02:16,750 This is a coherent state. 41 00:02:16,750 --> 00:02:19,800 And then they showed, and this is the subject of the paper, 42 00:02:19,800 --> 00:02:22,980 that the coherent state which has a well-defined phase, 43 00:02:22,980 --> 00:02:26,470 lost its phase through the Kerr medium. 44 00:02:26,470 --> 00:02:29,450 And you clearly see there is a big phase uncertainty. 45 00:02:29,450 --> 00:02:33,460 But then after certain time-- this is experiment 46 00:02:33,460 --> 00:02:36,380 and this is simulation-- there is a re-phasing, 47 00:02:36,380 --> 00:02:38,030 and the phase is back. 48 00:02:38,030 --> 00:02:40,280 There is a revival of the coherent state. 49 00:02:44,120 --> 00:02:45,030 All right. 50 00:03:05,480 --> 00:03:07,370 Now I want to address one question which 51 00:03:07,370 --> 00:03:11,410 Cody asked about the g2 function and fluctuations 52 00:03:11,410 --> 00:03:14,140 of single-mode light. 53 00:03:14,140 --> 00:03:15,630 Let me just summarize. 54 00:03:15,630 --> 00:03:20,880 I told you that if you do the thermodynamics 55 00:03:20,880 --> 00:03:24,010 of a single mode, we find Bose-Einstein distribution 56 00:03:24,010 --> 00:03:25,860 of photons. 57 00:03:25,860 --> 00:03:28,870 And we have a thermal distribution. 58 00:03:28,870 --> 00:03:32,270 And a thermal distribution means sometimes we have more photons, 59 00:03:32,270 --> 00:03:34,480 sometimes we have less photons, depending 60 00:03:34,480 --> 00:03:36,150 on the thermal distribution. 61 00:03:36,150 --> 00:03:39,810 And when we calculated what the intensity fluctuations were, 62 00:03:39,810 --> 00:03:44,290 we found they're characterized by a g2 function of 2. 63 00:03:44,290 --> 00:03:45,750 OK. 64 00:03:45,750 --> 00:03:48,810 Then a little bit later in this course-- actually, 65 00:03:48,810 --> 00:03:53,180 just this week-- I told you that single-mode light always 66 00:03:53,180 --> 00:03:55,470 has a g2 function of one. 67 00:03:55,470 --> 00:03:58,630 And what I meant here is the following, rather trivial. 68 00:03:58,630 --> 00:04:02,420 If you have a single mode, that means that you [? align ?] 69 00:04:02,420 --> 00:04:05,620 [? it ?] is simply e to the i omega t, 70 00:04:05,620 --> 00:04:07,130 the intensity is constant. 71 00:04:10,750 --> 00:04:13,170 There are no intensity fluctuations. 72 00:04:13,170 --> 00:04:16,950 And also, because everything is sort of predictable, 73 00:04:16,950 --> 00:04:20,589 it's just one wave. 74 00:04:20,589 --> 00:04:24,790 The Gn function factorizes into-- can 75 00:04:24,790 --> 00:04:26,870 be re-expressed by the g1 function. 76 00:04:26,870 --> 00:04:29,290 So therefore, this is the most trivial case. 77 00:04:29,290 --> 00:04:31,120 But the question now is, how do we 78 00:04:31,120 --> 00:04:34,700 reconcile those two statements, that a single mode, e to the i 79 00:04:34,700 --> 00:04:38,960 omega t, does not have intensity fluctuations? 80 00:04:38,960 --> 00:04:40,950 Therefore, is a g2 function of one. 81 00:04:40,950 --> 00:04:44,990 And our earlier treatment about single-mode black-body 82 00:04:44,990 --> 00:04:45,810 radiation. 83 00:04:45,810 --> 00:04:48,450 And of course, the answer is, what 84 00:04:48,450 --> 00:04:51,190 is the single mode in one context 85 00:04:51,190 --> 00:04:54,310 is different from the single mode in the other context. 86 00:04:54,310 --> 00:04:57,910 Maybe let me explain that. 87 00:04:57,910 --> 00:05:00,490 Let's just create an ensemble of cavities. 88 00:05:00,490 --> 00:05:03,010 We put them in thermal contact with a reservoir. 89 00:05:03,010 --> 00:05:06,270 And then we break the thermal contact with a reservoir. 90 00:05:06,270 --> 00:05:09,230 Each cavity has now a perfect single mode, e 91 00:05:09,230 --> 00:05:11,020 to the i omega t. 92 00:05:11,020 --> 00:05:14,470 But each cavity is filled with a different photon number 93 00:05:14,470 --> 00:05:17,730 according to the thermal statistics. 94 00:05:17,730 --> 00:05:19,965 So therefore, if you just look at one cavity, 95 00:05:19,965 --> 00:05:21,815 we find no intensity fluctuations. 96 00:05:21,815 --> 00:05:23,850 The g2 function is one. 97 00:05:23,850 --> 00:05:26,360 But if you extend the ensemble average 98 00:05:26,360 --> 00:05:28,350 over all the different cavities, we 99 00:05:28,350 --> 00:05:32,700 find that there are fluctuations in intensity. 100 00:05:32,700 --> 00:05:36,900 Well, we can now keep that in mind. 101 00:05:36,900 --> 00:05:38,970 But now we can say, well, let's just 102 00:05:38,970 --> 00:05:42,960 take one cavity which is weakly coupled to a thermal reservoir. 103 00:05:42,960 --> 00:05:45,040 And instead of looking at the ensemble average 104 00:05:45,040 --> 00:05:48,000 of many cavities, we look at the long-time average 105 00:05:48,000 --> 00:05:49,350 of this one cavity. 106 00:05:49,350 --> 00:05:52,770 And what will happen is thermal photons will be created, 107 00:05:52,770 --> 00:05:54,250 will disappear, and such. 108 00:05:54,250 --> 00:05:57,040 So now this one cavity fluctuates. 