1 00:00:00,050 --> 00:00:01,770 The following content is provided 2 00:00:01,770 --> 00:00:04,010 under a Creative Commons license. 3 00:00:04,010 --> 00:00:06,860 Your support will help MIT OpenCourseWare continue 4 00:00:06,860 --> 00:00:10,720 to offer high quality educational resources for free. 5 00:00:10,720 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,200 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,200 --> 00:00:17,825 at ocw.mit.edu. 8 00:00:22,490 --> 00:00:24,906 PROFESSOR: Good afternoon. 9 00:00:24,906 --> 00:00:25,406 Yes. 10 00:00:25,406 --> 00:00:29,466 Let's have an on time departure. 11 00:00:29,466 --> 00:00:32,748 The topic of last class was the ac Stark effect. 12 00:00:32,748 --> 00:00:35,616 And I said what I've been saying to you, 13 00:00:35,616 --> 00:00:40,874 was additional [INAUDIBLE] of the ac Stark difference 14 00:00:40,874 --> 00:00:42,800 from perturbation theory. 15 00:00:42,800 --> 00:00:46,650 But then I have three points for discussion. 16 00:00:46,650 --> 00:00:49,276 The first one was [INAUDIBLE] for the [INAUDIBLE] problem, 17 00:00:49,276 --> 00:00:50,240 [INAUDIBLE]. 18 00:00:50,240 --> 00:00:52,450 The second one was the oscillator strength, 19 00:00:52,450 --> 00:00:54,550 which we almost finished. 20 00:00:54,550 --> 00:00:59,760 And today, I want to say a few words about [INAUDIBLE] 21 00:00:59,760 --> 00:01:01,680 in [INAUDIBLE]. 22 00:01:01,680 --> 00:01:05,010 And eventually how the ac Stark effect 23 00:01:05,010 --> 00:01:08,815 leads to the absorption coefficient, which many of you 24 00:01:08,815 --> 00:01:10,174 use [INAUDIBLE]. 25 00:01:13,620 --> 00:01:22,710 So I mentioned that the oscillator strength is actually 26 00:01:22,710 --> 00:01:26,959 one way to parametrize the matrix element. 27 00:01:26,959 --> 00:01:29,344 We'll actually talk most of the lecture 28 00:01:29,344 --> 00:01:31,729 today about matrix element, but I 29 00:01:31,729 --> 00:01:34,340 come to that in a few minutes. 30 00:01:34,340 --> 00:01:36,700 From here, the matrix element is nothing, 31 00:01:36,700 --> 00:01:38,880 but you get on with [INAUDIBLE]. 32 00:01:38,880 --> 00:01:41,222 Sure, it's a dimension of the Bohr radius 33 00:01:41,222 --> 00:01:44,610 but we don't really think in terms of lengths. 34 00:01:44,610 --> 00:01:47,680 There are two ways how you want to use a matrix element. 35 00:01:47,680 --> 00:01:50,526 One is the matrix elements squared, 36 00:01:50,526 --> 00:01:52,234 having some other effect with which 37 00:01:52,234 --> 00:01:54,520 to [INAUDIBLE] this likeness, an actual likeness 38 00:01:54,520 --> 00:01:58,220 against a number you should know how to [INAUDIBLE]. 39 00:01:58,220 --> 00:02:01,606 Or another way to parametrize matrix elements 40 00:02:01,606 --> 00:02:04,480 is by this [INAUDIBLE] number, which 41 00:02:04,480 --> 00:02:07,354 is what we will see in this class. 42 00:02:07,354 --> 00:02:09,169 And that's, of course, also related 43 00:02:09,169 --> 00:02:11,180 to semi-classical physics. 44 00:02:11,180 --> 00:02:13,130 And I mentioned that in the last class, 45 00:02:13,130 --> 00:02:15,102 that the oscillator strength allows 46 00:02:15,102 --> 00:02:19,350 us to connect the quantum mechanical treatment of an atom 47 00:02:19,350 --> 00:02:22,330 to the response of a classical harmonic oscillator 48 00:02:22,330 --> 00:02:24,755 through the charge of an external [INAUDIBLE]. 49 00:02:37,680 --> 00:02:41,640 So I used this expression, it was the end of class last. 50 00:02:41,640 --> 00:02:45,112 I used this expression for the matrix element 51 00:02:45,112 --> 00:02:48,090 in terms of the oscillator strengths 52 00:02:48,090 --> 00:02:54,777 to give you a nice relationship for strong transition 53 00:02:54,777 --> 00:02:58,530 as in the alkaline E lines, where the oscillator strength 54 00:02:58,530 --> 00:03:00,010 is close to 1. 55 00:03:00,010 --> 00:03:04,240 You see that the matrix element is 56 00:03:04,240 --> 00:03:07,390 1 over square of 2 times the geometric mean 57 00:03:07,390 --> 00:03:09,840 of the complete wave lengths of the electron, 58 00:03:09,840 --> 00:03:12,626 and the wave lengths of the transition. 59 00:03:12,626 --> 00:03:16,500 So that's actually important for the dipole approximation, 60 00:03:16,500 --> 00:03:19,580 because the concurrent wave lengths of the electron 61 00:03:19,580 --> 00:03:21,730 is very, very small. 62 00:03:21,730 --> 00:03:23,850 So therefore if the matrix element 63 00:03:23,850 --> 00:03:26,650 is the geometric mean of the two, 64 00:03:26,650 --> 00:03:28,990 that means that the matrix element 65 00:03:28,990 --> 00:03:32,460 is smaller than the optical wave lengths. 66 00:03:32,460 --> 00:03:35,920 And therefore, the matrix element 67 00:03:35,920 --> 00:03:38,356 is the length scale, the relevant length 68 00:03:38,356 --> 00:03:39,730 scale of the atom for transition. 69 00:03:39,730 --> 00:03:42,570 It is smaller than the optical wave lengths. 70 00:03:42,570 --> 00:03:46,640 And that means-- and we'll talk about it today-- 71 00:03:46,640 --> 00:03:52,488 that in the expression E to the IKR, KR is really small. 72 00:03:52,488 --> 00:03:54,364 So we can do the dipole approximation. 73 00:03:57,180 --> 00:03:59,340 Now I have a question for you. 74 00:03:59,340 --> 00:04:01,510 I did it, frankly, because we don't 75 00:04:01,510 --> 00:04:04,160 have many questions to discuss with. 76 00:04:04,160 --> 00:04:10,405 But I mentioned that there is a sum rule for the oscillator 77 00:04:10,405 --> 00:04:11,300 strengths. 78 00:04:11,300 --> 00:04:14,030 And the sum rule limits it to 1. 79 00:04:14,030 --> 00:04:17,450 And if there's a really strong transition, 80 00:04:17,450 --> 00:04:19,410 it kind of exhausts the sum rule, 81 00:04:19,410 --> 00:04:21,769 and this is what I was talking about. 82 00:04:21,769 --> 00:04:27,400 But because the thought crossed my mind after last lecture, 83 00:04:27,400 --> 00:04:32,360 if I don't add something to it, you 84 00:04:32,360 --> 00:04:36,950 may actually use it as a proof, which would be wrong. 85 00:04:36,950 --> 00:04:41,430 That the matrix element cannot be larger than what you see 86 00:04:41,430 --> 00:04:43,430 on the right hand side. 87 00:04:43,430 --> 00:04:49,612 On the other hand, we discussed in the context of Rydberg atoms 88 00:04:49,612 --> 00:04:53,554 that Rydberg atoms can have huge matrix elements. 89 00:04:53,554 --> 00:05:04,735 Matrix elements which [INAUDIBLE] 90 00:05:04,735 --> 00:05:07,650 but they can really become huge. 91 00:05:07,650 --> 00:05:12,676 So how do you reconcile the fact that Rydberg atoms 92 00:05:12,676 --> 00:05:16,003 in large instates, in states with large strengths 93 00:05:16,003 --> 00:05:20,670 of [INAUDIBLE] numbers can have huge matrix elements. 94 00:05:20,670 --> 00:05:22,960 How do you reconcile it with our discussion 95 00:05:22,960 --> 00:05:25,385 of oscillator strengths and sum rules? 96 00:05:35,090 --> 00:05:36,659 The answer is simple but subtle. 97 00:05:41,450 --> 00:05:45,980 So is it a proof that the matrix elements are impossible? 98 00:05:45,980 --> 00:05:48,420 Or if big matrix elements, huge matrix elements, 99 00:05:48,420 --> 00:06:01,020 are possible, what property of the oscillator strengths 100 00:06:01,020 --> 00:06:03,012 did I not emphasize? 101 00:06:25,422 --> 00:06:27,470 I mean, keep watching. 102 00:06:27,470 --> 00:06:32,120 So first, are there atoms which have huge matrix elements? 103 00:06:32,120 --> 00:06:32,850 Yes. 104 00:06:32,850 --> 00:06:33,726 Rydberg atoms. 105 00:06:33,726 --> 00:06:36,510 So the left hand side can be really huge. 106 00:06:36,510 --> 00:06:38,750 On the right hand side, we have the wave lengths 107 00:06:38,750 --> 00:06:45,100 of the transition which can get large. 108 00:06:45,100 --> 00:06:46,730 You have to count the wave lengths. 109 00:06:46,730 --> 00:06:49,650 But then you have the oscillator strengths. 110 00:06:49,650 --> 00:06:54,540 And the oscillator strengths, the sum 111 00:06:54,540 --> 00:06:57,440 of all oscillator strengths is 1. 112 00:06:57,440 --> 00:07:00,810 But you have to be careful that not all oscillator 113 00:07:00,810 --> 00:07:02,210 strengths have to be possible. 114 00:07:21,810 --> 00:07:23,660 The definition of the oscillator strengths 115 00:07:23,660 --> 00:07:27,320 involves frequency from one state to the next. 116 00:07:27,320 --> 00:07:30,220 And depending whether you go up or down, 117 00:07:30,220 --> 00:07:33,390 the frequency has positive or negative side. 118 00:07:33,390 --> 00:07:35,520 So therefore we have a sum rule. 119 00:07:35,520 --> 00:07:38,490 For the ground state, all frequencies 120 00:07:38,490 --> 00:07:40,940 are positive because you can only go up. 121 00:07:40,940 --> 00:07:43,145 So therefore the sum rule is only 122 00:07:43,145 --> 00:07:44,980 very useful for the ground state, 123 00:07:44,980 --> 00:07:47,550 because all oscillator strengths are positive. 124 00:07:47,550 --> 00:07:50,635 And if one oscillator strength is 1, 125 00:07:50,635 --> 00:07:55,080 you know no other transition has any strength at all. 126 00:07:55,080 --> 00:07:57,020 However when you have an excited state 127 00:07:57,020 --> 00:07:59,900 and you find an oscillator strength which is 1, 128 00:07:59,900 --> 00:08:02,683 you could still have a lot of other strong transitions, 129 00:08:02,683 --> 00:08:05,016 but their oscillator strengths are positive and negative 130 00:08:05,016 --> 00:08:09,580 and they compensate for each other in the sum rule. 131 00:08:09,580 --> 00:08:11,600 So just be careful about that. 132 00:08:11,600 --> 00:08:14,575 For the ground state-- and this is what I've proven to you 133 00:08:14,575 --> 00:08:16,820 here, I think fairly rigorously-- 134 00:08:16,820 --> 00:08:22,260 if you are in the ground state, the transition matrix element 135 00:08:22,260 --> 00:08:24,560 cannot be larger than the geometric mean 136 00:08:24,560 --> 00:08:28,642 of the [INAUDIBLE] wave lengths and the [INAUDIBLE] wave 137 00:08:28,642 --> 00:08:29,160 lengths. 138 00:08:29,160 --> 00:08:31,190 But for excited states, you have to be careful, 139 00:08:31,190 --> 00:08:33,389 because the oscillator strengths can have cosines. 140 00:08:36,284 --> 00:08:36,784 OK. 141 00:08:40,768 --> 00:08:41,809 Any questions about that? 142 00:08:45,690 --> 00:09:11,010 Then let's talk quickly about the third point 143 00:09:11,010 --> 00:09:13,910 I wanted to discuss. 144 00:09:13,910 --> 00:09:17,895 And this is the relationship to the index of refraction. 145 00:09:22,650 --> 00:09:27,712 The polarizability here is responsible for the index 146 00:09:27,712 --> 00:09:28,656 of refraction. 147 00:09:33,380 --> 00:09:36,260 Well, the polarizability determines [INAUDIBLE] 148 00:09:36,260 --> 00:09:39,830 evaluate the constant and use whatever relation you want, 149 00:09:39,830 --> 00:09:41,990 but for a common physics purpose, 150 00:09:41,990 --> 00:09:43,880 this is sort of the useful relation. 151 00:09:43,880 --> 00:09:45,950 This is the index of refraction. 152 00:09:45,950 --> 00:09:47,980 It's related to the polarizability. 153 00:09:47,980 --> 00:09:51,014 And assuming that the polarizability is small, 154 00:09:51,014 --> 00:09:53,300 you get this, the parallel extension. 