1 00:00:00,060 --> 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 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,238 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,238 --> 00:00:17,863 at ocw.mit.edu. 8 00:00:21,490 --> 00:00:26,430 PROFESSOR: So the subject here is about 9 00:00:26,430 --> 00:00:30,830 if you want to understand interactions 10 00:00:30,830 --> 00:00:31,990 between neutral objects. 11 00:00:46,880 --> 00:00:50,870 So let's start in a very basic way 12 00:00:50,870 --> 00:00:57,380 by saying we have an atome A, we have an atom B, 13 00:00:57,380 --> 00:01:02,680 they are separated by a distance R, 14 00:01:02,680 --> 00:01:09,130 and we want to know what is the force between them. 15 00:01:09,130 --> 00:01:11,500 In a semiclassical picture, we would 16 00:01:11,500 --> 00:01:18,530 say that everything is-- the force which 17 00:01:18,530 --> 00:01:22,670 is responsible for that must be the Coulomb force. 18 00:01:22,670 --> 00:01:28,670 So the atom A and B consists of charges, 19 00:01:28,670 --> 00:01:31,160 so we should find the charge density 20 00:01:31,160 --> 00:01:35,262 in our system A, the charge density of B, 21 00:01:35,262 --> 00:01:44,880 we take the coulomb energy and we integrate over the volume. 22 00:01:44,880 --> 00:01:49,930 And this is how in e and m, in the most general way, 23 00:01:49,930 --> 00:01:55,760 you would to write down the electrostatic energy 24 00:01:55,760 --> 00:01:56,720 of a system. 25 00:01:56,720 --> 00:01:59,720 You take all charges, each charge 26 00:01:59,720 --> 00:02:02,270 has a coulomb potential with the other charge [INAUDIBLE] 27 00:02:02,270 --> 00:02:04,540 indicate over everything. 28 00:02:04,540 --> 00:02:10,080 But for objects which are localized 29 00:02:10,080 --> 00:02:15,810 and we can do multiple expansion, 30 00:02:15,810 --> 00:02:18,410 our objects are neutral, neutral atom. 31 00:02:18,410 --> 00:02:20,740 So therefore there is no coulomb term, 32 00:02:20,740 --> 00:02:25,090 and the next term is the dipole. 33 00:02:25,090 --> 00:02:31,900 So therefore, starting with classical physics, 34 00:02:31,900 --> 00:02:41,130 we would find that the interaction of those charge 35 00:02:41,130 --> 00:02:47,770 density distributions one for atom A, one for atom B, 36 00:02:47,770 --> 00:02:51,990 should actually interact with this Hamiltonian h prime. 37 00:03:03,930 --> 00:03:13,200 OK, but now, if you look at it classically, 38 00:03:13,200 --> 00:03:18,100 we find that classically, the expectation 39 00:03:18,100 --> 00:03:21,220 value for the dipole operator of each atom is zero. 40 00:03:24,030 --> 00:03:26,870 But you will also find-- if you said, 41 00:03:26,870 --> 00:03:29,650 well, OK, that's what I expect, because an atom 42 00:03:29,650 --> 00:03:31,280 is in isotopic space. 43 00:03:31,280 --> 00:03:33,000 How can it have a dipole moment? 44 00:03:33,000 --> 00:03:36,170 But maybe, what could happen is, if there 45 00:03:36,170 --> 00:03:38,810 is a fluctuation of a dipole moment here 46 00:03:38,810 --> 00:03:42,560 and a fluctuation there and the fluctuations are correlated, 47 00:03:42,560 --> 00:03:45,450 then the two systems may repel or attract each other. 48 00:03:45,450 --> 00:03:49,360 So therefore, when we are talking about fluctuations, 49 00:03:49,360 --> 00:03:53,990 we are talking about expectation values for d squared. 50 00:03:53,990 --> 00:03:59,090 But also in classical physics, this is exactly zero. 51 00:03:59,090 --> 00:04:02,210 So what classical physics tells us, 52 00:04:02,210 --> 00:04:04,990 it says, in classical physics, it 53 00:04:04,990 --> 00:04:10,450 costs more energy to create a dipole moment, or even 54 00:04:10,450 --> 00:04:13,554 a fluctuating dipole moment, than one gets back 55 00:04:13,554 --> 00:04:14,720 from the dipole interaction. 56 00:04:17,630 --> 00:04:20,240 So the system will never spontaneously form 57 00:04:20,240 --> 00:04:21,899 some form of dipole moment. 58 00:04:21,899 --> 00:04:24,600 And in classical physics, two neutral atoms 59 00:04:24,600 --> 00:04:26,598 will have no attraction at all. 60 00:04:29,920 --> 00:04:34,225 So it costs more energy. 61 00:05:03,110 --> 00:05:06,610 So now, we have to do quantum mechanics. 62 00:05:06,610 --> 00:05:11,120 And every so often, I have to go through a classical argument 63 00:05:11,120 --> 00:05:14,310 to make you wonder about the quantum mechanical result. 64 00:05:14,310 --> 00:05:16,840 Because the quantum mechanical result, you're used to it. 65 00:05:16,840 --> 00:05:19,230 You've heard about the van der Waals interaction. 66 00:05:19,230 --> 00:05:22,010 But now you know that there is really 67 00:05:22,010 --> 00:05:24,110 something quantum at work. 68 00:05:24,110 --> 00:05:32,350 In quantum mechanics, the expectation value 69 00:05:32,350 --> 00:05:37,080 of the square of the dipole operator is not zero. 70 00:05:37,080 --> 00:05:41,990 I mean, you can see that if you have an electron, which 71 00:05:41,990 --> 00:05:44,910 is in an harmonic oscillator, and if it 72 00:05:44,910 --> 00:05:53,760 is at the bottom of the oscillator-- well, 73 00:05:53,760 --> 00:05:57,150 the dipole moment-- what I want you to say 74 00:05:57,150 --> 00:06:00,300 is the dipole moment for a charged system 75 00:06:00,300 --> 00:06:04,210 is related to-- between electron and proton-- 76 00:06:04,210 --> 00:06:07,980 to the displacement operator, R. 77 00:06:07,980 --> 00:06:10,380 And even if you have an equilibrium position-- 78 00:06:10,380 --> 00:06:13,360 like in an harmonic oscillator, where R is equal to zero, 79 00:06:13,360 --> 00:06:16,480 R squared-- the expectation value-- is not zero, 80 00:06:16,480 --> 00:06:18,940 because whatever your stable equilibrium is, 81 00:06:18,940 --> 00:06:21,490 you will have zero point fluctuations around it. 82 00:06:21,490 --> 00:06:24,300 And so therefore, now in quantum physics, 83 00:06:24,300 --> 00:06:29,980 it's not that we have to provide energy to create a d squared. 84 00:06:29,980 --> 00:06:30,850 We have it. 85 00:06:30,850 --> 00:06:34,640 We have it by necessity, because of non-commuting operators, 86 00:06:34,640 --> 00:06:37,680 because of the nature of the quantum physics. 87 00:06:37,680 --> 00:06:40,920 So now, the fact that we have fluctuating dipole moments 88 00:06:40,920 --> 00:06:44,750 already, that means if now with the fluctuating dipole 89 00:06:44,750 --> 00:06:48,510 moments between the two atoms are synchronized, 90 00:06:48,510 --> 00:06:51,750 this will then lead to a non-vanishing average force 91 00:06:51,750 --> 00:06:53,230 between the atoms. 92 00:06:53,230 --> 00:06:55,100 So that's the idea. 93 00:06:55,100 --> 00:06:57,690 But the good thing is, you don't even 94 00:06:57,690 --> 00:07:00,780 have to know about it-- about fluctuations and all that. 95 00:07:00,780 --> 00:07:03,550 You just take-- and this is your homework-- 96 00:07:03,550 --> 00:07:09,060 you just take the operator H prime. 97 00:07:09,060 --> 00:07:12,160 And look what happens when you have an atom in the ground 98 00:07:12,160 --> 00:07:15,800 and excited state and what happens 99 00:07:15,800 --> 00:07:18,550 when you have both atoms in the ground state. 