109 00:05:57,040 --> 00:05:59,070 But technically, what that means now 110 00:05:59,070 --> 00:06:06,140 is it means that the sharp mode of the cavity 111 00:06:06,140 --> 00:06:08,500 is interacting with the environment, 112 00:06:08,500 --> 00:06:09,680 and it becomes broadened. 113 00:06:09,680 --> 00:06:12,750 It has a broadening delta omega. 114 00:06:12,750 --> 00:06:16,110 And this can be regarded as that we 115 00:06:16,110 --> 00:06:19,680 mix in modes of the environment. 116 00:06:19,680 --> 00:06:23,070 So in that case, strictly speaking, 117 00:06:23,070 --> 00:06:26,720 it's no longer a single-mode cavity. 118 00:06:26,720 --> 00:06:30,050 So you have to consider those things. 119 00:06:30,050 --> 00:06:35,090 And depending what point of view you want to take, 120 00:06:35,090 --> 00:06:36,470 you get the different result. 121 00:06:40,564 --> 00:06:41,230 Other questions? 122 00:06:48,700 --> 00:06:50,060 Then let me ask you a question. 123 00:06:50,060 --> 00:06:53,590 Last class, I explained to you-- well, at least, 124 00:06:53,590 --> 00:06:57,150 tried to explain to you that g2 function for bosons 125 00:06:57,150 --> 00:07:00,110 and fermions with the counting statistics, 126 00:07:00,110 --> 00:07:02,020 with permutations and such. 127 00:07:02,020 --> 00:07:04,510 I wasn't sure, at least from one question 128 00:07:04,510 --> 00:07:06,991 I got, whether this was completely clear. 129 00:07:06,991 --> 00:07:08,490 Do you have any question about that? 130 00:07:12,567 --> 00:07:13,294 [? Teroy. ?] 131 00:07:13,294 --> 00:07:14,710 AUDIENCE: This seems very obvious. 132 00:07:14,710 --> 00:07:16,480 But during class, I was trying to something 133 00:07:16,480 --> 00:07:17,040 with thermal state. 134 00:07:17,040 --> 00:07:18,980 What is the definition of our thermal state 135 00:07:18,980 --> 00:07:23,695 in terms of any basis, just generally speaking? 136 00:07:23,695 --> 00:07:25,820 We write it as-- I thought it'd be something like e 137 00:07:25,820 --> 00:07:28,365 to the minus beta light Hamiltonian. 138 00:07:28,365 --> 00:07:29,760 PROFESSOR: Yeah. 139 00:07:29,760 --> 00:07:33,450 So our definition of the thermal state-- 140 00:07:33,450 --> 00:07:35,670 when we had thermal light, we say 141 00:07:35,670 --> 00:07:43,370 that the statistical operator is given by that. 142 00:07:43,370 --> 00:07:48,560 And H is the Hamiltonian for a single-mode light, 143 00:07:48,560 --> 00:07:54,920 which is n plus 1/2 nn. 144 00:08:01,290 --> 00:08:05,920 Well, with suitable parentheses and summations. 145 00:08:08,879 --> 00:08:09,545 Other questions? 146 00:08:13,440 --> 00:08:13,940 OK. 147 00:08:20,790 --> 00:08:25,900 Then let's get to the main subject we want to discuss now. 148 00:08:25,900 --> 00:08:28,900 And these are actually Feynman diagrams. 149 00:08:28,900 --> 00:08:35,799 I wanted to give you an exact definition 150 00:08:35,799 --> 00:08:38,299 and a deep understanding, what does 151 00:08:38,299 --> 00:08:42,390 it mean when we talk about processes of absorption 152 00:08:42,390 --> 00:08:47,500 and emission, but also about absorption, emission processes 153 00:08:47,500 --> 00:08:49,820 which violate energy. 154 00:08:49,820 --> 00:08:53,440 And some people refer to them as virtual photons. 155 00:08:53,440 --> 00:08:59,480 The reason is that virtual photons cannot really exist 156 00:08:59,480 --> 00:09:01,490 for a long time. 157 00:09:01,490 --> 00:09:04,040 When you emit a virtual photon, another photon 158 00:09:04,040 --> 00:09:06,950 has to be absorbed immediately to reconcile energy 159 00:09:06,950 --> 00:09:09,560 conservation, as we want to see in a moment. 160 00:09:09,560 --> 00:09:13,980 So the goal of this presentation is 161 00:09:13,980 --> 00:09:18,320 I want you that you realize that each of those doodles 162 00:09:18,320 --> 00:09:20,610 has an exact mathematical meaning. 163 00:09:23,900 --> 00:09:28,465 Each of those diagrams represents one term, 164 00:09:28,465 --> 00:09:32,660 or a class of terms, in an exact solution 165 00:09:32,660 --> 00:09:36,780 for the time evolution of the system. 166 00:09:36,780 --> 00:09:49,870 So in other words, if you would ask me, 167 00:09:49,870 --> 00:09:52,360 we have a ground and excited state. 168 00:09:52,360 --> 00:09:56,210 Is it possible that the ground state emits a photon, 169 00:09:56,210 --> 00:09:58,860 goes to a virtual state, emits another photon 170 00:09:58,860 --> 00:10:04,400 of another frequency, and then somehow absorbs the photon, 171 00:10:04,400 --> 00:10:10,300 goes to here, and eventually takes another photon, 172 00:10:10,300 --> 00:10:12,700 and is back to the ground state. 173 00:10:12,700 --> 00:10:14,200 Is that a possibility? 174 00:10:14,200 --> 00:10:15,900 Can that happen? 175 00:10:15,900 --> 00:10:18,330 And I think what the message here 176 00:10:18,330 --> 00:10:21,130 is yes, everything happens. 