155 00:09:53,300 --> 00:09:56,430 So roughly saying n minus 1. 156 00:09:56,430 --> 00:09:59,290 The difference of the index of refraction from the vacuum 157 00:09:59,290 --> 00:10:01,400 is proportionate to the atomic density 158 00:10:01,400 --> 00:10:04,570 times the polarizability. 159 00:10:04,570 --> 00:10:05,100 OK. 160 00:10:05,100 --> 00:10:07,815 And we know this is what we derive from the perturbation 161 00:10:07,815 --> 00:10:09,440 theory, that the polarizability depends 162 00:10:09,440 --> 00:10:13,640 on a matrix element squared and 1 over the cube. 163 00:10:13,640 --> 00:10:17,643 Now I want to sort of show you that by just using that concept 164 00:10:17,643 --> 00:10:22,250 and putting in phenomenologically dissipation, 165 00:10:22,250 --> 00:10:26,040 we can get a full expression what happens to a laser 166 00:10:26,040 --> 00:10:30,400 beam crossing an atomic medium, like a [INAUDIBLE] condensate. 167 00:10:30,400 --> 00:10:32,180 How much of the laser beam is absorbed? 168 00:10:32,180 --> 00:10:34,326 And what is the feature of the laser beam. 169 00:10:40,130 --> 00:10:46,975 In order to get interesting or simple formula, 170 00:10:46,975 --> 00:10:50,430 I want to now parametrize the matrix element 171 00:10:50,430 --> 00:10:52,220 by this correlated gamma. 172 00:10:52,220 --> 00:10:54,870 This will turn out with an actual [INAUDIBLE], 173 00:10:54,870 --> 00:10:56,995 but right now, since we haven't talked 174 00:10:56,995 --> 00:10:59,140 about spontaneous emission, just say 175 00:10:59,140 --> 00:11:02,590 I replace the matrix element by gamma. 176 00:11:02,590 --> 00:11:04,805 And then I can write the index of refraction 177 00:11:04,805 --> 00:11:05,680 in the following way. 178 00:11:05,680 --> 00:11:11,210 But the matrix element squared is now parametrized by gamma. 179 00:11:11,210 --> 00:11:14,130 I have also introduced the coordinate sigma 180 00:11:14,130 --> 00:11:17,790 0, which will later be the lesson in absorption 181 00:11:17,790 --> 00:11:18,920 for section for atoms. 182 00:11:18,920 --> 00:11:22,160 But again here, just use it as a parametrization. 183 00:11:22,160 --> 00:11:24,035 I mean, I've not introduced any new concepts. 184 00:11:24,035 --> 00:11:29,260 I've just rewritten this expression in this way, 185 00:11:29,260 --> 00:11:35,340 involved sigma 0, which is a cross-section, and gamma, which 186 00:11:35,340 --> 00:11:36,814 is an actual light. 187 00:11:39,521 --> 00:11:40,020 OK. 188 00:11:56,180 --> 00:11:59,970 The polarizability depended-- that was perturbation 189 00:11:59,970 --> 00:12:03,615 theory-- 1 over the energy denominator, 190 00:12:03,615 --> 00:12:07,340 or 1 over the [INAUDIBLE]. 191 00:12:07,340 --> 00:12:13,390 Now we have used, with the oscillator strengths, 192 00:12:13,390 --> 00:12:16,664 the analogy to a classical harmonic oscillator. 193 00:12:16,664 --> 00:12:21,060 And at some level, you know that every harmonic oscillator 194 00:12:21,060 --> 00:12:23,880 must have little bit of damping. 195 00:12:23,880 --> 00:12:28,210 And so the same applies to atomic oscillators. 196 00:12:28,210 --> 00:12:30,620 And if you can account for the damping 197 00:12:30,620 --> 00:12:35,140 by putting into the frequency denominator of the [INAUDIBLE], 198 00:12:35,140 --> 00:12:36,550 1 over delta. 199 00:12:36,550 --> 00:12:39,626 Give it an imaginary part. 200 00:12:39,626 --> 00:12:42,440 So I'm just telling you here, yes. 201 00:12:42,440 --> 00:12:44,080 Every oscillator has some damping, 202 00:12:44,080 --> 00:12:47,420 and I can phenomenologically accounting for the damping 203 00:12:47,420 --> 00:12:50,970 by making the resonant frequency, or the tuning, 204 00:12:50,970 --> 00:12:54,020 slightly complex by adding imaginary part. 205 00:13:09,060 --> 00:13:12,370 Later on I want to express all the tunings normalized 206 00:13:12,370 --> 00:13:16,730 to this line which is parallel to a normalized [INAUDIBLE] 207 00:13:16,730 --> 00:13:19,550 imaginary part. 208 00:13:19,550 --> 00:13:20,050 OK. 209 00:13:20,050 --> 00:13:24,050 So we have the situation that this expression, gamma 210 00:13:24,050 --> 00:13:28,540 over delta-- this was the expression 211 00:13:28,540 --> 00:13:31,650 for the polarizability-- now acquires 212 00:13:31,650 --> 00:13:37,310 a small imaginary part, and that means it's a complex number. 213 00:13:37,310 --> 00:13:40,730 And I can now take this expression separate it 214 00:13:40,730 --> 00:13:46,190 into an imaginary part and a real part. 215 00:13:46,190 --> 00:13:47,960 So why do I do that? 216 00:13:47,960 --> 00:13:51,355 Well, the index of refraction appears 217 00:13:51,355 --> 00:13:53,650 in the propagation of the plane wave. 218 00:13:53,650 --> 00:13:56,480 If you're a plane wave, the key vector 219 00:13:56,480 --> 00:13:58,690 is no longer the key vector in vacuum. 220 00:13:58,690 --> 00:14:00,700 It's multiplied with the index of refraction, 221 00:14:00,700 --> 00:14:02,850 and that's what I do here. 222 00:14:02,850 --> 00:14:04,930 And now you see that immediately, 223 00:14:04,930 --> 00:14:09,640 an imaginary part of the index of refraction 224 00:14:09,640 --> 00:14:10,830 leads to absorption. 225 00:14:17,285 --> 00:14:19,185 And the real part leads to [INAUDIBLE]. 226 00:14:23,940 --> 00:14:24,450 OK. 227 00:14:24,450 --> 00:14:28,790 So what we have now is we have this expression for the plane 228 00:14:28,790 --> 00:14:31,550 wave after it is propagated through the medium. 229 00:14:31,550 --> 00:14:34,810 And we see there is exponential absorption. 230 00:14:34,810 --> 00:14:36,410 Exponential absorption. 231 00:14:36,410 --> 00:14:38,840 On resonance, we have an optimal density 232 00:14:38,840 --> 00:14:41,520 which is given by this expression. 233 00:14:41,520 --> 00:14:46,710 And the second term here give us a phase shift. 234 00:14:46,710 --> 00:14:49,210 And it's clear since we have one medium which 235 00:14:49,210 --> 00:14:54,485 has a certain thickness that the optical density and the phase 236 00:14:54,485 --> 00:14:57,630 shift are related. 237 00:14:57,630 --> 00:14:58,270 Yes. 238 00:14:58,270 --> 00:15:01,040 If you're on resonance, you have the maximum optical density 239 00:15:01,040 --> 00:15:03,250 and the maximum absorption. 240 00:15:03,250 --> 00:15:05,030 And the optical density, of course, 241 00:15:05,030 --> 00:15:07,450 is 1 over the [INAUDIBLE] squared. 242 00:15:07,450 --> 00:15:08,420 [INAUDIBLE] 243 00:15:08,420 --> 00:15:12,950 Whereas for large detuning, the phase shift goes as 1 244 00:15:12,950 --> 00:15:13,850 over delta. 245 00:15:13,850 --> 00:15:18,710 It's the dispersive scaling with the tuning, 246 00:15:18,710 --> 00:15:22,580 and this is also what we had when we started out. 247 00:15:22,580 --> 00:15:26,150 When we would have started out with the polarizability and not 248 00:15:26,150 --> 00:15:29,336 a decaying dissipation, because far away from resonance, 249 00:15:29,336 --> 00:15:32,646 you simply have a regional of dispersive effect, 250 00:15:32,646 --> 00:15:36,510 and dissipation doesn't occur. 251 00:15:36,510 --> 00:15:43,560 So anyway, I thought I would just 252 00:15:43,560 --> 00:15:47,110 show that to you because this is a full understanding of what 253 00:15:47,110 --> 00:15:50,141 happens to optical views when they pass the atomic 254 00:15:50,141 --> 00:15:50,640 [INAUDIBLE]. 255 00:15:50,640 --> 00:15:54,280 It's nothing else than the polarizability 256 00:15:54,280 --> 00:15:56,200 or the harmonic oscillator response 257 00:15:56,200 --> 00:16:00,480 of an atom to electromagnetic radiation. 258 00:16:00,480 --> 00:16:05,000 The only thing which is non-derivative here is-- 259 00:16:05,000 --> 00:16:07,750 and that's what we will discuss later in this course-- 260 00:16:07,750 --> 00:16:12,990 is in order to get the final result, which 261 00:16:12,990 --> 00:16:15,290 you may want to use in analyzing your data, 262 00:16:15,290 --> 00:16:18,680 you have to set the dissipative, the damping 263 00:16:18,680 --> 00:16:21,410 of the harmonic oscillator, equal with this gamma 264 00:16:21,410 --> 00:16:21,910 parameter. 265 00:16:21,910 --> 00:16:24,600 And that's something I cannot tell you without talking about 266 00:16:24,600 --> 00:16:27,530 spontaneous emission. 267 00:16:27,530 --> 00:16:30,575 So just to remind you, I introduce big gamma just 268 00:16:30,575 --> 00:16:33,710 as a parametrization of the matrix element. 269 00:16:33,710 --> 00:16:36,890 Then I introduce phenomenologically little gamma 270 00:16:36,890 --> 00:16:39,330 as just the damping of the harmonic oscillator 271 00:16:39,330 --> 00:16:42,030 of those two quantities for two levels [INAUDIBLE] 272 00:16:42,030 --> 00:16:43,016 are identical. 273 00:16:43,016 --> 00:16:45,650 And that's something I cannot show you here, 274 00:16:45,650 --> 00:16:48,876 because this requires discussion of spontaneous emission 275 00:16:48,876 --> 00:16:51,330 and the other modes of the electromagnetic field. 276 00:16:51,330 --> 00:16:54,047 This is not covered by a discussion of the ac 277 00:16:54,047 --> 00:16:54,672 polarizability. 278 00:16:59,112 --> 00:16:59,612 OK. 279 00:16:59,612 --> 00:17:03,070 Any questions? 280 00:17:03,070 --> 00:17:05,609 But still-- I know I'm repeating myself-- 281 00:17:05,609 --> 00:17:08,970 I always find it interesting that the whole physics 282 00:17:08,970 --> 00:17:11,134 of absorption, the beams are absorbed, 283 00:17:11,134 --> 00:17:13,212 you can pretty much pull it out of the response 284 00:17:13,212 --> 00:17:14,859 of a harmonic oscillator. 285 00:17:14,859 --> 00:17:18,450 It's only the damping rate of the harmonic oscillator 286 00:17:18,450 --> 00:17:21,760 that this is simply spontaneous emission. 287 00:17:21,760 --> 00:17:24,299 That's the only point where you have 288 00:17:24,299 --> 00:17:27,349 to go beyond the ac Stark shift and bring in the [INAUDIBLE] 289 00:17:27,349 --> 00:17:29,860 nature of the electromagnetic field 290 00:17:29,860 --> 00:17:31,840 and all the empty modes of the vacuum. 291 00:17:35,810 --> 00:17:42,050 Any questions about waves, or in general interactions 292 00:17:42,050 --> 00:17:45,515 of atoms with electric and magnetic fields? 293 00:17:53,880 --> 00:17:56,756 Well, then let's move on. 294 00:17:56,756 --> 00:18:01,295 We have talked about electric and magnetic fields 295 00:18:01,295 --> 00:18:03,640 intimately for very good reasons, 296 00:18:03,640 --> 00:18:07,568 because low frequency fields are either electric or magnetic 297 00:18:07,568 --> 00:18:08,560 in nature. 298 00:18:08,560 --> 00:18:11,470 But ultimately, we want to understand 299 00:18:11,470 --> 00:18:13,660 what happens when atoms interact with light. 300 00:18:18,430 --> 00:18:22,650 And this is our discussion for today. 301 00:18:30,579 --> 00:18:34,070 The outline of this chapter is as follows. 302 00:18:34,070 --> 00:18:45,201 Today, all I want to discuss is the coupling matrix in an atom. 303 00:18:45,201 --> 00:18:49,750 In other words, when we have a Hamiltonian between state A 304 00:18:49,750 --> 00:18:55,310 and B, we have an off-diagonal matrix element. 305 00:18:55,310 --> 00:18:58,700 And this off-diagonal matrix element causes transitions. 