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 --> 00:00:17,202 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,202 --> 00:00:17,827 at ocw.mit.edu. 8 00:00:20,632 --> 00:00:22,090 PROFESSOR: We want to start talking 9 00:00:22,090 --> 00:00:26,960 about a seemingly simple but very complex 10 00:00:26,960 --> 00:00:30,620 system in physics, the harmonic oscillator. 11 00:00:30,620 --> 00:00:35,400 So the next part is actually due to Professor Vladan Vuletic, 12 00:00:35,400 --> 00:00:40,320 who worked out the topic very nicely about, 13 00:00:40,320 --> 00:00:43,100 how precisely can you measure frequencies? 14 00:00:47,350 --> 00:00:49,860 And I don't need to remind you that some 15 00:00:49,860 --> 00:00:53,580 of the most accurate measurements in all of physics 16 00:00:53,580 --> 00:00:56,660 are done by measuring frequency. 17 00:00:56,660 --> 00:01:00,420 It's actually a kind of unwritten rule. 18 00:01:00,420 --> 00:01:02,370 If you want to measure something precisely, 19 00:01:02,370 --> 00:01:06,360 make sure that you find a way that this quantity can 20 00:01:06,360 --> 00:01:08,620 be measured in a frequency measurement. 21 00:01:08,620 --> 00:01:11,270 Because frequencies, that's what we 22 00:01:11,270 --> 00:01:14,660 can measure-- with synthesizers, with clocks, and such. 23 00:01:14,660 --> 00:01:16,990 So therefore, the question, how precisely can you 24 00:01:16,990 --> 00:01:20,270 measure frequency, is a question which is actually 25 00:01:20,270 --> 00:01:22,065 relevant for all precision measurements. 26 00:01:25,890 --> 00:01:28,860 Well, we know we have Fourier theorem. 27 00:01:28,860 --> 00:01:38,600 If you have an oscillator which oscillates for a time delta t, 28 00:01:38,600 --> 00:01:42,370 we have the Fourier theorem which 29 00:01:42,370 --> 00:01:50,450 says we have a finite width of the frequency spectrum delta 30 00:01:50,450 --> 00:02:00,150 omega, or there is the spread of frequency components involved 31 00:02:00,150 --> 00:02:06,750 in such a way that delta omega times delta t 32 00:02:06,750 --> 00:02:10,660 is large or equal than 1/2. 33 00:02:10,660 --> 00:02:15,370 The case of 1/2 is realized for Gaussian wave packets. 34 00:02:15,370 --> 00:02:24,580 And of course, Fourier theorem should also-- this Fourier 35 00:02:24,580 --> 00:02:30,035 limit should remind you of Heisenberg's uncertainty 36 00:02:30,035 --> 00:02:30,535 relation. 37 00:02:37,190 --> 00:02:40,375 Of course, Fourier limit and Heisenberg uncertainty relation 38 00:02:40,375 --> 00:02:43,550 are related because what Heisenberg expressed 39 00:02:43,550 --> 00:02:46,252 turns out to be simply the limit due to the wave 40 00:02:46,252 --> 00:02:46,960 nature of matter. 41 00:02:52,740 --> 00:02:53,290 OK. 42 00:02:53,290 --> 00:02:55,640 So I brought the clicker because I 43 00:02:55,640 --> 00:03:00,720 want you to think about some seemingly simple question. 44 00:03:00,720 --> 00:03:12,650 The first question is whether the uncertainty delta omega 45 00:03:12,650 --> 00:03:18,630 times delta t larger than 1/2, does this uncertainty 46 00:03:18,630 --> 00:03:22,790 hold for purely classical systems? 47 00:03:28,110 --> 00:03:33,464 So think about it and answer in your clicker. 48 00:03:33,464 --> 00:03:36,760 A is 1 and B-- oh, I should say that. 49 00:03:50,500 --> 00:03:54,190 So we now assume we have a purely classical-- you can even 50 00:03:54,190 --> 00:03:57,120 think a mechanical, large mechanical object, 51 00:03:57,120 --> 00:03:59,630 purely classical harmonic oscillator. 52 00:03:59,630 --> 00:04:02,350 You can observe it for time delta t, 53 00:04:02,350 --> 00:04:04,590 or it oscillates for time delta t. 54 00:04:04,590 --> 00:04:08,540 Can you determine its frequency better then 55 00:04:08,540 --> 00:04:10,650 this uncertainty suggests? 56 00:04:14,600 --> 00:04:16,050 19, 20. 57 00:04:16,050 --> 00:04:17,990 Just make up your mind. 58 00:04:17,990 --> 00:04:20,959 As I said, your responses are not 59 00:04:20,959 --> 00:04:25,130 recorded, so nothing to risk. 60 00:04:25,130 --> 00:04:25,630 OK. 61 00:04:29,980 --> 00:04:32,610 Yes. 62 00:04:32,610 --> 00:04:39,820 OK, the majority gave the right answer. 