177 00:10:21,130 --> 00:10:24,790 The system is trying out all of its possibility. 178 00:10:24,790 --> 00:10:27,820 And the two time evolution is the sum 179 00:10:27,820 --> 00:10:31,500 of all those possibilities, of all the amplitudes related 180 00:10:31,500 --> 00:10:33,270 to those diagrams. 181 00:10:33,270 --> 00:10:36,750 But what I want to show you is how sort of the weirder 182 00:10:36,750 --> 00:10:40,480 the diagrams get, the more you go in energy below the ground 183 00:10:40,480 --> 00:10:44,150 state, the more you go away from real atomic states, 184 00:10:44,150 --> 00:10:46,390 the bigger is your energy denominator. 185 00:10:46,390 --> 00:10:48,930 And that means those diagrams have 186 00:10:48,930 --> 00:10:50,840 a smaller and smaller weight. 187 00:10:50,840 --> 00:10:57,120 And in all practical calculations, we neglect those. 188 00:10:57,120 --> 00:11:01,030 But I want you sort of to be able to see that and realize, 189 00:11:01,030 --> 00:11:02,920 I exactly know what it means. 190 00:11:02,920 --> 00:11:05,800 It means this and this term in a summation 191 00:11:05,800 --> 00:11:07,802 over all the amplitudes which the quantum 192 00:11:07,802 --> 00:11:08,635 system is exploring. 193 00:11:11,065 --> 00:11:13,690 And I think with that, we really learn something about physics. 194 00:11:13,690 --> 00:11:16,830 We learn about what is actually inside the Schrodinger 195 00:11:16,830 --> 00:11:19,120 equation. 196 00:11:19,120 --> 00:11:22,990 A lot of people actually, before they take this class, 197 00:11:22,990 --> 00:11:24,460 think that this is just nonsense, 198 00:11:24,460 --> 00:11:26,060 that this has no physical reality. 199 00:11:26,060 --> 00:11:28,000 But I hope after this class, you see 200 00:11:28,000 --> 00:11:31,770 that pretty much everything you draw has physical reality. 201 00:11:31,770 --> 00:11:34,580 It's just-- it may not contribute a lot. 202 00:11:41,630 --> 00:11:46,600 So what we have done-- and let me just start here 203 00:11:46,600 --> 00:11:49,660 and invite your questions. 204 00:11:49,660 --> 00:11:53,340 We have figured out how an initial state evolves 205 00:11:53,340 --> 00:11:57,810 with a time evolution operator to another basis state, 206 00:11:57,810 --> 00:11:59,640 toward the final state. 207 00:11:59,640 --> 00:12:06,090 And formally, this is the formal solution and exact solution 208 00:12:06,090 --> 00:12:08,220 of the Schrodinger equation. 209 00:12:08,220 --> 00:12:14,491 We have to sum over all orders, orders in the interaction. 210 00:12:14,491 --> 00:12:16,130 Well, I will immediately tell you 211 00:12:16,130 --> 00:12:18,000 what is the first and second order. 212 00:12:18,000 --> 00:12:19,310 We are not going much higher. 213 00:12:19,310 --> 00:12:22,170 But if you want, here, you can. 214 00:12:22,170 --> 00:12:25,980 And then what you have to do is in the time evolution, 215 00:12:25,980 --> 00:12:28,540 you have to sum over intermediate times. 216 00:12:28,540 --> 00:12:31,910 You have to allow the system to propagate, to change its state, 217 00:12:31,910 --> 00:12:34,260 propagate again, change its state again. 218 00:12:34,260 --> 00:12:40,080 And the times where the change of state happens, 219 00:12:40,080 --> 00:12:43,570 that can happen at any time between your initial 220 00:12:43,570 --> 00:12:44,900 and your final time. 221 00:12:44,900 --> 00:12:49,470 And we integrate over all possible times. 222 00:12:49,470 --> 00:12:51,060 And I showed you-- and I think this 223 00:12:51,060 --> 00:12:54,520 was the very last thing we did on Wednesday. 224 00:12:54,520 --> 00:12:58,390 I showed you how this diagram can 225 00:12:58,390 --> 00:13:02,754 be translated into mathematical equation. 226 00:13:02,754 --> 00:13:04,670 And I think I picked the second order diagram. 227 00:13:04,670 --> 00:13:08,000 But I think from the way how I presented should be clear now 228 00:13:08,000 --> 00:13:13,580 how any such diagram can be translated into an equation. 229 00:13:13,580 --> 00:13:16,010 And eventually, you have to perform an integral 230 00:13:16,010 --> 00:13:17,647 over all intermediate times. 231 00:13:17,647 --> 00:13:19,855 And this is part of the time evolution of the system. 232 00:13:22,550 --> 00:13:23,530 Questions about that? 233 00:13:26,470 --> 00:13:28,920 AUDIENCE: We allowed these to be [INAUDIBLE] right? 234 00:13:28,920 --> 00:13:31,860 Because we haven't done anything with that? 235 00:13:34,800 --> 00:13:36,110 PROFESSOR: Good question. 236 00:13:36,110 --> 00:13:38,800 I assume that it's time-independent. 237 00:13:38,800 --> 00:13:43,620 Actually, right now, I assume, just to assume something, 238 00:13:43,620 --> 00:13:47,430 that it's a dipole interaction. 