306 00:18:58,700 --> 00:19:03,030 Or when we calculate it, the effect of electric field-- 307 00:19:03,030 --> 00:19:06,370 the ec Stark effect or the ac Stark effect, 308 00:19:06,370 --> 00:19:11,030 we assume there was a quantity, state 1, state 2, 309 00:19:11,030 --> 00:19:14,560 and we put some operator in between. 310 00:19:14,560 --> 00:19:19,940 So this was a matrix element which connects the two states. 311 00:19:19,940 --> 00:19:23,040 And so far, we have always assumed 312 00:19:23,040 --> 00:19:28,740 using the classical multiple expansion 313 00:19:28,740 --> 00:19:32,990 of electromagnetic field energy that this matrix element has 314 00:19:32,990 --> 00:19:36,820 the form electric field times dipole matrix element. 315 00:19:36,820 --> 00:19:39,830 And therefore, the relevant matrix element 316 00:19:39,830 --> 00:19:43,814 was the position operator of the electron connecting the two 317 00:19:43,814 --> 00:19:45,170 states. 318 00:19:45,170 --> 00:19:49,310 So in a way, we have used it all the time. 319 00:19:49,310 --> 00:19:53,336 But in any more advanced discussion of atomic physics, 320 00:19:53,336 --> 00:19:55,320 you have two states. 321 00:19:55,320 --> 00:19:58,630 And as long as they are coupled, you 322 00:19:58,630 --> 00:20:00,740 have something which you call H1 2 323 00:20:00,740 --> 00:20:03,300 or you call it matrix element N1 2, 324 00:20:03,300 --> 00:20:05,630 and you're not even asking where it comes from. 325 00:20:05,630 --> 00:20:08,880 When two levels are coupled, they undergo Rabi oscillations. 326 00:20:08,880 --> 00:20:09,879 You have transitions. 327 00:20:09,879 --> 00:20:12,420 If you're in the excited state and you have a matrix element, 328 00:20:12,420 --> 00:20:13,940 you get spontaneous emission. 329 00:20:13,940 --> 00:20:15,880 So a lot of the things we want to discuss 330 00:20:15,880 --> 00:20:18,260 about atomic physics, all they require 331 00:20:18,260 --> 00:20:21,125 a number which is a matrix element between state 332 00:20:21,125 --> 00:20:22,840 1 and state 2. 333 00:20:22,840 --> 00:20:29,690 And today, I want to talk to you about different mechanisms 334 00:20:29,690 --> 00:20:31,210 which can lead to matrix element. 335 00:20:31,210 --> 00:20:32,990 We talked about the dipole operator, 336 00:20:32,990 --> 00:20:36,230 but you also talk about higher order possibilities, 337 00:20:36,230 --> 00:20:40,840 magnetic dipole coupling, or electric [INAUDIBLE] coupling. 338 00:20:40,840 --> 00:20:42,900 And a little bit later-- this is an outlook-- 339 00:20:42,900 --> 00:20:45,240 we will talk about twofold uncoupling. 340 00:20:45,240 --> 00:20:48,550 But in the end, even if it's a twofold uncoupling, 341 00:20:48,550 --> 00:20:52,950 for a lot or phenomena, all you need is a coupling. 342 00:20:52,950 --> 00:20:54,445 A coupling means a Rabi frequency. 343 00:20:54,445 --> 00:20:58,410 A Rabi frequency means Rabi oscillation, and so on. 344 00:20:58,410 --> 00:21:02,310 So today the focus is, what is the structure, what 345 00:21:02,310 --> 00:21:06,190 are the principles behind this number, which 346 00:21:06,190 --> 00:21:10,060 is a relevant matrix element. 347 00:21:10,060 --> 00:21:10,560 OK. 348 00:21:10,560 --> 00:21:12,290 So that's today. 349 00:21:12,290 --> 00:21:15,510 Next week is spring break, but when we resume, 350 00:21:15,510 --> 00:21:19,860 we then want to talk about what is the matrix element doing? 351 00:21:19,860 --> 00:21:25,710 And there are two cases which we have to consider. 352 00:21:25,710 --> 00:21:32,620 One is we can see it's narrow band and broad band. 353 00:21:32,620 --> 00:21:34,670 This matrix element can couple just 354 00:21:34,670 --> 00:21:36,480 to one move of the electromagnetic field. 355 00:21:36,480 --> 00:21:38,430 Everything is coherent. 356 00:21:38,430 --> 00:21:41,520 We have coherent Rabi oscillation. 357 00:21:41,520 --> 00:21:44,330 Or you can couple to many modes, and then you 358 00:21:44,330 --> 00:21:47,580 have a broad band situation, which may be described 359 00:21:47,580 --> 00:21:49,890 [INAUDIBLE] and this [INAUDIBLE] unit. 360 00:21:49,890 --> 00:21:52,050 Very, very different behavior. 361 00:21:52,050 --> 00:21:54,220 So that's what we then want to discuss. 362 00:21:54,220 --> 00:21:56,220 When atoms interact with light, we 363 00:21:56,220 --> 00:21:59,680 have two very different limiting cases, narrow band 364 00:21:59,680 --> 00:22:00,402 and broad band. 365 00:22:03,290 --> 00:22:08,480 And eventually, we want to go beyond the semi-classical 366 00:22:08,480 --> 00:22:11,731 formulation of the electromagnetic field 367 00:22:11,731 --> 00:22:14,370 and we'll talk about the quantized electromagnetic 368 00:22:14,370 --> 00:22:14,870 field. 369 00:22:22,371 --> 00:22:22,870 OK. 370 00:22:22,870 --> 00:22:28,730 So let's talk about the coupling between atoms 371 00:22:28,730 --> 00:22:30,590 and electromagnetic fields. 372 00:22:36,920 --> 00:22:40,680 I want to give you very brief derivation 373 00:22:40,680 --> 00:22:48,920 of the canonical coupling between electromagnetic fields 374 00:22:48,920 --> 00:22:49,640 and atoms. 375 00:22:49,640 --> 00:22:51,710 I know this is covered in many textbooks, 376 00:22:51,710 --> 00:22:53,590 and we have a very deep discussion 377 00:22:53,590 --> 00:22:58,740 about it using the full QED formulas in A422, 378 00:22:58,740 --> 00:23:02,860 but I feel this course A421 would not 379 00:23:02,860 --> 00:23:05,070 be complete without a discussion like that. 380 00:23:05,070 --> 00:23:08,040 And secondly-- this is so important 381 00:23:08,040 --> 00:23:10,830 if you hear it twice from two different angles, that's 382 00:23:10,830 --> 00:23:15,550 probably useful-- I just want to remind you 383 00:23:15,550 --> 00:23:24,090 that if you use classical electromagnetism 384 00:23:24,090 --> 00:23:33,550 and the Lagrangian formalism, you find the very 385 00:23:33,550 --> 00:23:39,200 elegant result that the coupling to the electromagnetic field 386 00:23:39,200 --> 00:23:47,770 can be introduced by modifying the momentum. 387 00:23:47,770 --> 00:23:50,860 That the canonical momentum which appears in the Lagrangian 388 00:23:50,860 --> 00:23:53,950 equation is no longer the mechanical momentum 389 00:23:53,950 --> 00:23:56,240 which gives rise to kinetic energy, But? 390 00:23:56,240 --> 00:23:59,470 It is modified by the vector potential. 391 00:24:06,550 --> 00:24:14,449 So therefore the Hamiltonian-- at this point 392 00:24:14,449 --> 00:24:16,240 it's the classical Hamiltonian, but then we 393 00:24:16,240 --> 00:24:19,320 use the same expression in quantum mechanics. 394 00:24:19,320 --> 00:24:28,300 The Hamiltonian has kinetic energy and potential energy. 395 00:24:28,300 --> 00:24:35,400 But the kinetic energy is no longer p square. 396 00:24:35,400 --> 00:24:41,140 Because by p, I mean now the canonical momentum. 397 00:24:41,140 --> 00:24:44,410 We have to correct for the vector potential 398 00:24:44,410 --> 00:24:46,170 to go from the canonical momentum 399 00:24:46,170 --> 00:24:47,920 to the mechanical momentum. 400 00:24:47,920 --> 00:24:51,600 And if we squared it, this gives the kinetic energy. 401 00:24:51,600 --> 00:24:53,980 If you worry about this sign here, 402 00:24:53,980 --> 00:24:57,640 I will assume now that the electron has a charge. 403 00:24:57,640 --> 00:25:00,510 The charge q f the electron was minus e, 404 00:25:00,510 --> 00:25:03,590 so that's why we have a plus sign here and not a minus sign. 405 00:25:16,100 --> 00:25:16,900 OK. 406 00:25:16,900 --> 00:25:25,880 So we will use this fact that we have to substitute the momentum 407 00:25:25,880 --> 00:25:28,240 operator by the canonical momentum, which 408 00:25:28,240 --> 00:25:30,415 is no longer the mechanical momentum. 409 00:25:30,415 --> 00:25:33,370 We used it also in our quantum mechanical Hamiltonian 410 00:25:33,370 --> 00:25:35,430 in the Schrodinger equation. 411 00:25:35,430 --> 00:25:38,010 If you ask me, can you prove it? 412 00:25:38,010 --> 00:25:38,560 No. 413 00:25:38,560 --> 00:25:39,870 Nobody can prove it. 414 00:25:39,870 --> 00:25:43,250 We can never prove what a quantum mechanical equation is. 415 00:25:43,250 --> 00:25:46,180 We can just use physical understanding, 416 00:25:46,180 --> 00:25:47,630 physical analogies. 417 00:25:47,630 --> 00:25:50,540 And that's how quantum mechanics was developed in [INAUDIBLE]. 418 00:25:50,540 --> 00:25:53,950 Based on the classic quantum mechanical correspondence, 419 00:25:53,950 --> 00:25:57,890 this seems to be a very reasonable assumption. 420 00:25:57,890 --> 00:26:01,200 And ultimately, this very reasonable assumption 421 00:26:01,200 --> 00:26:05,030 has now withstood the test of time for many, many decades, 422 00:26:05,030 --> 00:26:06,790 over almost 100 years. 423 00:26:06,790 --> 00:26:11,530 So that's why we assume that doing the same substitution 424 00:26:11,530 --> 00:26:15,460 in the Schrodinger equation for the momentum operator 425 00:26:15,460 --> 00:26:18,400 gives the right result, but there is no way 426 00:26:18,400 --> 00:26:20,780 how you can rigorously derive fundamental equations 427 00:26:20,780 --> 00:26:21,470 in physics. 428 00:26:21,470 --> 00:26:24,020 You can observe nature, postulate them, 429 00:26:24,020 --> 00:26:26,350 and verify them. 430 00:26:26,350 --> 00:26:30,630 So therefore, this line, which was the classical Hamiltonian, 431 00:26:30,630 --> 00:26:33,170 by interpreting p as an operator, 432 00:26:33,170 --> 00:26:36,570 is now our quantum mechanical Hamiltonian. 433 00:26:36,570 --> 00:26:39,000 And the one thing we have to consider now 434 00:26:39,000 --> 00:26:41,175 is that once we have quantum mechanical operator, 435 00:26:41,175 --> 00:26:44,610 we have to be careful about compute order. 436 00:26:44,610 --> 00:26:48,200 That certain quantities have to be ordered. 437 00:26:48,200 --> 00:26:51,530 It matters which one we put first. 438 00:26:51,530 --> 00:26:54,595 What is convenient now is to use the Coulomb gauge. 439 00:26:58,050 --> 00:27:04,750 In the Coulomb gauge, diversions of A is zero. 440 00:27:04,750 --> 00:27:08,120 But the del operator, the deliberative operator, 441 00:27:08,120 --> 00:27:10,420 is the operator of the canonical momentum. 442 00:27:14,020 --> 00:27:17,390 And therefore, if you use the Coulomb gauge, 443 00:27:17,390 --> 00:27:25,230 then we have commutativity between the canonical momentum, 444 00:27:25,230 --> 00:27:28,228 p, and the vector potential. 445 00:27:28,228 --> 00:27:32,200 And that's sort of nice, because if you take this square 446 00:27:32,200 --> 00:27:37,170 and square it out, we get p dot A plus A dot p, 447 00:27:37,170 --> 00:27:41,740 but they are the same in the Coulomb gauge. 448 00:27:41,740 --> 00:27:47,805 So therefore our Hamiltonian has now the following terms. 449 00:27:58,170 --> 00:27:59,760 We call it H naught. 450 00:27:59,760 --> 00:28:03,484 And we assume that the Coulomb potential of the nucleus, 451 00:28:03,484 --> 00:28:05,400 which eventually gets the electronic structure 452 00:28:05,400 --> 00:28:07,880 of the atom, is included here. 453 00:28:07,880 --> 00:28:10,680 So this is the Hamiltonian part which 454 00:28:10,680 --> 00:28:13,300 describes the structure of atoms. 