100 00:07:18,550 --> 00:07:22,450 In the first case, you get the leading result 101 00:07:22,450 --> 00:07:25,600 in first order perturbation theory. 102 00:07:25,600 --> 00:07:30,190 For two ground state atoms, you have to go to second order. 103 00:07:30,190 --> 00:07:38,880 In first order, the potential is 1 over R cubed. 104 00:07:38,880 --> 00:07:42,420 And you will calculate the C3 coefficient. 105 00:07:42,420 --> 00:07:45,480 In second order, you get a van der Waals potential, 106 00:07:45,480 --> 00:07:48,180 which is 1 over R to the sixth. 107 00:07:48,180 --> 00:07:51,380 And it's fairly straightforward to calculate 108 00:07:51,380 --> 00:07:52,570 the C6 coefficient. 109 00:08:00,450 --> 00:08:01,110 Any questions? 110 00:08:07,870 --> 00:08:14,000 So let me reemphasize, the beauty of this perturbation 111 00:08:14,000 --> 00:08:15,950 result that it's so simple. 112 00:08:15,950 --> 00:08:17,640 But the downside is, you don't really 113 00:08:17,640 --> 00:08:18,850 understand what you're doing. 114 00:08:18,850 --> 00:08:20,891 I mean, you are solving the Schrodinger equation. 115 00:08:20,891 --> 00:08:22,710 You are finding an expectation value. 116 00:08:22,710 --> 00:08:25,490 But the nature of the effect-- how 117 00:08:25,490 --> 00:08:28,420 it is related to fluctuation, what is really behind it-- 118 00:08:28,420 --> 00:08:31,640 is sometimes obscured. 119 00:08:31,640 --> 00:08:34,031 So let me give you one interpretation, which I really 120 00:08:34,031 --> 00:08:34,530 like. 121 00:08:40,610 --> 00:08:43,379 And I learned that actually from Dan Kleppner. 122 00:08:47,660 --> 00:08:51,120 When he taught the atomic physics course at MIT 123 00:08:51,120 --> 00:08:55,620 20 years ago, that was one element of it. 124 00:08:55,620 --> 00:08:57,830 And I told you at the beginning that I'm actually 125 00:08:57,830 --> 00:09:01,310 proud that our atomic physics course is really 126 00:09:01,310 --> 00:09:04,200 kind of-- has hopefully kept-- the best of the decade 127 00:09:04,200 --> 00:09:07,550 long tradition of teachers like Norman Ramsey, Dan 128 00:09:07,550 --> 00:09:09,190 Kleppner, and Dave Pritchard. 129 00:09:09,190 --> 00:09:11,860 So this is now really due to Dan Kleppner. 130 00:09:11,860 --> 00:09:14,570 And I will later today actually post 131 00:09:14,570 --> 00:09:18,480 Dan Kleppner's original lecture notes about this effect. 132 00:09:18,480 --> 00:09:22,190 So we want to understand the van der Waals 133 00:09:22,190 --> 00:09:24,640 force in a completely different system. 134 00:09:24,640 --> 00:09:26,410 And I think that helps. 135 00:09:26,410 --> 00:09:31,025 We have two LC circuits. 136 00:09:34,634 --> 00:09:44,510 There's L, C, C, L. 137 00:09:44,510 --> 00:09:52,310 And what happens is, the two capacitors are closed. 138 00:09:52,310 --> 00:09:54,937 So there is this stray field of one capacitor reaching 139 00:09:54,937 --> 00:09:55,770 the other capacitor. 140 00:10:00,140 --> 00:10:03,590 So I won't go through the detailed math here, 141 00:10:03,590 --> 00:10:05,630 because it's not necessary. 142 00:10:05,630 --> 00:10:08,420 But what we have here is, we have 143 00:10:08,420 --> 00:10:12,440 two coupled systems-- two coupled capacitors. 144 00:10:15,450 --> 00:10:18,790 And the only thing I want to use is 145 00:10:18,790 --> 00:10:21,260 that the stray field of a capacitor 146 00:10:21,260 --> 00:10:24,360 has a diopolic characteristic. 147 00:10:24,360 --> 00:10:27,310 And it decays with 1 over R cubed. 148 00:10:31,230 --> 00:10:36,150 So what we have right now is, each LC circuit, 149 00:10:36,150 --> 00:10:38,870 when it's isolated, has a resonant frequency 150 00:10:38,870 --> 00:10:41,810 of omega naught. 151 00:10:41,810 --> 00:10:45,010 But now, when we have two pendulums-- two oscillators-- 152 00:10:45,010 --> 00:10:47,970 which are coupled, we get two modes, 153 00:10:47,970 --> 00:10:51,950 which are omega plus and minus. 154 00:10:51,950 --> 00:10:54,790 And if we simply solve for two modes, 155 00:10:54,790 --> 00:10:57,030 and they have some coupling, well, we 156 00:10:57,030 --> 00:11:00,690 get an upshift and downshift by the coupling. 157 00:11:00,690 --> 00:11:04,090 But if you look more carefully at the solution for two 158 00:11:04,090 --> 00:11:07,670 coupled modes, it has also a quadratic effect, 159 00:11:07,670 --> 00:11:11,830 which is sometimes neglected, but it's there. 160 00:11:11,830 --> 00:11:13,400 So this is, in general, what you will 161 00:11:13,400 --> 00:11:16,970 find, if you couple two modes. 162 00:11:16,970 --> 00:11:22,400 Now in quantum mechanics, when we 163 00:11:22,400 --> 00:11:29,830 ask for the oscillator in the ground state, 164 00:11:29,830 --> 00:11:32,295 we can find the zero point energy. 165 00:11:36,970 --> 00:11:39,650 And so the energy in the ground state 166 00:11:39,650 --> 00:11:45,325 is 1/2 the zero point energy in the mode plus, 167 00:11:45,325 --> 00:11:47,890 in the mode minus. 168 00:11:47,890 --> 00:11:53,730 The first order coupling term, the 1 over R cubed cancels out. 169 00:11:53,730 --> 00:11:57,670 But what remains is a contribution, 170 00:11:57,670 --> 00:11:59,150 which is 1 over R to the sixth. 171 00:12:05,810 --> 00:12:07,160 So we need quantum physics. 172 00:12:10,160 --> 00:12:16,050 Because in classical physics, the ground state 173 00:12:16,050 --> 00:12:19,910 of the LC circuit is nothing happens. 174 00:12:19,910 --> 00:12:22,022 No charge, no current, no nothing. 175 00:12:22,022 --> 00:12:23,730 I mean, this is a classical ground state, 176 00:12:23,730 --> 00:12:25,890 and you would not get any effect. 177 00:12:25,890 --> 00:12:31,480 So realize that this van der Waals potential requires 178 00:12:31,480 --> 00:12:42,980 quantum mechanics, and it is due to the zero point 179 00:12:42,980 --> 00:12:51,435 energy of the atomic oscillators. 180 00:13:03,000 --> 00:13:06,400 Just hold the thought, when we talk 181 00:13:06,400 --> 00:13:09,210 about atoms in the electromagnetic field, 182 00:13:09,210 --> 00:13:10,880 we have two oscillators. 183 00:13:10,880 --> 00:13:12,590 We have an harmonic oscillator, which 184 00:13:12,590 --> 00:13:16,000 is a fluctuating dipole moment on each side of the atom. 185 00:13:16,000 --> 00:13:19,190 But we have many, many harmonic oscillators in between, 186 00:13:19,190 --> 00:13:21,750 which is the electromagnetic field. 187 00:13:21,750 --> 00:13:26,870 And during the course of this lecture, 188 00:13:26,870 --> 00:13:30,520 I will refer to both the zero point 189 00:13:30,520 --> 00:13:34,340 energy of the atomic oscillator and the zero point 190 00:13:34,340 --> 00:13:38,110 energy of the oscillator, which is the electromagnetic field. 191 00:13:38,110 --> 00:13:42,840 And to give you the result at the end, 192 00:13:42,840 --> 00:13:46,110 I will sort of explain different effects 193 00:13:46,110 --> 00:13:49,380 by the zero point oscillation of the atoms, other effects 194 00:13:49,380 --> 00:13:53,060 by the zero point oscillation of the electromagnetic field. 