239 00:13:47,430 --> 00:13:52,080 My understanding is if it has an explicit time dependence, 240 00:13:52,080 --> 00:13:58,530 it would just appear there, and it would not change the-- 241 00:13:58,530 --> 00:14:00,600 would it? 242 00:14:00,600 --> 00:14:04,440 Wait, would it? 243 00:14:04,440 --> 00:14:07,030 Actually, when be derived the differential equation 244 00:14:07,030 --> 00:14:08,930 for the time evolution operator, did 245 00:14:08,930 --> 00:14:13,250 we assume that v was time-independent or not? 246 00:14:13,250 --> 00:14:14,150 I don't think we did. 247 00:14:16,820 --> 00:14:19,440 We integrate over time. 248 00:14:19,440 --> 00:14:21,100 I think the formal solution. 249 00:14:21,100 --> 00:14:23,100 Remember, I wrote down the differential equation 250 00:14:23,100 --> 00:14:24,974 for the time evolution operator and then say, 251 00:14:24,974 --> 00:14:28,140 this is a formal solution. 252 00:14:28,140 --> 00:14:29,910 My gut feeling is nothing changes 253 00:14:29,910 --> 00:14:32,990 when v is time-independent, but this one step 254 00:14:32,990 --> 00:14:35,284 should be confirmed. 255 00:14:35,284 --> 00:14:35,950 Other questions? 256 00:14:40,751 --> 00:14:41,250 OK. 257 00:14:41,250 --> 00:14:45,790 Then let me just spend a few minutes on connecting what we 258 00:14:45,790 --> 00:14:48,000 have done with standard first and second 259 00:14:48,000 --> 00:14:50,700 [? auto-perturbation ?] theory. 260 00:14:50,700 --> 00:14:55,720 I want to sort of throw a few definitions at you. 261 00:14:55,720 --> 00:14:57,810 S-matrix, T-matrix. 262 00:14:57,810 --> 00:15:00,130 But I'm not really going into any details. 263 00:15:00,130 --> 00:15:03,520 I just want to sort of wrap up the perturbative treatment 264 00:15:03,520 --> 00:15:07,100 by connecting it with the standard first and second 265 00:15:07,100 --> 00:15:10,040 [? auto-perturbation ?] theory. 266 00:15:10,040 --> 00:15:12,195 But after that in a few minutes, I 267 00:15:12,195 --> 00:15:15,650 want to have a discussion about the nature 268 00:15:15,650 --> 00:15:19,680 of intermediate and so-called virtual state, 269 00:15:19,680 --> 00:15:24,600 and then talk about the interaction v, whether it's 270 00:15:24,600 --> 00:15:28,725 d, dot, e; or the p minus e interaction. 271 00:15:28,725 --> 00:15:29,225 OK. 272 00:15:32,430 --> 00:15:35,050 So far, I've presented the formalism 273 00:15:35,050 --> 00:15:39,050 that we started in initial time and ended at final time. 274 00:15:39,050 --> 00:15:42,250 But usually, these are microscopic times. 275 00:15:42,250 --> 00:15:43,910 And in the experiment, we observe 276 00:15:43,910 --> 00:15:46,460 a system for macroscopic time. 277 00:15:46,460 --> 00:15:49,610 So for that purpose, we usually go to the limit 278 00:15:49,610 --> 00:15:53,685 that initial and final times are infinitely apart. 279 00:15:56,650 --> 00:16:01,210 And that actually means we have energy conservation. 280 00:16:01,210 --> 00:16:04,550 The initial and final energy has to be the same. 281 00:16:04,550 --> 00:16:07,450 And that can be, for instance, even if you restrict ourselves 282 00:16:07,450 --> 00:16:11,110 to second order, remember we had all those propagators, 283 00:16:11,110 --> 00:16:15,270 e to the i energy over H bar times t. 284 00:16:15,270 --> 00:16:18,060 And when we integrated over long times, 285 00:16:18,060 --> 00:16:21,200 it will just average out to zero unless 286 00:16:21,200 --> 00:16:23,672 the initial and final energy are the same. 287 00:16:23,672 --> 00:16:25,130 And technically, you have seen that 288 00:16:25,130 --> 00:16:27,535 in undergraduate derivation second order perturbation 289 00:16:27,535 --> 00:16:28,610 theory. 290 00:16:28,610 --> 00:16:31,650 You integrate the exponential function. 291 00:16:31,650 --> 00:16:35,700 And eventually, for sufficiently long time, 292 00:16:35,700 --> 00:16:37,320 capital T is the difference between 293 00:16:37,320 --> 00:16:38,700 initial and final times. 294 00:16:38,700 --> 00:16:40,630 It approaches the delta function. 295 00:16:40,630 --> 00:16:42,840 So that's how energy conservation comes in. 296 00:16:47,780 --> 00:16:51,480 So the fact that we have energy conservation 297 00:16:51,480 --> 00:16:56,990 is then used to define s and T-matrix. 298 00:16:56,990 --> 00:17:00,970 The transition amplitude from the initial to the final state, 299 00:17:00,970 --> 00:17:03,500 what we have just calculated and discussed, 300 00:17:03,500 --> 00:17:06,170 is called the S-matrix. 301 00:17:06,170 --> 00:17:09,890 It's just how it is called. 302 00:17:09,890 --> 00:17:15,140 In first order, of course, the time evolution 303 00:17:15,140 --> 00:17:16,970 is the unitary matrix. 304 00:17:16,970 --> 00:17:21,960 So therefore, we get [? conical ?] delta. 305 00:17:21,960 --> 00:17:26,619 And then we discussed that in the limit of large times, 306 00:17:26,619 --> 00:17:28,550 we have a delta function. 307 00:17:28,550 --> 00:17:31,670 So therefore, if we take the S-matrix, which 308 00:17:31,670 --> 00:17:34,350 is the transition amplitudes we have calculated, 309 00:17:34,350 --> 00:17:39,430 and sort of take out of the S-matrix, the unity matrix, 310 00:17:39,430 --> 00:17:42,690 and factor out the delta function, then what is left 311 00:17:42,690 --> 00:17:44,500 is the so-called T-matrix. 