455 00:28:13,300 --> 00:28:16,530 And then we have two terms which couple 456 00:28:16,530 --> 00:28:17,985 to the electromagnetic field. 457 00:28:20,930 --> 00:28:22,901 So this is p dot A or A dot p. 458 00:28:22,901 --> 00:28:24,525 It doesn't matter in the Coulomb gauge. 459 00:28:30,010 --> 00:28:33,440 We call this the interaction Hamiltonian. 460 00:28:33,440 --> 00:28:35,750 And now we get a second order term, 461 00:28:35,750 --> 00:28:37,130 which is the A square term. 462 00:28:46,250 --> 00:28:50,150 Well, since it's second order, we designate it. 463 00:28:50,150 --> 00:28:52,722 We use the symbol H2. 464 00:28:52,722 --> 00:28:55,690 In the following, we neglect the second order term. 465 00:28:55,690 --> 00:28:58,160 The argument is for weak fields. 466 00:28:58,160 --> 00:28:58,910 It doesn't matter. 467 00:28:58,910 --> 00:29:01,470 It's not so important because it's higher order. 468 00:29:01,470 --> 00:29:06,620 However, in A422, we look at it more carefully 469 00:29:06,620 --> 00:29:09,930 and we can actually do a canonical transformation which 470 00:29:09,930 --> 00:29:11,920 eliminates the A square term. 471 00:29:11,920 --> 00:29:14,155 So to drop it is not necessary. 472 00:29:16,690 --> 00:29:19,920 You can actually show with a canonical transformation 473 00:29:19,920 --> 00:29:23,260 that I'm doing is actually more exact than just 474 00:29:23,260 --> 00:29:25,950 saying small and delineated. 475 00:29:25,950 --> 00:29:31,460 But anyway, I don't want to spend too much time on it. 476 00:29:31,460 --> 00:29:36,900 All I wanted to remind you, what are the steps to obtain-- just 477 00:29:36,900 --> 00:29:41,600 give me one second-- to obtain the interaction Hamiltonian, 478 00:29:41,600 --> 00:29:44,000 and that's what I want to use for the remainder 479 00:29:44,000 --> 00:29:45,990 of this class. 480 00:29:45,990 --> 00:29:49,900 That we have the coupling of momentum 481 00:29:49,900 --> 00:29:52,440 to the vector potential. 482 00:29:52,440 --> 00:29:56,299 We assume that the vector potential-- we are not 483 00:29:56,299 --> 00:29:57,840 quantizing the electromagnetic field. 484 00:29:57,840 --> 00:29:59,970 It's a semi-classical field. 485 00:29:59,970 --> 00:30:02,080 It's therefore a classical vector. 486 00:30:05,800 --> 00:30:13,390 And we will investigate what is now 487 00:30:13,390 --> 00:30:17,560 the coupling between an atom for this Hamiltonian 488 00:30:17,560 --> 00:30:20,710 to a plane wave of the electromagnetic field. 489 00:30:20,710 --> 00:30:24,500 So therefore we are introducing for the vector potential 490 00:30:24,500 --> 00:30:27,400 the expression that the vector potential has an Hamiltonian 491 00:30:27,400 --> 00:30:28,220 naught. 492 00:30:28,220 --> 00:30:32,000 The polarization p hat, and then the plane wave factor 493 00:30:32,000 --> 00:30:35,234 e to the ikr minus i omega t. 494 00:30:38,580 --> 00:30:41,470 So this can be divided in a much more rigorous way, 495 00:30:41,470 --> 00:30:44,370 but this is now the starting point of our discussion. 496 00:30:44,370 --> 00:30:45,830 [INAUDIBLE] 497 00:30:45,830 --> 00:30:46,330 OK. 498 00:30:51,730 --> 00:30:52,962 Any questions about that? 499 00:30:52,962 --> 00:30:53,826 Yes. 500 00:30:53,826 --> 00:30:55,122 Matt. 501 00:30:55,122 --> 00:30:57,464 AUDIENCE: Is it a little bit weird 502 00:30:57,464 --> 00:31:01,360 that you're going to take the divergence of its particular A 503 00:31:01,360 --> 00:31:03,126 that's not 0? 504 00:31:03,126 --> 00:31:09,535 Since it's divergence of e to the ik dot r, just k dot A? 505 00:31:28,762 --> 00:31:30,241 PROFESSOR: Sorry. 506 00:31:30,241 --> 00:31:40,430 You said the divergence of A-- so you're 507 00:31:40,430 --> 00:31:42,710 asking what is the divergence of A, 508 00:31:42,710 --> 00:31:46,011 if A is a plane wave e to the ikr? 509 00:31:46,011 --> 00:31:49,304 AUDIENCE: [INAUDIBLE] the propagation of [INAUDIBLE]? 510 00:31:49,304 --> 00:31:49,970 PROFESSOR: Yeah. 511 00:31:49,970 --> 00:31:56,550 What happens is, the polarization is 512 00:31:56,550 --> 00:31:59,150 A, and if you take the divergence of the plane wave, 513 00:31:59,150 --> 00:32:01,800 you get a K vector, so you get this [INAUDIBLE] product of e 514 00:32:01,800 --> 00:32:02,750 dot k. 515 00:32:02,750 --> 00:32:06,280 And electromagnetic waves propagate. 516 00:32:06,280 --> 00:32:08,540 The polarizations [INAUDIBLE] propagation. 517 00:32:08,540 --> 00:32:15,690 So ultimately what you find is with Coulomb gauge 518 00:32:15,690 --> 00:32:17,300 and the radiation field is transverse. 519 00:32:17,300 --> 00:32:19,720 It is only polarization [INAUDIBLE] propagation. 520 00:32:25,670 --> 00:32:26,330 OK. 521 00:32:26,330 --> 00:32:30,220 So we want to talk about matrix element. 522 00:32:30,220 --> 00:32:35,000 So the matrix element between two states. 523 00:32:35,000 --> 00:32:40,670 Du to the interaction Hamiltonian for one plane wave 524 00:32:40,670 --> 00:32:46,200 has the e to the minus i omega t dependence, which is trivial. 525 00:32:46,200 --> 00:32:47,670 I mean, in terms of we will assume 526 00:32:47,670 --> 00:32:49,640 that there is a monochromatic wave. 527 00:32:49,640 --> 00:32:52,670 And what you want to discuss now is 528 00:32:52,670 --> 00:32:55,887 this time independent factor, H ba. 529 00:33:04,850 --> 00:33:06,575 Let me get the factors. 530 00:33:06,575 --> 00:33:10,300 Let me get the polarization. 531 00:33:10,300 --> 00:33:16,150 So your relevant operator which connects the two states A and B 532 00:33:16,150 --> 00:33:17,590 is the momentum operator. 533 00:33:21,730 --> 00:33:29,720 And now for the plane wave, we want 534 00:33:29,720 --> 00:33:41,900 to do an expansion in orders of kr. 535 00:33:41,900 --> 00:33:50,200 And the leading term will be the dipole approximation 536 00:33:50,200 --> 00:33:52,340 and the next order term will give rise 537 00:33:52,340 --> 00:33:56,750 to magnetic dipole and electric quadrupole transitions. 538 00:33:56,750 --> 00:34:00,480 So first, the fact that I can do this expansion 539 00:34:00,480 --> 00:34:06,200 requires that k dot A is much smaller than 1. 540 00:34:11,030 --> 00:34:22,150 So it's a long wave length approximation, 541 00:34:22,150 --> 00:34:26,406 which you can say is valid as long as the atom is smaller 542 00:34:26,406 --> 00:34:29,550 than the wave length we are talking about. 543 00:34:29,550 --> 00:34:33,300 And I will show you later when I discuss the next higher order 544 00:34:33,300 --> 00:34:36,719 term that for atoms in the ground state, 545 00:34:36,719 --> 00:34:39,650 this is actually also an expansion alpha. 546 00:34:39,650 --> 00:34:43,020 So again we retrieve the fine-structure constant. 547 00:34:43,020 --> 00:34:51,520 So when we do this expansion of this plane wave exponential, 548 00:34:51,520 --> 00:34:54,960 every term is smaller than the previous term 549 00:34:54,960 --> 00:35:09,240 by the fine-structure constant alpha, which is 1 over 137. 550 00:35:09,240 --> 00:35:09,870 OK. 551 00:35:09,870 --> 00:35:21,870 So the leading term is the 1, and this gives rise 552 00:35:21,870 --> 00:35:24,347 to the dipole approximation. 553 00:35:24,347 --> 00:35:26,180 So what I want to show you now in one or two 554 00:35:26,180 --> 00:35:29,830 minutes is that this is indeed the dipole approximation. 555 00:35:29,830 --> 00:35:31,500 It leads to a matrix element which 556 00:35:31,500 --> 00:35:36,720 is electric field times r, the dipole moment. 557 00:35:36,720 --> 00:35:40,420 What we have right now is A dot p, the vector potential, 558 00:35:40,420 --> 00:35:43,149 times the momentum. 559 00:35:43,149 --> 00:35:44,690 So in other words, I want to show you 560 00:35:44,690 --> 00:35:49,000 that the A dot p matrix element is equivalent to an e dot r 561 00:35:49,000 --> 00:35:51,800 matrix element. 562 00:35:51,800 --> 00:35:54,150 This can be done with a canonical transformation, which 563 00:35:54,150 --> 00:35:58,350 we'll do in A422, but here let me sort of just show you 564 00:35:58,350 --> 00:36:01,220 the elementary discussion. 565 00:36:01,220 --> 00:36:03,780 We want to replace the vector potential 566 00:36:03,780 --> 00:36:05,980 by the electric field. 567 00:36:05,980 --> 00:36:10,810 So therefore, we use the fact that the electric field 568 00:36:10,810 --> 00:36:15,650 is the derivative of the vector potential. 569 00:36:15,650 --> 00:36:18,280 And the derivative of the vector potential 570 00:36:18,280 --> 00:36:23,920 means since we have an E the minus i omega t dependence, 571 00:36:23,920 --> 00:36:28,210 that we simply multiply with a factor of omega 572 00:36:28,210 --> 00:36:29,130 of the frequency. 573 00:36:32,220 --> 00:36:37,180 And what we obtain here is then the amplitude 574 00:36:37,180 --> 00:36:40,710 of the electric field, E naught. 575 00:36:40,710 --> 00:36:45,260 So with that, we have a matrix element 576 00:36:45,260 --> 00:36:49,945 which now involves the electric field. 577 00:36:52,870 --> 00:36:57,635 But still the matrix element of the momentum operator. 578 00:37:00,390 --> 00:37:04,210 But we can easily go from the momentum operator 579 00:37:04,210 --> 00:37:08,080 to the position operator by taking the fact 580 00:37:08,080 --> 00:37:10,040 that the momentum operator is nothing 581 00:37:10,040 --> 00:37:17,110 else than the commutator of r with the Hamiltonian H naught. 582 00:37:17,110 --> 00:37:20,370 The kinetic energy in H naught is p square, 583 00:37:20,370 --> 00:37:26,500 and the commutator of r with p square is simply p. 584 00:37:26,500 --> 00:37:30,475 So this is what I'm using here. 585 00:37:30,475 --> 00:37:34,858 And these are the p factors. 586 00:37:40,710 --> 00:37:48,140 So therefore if you take the momentum operator between two 587 00:37:48,140 --> 00:37:59,400 states, we have this relation between the momentum operator 588 00:37:59,400 --> 00:38:02,690 and the position operator. 589 00:38:02,690 --> 00:38:06,340 But we have not the momentum operator. 590 00:38:06,340 --> 00:38:07,960 We have the commutator. 591 00:38:07,960 --> 00:38:11,910 And r H naught gives us r. 592 00:38:11,910 --> 00:38:17,090 But when H naught acts on the a, it gives the energy in a. 593 00:38:17,090 --> 00:38:19,300 For the other part of the commutator, which 594 00:38:19,300 --> 00:38:23,230 is H naught r, we can have H naught act on the left hand 595 00:38:23,230 --> 00:38:28,644 side on b, and this gives us the energy ab. 596 00:38:28,644 --> 00:38:32,460 For dividing by H bar, what we get 597 00:38:32,460 --> 00:38:36,330 is the energy difference, eb minus ea. 598 00:38:36,330 --> 00:38:39,950 Or in frequency units, omega ba. 599 00:38:39,950 --> 00:38:43,435 So this is now how we have implemented the commutator. 600 00:38:46,460 --> 00:38:59,220 So that by inserting this into this equation, 601 00:38:59,220 --> 00:39:04,890 we now finally obtain our result for the matrix element 602 00:39:04,890 --> 00:39:08,210 in the dipole approximation. 603 00:39:08,210 --> 00:39:15,210 We have e, the electric field amplitude, the polarization, 604 00:39:15,210 --> 00:39:22,890 the matrix element b r a, omega ba, 605 00:39:22,890 --> 00:39:27,380 and here we have an 1 over omega in the denominator, which 606 00:39:27,380 --> 00:39:31,451 came from the derivative of the vector potential da dt. 607 00:39:47,470 --> 00:39:51,915 This here is the matrix element of the dipole operator. 