195 00:13:53,060 --> 00:13:55,990 But at the end of this section, which is probably 196 00:13:55,990 --> 00:13:58,350 after spring break on Monday, I will actually 197 00:13:58,350 --> 00:14:04,180 tell you that you have to be careful. 198 00:14:04,180 --> 00:14:06,630 If you have quantum physics, you have 199 00:14:06,630 --> 00:14:09,450 to use a consistent description. 200 00:14:09,450 --> 00:14:14,330 And you always need that both the field and the atoms 201 00:14:14,330 --> 00:14:17,090 are quantized and give zero point fluctuations. 202 00:14:17,090 --> 00:14:20,650 And when I'm telling you now that the van der Waals 203 00:14:20,650 --> 00:14:22,590 potential is due only to zero point 204 00:14:22,590 --> 00:14:24,830 fluctuations of the atomic oscillator, 205 00:14:24,830 --> 00:14:27,880 there may be another way of getting the same result 206 00:14:27,880 --> 00:14:29,670 by looking at the zero point fluctuations 207 00:14:29,670 --> 00:14:32,570 of the electromagnetic field. 208 00:14:32,570 --> 00:14:36,100 So I just want you to take what I'm 209 00:14:36,100 --> 00:14:38,480 saying to you-- I'm not saying to you anything wrong. 210 00:14:38,480 --> 00:14:41,950 But it may not be the only way of expressing it. 211 00:14:41,950 --> 00:14:45,480 And I want you just to pay attention to it. 212 00:14:45,480 --> 00:14:49,880 But certain effects can be simply described 213 00:14:49,880 --> 00:14:52,390 by just simply looking at the atomic oscillators. 214 00:14:52,390 --> 00:14:55,190 Why do we need the quantized electromagnetic field 215 00:14:55,190 --> 00:14:57,410 when all we want to do is have a stray field 216 00:14:57,410 --> 00:14:58,830 coupling two capacitors. 217 00:14:58,830 --> 00:15:02,315 So we want to always learn about the simplest 218 00:15:02,315 --> 00:15:04,190 way to understanding an effect. 219 00:15:04,190 --> 00:15:08,810 But then in the end, there are maybe certain subtleties. 220 00:15:08,810 --> 00:15:09,430 OK. 221 00:15:09,430 --> 00:15:11,150 But let's keep it simple. 222 00:15:11,150 --> 00:15:15,100 So it's a zero point energy of the two oscillators. 223 00:15:15,100 --> 00:15:29,100 And maybe let me emphasize this by translating 224 00:15:29,100 --> 00:15:34,770 this description-- what we learned from the capacitors. 225 00:15:34,770 --> 00:15:38,130 Let me give you the Coulomb description of the atom. 226 00:15:38,130 --> 00:15:44,140 So an atom really never have in free space a spontaneous dipole 227 00:15:44,140 --> 00:15:46,500 moment. 228 00:15:46,500 --> 00:15:51,940 But because of zero point oscillation, 229 00:15:51,940 --> 00:15:55,400 it will have a dipole moment squared. 230 00:15:55,400 --> 00:16:01,240 So what happens now is, let's say at some moment, 231 00:16:01,240 --> 00:16:02,410 we have a fluctuation. 232 00:16:02,410 --> 00:16:05,090 One atom happens to have a fluctuation 233 00:16:05,090 --> 00:16:06,730 of the dipole moment. 234 00:16:06,730 --> 00:16:20,140 This creates an electric field at the position of the atom, b. 235 00:16:20,140 --> 00:16:21,790 So this is a fluctuating field. 236 00:16:26,090 --> 00:16:28,830 The fluctuating field, because there's 237 00:16:28,830 --> 00:16:39,490 a finite polarizability, induces now 238 00:16:39,490 --> 00:16:47,460 a dipole moment of the atom, db. 239 00:16:47,460 --> 00:16:49,610 I will later introduced polarizabilities. 240 00:16:49,610 --> 00:16:52,420 But here, it's just proportional to Eb. 241 00:16:52,420 --> 00:16:57,794 So that's proportional to da over R cubed. 242 00:17:00,460 --> 00:17:04,619 So now, we have no longer randomly fluctuating 243 00:17:04,619 --> 00:17:07,869 dipole moments with da db, which are out of phase. 244 00:17:07,869 --> 00:17:10,329 Positive and minus cancels out. 245 00:17:10,329 --> 00:17:13,550 We have a dipole moment, da, which 246 00:17:13,550 --> 00:17:18,280 creates-- through polarizability-- 247 00:17:18,280 --> 00:17:21,569 an aligned dipole moment, db. 248 00:17:21,569 --> 00:17:30,700 And for those, we have now the dipole-dipole interaction, 249 00:17:30,700 --> 00:17:36,370 which is db da over R cubed. 250 00:17:36,370 --> 00:17:39,310 And when db changes sign, da changes signs. 251 00:17:39,310 --> 00:17:42,300 So the product of them will always stay the same. 252 00:17:42,300 --> 00:17:47,110 And now, we find that we have an interaction. 253 00:17:47,110 --> 00:17:51,360 If you look, db was 1 over R cubed. 254 00:17:51,360 --> 00:17:54,190 And if you multiply that with the R 255 00:17:54,190 --> 00:17:57,500 cubed of the dipole-dipole interaction, 256 00:17:57,500 --> 00:18:01,070 we obtain, in a very different way, 257 00:18:01,070 --> 00:18:06,220 the van der Waals potential, which is 1 over R to the sixth. 258 00:18:06,220 --> 00:18:09,810 So therefore, what we can take away 259 00:18:09,810 --> 00:18:20,830 from that is that the 1 over R to the sixth potential 260 00:18:20,830 --> 00:18:34,970 is caused by the zero point fluctuations 261 00:18:34,970 --> 00:18:36,510 of the atomic dipole moments. 262 00:18:49,010 --> 00:18:49,510 OK. 263 00:18:54,550 --> 00:18:57,900 So now you have already three different ways 264 00:18:57,900 --> 00:18:59,840 to look at the van der Waals potential. 265 00:18:59,840 --> 00:19:01,480 One is just doing perturbation theory 266 00:19:01,480 --> 00:19:03,200 and not understanding anything. 267 00:19:03,200 --> 00:19:05,850 The second one is the ground state 268 00:19:05,850 --> 00:19:09,000 of two coupled oscillators, a la two capacitors. 269 00:19:09,000 --> 00:19:14,310 And finally, the fact that spontaneous fluctuations create 270 00:19:14,310 --> 00:19:16,730 stimulated fluctuations, and then there 271 00:19:16,730 --> 00:19:20,590 is a quadratic term, which doesn't vanish. 272 00:19:20,590 --> 00:19:22,230 OK. 273 00:19:22,230 --> 00:19:30,166 Well, let's now consider that the electromagnetic field-- 274 00:19:30,166 --> 00:19:31,790 that there is an electromagnetic field. 275 00:19:31,790 --> 00:19:34,215 Until now, we have not really used the fact 276 00:19:34,215 --> 00:19:37,410 that we have an electromagnetic field with photons. 277 00:19:37,410 --> 00:19:39,810 And the electromagnetic fields, each mode, 278 00:19:39,810 --> 00:19:41,300 is an harmonic oscillator. 279 00:19:41,300 --> 00:19:43,170 And each mode of the electromagnetic field 280 00:19:43,170 --> 00:19:47,000 has zero point fluctuation of itself. 281 00:19:47,000 --> 00:20:00,250 So let us now discuss what can the vacuum fluctuations 282 00:20:00,250 --> 00:20:02,300 of the electromagnetic field do for us. 283 00:20:05,780 --> 00:20:14,510 Well, if we have fluctuations-- momentary fluctuations-- 284 00:20:14,510 --> 00:20:17,710 of the electric field of the vacuum, 285 00:20:17,710 --> 00:20:24,680 the fluctuation at position, a, and position, b, 286 00:20:24,680 --> 00:20:31,104 will induce now dipole moments of the atoms, a and b, 287 00:20:31,104 --> 00:20:35,280 by multiplying with the atomic polarizability. 288 00:20:35,280 --> 00:20:38,350 And if you assume-- I'll say a little bit more about it-- 289 00:20:38,350 --> 00:20:40,830 that we're talking about the long wavelengths 290 00:20:40,830 --> 00:20:42,850 fluctuation of the electric field. 