312 00:17:48,740 --> 00:17:51,200 When we talk about transition amplitude, transition 313 00:17:51,200 --> 00:17:53,420 probabilities, we are asking, what 314 00:17:53,420 --> 00:17:55,600 is the probability that the system 315 00:17:55,600 --> 00:17:58,340 has gone from an initial state, maybe the ground-- 316 00:17:58,340 --> 00:18:00,660 the excited state to the ground state 317 00:18:00,660 --> 00:18:02,620 through spontaneous emission? 318 00:18:02,620 --> 00:18:05,400 Well, probability is an amplitude squared. 319 00:18:05,400 --> 00:18:11,640 So we take the matrix element of the S-matrix and square it. 320 00:18:11,640 --> 00:18:16,130 And from the line above, this is now 321 00:18:16,130 --> 00:18:20,310 involving the matrix element of the T-matrix squared. 322 00:18:23,210 --> 00:18:25,370 There's a delta function which becomes 323 00:18:25,370 --> 00:18:27,620 a delta function squared. 324 00:18:27,620 --> 00:18:31,540 But if you integrate over all final states, that's-- I mean, 325 00:18:31,540 --> 00:18:33,940 a delta function is always requiring that you do some 326 00:18:33,940 --> 00:18:34,870 integration later. 327 00:18:34,870 --> 00:18:37,360 Otherwise, the delta function doesn't make sense. 328 00:18:37,360 --> 00:18:39,490 The delta function squared. 329 00:18:39,490 --> 00:18:43,480 You can actually explicitly see from the sine function above. 330 00:18:43,480 --> 00:18:49,370 The delta function squared just turns into the time t. 331 00:18:49,370 --> 00:18:54,530 So therefore, if you divide the probability by the time, 332 00:18:54,530 --> 00:18:57,290 we have our transition probability. 333 00:18:57,290 --> 00:18:59,690 We have our transition rate. 334 00:18:59,690 --> 00:19:02,761 And what we obtain is the second order expression 335 00:19:02,761 --> 00:19:05,260 for the transition rate, which is essentially Fermi's Golden 336 00:19:05,260 --> 00:19:05,760 Rule. 337 00:19:16,880 --> 00:19:19,110 So anyway, this is just finishing 338 00:19:19,110 --> 00:19:27,430 the formal derivation. 339 00:19:27,430 --> 00:19:29,930 But now I want to discuss the nature 340 00:19:29,930 --> 00:19:32,420 of those intermediate states. 341 00:19:32,420 --> 00:19:34,690 And maybe what you should have in mind-- 342 00:19:34,690 --> 00:19:42,300 the intermediate state, which comes about when we have 343 00:19:42,300 --> 00:19:45,670 the system in the ground state, it emits a photon 344 00:19:45,670 --> 00:19:47,660 and goes down to this intermediate state, 345 00:19:47,660 --> 00:19:51,530 or this weird state which seems to be lower in energy 346 00:19:51,530 --> 00:19:53,280 than the ground state. 347 00:19:53,280 --> 00:19:57,850 Well, what happens is those intermediate states, when they 348 00:19:57,850 --> 00:20:08,900 appear after the vertex, they propagate with the energy. 349 00:20:08,900 --> 00:20:18,564 But if their energy is less than the initial energy, 350 00:20:18,564 --> 00:20:22,400 here, the energy-- the intermediate state k 351 00:20:22,400 --> 00:20:25,190 has a phase factor in its propagation 352 00:20:25,190 --> 00:20:28,380 which is determined by the difference of its energy 353 00:20:28,380 --> 00:20:30,070 with the initial energy. 354 00:20:30,070 --> 00:20:32,550 So when we violate energy conservation 355 00:20:32,550 --> 00:20:40,660 in intermediate state, delta Ek is non-zero, 356 00:20:40,660 --> 00:20:44,680 and it is the larger, the more we violate energy conservation. 357 00:20:44,680 --> 00:20:50,160 And then in the solution of the time evolution, 358 00:20:50,160 --> 00:20:54,140 we have to integrate over all intermediate times. 359 00:20:54,140 --> 00:20:58,140 So what we have here is we have an oscillating phase vector. 360 00:20:58,140 --> 00:21:00,460 And when we integrate it, when we integrate something 361 00:21:00,460 --> 00:21:03,530 oscillating over longer than an oscillation period, 362 00:21:03,530 --> 00:21:06,240 it averages out to zero. 363 00:21:06,240 --> 00:21:09,070 So therefore, those intermediate states 364 00:21:09,070 --> 00:21:13,590 which are off the energy shell, which seem to violate energy, 365 00:21:13,590 --> 00:21:16,280 can only noticeably contribute over 366 00:21:16,280 --> 00:21:21,620 a duration which is 1 over the energy defect, delta Ek. 367 00:21:24,540 --> 00:21:28,650 So it is correct to say that the system in its time 368 00:21:28,650 --> 00:21:33,370 evolution for short times can, so to speak, 369 00:21:33,370 --> 00:21:37,420 violate energy protected by Heisenberg's uncertainty 370 00:21:37,420 --> 00:21:38,570 relation. 371 00:21:38,570 --> 00:21:42,120 Or you can say, the system can do whatever it wants. 372 00:21:42,120 --> 00:21:45,200 It can spontaneously create 10 photons. 