63 00:04:43,880 --> 00:04:51,790 The situation is that the answer is yes, 64 00:04:51,790 --> 00:04:57,790 if you have a good signal to noise ratio. 65 00:05:04,270 --> 00:05:12,940 So what happens is, if you-- we have to bring in the fact 66 00:05:12,940 --> 00:05:15,600 that we have noise. 67 00:05:15,600 --> 00:05:21,780 So if you have a wave formed, there may be noise around it. 68 00:05:21,780 --> 00:05:33,680 And if you look at the spectrum, the spectral components 69 00:05:33,680 --> 00:05:35,520 have a certain width. 70 00:05:35,520 --> 00:05:39,980 And this width, delta omega, is given by Fourier's theorem 71 00:05:39,980 --> 00:05:44,490 by the time delta t you had for observation. 72 00:05:44,490 --> 00:05:51,230 But as you see, you can determine the center 73 00:05:51,230 --> 00:06:01,250 of this spectral peak with an accuracy delta omega, which 74 00:06:01,250 --> 00:06:05,210 may be much better than the [? width ?] delta omega. 75 00:06:05,210 --> 00:06:07,190 And the rule of thumb is that you 76 00:06:07,190 --> 00:06:11,630 can split a line by your signal to noise ratio. 77 00:06:11,630 --> 00:06:15,600 So typically, the accuracy of a measurement 78 00:06:15,600 --> 00:06:19,640 is whatever the width of the spectrum is, 79 00:06:19,640 --> 00:06:22,740 and then you can split the line by the signal to noise ratio. 80 00:06:31,860 --> 00:06:35,635 This is called splitting the line. 81 00:06:41,600 --> 00:06:48,486 Factor of hundred is usually regarded as straightforward. 82 00:06:51,600 --> 00:06:57,060 But if you want to go to larger than that, 83 00:06:57,060 --> 00:07:00,300 it becomes a challenge. 84 00:07:00,300 --> 00:07:03,440 Because even if you have a very good signal to noise ratio, 85 00:07:03,440 --> 00:07:06,550 you really have to make sure that you know the line shape, 86 00:07:06,550 --> 00:07:09,000 that you know that it is, for instance, symmetric, 87 00:07:09,000 --> 00:07:12,150 and that the line center of the observed shape 88 00:07:12,150 --> 00:07:14,640 is really where the frequency is. 89 00:07:20,760 --> 00:07:23,080 I will give you some examples in a few moments. 90 00:07:23,080 --> 00:07:29,720 But before that, I would like to continue our discussion 91 00:07:29,720 --> 00:07:32,700 whether we can measure the angle of frequency 92 00:07:32,700 --> 00:07:35,970 to better than the Heisenberg limit 93 00:07:35,970 --> 00:07:39,340 in case of a quantum-mechanical oscillator. 94 00:07:39,340 --> 00:07:49,830 So the question is, can you measure the frequency 95 00:07:49,830 --> 00:08:01,340 of a quantum-mechanical harmonic oscillator in a time delta t 96 00:08:01,340 --> 00:08:11,695 to an accuracy which is better than the limit of Heisenberg's 97 00:08:11,695 --> 00:08:12,195 uncertainty? 98 00:08:21,380 --> 00:08:22,570 OK, your vote, please. 99 00:08:49,000 --> 00:08:49,850 All right. 100 00:08:55,120 --> 00:08:56,220 OK. 101 00:08:56,220 --> 00:08:59,010 Why don't we hold that for a second 102 00:08:59,010 --> 00:09:06,150 and proceed to the next question which is the same? 103 00:09:06,150 --> 00:09:09,650 But now, instead of a quantum-mechanical harmonic 104 00:09:09,650 --> 00:09:12,710 oscillator, we take something we are 105 00:09:12,710 --> 00:09:17,300 very familiar with-- an optical laser-- 106 00:09:17,300 --> 00:09:24,190 and we observe a laser pulse lasting a duration delta t. 107 00:09:27,150 --> 00:09:30,500 So same situation, but instead of observing 108 00:09:30,500 --> 00:09:33,260 a quantum-mechanical harmonic oscillator, 109 00:09:33,260 --> 00:09:36,890 we observe a laser pulse. 110 00:09:36,890 --> 00:09:41,160 For the laser, can we measure the frequency 111 00:09:41,160 --> 00:09:46,950 of the optical radiation better than this equality tells us? 112 00:10:04,120 --> 00:10:04,620 OK. 113 00:10:04,620 --> 00:10:07,780 At least people are consistent because the first thing 114 00:10:07,780 --> 00:10:11,210 I wanted to tell you is that, is the laser actually 115 00:10:11,210 --> 00:10:15,700 a classical harmonic oscillator or quantum harmonic oscillator? 