291 00:20:42,850 --> 00:20:44,520 Then the two dipole moments, which 292 00:20:44,520 --> 00:20:46,800 are induced by those vacuum fluctuations, 293 00:20:46,800 --> 00:20:49,390 are actually in phase. 294 00:20:49,390 --> 00:20:53,700 So the key word is now that the vacuum fluctuations 295 00:20:53,700 --> 00:20:57,255 create correlated dipole moments. 296 00:21:02,210 --> 00:21:04,422 So the dipole moments are no longer independently 297 00:21:04,422 --> 00:21:05,505 fluctuating by themselves. 298 00:21:08,010 --> 00:21:26,260 So if you write down now the dipole-dipole interaction, 299 00:21:26,260 --> 00:21:31,414 we will find a result which is proportional to the product 300 00:21:31,414 --> 00:21:32,580 of the two polarizabilities. 301 00:21:37,330 --> 00:21:40,520 The dipole interaction has an intrinsic spatial dependence, 302 00:21:40,520 --> 00:21:43,200 which is 1 over R cubed. 303 00:21:43,200 --> 00:21:48,590 But now, the important term is that we 304 00:21:48,590 --> 00:22:03,430 have to calculate the correlation 305 00:22:03,430 --> 00:22:07,500 function of the electric field at position, 306 00:22:07,500 --> 00:22:08,722 a, and position, b. 307 00:22:08,722 --> 00:22:10,305 And these are the vacuum fluctuations. 308 00:22:19,212 --> 00:22:20,920 I'm not doing the calculation, because it 309 00:22:20,920 --> 00:22:22,640 would be somewhat messy. 310 00:22:22,640 --> 00:22:27,620 I really post on the web a fairly easy-to-read paper 311 00:22:27,620 --> 00:22:30,790 by Larry Spruch in Physics Today. 312 00:22:30,790 --> 00:22:33,320 And he gives you a little bit more details. 313 00:22:36,880 --> 00:22:39,320 But you have already the physics in this equation. 314 00:22:39,320 --> 00:22:41,165 What you should use now is, you have 315 00:22:41,165 --> 00:22:42,950 to use the density of modes. 316 00:22:46,480 --> 00:22:55,890 And for each mode, the zero point energy, 1/2 h-bar omega. 317 00:23:04,880 --> 00:23:12,690 And then, what you get is, you get an interaction, 318 00:23:12,690 --> 00:23:18,960 which is 1 over R to the seventh. 319 00:23:18,960 --> 00:23:20,103 Yes. 320 00:23:20,103 --> 00:23:22,075 AUDIENCE: When you're accounting here 321 00:23:22,075 --> 00:23:25,033 only the dipole moment, which are [INAUDIBLE]. 322 00:23:25,033 --> 00:23:27,498 So they're coming from the fluctuation 323 00:23:27,498 --> 00:23:30,949 of long wavelengths in your EM. 324 00:23:30,949 --> 00:23:33,907 And once they [INAUDIBLE] those but they 325 00:23:33,907 --> 00:23:38,837 are EM field fluctuation from vacuum. 326 00:23:38,837 --> 00:23:41,795 So what are the effect of those shorter waves? 327 00:23:41,795 --> 00:23:44,270 Are they just higher energies, or just-- 328 00:23:44,270 --> 00:23:45,730 PROFESSOR: Excellent question. 329 00:23:45,730 --> 00:23:48,800 Actually, I need that-- in 10 minutes-- 330 00:23:48,800 --> 00:23:51,410 I need exactly the answer to your question. 331 00:23:51,410 --> 00:23:53,740 So let me give you the answer right now. 332 00:23:53,740 --> 00:23:55,810 If you look at this expression, there 333 00:23:55,810 --> 00:23:58,460 are two things which come into place. 334 00:23:58,460 --> 00:24:06,030 One is we have now two atoms separated by a distance, R, 335 00:24:06,030 --> 00:24:09,060 which is ra minus rb . 336 00:24:09,060 --> 00:24:13,250 If you take, now, high frequency modes, which are very short 337 00:24:13,250 --> 00:24:15,730 wavelengths, and you integrate and sum over 338 00:24:15,730 --> 00:24:21,440 all of them, it's pretty clear that you will get 339 00:24:21,440 --> 00:24:24,780 plus and minus, which will completely average out. 340 00:24:24,780 --> 00:24:35,430 So it seems very clear that the modes with wavelengths, 341 00:24:35,430 --> 00:24:40,930 lambda, shorter-- definitely much shorter-- than R average 342 00:24:40,930 --> 00:24:41,430 out. 343 00:24:44,960 --> 00:24:47,800 So therefore, the bulk of the contribution 344 00:24:47,800 --> 00:24:50,960 will actually come from the modes with wavelengths 345 00:24:50,960 --> 00:24:53,310 smaller than R. 346 00:24:53,310 --> 00:24:55,080 But now, there is something else. 347 00:24:55,080 --> 00:24:58,375 The density of states for the electromagnetic field 348 00:24:58,375 --> 00:25:02,480 dramatically increases with frequency, omega. 349 00:25:02,480 --> 00:25:05,820 So we have many more modes at short wavelengths. 350 00:25:05,820 --> 00:25:09,130 So therefore, the argument says, well, 351 00:25:09,130 --> 00:25:11,480 the long wavelength modes, up to wavelengths, 352 00:25:11,480 --> 00:25:13,850 R, are the dominant ones. 353 00:25:13,850 --> 00:25:15,540 They don't average out. 354 00:25:15,540 --> 00:25:18,730 But because the density of modes increases, 355 00:25:18,730 --> 00:25:20,780 the shorter the wavelength is, by 356 00:25:20,780 --> 00:25:22,865 far the dominant contribution, will 357 00:25:22,865 --> 00:25:32,350 come from the modes, which are centered around lambda 358 00:25:32,350 --> 00:25:34,840 equals R. 359 00:25:34,840 --> 00:25:41,490 So the density of modes implies now 360 00:25:41,490 --> 00:25:43,130 that those modes will dominate. 361 00:25:45,670 --> 00:25:47,480 So that's the answer. 362 00:25:47,480 --> 00:25:51,580 But ultimately, you have to put everything into an equation, 363 00:25:51,580 --> 00:25:53,410 have an integral, solve the integral, 364 00:25:53,410 --> 00:25:54,346 do some approximation. 365 00:25:54,346 --> 00:25:57,720 And you will exactly see from the density of modes 366 00:25:57,720 --> 00:26:00,210 that the very long wavelengths don't contribute. 367 00:26:00,210 --> 00:26:03,810 And you will also see that because of rapid averaging of E 368 00:26:03,810 --> 00:26:07,660 to the ikr phase factors, the high frequency modes 369 00:26:07,660 --> 00:26:08,616 will not contribute. 370 00:26:14,270 --> 00:26:14,770 OK. 371 00:26:14,770 --> 00:26:18,520 So we have now two different power laws. 372 00:26:21,040 --> 00:26:23,950 One is 1 over R to the sixth. 373 00:26:23,950 --> 00:26:27,490 And one is 1 over R to the seventh. 374 00:26:35,860 --> 00:26:37,230 So let's explain that now. 375 00:26:41,450 --> 00:26:52,790 So we obtain the 1 over R sixth by using the uncertainty 376 00:26:52,790 --> 00:26:53,950 principle for atoms. 377 00:26:57,530 --> 00:27:01,360 And here, we have used the uncertainty principle, the zero 378 00:27:01,360 --> 00:27:04,340 point fluctuations for electromagnetic waves. 379 00:27:12,680 --> 00:27:17,350 When we derive the 1 over R sixth potential, 380 00:27:17,350 --> 00:27:23,970 we actually used simply the electrostatic instantaneous 381 00:27:23,970 --> 00:27:26,410 Coulomb field dipole field. 382 00:27:26,410 --> 00:27:31,560 And this is only valid for short range when-- you know, 383 00:27:31,560 --> 00:27:33,830 I said we have a fluctuation and the fluctuation 384 00:27:33,830 --> 00:27:36,000 created another one, which was in phase. 385 00:27:36,000 --> 00:27:38,320 But if one fluctuation has to send out 386 00:27:38,320 --> 00:27:41,660 an electromagnetic wave, the second fluctuation 387 00:27:41,660 --> 00:27:42,990 may not be in. 388 00:27:42,990 --> 00:27:45,660 When propagation effects come into play, 389 00:27:45,660 --> 00:27:48,560 there's a time lag between the two oscillators 390 00:27:48,560 --> 00:27:50,370 and may actually be out of phase. 