373 00:21:45,200 --> 00:21:49,230 But this is pretty much like taking money, taking a deposit 374 00:21:49,230 --> 00:21:50,750 out of Heisenberg's bank. 375 00:21:50,750 --> 00:21:52,346 And after very, very short time, you 376 00:21:52,346 --> 00:21:55,620 have to pay it back by having another process which 377 00:21:55,620 --> 00:22:00,840 brings you back to the correct energy. 378 00:22:00,840 --> 00:22:02,870 I should put it under quotation marks when 379 00:22:02,870 --> 00:22:06,990 I said we violate energy, because energy is not 380 00:22:06,990 --> 00:22:11,240 sort of defined or measurable in this intermediate times. 381 00:22:11,240 --> 00:22:13,630 We start a process with a quantum system, 382 00:22:13,630 --> 00:22:15,310 we look what happens afterwards. 383 00:22:15,310 --> 00:22:17,960 And whenever we assess energy, it 384 00:22:17,960 --> 00:22:24,230 is assessed when the final time is 385 00:22:24,230 --> 00:22:26,210 much, much larger than the initial time. 386 00:22:26,210 --> 00:22:29,380 And I just showed you that eventually, the system 387 00:22:29,380 --> 00:22:33,220 has to be back to a final energy, Ef, 388 00:22:33,220 --> 00:22:36,491 which is identical to the initial time, Ei. 389 00:22:36,491 --> 00:22:38,740 So in other words, I just want you to keep it in mind. 390 00:22:38,740 --> 00:22:41,200 I've proven to you energy conservation 391 00:22:41,200 --> 00:22:46,400 in the limit that t final minus t initial is large. 392 00:22:46,400 --> 00:22:49,900 And when I'm now talking about non-conservation of energy, 393 00:22:49,900 --> 00:22:52,330 I do that in quotation marks, because we 394 00:22:52,330 --> 00:22:54,650 know energy is conserved in the end. 395 00:22:54,650 --> 00:22:58,220 It's just that for very, very short times in the time 396 00:22:58,220 --> 00:23:04,110 evolution of the system, there appear virtual states which 397 00:23:04,110 --> 00:23:06,146 seem to violate energy. 398 00:23:06,146 --> 00:23:07,770 But you can think about it in that way. 399 00:23:10,910 --> 00:23:12,030 Questions about that? 400 00:23:21,420 --> 00:23:27,410 Finally, let me now address the question, 401 00:23:27,410 --> 00:23:31,140 is everything I'm explaining to you really happening? 402 00:23:31,140 --> 00:23:34,200 Is it really happening in a physical system? 403 00:23:34,200 --> 00:23:37,550 Well, the first answer is, I wouldn't tell you 404 00:23:37,550 --> 00:23:39,750 about it if it had no reality. 405 00:23:39,750 --> 00:23:42,290 So yes. 406 00:23:42,290 --> 00:23:46,080 You can imagine that this is what your atom is doing. 407 00:23:46,080 --> 00:23:50,770 You can imagine that the hydrogen atom that's 408 00:23:50,770 --> 00:23:52,880 [INAUDIBLE] [? shift ?] is permanently 409 00:23:52,880 --> 00:23:56,400 emitting and absorbing photons and all sorts 410 00:23:56,400 --> 00:23:58,380 of weird photons from the ground state, 411 00:23:58,380 --> 00:24:01,080 lower, up, and back again, and such. 412 00:24:01,080 --> 00:24:02,760 Yes. 413 00:24:02,760 --> 00:24:06,210 This is the way how we derive some 414 00:24:06,210 --> 00:24:11,930 of the most precise predictions in physics, namely QED. 415 00:24:11,930 --> 00:24:17,830 However, we can represent systems in different gauges, 416 00:24:17,830 --> 00:24:20,040 in different representations. 417 00:24:20,040 --> 00:24:23,870 And we had discussed earlier that we often 418 00:24:23,870 --> 00:24:29,360 take the dipole representation for the light atom interaction. 419 00:24:29,360 --> 00:24:33,250 But there's also the p minus a representation. 420 00:24:33,250 --> 00:24:39,800 And if you look at the two, both the e, dot, d and the a, dot, p 421 00:24:39,800 --> 00:24:45,320 interaction are the product of something 422 00:24:45,320 --> 00:24:48,830 which creates and annihilates single photons-- a plus a 423 00:24:48,830 --> 00:24:53,130 [? dega. ?] And then the p operator or the d operator 424 00:24:53,130 --> 00:24:56,300 connect the ground with the excited state. 425 00:24:56,300 --> 00:24:58,800 So those matrix element would tell you, 426 00:24:58,800 --> 00:25:01,370 you can only emit and absorb a photon 427 00:25:01,370 --> 00:25:04,870 when you go from the ground to the excited state. 428 00:25:04,870 --> 00:25:09,750 However, in the p minus a representation, 429 00:25:09,750 --> 00:25:12,070 we also have the a squared term. 430 00:25:12,070 --> 00:25:15,030 And the a squared term allows you-- 431 00:25:15,030 --> 00:25:17,750 because there is no atomic operator in front of it-- 432 00:25:17,750 --> 00:25:20,970 allows you to scatter two photons 433 00:25:20,970 --> 00:25:25,910 without changing the quantum state of the atom. 434 00:25:25,910 --> 00:25:28,860 But this does not contradict anything. 435 00:25:28,860 --> 00:25:30,946 You can take either approach. 436 00:25:30,946 --> 00:25:32,570 And in your homework, you will actually 437 00:25:32,570 --> 00:25:35,350 do that when you calculate Rayleigh scattering and Thomson 438 00:25:35,350 --> 00:25:36,070 scattering. 439 00:25:36,070 --> 00:25:39,420 And you will find out that the two approaches 440 00:25:39,420 --> 00:25:42,150 give identical results. 