608 00:39:55,790 --> 00:40:01,610 So in other words, I have derived for you 609 00:40:01,610 --> 00:40:04,862 that the interaction matrix element 610 00:40:04,862 --> 00:40:15,420 in the dipole approximation is the dipole operator 611 00:40:15,420 --> 00:40:21,720 times the electric field as long as I make the approximation 612 00:40:21,720 --> 00:40:26,140 that this frequency factor's on the order of [INAUDIBLE]. 613 00:40:26,140 --> 00:40:32,530 Now this is clearly the case in your resonance, 614 00:40:32,530 --> 00:40:35,240 so you should be safe. 615 00:40:38,840 --> 00:40:43,545 However, if you do the more rigorous derivation 616 00:40:43,545 --> 00:40:48,770 of the dipole Hamiltonian using canonical transformation, 617 00:40:48,770 --> 00:40:52,960 this factor of omega ba over omega does not appear. 618 00:40:52,960 --> 00:40:55,096 So again, the dipole approximations 619 00:40:55,096 --> 00:40:57,460 is better than I presented today. 620 00:40:57,460 --> 00:41:02,070 The two approximations topping the A squared term 621 00:41:02,070 --> 00:41:06,340 and approximating this ratio of frequencies with 1 622 00:41:06,340 --> 00:41:10,619 is not necessary if you use the other methods using 623 00:41:10,619 --> 00:41:11,660 canonical transformation. 624 00:41:23,130 --> 00:41:24,790 Any questions about that? 625 00:41:27,780 --> 00:41:29,190 Let me make one comment. 626 00:41:29,190 --> 00:41:31,510 I've shown you in this derivation 627 00:41:31,510 --> 00:41:38,540 that the A dot p term, the A dot p interaction 628 00:41:38,540 --> 00:41:43,180 within the dipole approximation is identical to the e dot r 629 00:41:43,180 --> 00:41:44,770 interaction. 630 00:41:44,770 --> 00:41:49,190 So these are the same operators, and if you would exactly 631 00:41:49,190 --> 00:41:52,170 calculate them in atomic structure calculation, 632 00:41:52,170 --> 00:41:55,740 you would get the same coupling, the same matrix element. 633 00:41:55,740 --> 00:41:59,330 However, in practice, there are important differences. 634 00:41:59,330 --> 00:42:03,450 Because the E dot r operator, the r operator, 635 00:42:03,450 --> 00:42:06,550 has a lot of weight where the electron is 636 00:42:06,550 --> 00:42:10,100 far away from the nucleus because it's multiplied with r. 637 00:42:10,100 --> 00:42:14,130 Whereas the p operator emphasizes the derivative 638 00:42:14,130 --> 00:42:16,580 of the wave function, and this is usually 639 00:42:16,580 --> 00:42:19,260 strong at close distances. 640 00:42:19,260 --> 00:42:22,840 So in the end, the results have to be the same. 641 00:42:22,840 --> 00:42:25,670 But if you make an approximation to your wave function, 642 00:42:25,670 --> 00:42:30,130 one formulation may be numerically much more accurate 643 00:42:30,130 --> 00:42:32,040 than the other one. 644 00:42:32,040 --> 00:42:35,430 But fundamentally, the two terms are the same 645 00:42:35,430 --> 00:42:39,988 and they are related the way how I derived it for you. 646 00:42:39,988 --> 00:42:40,488 Yes? 647 00:42:40,488 --> 00:42:42,958 AUDIENCE: So when we say it's the dipole approximation, 648 00:42:42,958 --> 00:42:46,190 does it mean that it's a long wave length, 649 00:42:46,190 --> 00:42:49,310 or be the small field or everything together. 650 00:42:49,310 --> 00:42:53,574 So what's the absolute fundamental assumption of this? 651 00:42:53,574 --> 00:42:55,490 PROFESSOR: The absolute fundamental assumption 652 00:42:55,490 --> 00:42:58,410 of the dipole approximation is that we 653 00:42:58,410 --> 00:43:00,800 assume-- let me just scroll up. 654 00:43:06,170 --> 00:43:08,190 In the equation above, we had an expression 655 00:43:08,190 --> 00:43:10,240 for the vector potential. 656 00:43:10,240 --> 00:43:18,570 And the dipole approximation means 657 00:43:18,570 --> 00:43:21,270 that we can neglect the position dependence of the vector 658 00:43:21,270 --> 00:43:25,160 potential A of r, so we approximate A 659 00:43:25,160 --> 00:43:31,610 of r as being A evaluated at the origin of the atom. 660 00:43:31,610 --> 00:43:35,540 And over the extent, over the size of the atom, 661 00:43:35,540 --> 00:43:37,710 we do not have to consider a spacial dependence 662 00:43:37,710 --> 00:43:40,380 of the vector potential. 663 00:43:40,380 --> 00:43:42,490 So this is sufficient. 664 00:43:42,490 --> 00:43:45,740 The other things that e square is small 665 00:43:45,740 --> 00:43:49,760 and that we're near resonance-- I 666 00:43:49,760 --> 00:43:51,480 have to make those assumptions if I 667 00:43:51,480 --> 00:43:53,170 want to use this elementary derivation. 668 00:43:53,170 --> 00:43:56,590 But with a canonical transformation, 669 00:43:56,590 --> 00:43:59,790 as is discussed in atom-photon interactions, 670 00:43:59,790 --> 00:44:03,190 you do not need those additional assumptions. 671 00:44:03,190 --> 00:44:05,870 The only assumption behind the dipole approximation 672 00:44:05,870 --> 00:44:10,750 is that one, and this means that the extent of the atom 673 00:44:10,750 --> 00:44:13,092 is much smaller than the optical wave lengths. 674 00:44:25,960 --> 00:44:26,510 OK. 675 00:44:26,510 --> 00:44:33,140 But there are situations where the leading approximation 676 00:44:33,140 --> 00:44:34,050 vanishes. 677 00:44:34,050 --> 00:44:37,930 For instance, if two levels have the same parity, 678 00:44:37,930 --> 00:44:41,820 then the dipole operator between the two of them is 0. 679 00:44:41,820 --> 00:44:44,966 And then if you want to have a transition between, 680 00:44:44,966 --> 00:44:47,340 or if you want to consider a transition between those two 681 00:44:47,340 --> 00:44:52,410 levels, it will come from next order terms. 682 00:44:52,410 --> 00:45:00,960 So let's now discuss higher order radiation processes. 683 00:45:12,410 --> 00:45:15,190 So the motivation why I want to discuss higher order radiation 684 00:45:15,190 --> 00:45:19,420 processes is because in some basic courses, 685 00:45:19,420 --> 00:45:21,830 you only need the dipole approximation. 686 00:45:21,830 --> 00:45:24,145 And you think the dipole coupling 687 00:45:24,145 --> 00:45:26,630 is the only coupling which exists in the world. 688 00:45:26,630 --> 00:45:28,447 And by going to higher order, I want 689 00:45:28,447 --> 00:45:31,240 to show you that this is not the case. 690 00:45:31,240 --> 00:45:35,050 Also I want to sort of give you an idea what 691 00:45:35,050 --> 00:45:38,860 it means if you have leading order 692 00:45:38,860 --> 00:45:40,310 transitions are forbidden. 693 00:45:40,310 --> 00:45:42,910 I want to sort of show you how other terms come 694 00:45:42,910 --> 00:45:46,150 in which can couple two levels. 695 00:45:46,150 --> 00:45:53,600 And also, actually, when you drive transitions 696 00:45:53,600 --> 00:45:56,950 within the hyper-fine structure using radio frequency fields, 697 00:45:56,950 --> 00:45:59,780 you are not driving them with the electric dipole 698 00:45:59,780 --> 00:46:00,546 approximation. 699 00:46:00,546 --> 00:46:03,110 You're driving them with a magnetic dipole, 700 00:46:03,110 --> 00:46:07,820 and this is actually the next approximation. 701 00:46:07,820 --> 00:46:09,610 So there's a number of reasons why 702 00:46:09,610 --> 00:46:13,100 I want to show you what the next order terms are, 703 00:46:13,100 --> 00:46:16,050 and how they actually lead to beautiful result, 704 00:46:16,050 --> 00:46:19,222 magnetic dipole, and electric quadrupole transitions. 705 00:46:35,090 --> 00:46:39,290 So our coupling term-- let me just rewrite the equation. 706 00:46:46,450 --> 00:46:49,060 Let me now simplify the notation by assuming 707 00:46:49,060 --> 00:46:52,075 that the polarization is in the z direction 708 00:46:52,075 --> 00:46:54,135 and the propagation is in the x direction. 709 00:46:56,910 --> 00:47:03,880 So then the coupling had the dipole term. 710 00:47:03,880 --> 00:47:08,170 And the next order term is ikx, and this 711 00:47:08,170 --> 00:47:10,150 is what we want to investigate now. 712 00:47:16,670 --> 00:47:26,450 So the term kx or kr is smaller. 713 00:47:26,450 --> 00:47:28,626 And now I want to show you explicitly 714 00:47:28,626 --> 00:47:31,380 that it's smaller by alpha. 715 00:47:31,380 --> 00:47:37,140 The k vector of the photon is related to the frequency H bar 716 00:47:37,140 --> 00:47:40,650 omega divided by H bar z. 717 00:47:40,650 --> 00:47:44,126 If I approximate for r, the relevant r 718 00:47:44,126 --> 00:47:46,910 when we indicate over the wave function 719 00:47:46,910 --> 00:47:48,412 will be the Bohr radius. 720 00:48:00,860 --> 00:48:07,040 The relevant frequency, H bar omega, well, it's a Rydberg. 721 00:48:07,040 --> 00:48:13,100 And the Rydberg is e square over the Bohr radius. 722 00:48:13,100 --> 00:48:20,700 So now if I insert it, the Bohr radius cancels out. 723 00:48:20,700 --> 00:48:23,770 And what we find is, well, kr is dimensionless. 724 00:48:23,770 --> 00:48:27,130 We have to find something which is dimensionless, 725 00:48:27,130 --> 00:48:30,620 but expressed by the fundamental constants of atomic physics. 726 00:48:30,620 --> 00:48:34,287 And the only quantity which is available for that-- it's not 727 00:48:34,287 --> 00:48:35,430 a surprise-- is alpha. 728 00:48:35,430 --> 00:48:37,960 The fine-structure constant. 729 00:48:37,960 --> 00:48:41,340 So therefore the dipole approximation 730 00:48:41,340 --> 00:48:46,110 is actually the result of an expansion of the plane wave 731 00:48:46,110 --> 00:48:50,185 e the ikr factor in units of expansion 732 00:48:50,185 --> 00:48:51,536 in the fine-structure constant. 733 00:48:58,790 --> 00:49:00,180 OK. 734 00:49:00,180 --> 00:49:02,730 We want to look now at the second term. 735 00:49:10,440 --> 00:49:14,050 And well, sometimes if you deal with a term, 736 00:49:14,050 --> 00:49:16,800 we first make it more complicated 737 00:49:16,800 --> 00:49:20,220 and then we simplify it. 738 00:49:20,220 --> 00:49:24,420 I want to sort of symmetrize and anti-symmetrize it 739 00:49:24,420 --> 00:49:26,570 in the following way. 740 00:49:26,570 --> 00:49:30,150 This is p z x. 741 00:49:30,150 --> 00:49:35,890 So let me subtract z p x. 742 00:49:35,890 --> 00:49:39,000 But then, of course, I have to add it. 743 00:49:45,940 --> 00:49:48,360 So now we have two terms. 744 00:49:48,360 --> 00:49:52,020 1 has a minus sign, one has a plus sign. 745 00:49:52,020 --> 00:49:56,940 And as will see in just a minute, 746 00:49:56,940 --> 00:49:59,690 is the first one is magnetic dipole transition. 747 00:49:59,690 --> 00:50:01,780 The second one is electric quadrupole transition. 748 00:50:04,550 --> 00:50:11,500 And we see that the first one can be regarded as-- p 749 00:50:11,500 --> 00:50:14,890 z x is like p dot r. 750 00:50:14,890 --> 00:50:18,250 Its the y component of the vector product and 751 00:50:18,250 --> 00:50:21,334 is therefore the y component of the orbital angular momentum. 752 00:50:30,280 --> 00:50:32,760 OK. 753 00:50:32,760 --> 00:50:40,070 So let's focus for now on this part. 754 00:50:40,070 --> 00:50:42,850 The second term, which is the electric quadrupole term, 755 00:50:42,850 --> 00:50:44,255 we'll do in a few moments. 