391 00:27:50,370 --> 00:27:52,470 So we really assumed in the derivation, 392 00:27:52,470 --> 00:27:54,550 without ever saying it explicitly, 393 00:27:54,550 --> 00:27:57,160 that there are no propagation effects. 394 00:27:57,160 --> 00:28:01,200 And this is called the short range potential. 395 00:28:01,200 --> 00:28:03,690 Whereas, for the second argument that we said, 396 00:28:03,690 --> 00:28:06,570 we have two distant atoms, which get synchronized 397 00:28:06,570 --> 00:28:11,800 by being driven by fluctuations of the electromagnetic wave 398 00:28:11,800 --> 00:28:15,900 at wavelengths, lambda equals R. This 399 00:28:15,900 --> 00:28:18,720 is what happens at long range. 400 00:28:21,580 --> 00:28:25,720 So this is a famous result, that we have a van der Waals 401 00:28:25,720 --> 00:28:29,600 force, which is the instantaneous force. 402 00:28:29,600 --> 00:28:34,510 And when propagation effects come into play, 403 00:28:34,510 --> 00:28:37,560 this goes by the name Casimir-Polder potential. 404 00:28:37,560 --> 00:28:40,570 The moment when propagation effects come into play-- 405 00:28:40,570 --> 00:28:44,420 when radiation plays a role-- we get a different power law. 406 00:28:53,700 --> 00:28:58,140 So again, I'm just playing with ideas. 407 00:28:58,140 --> 00:29:00,810 I'm taking a train of thought, atoms 408 00:29:00,810 --> 00:29:03,470 fluctuate; another train of thought, electromagnetic fields 409 00:29:03,470 --> 00:29:05,702 propagate; and just see, what are the ramifications? 410 00:29:05,702 --> 00:29:06,785 What are the consequences? 411 00:29:11,570 --> 00:29:15,660 So now, I want to eventually give you 412 00:29:15,660 --> 00:29:19,936 a treatment, which has both aspects in one. 413 00:29:19,936 --> 00:29:21,560 And this is our diagrammatic treatment. 414 00:29:24,630 --> 00:29:30,210 First, before I do with that, do you 415 00:29:30,210 --> 00:29:35,650 have any idea what distinguishes the short range 416 00:29:35,650 --> 00:29:37,530 from the long range potential? 417 00:29:37,530 --> 00:29:38,860 So you have two atoms. 418 00:29:38,860 --> 00:29:40,110 You pull them apart. 419 00:29:40,110 --> 00:29:44,720 And what distance do we have the 1 over R to the sixth physics? 420 00:29:44,720 --> 00:29:46,660 At what distance do you have the 1 over R 421 00:29:46,660 --> 00:29:47,670 to the seventh physics? 422 00:29:53,442 --> 00:29:55,860 AUDIENCE: When [INAUDIBLE] transitions? 423 00:29:55,860 --> 00:29:58,310 PROFESSOR: When the distance is the wavelengths 424 00:29:58,310 --> 00:29:59,360 of the atomic transition. 425 00:29:59,360 --> 00:30:02,200 That's the only thing which matters. 426 00:30:02,200 --> 00:30:05,250 You don't get it immediately from here. 427 00:30:05,250 --> 00:30:09,280 Because-- I mean, I used an instantaneous potential, 428 00:30:09,280 --> 00:30:10,660 which didn't have any scale. 429 00:30:10,660 --> 00:30:13,090 And here, I said we have vacuum fluctuations 430 00:30:13,090 --> 00:30:14,300 of all wavelengths. 431 00:30:14,300 --> 00:30:16,600 So here, I had a picture, which clearly 432 00:30:16,600 --> 00:30:18,830 works at long separation. 433 00:30:18,830 --> 00:30:21,020 And propagation effects are built in, 434 00:30:21,020 --> 00:30:23,170 because it's a vacuum fluctuation, which 435 00:30:23,170 --> 00:30:26,060 sort of act at position, a and b. 436 00:30:26,060 --> 00:30:28,240 And the wavelengths-- their propagation-- 437 00:30:28,240 --> 00:30:30,430 is part of the formulation. 438 00:30:30,430 --> 00:30:35,642 But yes, Bohr's conjecture that it 439 00:30:35,642 --> 00:30:39,000 is the wavelengths of the atom-- the wavelengths 440 00:30:39,000 --> 00:30:41,040 of the resonant radiation-- is correct. 441 00:30:41,040 --> 00:30:42,780 But I want to show you now how we 442 00:30:42,780 --> 00:30:45,130 can obtain this result in a, I think, 443 00:30:45,130 --> 00:30:47,500 really elegant and beautiful way by looking 444 00:30:47,500 --> 00:30:49,390 at the diagrams we have just learned. 445 00:30:57,340 --> 00:30:58,560 OK. 446 00:30:58,560 --> 00:31:01,940 What did we just have? 447 00:31:01,940 --> 00:31:03,130 Time to do that, yes. 448 00:31:03,130 --> 00:31:04,040 Any questions here? 449 00:31:11,020 --> 00:31:13,330 The reason why I want to show it diagrammatically 450 00:31:13,330 --> 00:31:14,580 is actually two fold. 451 00:31:14,580 --> 00:31:17,430 One is, it's really beautiful how it comes out. 452 00:31:17,430 --> 00:31:21,910 But the second thing is, when I introduced the Feynman diagrams 453 00:31:21,910 --> 00:31:27,620 and all of these formulation in terms of propagators, 454 00:31:27,620 --> 00:31:29,510 I gave you a wonderful picture. 455 00:31:29,510 --> 00:31:32,170 But in the end, I reduced it to first and second order 456 00:31:32,170 --> 00:31:33,420 perturbation theory. 457 00:31:33,420 --> 00:31:37,410 So I want to show you at least one nontrivial example 458 00:31:37,410 --> 00:31:42,280 where you would not get the result easily 459 00:31:42,280 --> 00:31:43,910 without this formalism. 460 00:31:43,910 --> 00:31:47,910 So enjoy one nontrivial application of diagrams. 461 00:32:04,730 --> 00:32:16,440 So if you follow the systematic way, how we have set up 462 00:32:16,440 --> 00:32:19,230 our formulation of quantum electrodynamic 463 00:32:19,230 --> 00:32:22,370 with a photon field-- the quantized photon field-- 464 00:32:22,370 --> 00:32:28,850 then the two neutral atoms have no direct Coulomb interaction. 465 00:32:34,250 --> 00:32:35,500 I just want to point that out. 466 00:32:35,500 --> 00:32:37,920 Because a few minutes ago, when I 467 00:32:37,920 --> 00:32:43,130 said a classical system is just Coulomb field-- each charge 468 00:32:43,130 --> 00:32:46,050 element has a Coulomb potential with the other charge element-- 469 00:32:46,050 --> 00:32:48,250 this was just the opposite approach. 470 00:32:48,250 --> 00:32:52,140 But in our QED formulation, we have sold our soul. 471 00:32:52,140 --> 00:32:56,100 We have said we only really got the near field, 472 00:32:56,100 --> 00:32:59,430 the longitudinal field, as belonging to the atoms. 473 00:32:59,430 --> 00:33:01,780 Everything which happens at longer range 474 00:33:01,780 --> 00:33:06,740 is included in our atom-photon Hamiltonian. 475 00:33:06,740 --> 00:33:09,590 So therefore, there is no direct Coulomb interaction 476 00:33:09,590 --> 00:33:11,430 for the longitudinal field. 477 00:33:11,430 --> 00:33:17,790 Everything, everything now, has to come from the interaction-- 478 00:33:17,790 --> 00:33:23,290 the perturbation operator-- for the quantized radiation field. 479 00:33:23,290 --> 00:33:28,100 And therefore, we must get it out. 480 00:33:28,100 --> 00:33:31,030 We have one atom; it's d1 at d prime. 481 00:33:31,030 --> 00:33:37,480 I think that's how it's-- I'm using now the nomenclature, 482 00:33:37,480 --> 00:33:44,450 which is used in atom-photon interaction, pages 121 to 126. 483 00:33:44,450 --> 00:33:48,260 And so, there is one atom at the unprimed location, 484 00:33:48,260 --> 00:33:51,200 one atom at the time primed location. 