441 00:25:42,150 --> 00:25:44,980 So if you know ask the philosophical question, 442 00:25:44,980 --> 00:25:48,720 can an atom scatter a photon without changing its quantum 443 00:25:48,720 --> 00:25:52,210 state, well, the answer to this question 444 00:25:52,210 --> 00:25:53,520 is actually gauge-dependent. 445 00:25:56,150 --> 00:25:59,910 But maybe to lift away some of the confusion, 446 00:25:59,910 --> 00:26:02,610 one can, of course, also say, the a 447 00:26:02,610 --> 00:26:04,610 squared term is really important. 448 00:26:04,610 --> 00:26:07,710 And it's a simple description of Thomson scattering. 449 00:26:07,710 --> 00:26:14,120 Thomson scattering is about photons 450 00:26:14,120 --> 00:26:16,300 which have much, much more energy than the energy 451 00:26:16,300 --> 00:26:18,770 difference between the ground and excited state. 452 00:26:18,770 --> 00:26:21,750 So therefore, if you want to completely describe 453 00:26:21,750 --> 00:26:24,430 the system-- and you can, it's just more complicated-- 454 00:26:24,430 --> 00:26:28,230 with a dipole approximation, we have to scatter a photon. 455 00:26:31,470 --> 00:26:34,430 But because the photon has so much more energy, 456 00:26:34,430 --> 00:26:37,790 it's so far away from resonance, this intermediate state 457 00:26:37,790 --> 00:26:39,640 has a huge energy defect. 458 00:26:39,640 --> 00:26:42,900 And that means what I just explained to you-- 459 00:26:42,900 --> 00:26:44,820 the system wants to immediately pay back 460 00:26:44,820 --> 00:26:46,620 the money to Heisenberg's bank. 461 00:26:46,620 --> 00:26:50,440 So it will remain in this state only for very, very short time. 462 00:26:50,440 --> 00:26:52,220 And now the other gauge tells you, 463 00:26:52,220 --> 00:26:55,800 this very, very short time can also be zero. 464 00:26:55,800 --> 00:27:01,290 So see, it's not as dramatically different as you might assume. 465 00:27:01,290 --> 00:27:03,530 But this is just something which is common 466 00:27:03,530 --> 00:27:05,870 that, if you have different representations, 467 00:27:05,870 --> 00:27:09,250 the physics tastes different, but you always 468 00:27:09,250 --> 00:27:12,860 have to remind you that when you calculate an observable result, 469 00:27:12,860 --> 00:27:16,345 the two different gauges must exactly agree. 470 00:27:22,150 --> 00:27:22,650 Questions? 471 00:27:29,236 --> 00:27:29,735 All right. 472 00:27:32,650 --> 00:27:42,530 So this is all I wanted to say about the diagrammatic solution 473 00:27:42,530 --> 00:27:48,030 for the time evolution of a quantum system. 474 00:27:48,030 --> 00:27:49,150 But I want to use it now. 475 00:27:49,150 --> 00:27:50,710 I think this is interesting stuff. 476 00:27:50,710 --> 00:27:52,110 It's a powerful method. 477 00:27:52,110 --> 00:27:55,040 And I want to illustrate to you how this method can 478 00:27:55,040 --> 00:27:57,430 be used in the next two sections. 479 00:28:06,270 --> 00:28:09,645 Our next section is van der Waal's interactions. 480 00:28:33,140 --> 00:28:40,520 So the chapter on van der Waals interaction 481 00:28:40,520 --> 00:28:46,500 is quite interesting for the following reason. 482 00:28:46,500 --> 00:28:49,460 It really tells us something about the vacuum. 483 00:28:49,460 --> 00:28:52,160 I think modern physics has come to the conclusion 484 00:28:52,160 --> 00:28:55,490 that the vacuum is one of the most interesting subjects 485 00:28:55,490 --> 00:28:58,470 to study, because the vacuum is alive. 486 00:28:58,470 --> 00:29:03,860 It's filled by virtual photons, virtual particles. 487 00:29:03,860 --> 00:29:07,640 And as we know now, by a condensate of Higgs boson. 488 00:29:07,640 --> 00:29:10,470 So there is a lot of stuff, a lot of structure, 489 00:29:10,470 --> 00:29:14,560 a lot of phenomena associated in the vacuum. 490 00:29:14,560 --> 00:29:17,630 And the subject of van der Waals interactions 491 00:29:17,630 --> 00:29:21,180 is it's a nice way to talk about the vacuum, 492 00:29:21,180 --> 00:29:25,180 but it's nice also for the following reason. 493 00:29:25,180 --> 00:29:30,220 There is a completely, I would say, semi-classical way. 494 00:29:30,220 --> 00:29:32,110 Just use the Schrodinger equation 495 00:29:32,110 --> 00:29:34,180 and calculate what is the Van der Waals 496 00:29:34,180 --> 00:29:36,770 interaction between two atoms. 497 00:29:36,770 --> 00:29:39,470 So you can just calculate in perturbation theory. 498 00:29:39,470 --> 00:29:42,690 And in your whole calculation, you never 499 00:29:42,690 --> 00:29:45,840 use the quantized electromagnetic field. 500 00:29:45,840 --> 00:29:47,790 Photons never appear. 501 00:29:47,790 --> 00:29:51,026 You just use perturbation theory. 502 00:29:51,026 --> 00:29:52,400 And you do that in your homework. 503 00:29:52,400 --> 00:29:53,660 It's rather straightforward. 504 00:29:53,660 --> 00:29:56,870 And you could have Van der Waals interaction. 