756 00:50:48,040 --> 00:50:54,600 The relevant matrix element is now the matrix element 757 00:50:54,600 --> 00:50:58,130 of the orbital angular momentum Ly. 758 00:51:11,770 --> 00:51:14,930 The p factor-- let me just collect 759 00:51:14,930 --> 00:51:27,040 all of the constant imaginary unit ehr AK over 2 mc. 760 00:51:27,040 --> 00:51:29,340 That looks complicated, but it immediately 761 00:51:29,340 --> 00:51:36,855 simplifies when we realize that this here is the Bohr magneton. 762 00:51:39,640 --> 00:51:44,132 And well, we still have the vector potential. 763 00:51:44,132 --> 00:51:52,040 But the magnetic field is the curl of the vector potential, 764 00:51:52,040 --> 00:51:55,120 and that means we will assume that we have a vector 765 00:51:55,120 --> 00:51:59,270 potential propagating in x, polarized in z. 766 00:51:59,270 --> 00:52:07,230 That means that our magnetic field is that. 767 00:52:07,230 --> 00:52:13,650 So therefore ka, which appears in our expression 768 00:52:13,650 --> 00:52:18,160 for the coupling, is just the magnetic field. 769 00:52:18,160 --> 00:52:21,720 So therefore we find the result that the next order 770 00:52:21,720 --> 00:52:23,940 term in the coupling of the atoms 771 00:52:23,940 --> 00:52:27,432 to the electromagnetic field have the simple form 772 00:52:27,432 --> 00:52:28,640 to the electromagnetic field. 773 00:52:28,640 --> 00:52:31,660 That it is the magnetic field part 774 00:52:31,660 --> 00:52:38,240 of the electromagnetic wave times the Bohr magneton 775 00:52:38,240 --> 00:52:47,480 times the matrix element, due to the orbital angular momentum 776 00:52:47,480 --> 00:52:47,980 operator. 777 00:52:50,500 --> 00:52:55,510 And actually, if you take the orbital angular momentum 778 00:52:55,510 --> 00:52:58,890 and multiply it with the Bohr magneton, 779 00:52:58,890 --> 00:53:04,487 this is actually the operator of the magnetic moment. 780 00:53:04,487 --> 00:53:06,820 Well, with a minus sign because the electron is negative 781 00:53:06,820 --> 00:53:08,650 charged. 782 00:53:08,650 --> 00:53:11,750 Remember we had introduced the operator 783 00:53:11,750 --> 00:53:12,750 for the magnetic moment. 784 00:53:12,750 --> 00:53:17,150 And the magnetic moment was the g factor 785 00:53:17,150 --> 00:53:20,320 times the Bohr magneton times the orbital angular momentum. 786 00:53:20,320 --> 00:53:24,320 And the g factor for the orbital motion is 1. 787 00:53:27,270 --> 00:53:32,180 So therefore the interaction we are talking about 788 00:53:32,180 --> 00:53:37,415 is the Bohr magneton times the magnetic field. 789 00:53:40,540 --> 00:53:44,440 And, of course, what we realize is the operator-- maybe 790 00:53:44,440 --> 00:53:48,230 I should back up for a second and say what we actually 791 00:53:48,230 --> 00:53:51,625 realize is that operator which couples 792 00:53:51,625 --> 00:53:59,640 to the electromagnetic field has actually the form u dot b. 793 00:53:59,640 --> 00:54:04,070 This is exactly what we use for the Zeeman effect in a DC 794 00:54:04,070 --> 00:54:05,570 magnetic field. 795 00:54:05,570 --> 00:54:09,720 But now the same form u dot b appears 796 00:54:09,720 --> 00:54:12,414 for time dependent magnetic field. 797 00:54:12,414 --> 00:54:13,830 And time depending magnetic fields 798 00:54:13,830 --> 00:54:16,160 are not only creating level shifts. 799 00:54:16,160 --> 00:54:20,160 They can also use transitions through the matrix element. 800 00:54:20,160 --> 00:54:23,540 So in other words, this whole exercise 801 00:54:23,540 --> 00:54:29,010 show you that the form u dot b, which appeared naturally 802 00:54:29,010 --> 00:54:31,520 in the formulation for DC magnetic field, 803 00:54:31,520 --> 00:54:34,046 also applies to AC magnetic fields. 804 00:54:37,990 --> 00:54:41,040 But with that, I can say, wait a moment. 805 00:54:41,040 --> 00:54:43,910 There are now two sources for the magnetic movement 806 00:54:43,910 --> 00:54:45,270 of the atom. 807 00:54:45,270 --> 00:54:52,040 One is due to orbital angular momentum, 808 00:54:52,040 --> 00:54:55,470 and the other one is due to spin angular momentum. 809 00:54:55,470 --> 00:54:57,510 But the spin angular momentum has 810 00:54:57,510 --> 00:54:58,995 a g factor, which is different. 811 00:55:01,730 --> 00:55:06,220 The approximation of the [INAUDIBLE] creation is 2. 812 00:55:06,220 --> 00:55:10,050 So therefore I mean, we will never 813 00:55:10,050 --> 00:55:13,000 get spin out of a semi-classical discussion. 814 00:55:13,000 --> 00:55:16,945 Remember, we started with a classical canonical treatment 815 00:55:16,945 --> 00:55:20,360 of the electromagnetic field, canonical momentum. 816 00:55:20,360 --> 00:55:22,190 And now we are running with it and we 817 00:55:22,190 --> 00:55:25,695 find that there is a coupling between the magnetic field 818 00:55:25,695 --> 00:55:28,076 and the magnetic moment of the atom. 819 00:55:28,076 --> 00:55:31,325 But, of course, we only get the magnetic moment to the extent 820 00:55:31,325 --> 00:55:33,820 that it comes from orbital motion. 821 00:55:33,820 --> 00:55:37,860 But in a semi-classical way, I'm waving my hands now and say, 822 00:55:37,860 --> 00:55:39,600 well, what is valid for the coupling 823 00:55:39,600 --> 00:55:43,300 to orbit to the magnetic moment of the orbital angular momentum 824 00:55:43,300 --> 00:55:46,340 also applies to the speed. 825 00:55:46,340 --> 00:55:50,300 And I'm simply adding the spin here. 826 00:55:50,300 --> 00:55:54,680 And with that, I've derived for you 827 00:55:54,680 --> 00:55:58,810 the expression for the interaction matrix element 828 00:55:58,810 --> 00:56:02,390 called M1. 829 00:56:02,390 --> 00:56:08,610 M1 is the magnetic dipole transitions. 830 00:56:16,290 --> 00:56:17,860 Let me just write it down here. 831 00:56:17,860 --> 00:56:25,640 So this is nothing else than u dot b. 832 00:56:35,440 --> 00:56:36,420 Any questions? 833 00:56:43,280 --> 00:56:44,500 Let me just summarize. 834 00:56:44,500 --> 00:56:46,580 You may find it sort of interesting, 835 00:56:46,580 --> 00:56:50,450 when we discussed static electric and static magnetic 836 00:56:50,450 --> 00:56:50,950 fields. 837 00:56:50,950 --> 00:56:52,900 For the static electric field, we 838 00:56:52,900 --> 00:56:55,220 had an electric field times the dipole. 839 00:56:55,220 --> 00:56:58,590 And we find this now as a time independent term 840 00:56:58,590 --> 00:57:04,760 which can drive transitions for interactions with time 841 00:57:04,760 --> 00:57:07,310 dependent electromagnetic field, but we find it 842 00:57:07,310 --> 00:57:09,110 in the dipole approximation. 843 00:57:09,110 --> 00:57:12,980 The magnetic part, u dot b, we find 844 00:57:12,980 --> 00:57:14,490 when we go to the next order. 845 00:57:14,490 --> 00:57:17,740 We find it as a magnetic dipole term. 846 00:57:17,740 --> 00:57:20,940 But there are more terms, and I just 847 00:57:20,940 --> 00:57:22,560 want you to illustrate that and then 848 00:57:22,560 --> 00:57:25,320 I'll stop with that multiple expansion. 849 00:57:25,320 --> 00:57:40,815 We had the second term, the kr term, was this one. 850 00:57:40,815 --> 00:57:42,440 But then we sort of anti-symmetrized it 851 00:57:42,440 --> 00:57:44,380 and symmetrized it. 852 00:57:44,380 --> 00:57:48,800 The first one here, we could relate to orbital angular 853 00:57:48,800 --> 00:57:51,065 momentum and to the magnetic moment. 854 00:57:51,065 --> 00:57:53,065 And now I would want to discuss the second term. 855 00:57:56,530 --> 00:58:07,540 So that term uses a mixture of position and momentum 856 00:58:07,540 --> 00:58:08,040 operators. 857 00:58:15,520 --> 00:58:22,860 But we know already how we can get rid of momentum operators, 858 00:58:22,860 --> 00:58:28,076 namely by expressing momentum operators as commutators 859 00:58:28,076 --> 00:58:28,950 with the Hamiltonian. 860 00:58:33,670 --> 00:58:43,820 So this gives us the commutator of z with H naught times x 861 00:58:43,820 --> 00:58:50,380 plus z times the commutator of x with H naught. 862 00:59:05,070 --> 00:59:05,570 OK. 863 00:59:05,570 --> 00:59:06,870 So we have two commutators. 864 00:59:06,870 --> 00:59:09,910 Each of them has two terms. 865 00:59:09,910 --> 00:59:12,260 That means a total of four terms. 866 00:59:12,260 --> 00:59:13,720 And if you write them down, you see 867 00:59:13,720 --> 00:59:16,690 that two are opposite but equal and cancel. 868 00:59:16,690 --> 00:59:22,970 So therefore we are left with two terms, which 869 00:59:22,970 --> 00:59:31,370 are minus H naught zx plus zx times H naught. 870 00:59:36,660 --> 00:59:43,480 So this term has now the following contribution 871 00:59:43,480 --> 00:59:55,890 to the coupling between the levels A and B. 872 00:59:55,890 --> 00:59:59,850 So we have this coupling matrix element. 873 01:00:08,455 --> 01:00:13,495 We have b factor mc Am. 874 01:00:19,360 --> 01:00:20,010 OK. 875 01:00:20,010 --> 01:00:26,065 So using the same approach we used for the dipole matrix 876 01:00:26,065 --> 01:00:30,630 element, the H naught can act on the state A on the right side, 877 01:00:30,630 --> 01:00:34,000 and can act on the state B on the left hand side. 878 01:00:34,000 --> 01:00:36,810 So this gives us simply the energy difference. 879 01:00:40,540 --> 01:00:44,620 And what is left is now a matrix element 880 01:00:44,620 --> 01:00:48,570 which only involves position operators, 881 01:00:48,570 --> 01:00:49,980 but it uses a product of 2. 882 01:01:01,740 --> 01:01:07,320 And now, again, we express the vector potential 883 01:01:07,320 --> 01:01:13,660 by the electric field, as we had done before. 884 01:01:13,660 --> 01:01:27,270 And therefore we have now expressed our coupling yes, 885 01:01:27,270 --> 01:01:29,620 by an electric field. 886 01:01:29,620 --> 01:01:33,430 And we assume that in the near resonant approximation, 887 01:01:33,430 --> 01:01:35,520 as we had used for the dipole coupling, 888 01:01:35,520 --> 01:01:37,195 that this is the frequency omega. 889 01:01:44,390 --> 01:01:46,570 OK. 890 01:01:46,570 --> 01:01:50,440 So what we are realizing is that we 891 01:01:50,440 --> 01:01:54,660 have one part of the interaction Hamiltonian 892 01:01:54,660 --> 01:01:59,440 which couples levels A and B, has-- 893 01:01:59,440 --> 01:02:04,670 we call it E2, this electric quadrupole 894 01:02:04,670 --> 01:02:11,945 because it involves elements of the quadrupole tensor, 895 01:02:11,945 --> 01:02:18,262 or products of coordinates x, y, y, z, x, z. 896 01:02:18,262 --> 01:02:21,870 And for this geometry, which we assume with our plane wave, 897 01:02:21,870 --> 01:02:22,680 it is z, x. 898 01:02:26,026 --> 01:02:27,980 It couples to the electric field. 899 01:02:32,830 --> 01:02:36,460 And this is the p factor. 900 01:02:36,460 --> 01:02:40,410 For a more general geometry of plane waves 901 01:02:40,410 --> 01:02:43,740 with different polarization going in different directions, 902 01:02:43,740 --> 01:02:51,570 we would have obtained different products of coordinates. 903 01:02:51,570 --> 01:02:56,890 So let me indicate that, that what we have picked out here 904 01:02:56,890 --> 01:03:00,290 is one specific component of a tensor, which 905 01:03:00,290 --> 01:03:06,590 is sort of the tensor form by using the position vector 906 01:03:06,590 --> 01:03:07,520 r twice. 