485 00:33:53,940 --> 00:33:56,680 And everything has to come out now 486 00:33:56,680 --> 00:34:01,650 of the formulation of the quantized radiation field. 487 00:34:01,650 --> 00:34:03,240 And we have discussed that lengths 488 00:34:03,240 --> 00:34:05,920 at the transverse electric field become operators, 489 00:34:05,920 --> 00:34:11,460 a plus a dagger-- the symbol for the dipole operators 490 00:34:11,460 --> 00:34:13,857 are acting on the atoms. 491 00:34:13,857 --> 00:34:14,690 We've done all that. 492 00:34:19,480 --> 00:34:23,070 So then the only thing which happens is-- 493 00:34:23,070 --> 00:34:25,270 and this is described by a and a dagger-- 494 00:34:25,270 --> 00:34:28,030 is the exchange of photons. 495 00:34:28,030 --> 00:34:32,060 So the only way how those two neutral objects can interact 496 00:34:32,060 --> 00:34:33,560 is through the radiation field. 497 00:34:33,560 --> 00:34:35,560 And through the radiation field means 498 00:34:35,560 --> 00:34:37,920 they have to emit photons. 499 00:34:37,920 --> 00:34:40,130 So I want to show you now that the van der Waals 500 00:34:40,130 --> 00:34:44,340 interaction can be thought of as one atom emitting a photon. 501 00:34:44,340 --> 00:34:46,510 But this virtual photon is absorbed 502 00:34:46,510 --> 00:34:48,090 by the other atom and vice versa. 503 00:34:51,320 --> 00:34:55,940 And you may have heard quite often-- 504 00:34:55,940 --> 00:34:58,560 but I'm not sure if you've seen it explicitly-- 505 00:34:58,560 --> 00:35:01,280 that a lot of interactions in physics 506 00:35:01,280 --> 00:35:05,120 are actually mediated by the exchange of virtual particles. 507 00:35:05,120 --> 00:35:08,180 The famous example is that the nuclear force 508 00:35:08,180 --> 00:35:10,295 comes from a virtual exchange of pions. 509 00:35:12,910 --> 00:35:15,490 But then, people say, the Coulomb force 510 00:35:15,490 --> 00:35:17,840 comes from a virtual exchange of photons. 511 00:35:17,840 --> 00:35:19,490 But usually, we don't really show you 512 00:35:19,490 --> 00:35:21,870 how this virtual exchange of photons work. 513 00:35:21,870 --> 00:35:23,590 So in that sense, I'm proud that I 514 00:35:23,590 --> 00:35:28,350 can show you, at least the basic outline, how 515 00:35:28,350 --> 00:35:32,170 virtual exchange of photons between two neutral atoms 516 00:35:32,170 --> 00:35:34,986 leads to the van der Waals potential. 517 00:35:34,986 --> 00:35:35,860 So that's our agenda. 518 00:35:44,530 --> 00:35:45,030 OK. 519 00:35:49,970 --> 00:35:57,202 Now the fact that this takes six pages in the book-- 520 00:35:57,202 --> 00:35:59,760 and the book is not solving all equations-- 521 00:35:59,760 --> 00:36:02,570 means if you really want to do it quantitatively, 522 00:36:02,570 --> 00:36:04,520 it has a certain complexity. 523 00:36:04,520 --> 00:36:09,110 But what I've done is, I've sort of distilled out the grand idea 524 00:36:09,110 --> 00:36:11,520 and which also shows you what happens 525 00:36:11,520 --> 00:36:13,560 when you go from short range to long range. 526 00:36:13,560 --> 00:36:16,140 And this is what I want to present you. 527 00:36:16,140 --> 00:36:20,200 And therefore, I take [? Benji's ?] question 528 00:36:20,200 --> 00:36:22,760 and will not discuss all the modes 529 00:36:22,760 --> 00:36:25,330 of the electromagnetic field. 530 00:36:25,330 --> 00:36:34,630 I will immediately use the fact that the dominant contribution 531 00:36:34,630 --> 00:36:42,700 comes from modes. 532 00:36:42,700 --> 00:36:44,450 And since we don't have real photons, 533 00:36:44,450 --> 00:36:47,220 we don't have energy to create real photons when everything 534 00:36:47,220 --> 00:36:48,310 is in the current state. 535 00:36:48,310 --> 00:36:51,281 We call them virtual photons. 536 00:36:51,281 --> 00:36:53,280 Virtual photons are photons which are immediate. 537 00:36:53,280 --> 00:36:55,696 But a short time later, because of Heisenberg uncertainty, 538 00:36:55,696 --> 00:36:56,990 have to disappear. 539 00:36:56,990 --> 00:37:04,420 So the dominant contribution comes from virtual photons 540 00:37:04,420 --> 00:37:09,920 at the wavelengths, which is D, the distance between atoms. 541 00:37:16,230 --> 00:37:19,500 So the frequency, which will play an important role, 542 00:37:19,500 --> 00:37:22,470 is simply the frequency of those virtual photons. 543 00:37:28,250 --> 00:37:31,130 And I gave you already the reason-- the density of states 544 00:37:31,130 --> 00:37:33,550 which favor higher frequencies. 545 00:37:33,550 --> 00:37:36,700 But higher frequencies cancel, because of the E 546 00:37:36,700 --> 00:37:38,740 to the ikr terms. 547 00:37:38,740 --> 00:37:40,080 So we only focus on those. 548 00:37:45,340 --> 00:37:50,250 The second thing I need is that when 549 00:37:50,250 --> 00:38:01,010 we do perturbation theory, the energy shift in perturbation 550 00:38:01,010 --> 00:38:09,900 theory-- well, if you do second order, 551 00:38:09,900 --> 00:38:15,460 you go from the initial state to an intermediate state. 552 00:38:15,460 --> 00:38:18,660 And then, you go from the intermediate state, one, 553 00:38:18,660 --> 00:38:20,700 to the final state. 554 00:38:20,700 --> 00:38:22,900 And what you have to do is, you have 555 00:38:22,900 --> 00:38:26,745 to divide by the energy denominator. 556 00:38:29,410 --> 00:38:34,790 If you do higher and higher order perturbation theory, 557 00:38:34,790 --> 00:38:37,010 in our diagrammatic presentation, 558 00:38:37,010 --> 00:38:41,760 that means we have propagators at intermediate energies. 559 00:38:41,760 --> 00:38:45,020 And I told you that the time an atom can 560 00:38:45,020 --> 00:38:49,720 spend in an intermediate state is 1 over the energy defect. 561 00:38:49,720 --> 00:38:58,400 And if you integrate the E to the i energy defect 562 00:38:58,400 --> 00:39:00,950 factor over time, you get something 563 00:39:00,950 --> 00:39:04,300 which is proportional to 1 over delta E. 564 00:39:04,300 --> 00:39:08,430 So therefore, if you perform the time integration 565 00:39:08,430 --> 00:39:12,210 of the perturbation analysis, you 566 00:39:12,210 --> 00:39:16,710 find by integrating over the short time particles spend 567 00:39:16,710 --> 00:39:18,490 in the intermediate state, you actually 568 00:39:18,490 --> 00:39:21,030 generate those energy denominators. 569 00:39:21,030 --> 00:39:24,860 That's actually the relationship between the time propagation 570 00:39:24,860 --> 00:39:26,100 I showed you earlier. 571 00:39:26,100 --> 00:39:28,630 And when you integrate over the time, 572 00:39:28,630 --> 00:39:31,410 you get for this intermediate state, a contribution, 573 00:39:31,410 --> 00:39:33,780 which is 1 over the energy defect 574 00:39:33,780 --> 00:39:36,630 over the intermediate state. 575 00:39:36,630 --> 00:39:41,650 So therefore, if you go now to higher order perturbation 576 00:39:41,650 --> 00:39:51,700 theory, we simply have a product of matrix elements 577 00:39:51,700 --> 00:39:56,040 in the numerator. 578 00:39:56,040 --> 00:39:59,735 And in the denominator, we have a product of energy defects. 