116 00:10:15,700 --> 00:10:18,990 Well, we use a quantum description 117 00:10:18,990 --> 00:10:23,120 of light and the laser is the population of photons 118 00:10:23,120 --> 00:10:26,070 in the single note of the electromagnetic field. 119 00:10:26,070 --> 00:10:29,340 So in that sense, the laser is fully quantum. 120 00:10:29,340 --> 00:10:31,430 But in the limit that the laser is many, 121 00:10:31,430 --> 00:10:35,300 many photons-- and some of you know about coherence states. 122 00:10:35,300 --> 00:10:39,890 If there's a coherent state with a large photon number, 123 00:10:39,890 --> 00:10:41,910 the laser is actually the classical limit 124 00:10:41,910 --> 00:10:44,280 of an electromagnetic field. 125 00:10:44,280 --> 00:10:47,300 So maybe that tells me that the answers 126 00:10:47,300 --> 00:10:51,590 to questions 2 and question 3 should probably be the same. 127 00:10:51,590 --> 00:10:53,620 And I want to say more about it. 128 00:10:53,620 --> 00:10:55,680 So at least in this class, you were consistent. 129 00:10:55,680 --> 00:10:57,390 I have often seen a big discrepancy 130 00:10:57,390 --> 00:10:59,550 in the answers between question 2 and question 3. 131 00:11:02,930 --> 00:11:09,830 OK, so the answer is yes. 132 00:11:09,830 --> 00:11:13,510 So both for the laser-- I first explain the laser to you, 133 00:11:13,510 --> 00:11:17,027 and then we go back to the pure quantum system, thing 134 00:11:17,027 --> 00:11:18,360 has actually certain subtleties. 135 00:11:20,890 --> 00:11:25,130 But for the laser, it is obvious, 136 00:11:25,130 --> 00:11:28,180 at least if I tell you how I want to measure it, 137 00:11:28,180 --> 00:11:32,440 because I can take the laser and create a beat 138 00:11:32,440 --> 00:11:44,210 note with another very stable laser. 139 00:11:44,210 --> 00:11:46,460 And I record this beat note on a photodiode. 140 00:11:51,230 --> 00:11:54,145 I can realize by making this other laser, 141 00:11:54,145 --> 00:11:57,340 the local oscillator stronger and stronger, 142 00:11:57,340 --> 00:11:59,390 I can create a beat note which is 143 00:11:59,390 --> 00:12:01,860 larger and larger and corresponds 144 00:12:01,860 --> 00:12:04,400 to a macroscopic electric current which 145 00:12:04,400 --> 00:12:06,280 can be measured with very high precision. 146 00:12:10,060 --> 00:12:13,600 So you can realize an arbitrarily high signal 147 00:12:13,600 --> 00:12:22,000 to noise ratio by using a strong local oscillator. 148 00:12:27,120 --> 00:12:30,580 And then you can actually say the photocurrent, which 149 00:12:30,580 --> 00:12:34,020 comes out of the photodiode, is actually-- 150 00:12:34,020 --> 00:12:37,440 you can regard this microscopic current as purely classical. 151 00:12:37,440 --> 00:12:40,440 And then of course, the answer to the first question applies. 152 00:12:43,520 --> 00:12:47,740 So that takes care of question 3 by mapping it actually 153 00:12:47,740 --> 00:12:49,540 on question 2. 154 00:12:49,540 --> 00:12:51,920 But now by saying that the laser also 155 00:12:51,920 --> 00:12:55,420 has a quantum-mechanical limit and I'm not changing anything, 156 00:12:55,420 --> 00:12:58,390 we realize that probably the answer to question 2 157 00:12:58,390 --> 00:13:00,750 should also be yes. 158 00:13:00,750 --> 00:13:08,160 So let's, therefore, ask our self, what 159 00:13:08,160 --> 00:13:13,530 is the situation when the Heisenberg uncertainty 160 00:13:13,530 --> 00:13:14,325 relation applies? 161 00:13:25,550 --> 00:13:32,440 Well, one is we have to be really careful. 162 00:13:32,440 --> 00:13:37,070 It predicts the outcome of a single measurement 163 00:13:37,070 --> 00:13:39,806 on a single quantum system. 164 00:13:39,806 --> 00:13:41,290 Let me write that down. 165 00:13:44,180 --> 00:13:46,230 Or if the Heisenberg uncertainty relation 166 00:13:46,230 --> 00:13:49,670 sets a limit, how well we prepare a quantum system. 