907 01:03:11,710 --> 01:03:17,340 So in this derivation, we found when 908 01:03:17,340 --> 01:03:19,730 we go beyond the dipole approximation, 909 01:03:19,730 --> 01:03:26,202 when we take the kr term in leading order, that we 910 01:03:26,202 --> 01:03:27,160 have two contributions. 911 01:03:30,580 --> 01:03:35,280 One is M1, magnetic dipole. 912 01:03:35,280 --> 01:03:37,761 The other one was electric quadrupole. 913 01:03:42,820 --> 01:03:47,420 We realized-- I tried to keep of all the p factors-- 914 01:03:47,420 --> 01:03:57,840 that the electric quadrupole matrix element is imaginary, 915 01:03:57,840 --> 01:04:01,350 whereas the magnetic dipole was u dot b. 916 01:04:01,350 --> 01:04:02,850 There was no imaginary unit. 917 01:04:02,850 --> 01:04:03,350 It's real. 918 01:04:06,090 --> 01:04:13,620 That means that you will never have any interference 919 01:04:13,620 --> 01:04:18,040 effect between magnetic dipole and electric quadrupole. 920 01:04:18,040 --> 01:04:21,420 Or in other words, when we have processes 921 01:04:21,420 --> 01:04:24,240 like spontaneous emission where we 922 01:04:24,240 --> 01:04:26,670 take the square of the matrix element, 923 01:04:26,670 --> 01:04:31,230 the square of the matrix element will 924 01:04:31,230 --> 01:04:38,370 be the sum of the squares of the matrix element 925 01:04:38,370 --> 01:04:41,766 for magnetic dipole and electric quadrupole transitions. 926 01:04:51,700 --> 01:04:52,878 So let me summarize. 927 01:04:58,980 --> 01:05:05,630 We have discussed three different ways 928 01:05:05,630 --> 01:05:12,100 how we can have coupling matrix elements between two 929 01:05:12,100 --> 01:05:17,551 states, electric dipole, magnetic dipole, 930 01:05:17,551 --> 01:05:24,140 electric quadrupole, E1, M1, E2. 931 01:05:30,355 --> 01:05:35,640 The operator was the electric dipole operator. 932 01:05:38,300 --> 01:05:42,660 Here it was the operator of the magnetic movement, which 933 01:05:42,660 --> 01:05:48,430 is orbital angular momentum and spin angular momentum. 934 01:05:48,430 --> 01:05:53,430 And for the quadrupole, it was the quadratic expression 935 01:05:53,430 --> 01:05:55,815 in the spatial coordinates. 936 01:05:59,220 --> 01:06:03,690 You can also ask what is the parity? 937 01:06:03,690 --> 01:06:08,060 The electric dipole operator connects 938 01:06:08,060 --> 01:06:13,090 states with opposite parity, whereas both magnetic dipole 939 01:06:13,090 --> 01:06:15,900 and electric quadrupole connect states 940 01:06:15,900 --> 01:06:17,828 even with the same parity. 941 01:06:23,620 --> 01:06:31,162 Magnetic dipole and electric quadrupole transition 942 01:06:31,162 --> 01:06:34,310 are often called forbidden transitions. 943 01:06:39,380 --> 01:06:41,520 Well, you would say it's a misnomer because they 944 01:06:41,520 --> 01:06:44,790 are transitions, so they are allowed, but they are weak. 945 01:06:44,790 --> 01:06:46,550 But this is the language we use. 946 01:06:46,550 --> 01:06:48,750 Weak transitions are forbidden, which 947 01:06:48,750 --> 01:06:50,900 simply means they don't appear. 948 01:06:50,900 --> 01:06:52,520 They are forbidden in leading order, 949 01:06:52,520 --> 01:06:55,124 but when you go to higher order, they are allowed. 950 01:06:55,124 --> 01:06:57,540 You can, of course, say if they were completely forbidden, 951 01:06:57,540 --> 01:06:59,520 there would be no need to discuss them. 952 01:06:59,520 --> 01:07:01,800 But since they are only forbidden 953 01:07:01,800 --> 01:07:04,035 at a certain level, then, of course, 954 01:07:04,035 --> 01:07:05,410 it's interesting to discuss them. 955 01:07:05,410 --> 01:07:07,310 And a lot of narrow transitions which 956 01:07:07,310 --> 01:07:10,035 are relevant for atomic clocks are highly forbidden 957 01:07:10,035 --> 01:07:10,535 transitions. 958 01:07:14,220 --> 01:07:18,490 The strength of them, which usually scales with transitions 959 01:07:18,490 --> 01:07:21,270 with the square of the matrix element, 960 01:07:21,270 --> 01:07:25,410 is on the order of 5 times 10 to the minus 5. 961 01:07:25,410 --> 01:07:28,570 So those transitions are four or five orders 962 01:07:28,570 --> 01:07:38,400 of magnitude weaker Yeah than an allowed E1 transition. 963 01:07:44,822 --> 01:07:46,798 So that's what they are. 964 01:07:57,166 --> 01:07:57,666 Questions? 965 01:08:06,064 --> 01:08:09,028 AUDIENCE: When people talk about highly forbidden transitions, 966 01:08:09,028 --> 01:08:13,026 does this mean that's like a optical transition, or how 967 01:08:13,026 --> 01:08:15,814 do we distinguish it from just regularly forbidden? 968 01:08:17,351 --> 01:08:19,559 PROFESSOR: Actually I would say forbidden transitions 969 01:08:19,559 --> 01:08:22,590 are weaker by alpha to the power n. 970 01:08:22,590 --> 01:08:24,800 Here we have situations where the matrix 971 01:08:24,800 --> 01:08:26,700 element is just smaller by alpha. 972 01:08:26,700 --> 01:08:27,725 But, yes. 973 01:08:36,630 --> 01:08:38,474 Sometimes, yes. 974 01:08:41,300 --> 01:08:43,180 Some transitions are highly forbidden. 975 01:08:43,180 --> 01:08:45,310 For instance-- I try to remember-- 976 01:08:45,310 --> 01:08:48,620 if you have hydrogen 1s and 2s. 977 01:08:48,620 --> 01:08:52,787 Because of s states-- actually, you 978 01:08:52,787 --> 01:08:55,120 are asking question about the next chapter, namely about 979 01:08:55,120 --> 01:08:57,285 selection rules. 980 01:08:57,285 --> 01:08:59,029 Let me give it in words. 981 01:08:59,029 --> 01:09:01,779 The s, if you connect two s states, 982 01:09:01,779 --> 01:09:04,076 they have both 0 angular momentum. 983 01:09:04,076 --> 01:09:04,770 AUDIENCE: Yeah. 984 01:09:04,770 --> 01:09:07,353 PROFESSOR: So you cannot have a quadrupole operator connecting 985 01:09:07,353 --> 01:09:08,290 the two. 986 01:09:08,290 --> 01:09:12,060 It would violate the triangle rule. 987 01:09:12,060 --> 01:09:14,840 You cannot have angular momentum of 0 and angular momentum of 2 988 01:09:14,840 --> 01:09:16,840 and get angular momentum of 0. 989 01:09:16,840 --> 01:09:19,010 So therefore you have a transition 990 01:09:19,010 --> 01:09:20,970 between two states of the same parity. 991 01:09:20,970 --> 01:09:22,990 There is no dipole operator. 992 01:09:22,990 --> 01:09:24,410 There is no quadrupole operator. 993 01:09:24,410 --> 01:09:27,086 So you soon run into a situation where it's highly forbidden. 994 01:09:31,750 --> 01:09:35,330 Sometimes you have the situation that something is forbidden 995 01:09:35,330 --> 01:09:37,877 in non-relativistic physics, but they're 996 01:09:37,877 --> 01:09:39,710 a relativistic term, which makes it allowed. 997 01:09:43,116 --> 01:09:46,270 Well, there's also that relativistic terms 998 01:09:46,270 --> 01:09:49,859 are fine-structure terms is also an expansion alpha squared. 999 01:09:49,859 --> 01:09:52,149 So the symmetry may allow it, but only 1000 01:09:52,149 --> 01:09:54,730 in connection with relativistic terms. 1001 01:09:54,730 --> 01:09:56,610 I'm not an expert on forbidden transitions, 1002 01:09:56,610 --> 01:10:01,700 but usually you have transitions which are multiply forbidden. 1003 01:10:01,700 --> 01:10:03,860 They are forbidden by spacial symmetry. 1004 01:10:03,860 --> 01:10:05,640 For instance, in the helium atom, 1005 01:10:05,640 --> 01:10:09,640 singular triplet are forbidden by spin symmetry. 1006 01:10:09,640 --> 01:10:13,060 So if you have multiple layers of being forbidden, 1007 01:10:13,060 --> 01:10:15,400 then you get extremely weak transitions. 1008 01:10:15,400 --> 01:10:20,786 And one example, actually, are the singular triplet transition 1009 01:10:20,786 --> 01:10:25,260 in helium, or the 1s to 2s transition in hydrogen. 1010 01:10:25,260 --> 01:10:27,650 They are not allowed by simply going 1011 01:10:27,650 --> 01:10:32,050 to the next order in a multiple expansion. 1012 01:10:32,050 --> 01:10:32,550 Yes? 1013 01:10:32,550 --> 01:10:35,022 AUDIENCE: In ordinary situations when 1014 01:10:35,022 --> 01:10:37,827 you do have to possibly do at least even more [INAUDIBLE] 1015 01:10:37,827 --> 01:10:38,910 for the other transitions? 1016 01:10:42,800 --> 01:10:44,930 PROFESSOR: I'm not an expert on that. 1017 01:10:44,930 --> 01:10:47,450 I'm not sure if there is an atom which 1018 01:10:47,450 --> 01:10:50,550 has a relevant transition which is a [INAUDIBLE] transition. 1019 01:10:50,550 --> 01:10:53,380 I've not heard about that. 1020 01:10:53,380 --> 01:10:57,300 At least relevant examples which are the fundamental atoms, 1021 01:10:57,300 --> 01:10:58,600 helium and hydrogen. 1022 01:10:58,600 --> 01:11:02,580 For hydrogen, at the 2s to 1 transition, 1023 01:11:02,580 --> 01:11:05,060 the leading order is the emission, 1024 01:11:05,060 --> 01:11:08,670 the simultaneous emission of two photons. 1025 01:11:08,670 --> 01:11:10,730 So you have not just one photon, you 1026 01:11:10,730 --> 01:11:14,636 have two photons which, of course, requires 1027 01:11:14,636 --> 01:11:17,928 in the perturbation expansion the immediate step. 1028 01:11:17,928 --> 01:11:19,302 We discuss two photon transitions 1029 01:11:19,302 --> 01:11:20,960 at the end of this course. 1030 01:11:20,960 --> 01:11:22,930 So in that case, it's not in higher order 1031 01:11:22,930 --> 01:11:26,010 in the single photon multiple expansion, 1032 01:11:26,010 --> 01:11:29,990 it becomes a multi-photon transition. 1033 01:11:29,990 --> 01:11:31,865 So this is one relevant case. 1034 01:11:31,865 --> 01:11:37,580 And for helium, triplet to singlet, 1035 01:11:37,580 --> 01:11:39,385 this involves relativistic physics. 1036 01:11:42,258 --> 01:11:45,300 Actually, I mentioned it in the other class on helium. 1037 01:11:45,300 --> 01:11:47,739 I tried to look it up and I want to show here, 1038 01:11:47,739 --> 01:11:49,780 you have to go to this order to get a transition. 1039 01:11:49,780 --> 01:11:51,820 But when I tried to look into the literature, 1040 01:11:51,820 --> 01:11:53,650 I couldn't find a clear answer. 1041 01:11:53,650 --> 01:11:56,420 Ultimately, it was a relativistic term 1042 01:11:56,420 --> 01:11:58,410 in the fully relativistic formulation 1043 01:11:58,410 --> 01:12:01,620 of the coupling of electromagnetic fields 1044 01:12:01,620 --> 01:12:04,880 to the atom. 1045 01:12:04,880 --> 01:12:07,550 I'm not sure if you can put a label on it and it would say, 1046 01:12:07,550 --> 01:12:09,210 this is this and this order term. 1047 01:12:09,210 --> 01:12:11,510 It may actually involve-- and we know 1048 01:12:11,510 --> 01:12:13,600 that this happens in the [INAUDIBLE] creation-- 1049 01:12:13,600 --> 01:12:18,020 that spin and spacial degrees become treated together 1050 01:12:18,020 --> 01:12:21,226 in the [INAUDIBLE] creation, and maybe it's one of those terms. 1051 01:12:24,031 --> 01:12:24,530 Yes? 1052 01:12:24,530 --> 01:12:26,113 AUDIENCE: So what's the actual meaning 1053 01:12:26,113 --> 01:12:28,877 of interacting with different Hamiltonians? 1054 01:12:28,877 --> 01:12:32,947 Like, you wrote square root of Hamiltonians and then-- 1055 01:12:32,947 --> 01:12:33,530 PROFESSOR: No. 1056 01:12:33,530 --> 01:12:37,585 I meant actually the square of the matrix element. 