579 00:40:04,640 --> 00:40:08,000 You are using that all the time for second order perturbation 580 00:40:08,000 --> 00:40:11,590 theory, but the structure is that you can just 581 00:40:11,590 --> 00:40:13,556 sort of daisy chain the expression. 582 00:40:13,556 --> 00:40:15,430 And you get higher order perturbation theory. 583 00:40:19,760 --> 00:40:25,660 So therefore, all we will have to understand to figure out 584 00:40:25,660 --> 00:40:28,530 what happens when virtual photons are exchanged, 585 00:40:28,530 --> 00:40:31,740 I will just show you two relevant diagrams. 586 00:40:31,740 --> 00:40:34,560 And we just look-- we stare at the diagrams-- 587 00:40:34,560 --> 00:40:37,090 and figure out what are these denominators? 588 00:40:37,090 --> 00:40:38,795 What are those energy defects? 589 00:40:38,795 --> 00:40:40,545 And we will find that there's a difference 590 00:40:40,545 --> 00:40:42,190 at long range and short range. 591 00:40:42,190 --> 00:40:43,140 So that's the agenda. 592 00:40:46,280 --> 00:40:48,760 So we will need two energies. 593 00:40:48,760 --> 00:40:51,840 One energy is the energy of the virtual photon, 594 00:40:51,840 --> 00:40:54,580 which is exchanged. 595 00:40:54,580 --> 00:40:58,080 And the second energy, or frequency, 596 00:40:58,080 --> 00:41:05,141 is the resonant frequency of the atom. 597 00:41:20,870 --> 00:41:25,820 So let's look at two relevant diagrams. 598 00:41:34,540 --> 00:41:38,590 So one is atom, a; one is atom, b. 599 00:41:38,590 --> 00:41:41,200 We are interested in interactions 600 00:41:41,200 --> 00:41:43,740 between atoms in the ground state. 601 00:41:43,740 --> 00:41:46,500 And after a sufficiently long time, 602 00:41:46,500 --> 00:41:48,580 both atoms have to be back in the ground state. 603 00:41:58,420 --> 00:42:03,550 Let me just get this organized, one, two, three, four. 604 00:42:10,530 --> 00:42:14,750 And the result is obtained in fourth order perturbation 605 00:42:14,750 --> 00:42:16,590 theory. 606 00:42:16,590 --> 00:42:18,970 It has to be fourth order, because we 607 00:42:18,970 --> 00:42:20,870 need photon exchange. 608 00:42:20,870 --> 00:42:24,340 To emit a photon here, absorb it here, is second order. 609 00:42:28,340 --> 00:42:30,520 But then, one atom is in the excited state 610 00:42:30,520 --> 00:42:31,630 and this can't be. 611 00:42:31,630 --> 00:42:34,430 So a second photon is required to undo the effect. 612 00:42:34,430 --> 00:42:37,850 So we need exchange of photon pairs. 613 00:42:37,850 --> 00:42:40,780 Each exchange means absorption and emission. 614 00:42:40,780 --> 00:42:42,200 That means two vertices. 615 00:42:42,200 --> 00:42:43,980 And photon pairs means another two. 616 00:42:43,980 --> 00:42:47,410 So we need fourth order perturbation theory. 617 00:42:47,410 --> 00:42:47,910 OK. 618 00:42:47,910 --> 00:42:52,590 But it's much, much easier than you think. 619 00:42:52,590 --> 00:42:54,640 So we have a ground state atom. 620 00:42:54,640 --> 00:42:56,430 And now the action starts. 621 00:42:56,430 --> 00:42:57,920 This is the first vertex. 622 00:42:57,920 --> 00:43:01,240 It emits a photon. 623 00:43:01,240 --> 00:43:05,380 The photon is absorbed by the other atom, which 624 00:43:05,380 --> 00:43:08,990 puts the atom into the excited state. 625 00:43:08,990 --> 00:43:11,490 This atom is also in the excited state, of course. 626 00:43:24,790 --> 00:43:28,880 And then, we know the atom cannot stay in the excited 627 00:43:28,880 --> 00:43:32,210 state for a long time. 628 00:43:32,210 --> 00:43:37,920 It can now emit a second photon, which brings it back 629 00:43:37,920 --> 00:43:40,030 to the ground state. 630 00:43:40,030 --> 00:43:45,325 And the second photon is absorbed by atom, b, 631 00:43:45,325 --> 00:43:47,200 and puts the atom back into the ground state. 632 00:43:51,000 --> 00:43:53,520 So this is one relevant diagram. 633 00:43:53,520 --> 00:44:01,790 And let me analyze it in the energy defects. 634 00:44:01,790 --> 00:44:05,320 Here, in the first intermediate step, 635 00:44:05,320 --> 00:44:09,670 the energy defect is-- well, we have ground-ground. 636 00:44:09,670 --> 00:44:12,910 This is sort our reference. 637 00:44:12,910 --> 00:44:17,070 But now, we have put one atom into the excited state. 638 00:44:17,070 --> 00:44:19,630 That causes energy, omega naught. 639 00:44:19,630 --> 00:44:23,290 And we have created one photon, omega. 640 00:44:23,290 --> 00:44:27,100 Here in that time step, there are no photons, 641 00:44:27,100 --> 00:44:29,740 but there are two excited atoms. 642 00:44:29,740 --> 00:44:32,700 And in the third time step, we have still one atom 643 00:44:32,700 --> 00:44:36,280 in the excited state and one photon. 644 00:44:39,810 --> 00:44:43,250 Now, this is one relevant class. 645 00:44:43,250 --> 00:44:46,200 I mean, you can now use permutations at atom, b, 646 00:44:46,200 --> 00:44:47,930 excites photons first and such. 647 00:44:47,930 --> 00:44:50,570 But the structure of many of these diagrams 648 00:44:50,570 --> 00:44:55,360 will be the same that we have those three energy defects. 649 00:44:55,360 --> 00:44:57,540 But now, we have another possibility. 650 00:44:57,540 --> 00:45:01,420 And this is, one atom emits a photon, 651 00:45:01,420 --> 00:45:06,570 but it feels it doesn't want to be excited, 652 00:45:06,570 --> 00:45:09,930 because the excitation energy is so precious. 653 00:45:09,930 --> 00:45:12,760 It wants to immediately return to the ground state. 654 00:45:12,760 --> 00:45:16,170 So we could have a situation that the atom fires 655 00:45:16,170 --> 00:45:21,140 a second photon before the first photon is absorbed. 656 00:45:23,860 --> 00:45:29,370 So the diagram-- the relevant diagram-- is now this. 657 00:45:33,660 --> 00:45:36,220 So the atom started in the ground state, 658 00:45:36,220 --> 00:45:39,460 is in the excited state, but as soon as possible, 659 00:45:39,460 --> 00:45:42,550 goes back to the ground state. 660 00:45:42,550 --> 00:45:46,080 And the other atom starts out in ground state, 661 00:45:46,080 --> 00:45:49,470 is in the excited state, and is back in the ground state. 662 00:45:49,470 --> 00:45:55,360 So what we have here is, we have again three intermediate steps. 663 00:45:55,360 --> 00:45:58,990 In the first step, of course, we have an excited atom 664 00:45:58,990 --> 00:46:03,800 and a photon. 665 00:46:03,800 --> 00:46:05,810 And we have the same here. 666 00:46:05,810 --> 00:46:09,450 But now the difference is, that in the intermediate step, 667 00:46:09,450 --> 00:46:13,890 our energy defect is not two atomic excitations, 668 00:46:13,890 --> 00:46:15,755 it is two photonic excitations. 669 00:46:20,620 --> 00:46:25,620 And now we want to compare those two diagrams 670 00:46:25,620 --> 00:46:31,565 for short and for long range. 671 00:46:38,790 --> 00:46:42,360 And it becomes clear that short range 672 00:46:42,360 --> 00:46:48,170 is where the frequency of the photon-- 673 00:46:48,170 --> 00:46:49,680 its wavelength is lambda. 674 00:46:49,680 --> 00:46:52,410 Short range means high frequency. 675 00:46:52,410 --> 00:46:54,450 The frequency of the photon is larger 676 00:46:54,450 --> 00:46:56,980 than the atomic excitation energy. 