167 00:13:49,670 --> 00:13:53,132 It's about a single quantum system, 168 00:13:53,132 --> 00:13:54,840 and then we perform a single measurement. 169 00:14:17,110 --> 00:14:23,680 So in a sense, if we would say all 170 00:14:23,680 --> 00:14:30,690 you have is a single photon, which is a very special quantum 171 00:14:30,690 --> 00:14:31,480 system. 172 00:14:31,480 --> 00:14:35,740 You have a single photon and you measure 173 00:14:35,740 --> 00:14:39,010 the frequency of the photon only once. 174 00:14:39,010 --> 00:14:43,520 Then your will find the limit, which is the Heisenberg limit. 175 00:14:43,520 --> 00:14:48,040 You cannot, with a single measurement on a single photon, 176 00:14:48,040 --> 00:14:57,540 determine the accuracy of the frequency better than this. 177 00:14:57,540 --> 00:15:00,890 And of course, you can get higher accuracy 178 00:15:00,890 --> 00:15:12,400 by doing repeated measurements or by using many photons. 179 00:15:21,470 --> 00:15:27,630 We talk about it more in 8.422, but I just want to remind you, 180 00:15:27,630 --> 00:15:32,500 if you have n uncorrelated photons. 181 00:15:37,670 --> 00:15:40,810 In other words, we perform n measurements 182 00:15:40,810 --> 00:15:45,530 on n different objects, then the signal to noise ratio 183 00:15:45,530 --> 00:15:51,510 is-- just by Poisson distribution, square root n. 184 00:15:51,510 --> 00:15:54,790 And therefore, the resolution for the frequency 185 00:15:54,790 --> 00:15:59,970 of the photons is better than the Heisenberg or the Fourier 186 00:15:59,970 --> 00:16:02,300 limit by 1 over the square root n. 187 00:16:06,730 --> 00:16:09,520 Some of you-- and actually, in Professor Vuletic's group, 188 00:16:09,520 --> 00:16:14,810 there is research on it that if you have correlated-- well, 189 00:16:14,810 --> 00:16:16,500 in his case, correlated atoms. 190 00:16:16,500 --> 00:16:21,090 But if you had correlated photons, 191 00:16:21,090 --> 00:16:24,090 then you can even do better. 192 00:16:24,090 --> 00:16:29,340 You can reach what is sometimes called the Heisenberg limit 193 00:16:29,340 --> 00:16:32,260 where you are better than the limitation 194 00:16:32,260 --> 00:16:34,950 given by Fourier's theorem or by the Heisenberg uncertainty 195 00:16:34,950 --> 00:16:37,390 relation by a factor of 1/n. 196 00:17:13,869 --> 00:17:14,369 OK. 197 00:17:14,369 --> 00:17:23,200 So as far as the question 2 where the quantum harmonic 198 00:17:23,200 --> 00:17:33,540 oscillator is concerned, we would 199 00:17:33,540 --> 00:17:46,860 say the answer is yes, if you have 200 00:17:46,860 --> 00:17:57,785 is single photon at frequency omega 0, 201 00:17:57,785 --> 00:18:08,360 which interacts with the quantum harmonic oscillator 202 00:18:08,360 --> 00:18:09,650 at frequency omega 0. 203 00:18:09,650 --> 00:18:16,710 However, the answer would, even in that case, 204 00:18:16,710 --> 00:18:23,270 be no if you have harmonic oscillator levels 205 00:18:23,270 --> 00:18:29,740 and you take a photon and by a nonlinear process 206 00:18:29,740 --> 00:18:32,250 excite the n-th level. 207 00:18:32,250 --> 00:18:33,995 So you have a single photon now. 208 00:18:38,620 --> 00:18:44,850 You can resolve the energy delta E of this level. 209 00:18:44,850 --> 00:18:47,960 A single photon, a single quantum 210 00:18:47,960 --> 00:18:55,382 object, you can define the energy-- 211 00:18:55,382 --> 00:18:57,090 that's Heisenberg's uncertainty relation. 212 00:18:57,090 --> 00:19:01,160 The energy is determined to that precision. 213 00:19:04,140 --> 00:19:17,810 But your frequency omega of the optical pulse 214 00:19:17,810 --> 00:19:22,230 is n times omega 0 using nonlinear process. 215 00:19:22,230 --> 00:19:26,170 And then you can determine the frequency of the harmonic 216 00:19:26,170 --> 00:19:28,770 oscillator, even for a single quantum 217 00:19:28,770 --> 00:19:38,450 system and a single photon with a precision 218 00:19:38,450 --> 00:19:41,510 which is 1/n times better. 