1057 01:12:37,585 --> 01:12:38,126 AUDIENCE: Oh. 1058 01:12:38,126 --> 01:12:38,625 Sorry. 1059 01:12:38,625 --> 01:12:43,630 PROFESSOR: If you have, for instance-- actually 1060 01:12:43,630 --> 01:12:45,650 I will talk about it in the next lecture when 1061 01:12:45,650 --> 01:12:47,880 we discuss Fermi's golden rule. 1062 01:12:47,880 --> 01:12:50,400 The transition strength in Fermi's golden rule 1063 01:12:50,400 --> 01:12:53,710 is proportionate to the square of the matrix element. 1064 01:12:53,710 --> 01:12:55,690 But now you would ask the question, well, 1065 01:12:55,690 --> 01:12:58,310 could we have some interference between the two 1066 01:12:58,310 --> 01:12:59,670 different processes? 1067 01:12:59,670 --> 01:13:01,870 And I wanted to point out at this level 1068 01:13:01,870 --> 01:13:05,670 that we don't, because one matrix element is imaginary, 1069 01:13:05,670 --> 01:13:07,100 the other one is real. 1070 01:13:07,100 --> 01:13:10,100 And if you take the complex matrix element between two 1071 01:13:10,100 --> 01:13:14,210 states and calculate the square, the square 1072 01:13:14,210 --> 01:13:17,950 of the complex matrix element is the square of the real part 1073 01:13:17,950 --> 01:13:19,680 plus the square of the imaginary part, 1074 01:13:19,680 --> 01:13:22,730 and there's no interference trend between the two. 1075 01:13:22,730 --> 01:13:26,410 So in other words, if you have an atom which has a weak decay 1076 01:13:26,410 --> 01:13:29,950 through M1 and a weak decay through quadrupole, 1077 01:13:29,950 --> 01:13:33,380 the two parts cannot destructively interfere, 1078 01:13:33,380 --> 01:13:35,575 because one is real, one is imaginary. 1079 01:13:35,575 --> 01:13:39,429 They add up in quadrature, That's what I wanted to say. 1080 01:13:43,410 --> 01:13:44,790 OK. 1081 01:13:44,790 --> 01:13:49,040 So these were examples of higher order transitions. 1082 01:13:49,040 --> 01:13:55,420 And as the questions have shown, this 1083 01:13:55,420 --> 01:13:59,130 leads us to a discussion of selection rules. 1084 01:14:01,790 --> 01:14:06,510 Selection rules is nothing else than a classification 1085 01:14:06,510 --> 01:14:10,120 of possible transitions according to symmetry. 1086 01:14:10,120 --> 01:14:17,300 And it's a way of using well, [INAUDIBLE] coefficients, 1087 01:14:17,300 --> 01:14:20,650 angular momentum coupling, or using, you would say, 1088 01:14:20,650 --> 01:14:25,140 symmetry to figure out if matrix elements are 1089 01:14:25,140 --> 01:14:26,740 non-vanishing or vanishing. 1090 01:14:26,740 --> 01:14:28,540 And I gave you already one example, 1091 01:14:28,540 --> 01:14:30,520 and I want to formalize it now. 1092 01:14:30,520 --> 01:14:34,230 If you go between two s states which have 0 angular momentum, 1093 01:14:34,230 --> 01:14:37,400 you cannot have an operator which is a quadrupole operator 1094 01:14:37,400 --> 01:14:39,910 because-- and this is what I want to tell you now-- 1095 01:14:39,910 --> 01:14:43,385 the quadrupole operator is a spherical tensor with two units 1096 01:14:43,385 --> 01:14:45,056 of angular momentum. 1097 01:14:45,056 --> 01:14:47,730 And this would forbid the triangle rule. 1098 01:14:47,730 --> 01:14:52,440 This would forbid "conservation" of angular momentum. 1099 01:14:52,440 --> 01:14:55,080 So that's what I want to discuss now in the next chapter, 1100 01:14:55,080 --> 01:14:58,470 or at least get started for the next five minutes 1101 01:14:58,470 --> 01:15:02,110 by discussing selection rules. 1102 01:15:02,110 --> 01:15:04,960 So the introduction to selection rules 1103 01:15:04,960 --> 01:15:08,750 is that we have forbidden transitions. 1104 01:15:08,750 --> 01:15:13,480 Forbidden transitions are suppressed 1105 01:15:13,480 --> 01:15:15,890 because we are forced to go to higher order, 1106 01:15:15,890 --> 01:15:18,710 and this is usually higher order in alpha. 1107 01:15:18,710 --> 01:15:23,680 So forbidden transitions are weaker by some power of alpha. 1108 01:15:23,680 --> 01:15:29,020 And that means they require higher approximations. 1109 01:15:34,310 --> 01:15:37,410 And, of course, the comparison is always 1110 01:15:37,410 --> 01:15:38,630 the dipole transition. 1111 01:15:38,630 --> 01:15:40,785 This is the dominant transition. 1112 01:15:40,785 --> 01:15:44,030 This is the industrial strengths transition. 1113 01:15:44,030 --> 01:15:47,470 And from there on, it can get weaker. 1114 01:15:47,470 --> 01:15:52,398 So if can get weaker by multiple expansion. 1115 01:15:56,520 --> 01:15:59,390 We've just discussed that. 1116 01:15:59,390 --> 01:16:03,750 It can get weaker because you have 1117 01:16:03,750 --> 01:16:07,810 to have a cascade of dipole transitions. 1118 01:16:07,810 --> 01:16:10,800 This would be multi-photon processes, 1119 01:16:10,800 --> 01:16:13,640 as we discuss later in the course. 1120 01:16:13,640 --> 01:16:16,810 It can get weaker because they are exactly 1121 01:16:16,810 --> 01:16:20,670 0 in a non-relativistic approximation 1122 01:16:20,670 --> 01:16:24,120 and require relativistic effects. 1123 01:16:27,920 --> 01:16:30,740 The example we have encountered in this course 1124 01:16:30,740 --> 01:16:33,560 is the singlet to triplet transition in helium. 1125 01:16:37,760 --> 01:16:42,410 Or there are transitions which would not be allowed just 1126 01:16:42,410 --> 01:16:52,260 for the electron, but if we invoke hyperfine interactions 1127 01:16:52,260 --> 01:16:54,070 with a nucleus, then they become allowed. 1128 01:16:58,390 --> 01:17:03,720 So it's a rich subject, and I'm not an expert and I cannot do 1129 01:17:03,720 --> 01:17:05,150 full justice to it. 1130 01:17:05,150 --> 01:17:15,220 But I want to at least give you some general rules how 1131 01:17:15,220 --> 01:17:16,921 we discussed matrix elements. 1132 01:17:24,460 --> 01:17:27,880 So what is always a good quantum number, what is always 1133 01:17:27,880 --> 01:17:31,100 a label for our atomic states is angular momentum. 1134 01:17:31,100 --> 01:17:33,070 Because atoms [INAUDIBLE] through space, 1135 01:17:33,070 --> 01:17:35,930 and there is rotation in variance. 1136 01:17:35,930 --> 01:17:41,600 So we always categorize our atoms with angular momentum 1137 01:17:41,600 --> 01:17:44,740 with the quantum numbers JM. 1138 01:17:44,740 --> 01:17:46,980 And we are asking, are there transitions 1139 01:17:46,980 --> 01:17:51,670 between a state JM to a state J prime M prime? 1140 01:17:51,670 --> 01:17:54,370 And all other quantum numbers, we 1141 01:17:54,370 --> 01:17:56,520 can now summarize with a label n. 1142 01:17:59,160 --> 01:18:01,220 And now we have an operator. 1143 01:18:01,220 --> 01:18:03,420 I gave you examples for the operator. 1144 01:18:03,420 --> 01:18:06,745 The magnetic moment, the electric dipole, 1145 01:18:06,745 --> 01:18:09,010 the quadrupole operator. 1146 01:18:09,010 --> 01:18:13,100 But in general, every operator can be written, 1147 01:18:13,100 --> 01:18:17,706 can be expanded, in a sum of spherical tensors. 1148 01:18:17,706 --> 01:18:19,456 So what is discussed in the classification 1149 01:18:19,456 --> 01:18:21,450 of matrix elements. 1150 01:18:21,450 --> 01:18:26,220 Our matrix elements involving components, 1151 01:18:26,220 --> 01:18:30,870 the operator are components of a spherical tensor. 1152 01:18:30,870 --> 01:18:41,130 So T is a spherical tensor of rank l. 1153 01:18:41,130 --> 01:18:44,300 And if you want a simple definition 1154 01:18:44,300 --> 01:18:46,420 of what is a spherical tensor, you 1155 01:18:46,420 --> 01:18:53,400 try to write an operator-- like the position operator 1156 01:18:53,400 --> 01:18:58,100 r-- you try to write it as a sum of terms, 1157 01:18:58,100 --> 01:19:01,980 and each term transforms like a spherical harmonic Ylm. 1158 01:19:05,100 --> 01:19:08,370 So in other words, we can write every operator 1159 01:19:08,370 --> 01:19:11,230 as a sum of spherical tensors. 1160 01:19:11,230 --> 01:19:13,730 And a spherical tensor is characterized 1161 01:19:13,730 --> 01:19:25,160 that it transforms under rotations exactly 1162 01:19:25,160 --> 01:19:32,480 as the spherical harmonics Ylm. 1163 01:19:39,540 --> 01:19:46,385 so every operator is now a sum of spherical tensors. 1164 01:19:51,610 --> 01:19:54,200 I don't want to get too much into symmetry classification, 1165 01:19:54,200 --> 01:19:57,859 but the story is that you know the Ylm functions are 1166 01:19:57,859 --> 01:19:58,775 compensator functions. 1167 01:19:58,775 --> 01:20:01,880 Every function can be expanded and swell to harmonics. 1168 01:20:01,880 --> 01:20:05,330 And similarly, if you have an operator, 1169 01:20:05,330 --> 01:20:08,770 you can decompose it into objects 1170 01:20:08,770 --> 01:20:12,410 which transform under rotation, and it's the Ylms. 1171 01:20:12,410 --> 01:20:15,810 So you have a part which transforms 1172 01:20:15,810 --> 01:20:18,380 according to an object. 1173 01:20:18,380 --> 01:20:21,420 And Ylm is the classification of wave functions 1174 01:20:21,420 --> 01:20:23,220 with angular momentum. 1175 01:20:23,220 --> 01:20:27,550 And so therefore each operator may not have a specific angular 1176 01:20:27,550 --> 01:20:31,350 momentum, but can be written as the sum of operators, 1177 01:20:31,350 --> 01:20:34,660 each of which has the same symmetry, 1178 01:20:34,660 --> 01:20:36,690 the same transformation properties, 1179 01:20:36,690 --> 01:20:38,676 as a state of angular momentum. 1180 01:20:41,610 --> 01:20:43,330 And I think I should stop here, but let 1181 01:20:43,330 --> 01:20:47,030 me just give you the final message. 1182 01:20:47,030 --> 01:20:51,510 So by doing that, by separating the operator 1183 01:20:51,510 --> 01:20:54,885 into a sum of spherical tensors, we are actually now back 1184 01:20:54,885 --> 01:20:57,010 in angular momentum. 1185 01:20:57,010 --> 01:20:58,815 If you have a matrix element-- and just 1186 01:20:58,815 --> 01:21:02,690 think how you calculate it in the Schrodinger representation. 1187 01:21:02,690 --> 01:21:04,260 You have a wave function, operator, 1188 01:21:04,260 --> 01:21:06,850 and a wave function we indicate go by. 1189 01:21:06,850 --> 01:21:11,680 So what you have is, you have the product of the objects. 1190 01:21:11,680 --> 01:21:20,070 And now we can classify them by angular momentum. 1191 01:21:20,070 --> 01:21:24,820 And this is what I will show you in the next class, 1192 01:21:24,820 --> 01:21:28,720 is that ultimately the question whether this matrix element is 1193 01:21:28,720 --> 01:21:31,410 0 or not will boil down to the question 1194 01:21:31,410 --> 01:21:35,000 whether J prime and M prime angular momentum 1195 01:21:35,000 --> 01:21:36,960 can be added to l and m. 1196 01:21:36,960 --> 01:21:40,116 And there is overlap with angular momentum of JM. 1197 01:21:40,116 --> 01:21:44,209 So we are back to the rules for adding angular momentum. 1198 01:21:44,209 --> 01:21:45,250 We get the triangle rule. 1199 01:21:45,250 --> 01:21:47,502 We get the [INAUDIBLE] quote unquote [INAUDIBLE]. 1200 01:21:50,420 --> 01:21:51,450 OK. 1201 01:21:51,450 --> 01:21:53,380 Enjoy the spring break, and we meet on Monday 1202 01:21:53,380 --> 01:21:55,530 after the spring break.