505 00:29:56,870 --> 00:30:05,010 On the other hand, I started to derive for you the theory 506 00:30:05,010 --> 00:30:07,610 of the quantized electromagnetic field by saying, 507 00:30:07,610 --> 00:30:09,880 we really look at everything. 508 00:30:09,880 --> 00:30:14,940 We have charges, and we have the electric and magnetic field. 509 00:30:14,940 --> 00:30:16,810 And we described everything. 510 00:30:16,810 --> 00:30:19,500 And the way how we separate it is we 511 00:30:19,500 --> 00:30:23,560 had this Hamiltonian H naught, which gives us the atom. 512 00:30:23,560 --> 00:30:27,140 And the longitudinal Coulomb field became part of it. 513 00:30:27,140 --> 00:30:29,260 So this is the degree of freedom of the atom. 514 00:30:29,260 --> 00:30:32,180 And we have H naught for the other atom. 515 00:30:32,180 --> 00:30:37,790 And then we had radiation field and the coupling 516 00:30:37,790 --> 00:30:42,330 to the atom described by our operator V, which 517 00:30:42,330 --> 00:30:44,630 can be the dipole operator or can be written down 518 00:30:44,630 --> 00:30:45,910 in other gauge. 519 00:30:45,910 --> 00:30:50,690 But what I'm telling you is the way how we have fundamentally 520 00:30:50,690 --> 00:30:54,600 divided the world into atoms, atom's neutral objects, 521 00:30:54,600 --> 00:30:58,690 and the rest is interactions with the radiation field. 522 00:30:58,690 --> 00:31:01,290 That should tell you that the Van der Waals 523 00:31:01,290 --> 00:31:08,600 interaction between two atoms must have a description in QED 524 00:31:08,600 --> 00:31:11,200 where the Van der Waals indirection between atoms 525 00:31:11,200 --> 00:31:13,060 comes from the exchange or photons. 526 00:31:16,690 --> 00:31:19,590 So there are two very simple pictures at the same physics. 527 00:31:19,590 --> 00:31:21,590 One is you don't even know that they're photons, 528 00:31:21,590 --> 00:31:23,500 you just do perturbation theory. 529 00:31:23,500 --> 00:31:26,860 But in a more comprehensive description 530 00:31:26,860 --> 00:31:28,920 where we include the photons, you 531 00:31:28,920 --> 00:31:32,000 should be able to understand the Van der Waals 532 00:31:32,000 --> 00:31:36,040 action between two atoms due to the exchange 533 00:31:36,040 --> 00:31:37,920 of virtual photons. 534 00:31:37,920 --> 00:31:40,910 In other words, one atom is in the ground state, 535 00:31:40,910 --> 00:31:47,380 emits a photon that's going down from the ground state. 536 00:31:47,380 --> 00:31:51,540 But until now, if you have only one atom, 537 00:31:51,540 --> 00:31:53,817 it has to reabsorb the photon again and be back 538 00:31:53,817 --> 00:31:54,650 in the ground state. 539 00:31:54,650 --> 00:31:56,570 Otherwise, energy would be violated. 540 00:31:56,570 --> 00:32:00,810 But if you have two atoms, one atom can emit a photon, 541 00:32:00,810 --> 00:32:02,850 and the other atom can absorb it. 542 00:32:02,850 --> 00:32:05,980 And then the other atom can send a photon back. 543 00:32:05,980 --> 00:32:08,340 And if we consider that process, we 544 00:32:08,340 --> 00:32:12,880 will actually find the Van der Waals interaction. 545 00:32:12,880 --> 00:32:17,320 So I hope this is showing interesting physics from two 546 00:32:17,320 --> 00:32:19,330 angles-- that something which looked maybe 547 00:32:19,330 --> 00:32:22,860 trivial a long time ago now looks 548 00:32:22,860 --> 00:32:28,370 much richer, because those forces are really 549 00:32:28,370 --> 00:32:33,000 mediated by virtual photon pairs. 550 00:32:33,000 --> 00:32:39,910 So that's sort of the discussion I want to go through. 551 00:32:39,910 --> 00:32:41,580 There is another aspect to it. 552 00:32:41,580 --> 00:32:44,640 And this is when we go from the Van der Waals force 553 00:32:44,640 --> 00:32:47,720 to the Casimir force. 554 00:32:47,720 --> 00:32:50,670 The Casimir force has one exact derivation, 555 00:32:50,670 --> 00:32:53,520 which I want to share with you, which relates the Casimir 556 00:32:53,520 --> 00:32:57,980 force to the vacuum fluctuations of the electromagnetic field. 557 00:32:57,980 --> 00:33:01,870 So eventually, for those forces between two metal 558 00:33:01,870 --> 00:33:04,370 plates-- a neutral atom in the plate, 559 00:33:04,370 --> 00:33:08,160 two neutral atoms-- we will have three different pictures. 560 00:33:08,160 --> 00:33:10,420 One is we use the semi-classical dipole field 561 00:33:10,420 --> 00:33:11,809 as a perturbation operator. 562 00:33:11,809 --> 00:33:12,850 You don't think about it. 563 00:33:12,850 --> 00:33:15,130 It's trivial, and you check it off in your homework. 564 00:33:15,130 --> 00:33:17,530 The second one is you look at the exchange 565 00:33:17,530 --> 00:33:19,180 of virtual photons. 566 00:33:19,180 --> 00:33:20,980 And the third one is you only look 567 00:33:20,980 --> 00:33:24,950 at the zero-point fluctuations of the electromagnetic field. 568 00:33:24,950 --> 00:33:26,230 What is real here? 569 00:33:26,230 --> 00:33:27,220 What causes it? 570 00:33:30,440 --> 00:33:32,170 We'll see.