219 00:19:41,510 --> 00:19:44,470 So you have to be also careful, but I 220 00:19:44,470 --> 00:19:46,120 don't want to beat it to death now, 221 00:19:46,120 --> 00:19:50,010 to distinguish between the accuracy at which Heisenberg's 222 00:19:50,010 --> 00:19:52,520 uncertainty relation maybe limits 223 00:19:52,520 --> 00:19:54,530 the measurement of an energy level. 224 00:19:54,530 --> 00:19:57,340 And how this is related to the frequency 225 00:19:57,340 --> 00:19:58,800 of the harmonic oscillator. 226 00:19:58,800 --> 00:20:02,100 And by going sort of immediately to the n-th level, 227 00:20:02,100 --> 00:20:04,721 you can, of course, measure the distance between two levels 228 00:20:04,721 --> 00:20:07,220 more accurately because you have increased your [INAUDIBLE]. 229 00:20:14,770 --> 00:20:15,826 Any questions? 230 00:20:15,826 --> 00:20:17,700 AUDIENCE: Sorry, I just got a little confused 231 00:20:17,700 --> 00:20:22,630 about when yes means one thing and no means one thing. 232 00:20:22,630 --> 00:20:26,642 So you're saying that you can beat the uncertainty relation 233 00:20:26,642 --> 00:20:31,362 in questions 1, 2, and 3 if you can put it in a way 234 00:20:31,362 --> 00:20:34,210 where you get good signal to noise? 235 00:20:34,210 --> 00:20:37,240 PROFESSOR: OK, I gave you-- sorry for being complicated, 236 00:20:37,240 --> 00:20:38,760 but the physics is complicated. 237 00:20:38,760 --> 00:20:40,260 I try to give it to you in different layers. 238 00:20:40,260 --> 00:20:41,843 I first looked at the classical limit, 239 00:20:41,843 --> 00:20:43,190 which is pretty clear-cut. 240 00:20:43,190 --> 00:20:44,820 Then, I used a laser. 241 00:20:44,820 --> 00:20:46,500 The laser has a classical limit where 242 00:20:46,500 --> 00:20:48,700 the answer is the same as in the classical limit. 243 00:20:48,700 --> 00:20:50,610 But then we can talk also in the laser 244 00:20:50,610 --> 00:20:52,680 in the limit of single photons. 245 00:20:52,680 --> 00:20:55,490 And then I said, OK, the single photon interacting 246 00:20:55,490 --> 00:20:57,460 with a single quantum system, this 247 00:20:57,460 --> 00:20:59,200 is really when it is quantum. 248 00:20:59,200 --> 00:21:01,900 And if you have a two-level system and you [INAUDIBLE] 249 00:21:01,900 --> 00:21:04,040 it with a single photon, then you 250 00:21:04,040 --> 00:21:08,690 can make a measurement which is limited by this inequality. 251 00:21:08,690 --> 00:21:10,870 But then I said there is a caveat. 252 00:21:10,870 --> 00:21:24,000 And this is if you bring in a nonlinear process. 253 00:21:31,460 --> 00:21:35,630 So if your bring in a nonlinear process, we can go up n steps. 254 00:21:35,630 --> 00:21:38,366 We can drive-- we can, by some nonlinearity, 255 00:21:38,366 --> 00:21:40,490 drive the harmonic oscillator from the ground state 256 00:21:40,490 --> 00:21:41,900 to the n state. 257 00:21:41,900 --> 00:21:44,600 Then, everything we have said about a single photon 258 00:21:44,600 --> 00:21:46,740 and the measurement of the resonance and such 259 00:21:46,740 --> 00:21:48,860 applies to this photon. 260 00:21:48,860 --> 00:21:51,650 But the energy level of the harmonic oscillator 261 00:21:51,650 --> 00:21:53,880 has now been measured with n times higher precision 262 00:21:53,880 --> 00:21:56,340 because we can divide by n. 263 00:21:56,340 --> 00:22:02,290 So the answer is yes, yes, then quantum mechanics we cannot 264 00:22:02,290 --> 00:22:05,500 make it more accurate unless we pull some tricks. 265 00:22:05,500 --> 00:22:08,000 And nonlinear physics would be a trick. 266 00:22:08,000 --> 00:22:10,860 So in general, the situation where you really 267 00:22:10,860 --> 00:22:15,470 limit it by this inequality where your precision is limited 268 00:22:15,470 --> 00:22:20,040 would really only apply to a single photon, a single quantum 269 00:22:20,040 --> 00:22:23,750 physics, and linear physics. 270 00:22:23,750 --> 00:22:24,450 Other questions? 271 00:22:24,450 --> 00:22:25,422 AUDIENCE: Yes. 272 00:22:25,422 --> 00:22:27,366 Maybe just the phrasing. 