677 00:46:56,980 --> 00:46:59,480 Or that means, the distance is smaller 678 00:46:59,480 --> 00:47:04,570 than the resonant wavelength of the photon. 679 00:47:04,570 --> 00:47:09,750 Whereas, long range is that the atomic excitation 680 00:47:09,750 --> 00:47:11,835 is more precious than the photonic excitation. 681 00:47:18,180 --> 00:47:18,780 OK. 682 00:47:18,780 --> 00:47:25,330 So I need three more minutes. 683 00:47:25,330 --> 00:47:30,390 So it just fits into the class time. 684 00:47:30,390 --> 00:47:33,980 So we are here in the limit that omega 685 00:47:33,980 --> 00:47:36,360 is larger than omega naught. 686 00:47:36,360 --> 00:47:40,580 So that means, if we take now the product of the three energy 687 00:47:40,580 --> 00:47:44,400 denominators, omega dominates over omega naught. 688 00:47:44,400 --> 00:47:47,400 So we'll have this structure omega, omega naught, omega. 689 00:47:53,270 --> 00:47:55,435 Omega, omega naught, omega. 690 00:47:58,110 --> 00:48:01,950 Whereas, at long range, omega naught dominates. 691 00:48:01,950 --> 00:48:05,030 We have omega naught, omega naught, omega naught. 692 00:48:05,030 --> 00:48:10,950 So the structure of the product of energy denominators 693 00:48:10,950 --> 00:48:13,616 is 1 over omega naught. 694 00:48:13,616 --> 00:48:17,890 If you look on the right hand side, 695 00:48:17,890 --> 00:48:22,550 we have replaced-- in the intermediate part 696 00:48:22,550 --> 00:48:26,100 in the propagator-- the energy defect by omega naught 697 00:48:26,100 --> 00:48:28,100 by omega. 698 00:48:28,100 --> 00:48:34,370 So therefore, in this case, we have 1 over omega cubed 699 00:48:34,370 --> 00:48:39,000 and 1 over omega naught squared, omega. 700 00:48:39,000 --> 00:48:42,110 So we know that when we sum up over diagrams, 701 00:48:42,110 --> 00:48:45,680 we have to sum up over all possibly diagrams. 702 00:48:45,680 --> 00:48:51,040 And the ones which dominate at short range are those ones 703 00:48:51,040 --> 00:48:53,110 and at long range are the other ones. 704 00:48:58,290 --> 00:49:01,600 So in other words, at short range, 705 00:49:01,600 --> 00:49:05,620 one photon is exchanged and disappears. 706 00:49:05,620 --> 00:49:07,590 And then, the next photon comes. 707 00:49:07,590 --> 00:49:11,860 But at long range, we have an exchange 708 00:49:11,860 --> 00:49:16,480 that one atom sends out two photons almost simultaneously. 709 00:49:16,480 --> 00:49:19,040 And the other atom absorbs them. 710 00:49:19,040 --> 00:49:21,570 Because it sends them out almost simultaneously. 711 00:49:21,570 --> 00:49:25,470 Because here, we are talking about long wavelength photons, 712 00:49:25,470 --> 00:49:28,260 and photons are cheaper. 713 00:49:28,260 --> 00:49:31,260 Rather send out two photons before you spend too much time 714 00:49:31,260 --> 00:49:33,920 in the excited state. 715 00:49:33,920 --> 00:49:34,420 OK. 716 00:49:34,420 --> 00:49:35,320 Let's wrap it up. 717 00:49:37,940 --> 00:49:44,160 Since omega is proportional to 1 over the wavelengths, 718 00:49:44,160 --> 00:49:47,540 which was-- and I said the photons we are concentrating, 719 00:49:47,540 --> 00:49:59,566 is the photons which have a wavelength of the distance, D . 720 00:49:59,566 --> 00:50:03,630 So what we are realizing now is that, if you just 721 00:50:03,630 --> 00:50:11,840 compare those diagrams at long range, 722 00:50:11,840 --> 00:50:17,750 we go from 1 over omega squared to 1 over omega. 723 00:50:17,750 --> 00:50:34,540 So at long range-- omega is-- we have an additional factor 724 00:50:34,540 --> 00:50:41,340 of omega, which is 1 over D. 725 00:50:41,340 --> 00:50:45,180 And so now, if you assume-- I can't show you 726 00:50:45,180 --> 00:50:47,220 that without solving the real problem-- 727 00:50:47,220 --> 00:50:50,070 but if you assume that at short range, 728 00:50:50,070 --> 00:50:55,860 we have 1 over R to the sixth-- van der Waals potential-- 729 00:50:55,860 --> 00:50:59,590 then at least, I've proven to you that at long range, 730 00:50:59,590 --> 00:51:01,690 you get one more power in the distance. 731 00:51:05,040 --> 00:51:09,500 And this is a transition from an instantaneous potential, where 732 00:51:09,500 --> 00:51:11,080 propagation effects do not matter, 733 00:51:11,080 --> 00:51:13,905 to what is called retarded potential. 734 00:51:24,340 --> 00:51:25,790 So that's all I wanted to show you 735 00:51:25,790 --> 00:51:29,610 about interactions between two neutral atoms. 736 00:51:29,610 --> 00:51:32,460 Are there any questions? 737 00:51:32,460 --> 00:51:32,960 Timor? 738 00:51:32,960 --> 00:51:36,160 AUDIENCE: Does it matter the direction of both the photons 739 00:51:36,160 --> 00:51:36,810 that we drew? 740 00:51:36,810 --> 00:51:40,080 For example, can atom, a, emit to b, b 741 00:51:40,080 --> 00:51:44,564 absorbs, and b emits to a, rather than-- 742 00:51:44,564 --> 00:51:46,260 PROFESSOR: In practice, it matters. 743 00:51:46,260 --> 00:51:50,340 When you solve it, you have to sum over all possibilities. 744 00:51:50,340 --> 00:51:54,307 Remember, quantum mechanics in the system does everything. 745 00:51:54,307 --> 00:51:56,140 It doesn't care whether it's allowed or not. 746 00:51:56,140 --> 00:51:59,150 It tries everything out, whether it violates energy or not. 747 00:51:59,150 --> 00:52:00,910 And you have to sum over everything. 748 00:52:00,910 --> 00:52:04,550 But if you then analyze all the possible diagrams, 749 00:52:04,550 --> 00:52:09,530 you will actually figure out that they can-- no matter 750 00:52:09,530 --> 00:52:13,030 what you do, what permutation of photons and atoms you 751 00:52:13,030 --> 00:52:16,470 have-- the fourth order diagrams will be such 752 00:52:16,470 --> 00:52:19,190 that they are always distinguished 753 00:52:19,190 --> 00:52:23,630 by in this intermediate zone, whether you have 754 00:52:23,630 --> 00:52:27,270 two atomic excitations or photonic excitations. 755 00:52:27,270 --> 00:52:30,430 And so what I do for you were the two diagrams, 756 00:52:30,430 --> 00:52:37,590 which are representative for a whole class of diagrams here. 757 00:52:37,590 --> 00:52:40,311 But absolutely, yes, you are right. 758 00:52:40,311 --> 00:52:40,810 OK. 759 00:52:40,810 --> 00:52:42,330 So that's all about neutral atoms. 760 00:52:42,330 --> 00:52:46,129 I know you have spring break next week-- MIT spring break. 761 00:52:46,129 --> 00:52:48,045 Maybe that's good news for the Harvard people, 762 00:52:48,045 --> 00:52:50,311 who-- when is Harvard's spring break? 763 00:52:50,311 --> 00:52:52,566 AUDIENCE: We just finished. 764 00:52:52,566 --> 00:52:53,430 PROFESSOR: OK. 765 00:52:53,430 --> 00:52:56,360 So you have-- at least for this class, 766 00:52:56,360 --> 00:52:59,270 you have a second spring break. 767 00:52:59,270 --> 00:53:02,240 So we are not yet finished with van der Waals and passing 768 00:53:02,240 --> 00:53:04,950 interactions, but we finished the neutral atoms. 769 00:53:04,950 --> 00:53:07,210 And on Monday after spring break, 770 00:53:07,210 --> 00:53:09,950 we talk about interactions between metal plate 771 00:53:09,950 --> 00:53:12,550 and atoms in two metal plates.