273 00:22:27,366 --> 00:22:31,254 AUDIENCE: What does delta t mean for a single photon 274 00:22:31,254 --> 00:22:32,240 measurement? 275 00:22:32,240 --> 00:22:34,980 PROFESSOR: Delta t could be the time 276 00:22:34,980 --> 00:22:36,790 you allow yourself to make the measurement. 277 00:22:39,490 --> 00:22:41,110 You have a measurement apparatus. 278 00:22:41,110 --> 00:22:43,020 You switch it on, you switch it off. 279 00:22:45,262 --> 00:22:47,220 Eventually, you want to get out of grad school. 280 00:22:47,220 --> 00:22:49,270 I mean, you don't want to take an infinite amount of time 281 00:22:49,270 --> 00:22:50,145 for the measurements. 282 00:22:50,145 --> 00:22:51,800 There's always a window, delta t, 283 00:22:51,800 --> 00:22:53,490 and there's a fundamental limit. 284 00:22:53,490 --> 00:22:55,859 The duration of the measurement limits the precision 285 00:22:55,859 --> 00:22:56,650 of the measurement. 286 00:23:02,910 --> 00:23:03,410 OK. 287 00:23:07,910 --> 00:23:17,900 The next thing I want to discuss is the analogy, but also 288 00:23:17,900 --> 00:23:21,610 the differences between a harmonic oscillator 289 00:23:21,610 --> 00:23:23,130 and a two-level system. 290 00:23:33,420 --> 00:23:37,790 So what is a two-level system? 291 00:23:37,790 --> 00:23:40,980 Well, it's a system with two levels. 292 00:23:40,980 --> 00:23:43,760 What is a harmonic oscillator? 293 00:23:43,760 --> 00:23:47,650 Well, it's a system which has an infinite number 294 00:23:47,650 --> 00:23:49,180 of equidistant levels. 295 00:23:53,150 --> 00:23:55,480 I will tell you tell you later in this course when 296 00:23:55,480 --> 00:24:00,330 we talk about the AC and DC stark effect, 297 00:24:00,330 --> 00:24:02,630 you talk about the polarizability and light 298 00:24:02,630 --> 00:24:07,650 scattering that you can regard the atom or the electron 299 00:24:07,650 --> 00:24:09,370 in the atom as a harmonic oscillator. 300 00:24:09,370 --> 00:24:14,460 An atom scatters light exactly in the same way 301 00:24:14,460 --> 00:24:19,200 as the charge which is connected to some support structure 302 00:24:19,200 --> 00:24:20,200 with a spring. 303 00:24:20,200 --> 00:24:24,690 How an oscillating charge would scatter light? 304 00:24:24,690 --> 00:24:27,940 Well, you know of course, the atom is a two-level system. 305 00:24:27,940 --> 00:24:32,890 And the sort of model I make for the electron as a harmonic 306 00:24:32,890 --> 00:24:36,300 oscillator at a single resonance frequency, which 307 00:24:36,300 --> 00:24:42,380 is 100% exact in the limit of [? low ?] laser power. 308 00:24:42,380 --> 00:24:46,210 Well, this realize is a harmonic oscillator. 309 00:24:46,210 --> 00:24:49,370 So therefore, what I'm telling you by this example, 310 00:24:49,370 --> 00:24:52,860 that there are situations where a two-level system 311 00:24:52,860 --> 00:24:58,300 and a harmonic oscillator are the same. 312 00:24:58,300 --> 00:25:01,970 Or, create the same [? observance, ?] 313 00:25:01,970 --> 00:25:03,990 create the same physics. 314 00:25:03,990 --> 00:25:09,500 Do you have any idea when the two systems may look the same 315 00:25:09,500 --> 00:25:13,140 or when the two systems react exactly in the same way 316 00:25:13,140 --> 00:25:14,985 to, for instance, external radiation? 317 00:25:21,375 --> 00:25:22,850 AUDIENCE: At very low temperatures? 318 00:25:22,850 --> 00:25:24,340 PROFESSOR: At very low temperature? 319 00:25:24,340 --> 00:25:25,526 Well-- 320 00:25:25,526 --> 00:25:26,924 AUDIENCE: [INAUDIBLE]. 321 00:25:26,924 --> 00:25:27,860 PROFESSOR: Yes. 322 00:25:27,860 --> 00:25:29,977 Well, we assume these are atoms and we always 323 00:25:29,977 --> 00:25:31,060 start in the common state. 324 00:25:31,060 --> 00:25:33,480 So let's assume we have 0 temperature. 325 00:25:33,480 --> 00:25:34,510 We have an atom. 326 00:25:34,510 --> 00:25:38,940 And maybe what I'm asking is, if I excite the atom and I said, 327 00:25:38,940 --> 00:25:42,240 there may be a situation where the atom is a two-level system 328 00:25:42,240 --> 00:25:46,780 but it reacts like a harmonic oscillator, 329 00:25:46,780 --> 00:25:49,269 when does it break down? 330 00:25:49,269 --> 00:25:50,560 AUDIENCE: When the [INAUDIBLE]. 331 00:25:54,544 --> 00:25:55,335 PROFESSOR: Exactly. 332 00:25:58,160 --> 00:26:00,050 When we go beyond the perturbative 333 00:26:00,050 --> 00:26:02,550 limit when we use a strong excitation. 334 00:26:02,550 --> 00:26:05,460 So in other words-- and I like to give you the answer 335 00:26:05,460 --> 00:26:08,260 before I give you the full explanation, which now comes. 336 00:26:08,260 --> 00:26:11,790 If you start out in the ground state, 337 00:26:11,790 --> 00:26:15,580 you can see at 0 temperature, we have mainly all the population 338 00:26:15,580 --> 00:26:16,080 there. 339 00:26:17,860 --> 00:26:20,580 If you start now driving the system, 340 00:26:20,580 --> 00:26:23,720 we put-- and I will say a little bit more about it. 341 00:26:23,720 --> 00:26:27,200 A little bit into the excited state. 342 00:26:27,200 --> 00:26:30,380 But it is the nature of a harmonic oscillator when 343 00:26:30,380 --> 00:26:32,960 you put something into the excited state 344 00:26:32,960 --> 00:26:36,280 that immediately a little bit goes into the second excited 345 00:26:36,280 --> 00:26:37,540 state. 346 00:26:37,540 --> 00:26:41,100 And this is, of course, something which you can only do 347 00:26:41,100 --> 00:26:45,080 in a harmonic oscillator but you cannot do in a level system. 348 00:26:45,080 --> 00:26:49,410 So to the extent that we have weak excitation 349 00:26:49,410 --> 00:26:55,600 and we can neglect the excitation in higher levels. 350 00:26:55,600 --> 00:26:58,475 To that extent, a two-level system and harmonic oscillator 351 00:26:58,475 --> 00:27:00,870 are identical. 352 00:27:00,870 --> 00:27:02,910 Actually, what I'm saying appears trivial, 353 00:27:02,910 --> 00:27:05,210 but I really want you to think about it. 354 00:27:05,210 --> 00:27:08,090 It's actually a very profound statement. 355 00:27:08,090 --> 00:27:10,730 When can you describe a quantum-mechanical system 356 00:27:10,730 --> 00:27:12,090 as a harmonic oscillator? 357 00:27:12,090 --> 00:27:15,290 For weak excitation when all what matters 358 00:27:15,290 --> 00:27:19,250 is that you have put a small fraction of the system 359 00:27:19,250 --> 00:27:22,150 into the first excited state. 360 00:27:22,150 --> 00:27:27,010 And you immediately realize that the feature which distinguishes 361 00:27:27,010 --> 00:27:30,360 a two-level system from a harmonic oscillator 362 00:27:30,360 --> 00:27:32,860 is the phenomenon of, let's say, saturation. 363 00:27:32,860 --> 00:27:35,000 You cannot go higher. 364 00:27:35,000 --> 00:27:38,600 If you do not saturate a two-level system, 365 00:27:38,600 --> 00:27:41,380 it behaves like a harmonic oscillator. 366 00:27:41,380 --> 00:27:45,710 And therefore, it behaves completely classical. 367 00:27:45,710 --> 00:27:47,790 OK, let's work that out a little bit. 368 00:27:47,790 --> 00:27:50,660 I'll give you some examples. 369 00:27:50,660 --> 00:27:59,380 So the phenomenon of a two-level system is it has saturation. 370 00:27:59,380 --> 00:28:02,115 The maximum energy you can put in is one quantum. 371 00:28:02,115 --> 00:28:06,780 Whereas, a harmonic oscillator can never be saturated. 372 00:28:09,620 --> 00:28:13,850 Just think of the harmonic oscillator potential parabola. 373 00:28:13,850 --> 00:28:16,920 You can drive the system as high as you want. 374 00:28:16,920 --> 00:28:21,460 So you can go in this classical language 375 00:28:21,460 --> 00:28:24,380 to arbitrarily large amplitudes. 376 00:28:33,280 --> 00:28:38,450 So what I just mentioned where the equivalence holds-- 377 00:28:38,450 --> 00:28:40,770 I always want you to have an example in mind-- 378 00:28:40,770 --> 00:28:52,090 is the Lorentz model for an atom where you describe the atom 379 00:28:52,090 --> 00:29:03,940 as an electron connected with a spring to the nucleus. 380 00:29:03,940 --> 00:29:07,160 And as we will see in a few weeks, 381 00:29:07,160 --> 00:29:11,990 this model gives the identical answer, 382 00:29:11,990 --> 00:29:14,910 identical to the quantum-mechanical treatment 383 00:29:14,910 --> 00:29:22,940 for probabilities like the polarizability 384 00:29:22,940 --> 00:29:28,120 and the index of refraction for gas of atoms or molecules. 385 00:29:41,590 --> 00:29:46,450 So if you have a two-level system, 386 00:29:46,450 --> 00:29:49,370 we can often think we have an s, ground state, and then 387 00:29:49,370 --> 00:29:51,620 p, excited state. 388 00:29:51,620 --> 00:29:55,890 And if you do a weak excitation, we 389 00:29:55,890 --> 00:30:01,480 have sort of a wave function, which is all, 390 00:30:01,480 --> 00:30:05,050 lets say, positive one side-- one sign. 391 00:30:05,050 --> 00:30:21,820 And then we [INAUDIBLE] a p orbital, 392 00:30:21,820 --> 00:30:24,650 which has a note which is positive and negative. 393 00:30:24,650 --> 00:30:27,890 And now we have the positive-negative 394 00:30:27,890 --> 00:30:32,100 and the positive and the-- there is the resonance 395 00:30:32,100 --> 00:30:33,740 frequency between the two. 396 00:30:33,740 --> 00:30:38,240 And that together results in an oscillating dipole. 397 00:30:41,420 --> 00:30:44,880 So the simple model of superimposing 398 00:30:44,880 --> 00:30:48,980 an s state and a p state at a certain frequency 399 00:30:48,980 --> 00:30:52,810 gives us an oscillating dipole, which 400 00:30:52,810 --> 00:30:56,990 is the realization of a harmonic oscillator. 401 00:30:56,990 --> 00:31:01,010 But the harmonic oscillator, it oscillates. 402 00:31:01,010 --> 00:31:06,195 And this is, of course, valid for sufficiently small 403 00:31:06,195 --> 00:31:06,695 excitation. 404 00:31:09,550 --> 00:31:12,700 So the question we have already addressed 405 00:31:12,700 --> 00:31:15,415 in the discussion, what is small? 406 00:31:19,140 --> 00:31:38,670 So "small" means population of higher-excited states 407 00:31:38,670 --> 00:31:39,260 is negligible. 408 00:31:43,670 --> 00:31:47,750 So in other words, as long as the excitation of the first 409 00:31:47,750 --> 00:31:57,890 excited state is small, then we can neglect the excitation 410 00:31:57,890 --> 00:32:01,440 in the second excited state, which is even smaller. 411 00:32:18,860 --> 00:32:23,580 So let me kind of bring out the difference 412 00:32:23,580 --> 00:32:26,475 between a two-level system and a harmonic oscillator 413 00:32:26,475 --> 00:32:33,099 a little bit more by discussing the situational of cavity QED. 414 00:32:48,370 --> 00:33:05,190 Let's assume we want 100% population in the first excited 415 00:33:05,190 --> 00:33:06,826 state. 416 00:33:06,826 --> 00:33:08,200 If you have a harmonic oscillator 417 00:33:08,200 --> 00:33:12,660 and the system is prepared in the first excited state, 418 00:33:12,660 --> 00:33:18,330 this is also called Fock state with one quantum of excitation. 419 00:33:18,330 --> 00:33:22,670 And it's a rather special state where people have worked hard 420 00:33:22,670 --> 00:33:28,770 to generate it because you cannot realize it in a harmonic 421 00:33:28,770 --> 00:33:30,000 oscillator. 422 00:33:30,000 --> 00:33:33,390 And let me sort of explain that in the following way. 423 00:33:33,390 --> 00:33:36,450 If you have a harmonic oscillator, 424 00:33:36,450 --> 00:33:40,420 you start and you would drive it. 425 00:33:40,420 --> 00:33:49,660 And you try to put 100% in the n equals 1 state. 426 00:33:49,660 --> 00:33:54,210 Before you have accumulated 100% in the n equals 1 state, 427 00:33:54,210 --> 00:33:57,660 you drive it already to higher states. 428 00:33:57,660 --> 00:34:00,210 And of course, you know when you start 429 00:34:00,210 --> 00:34:02,630 driving a harmonic oscillator, classical 430 00:34:02,630 --> 00:34:06,060 or quantum-mechanically, you create a coherent state, which 431 00:34:06,060 --> 00:34:09,469 is a superposition of excited states. 432 00:34:09,469 --> 00:34:15,049 So we would say an n equals 1 state cannot be excited. 433 00:34:21,199 --> 00:34:27,630 We usually get a coherent state, which 434 00:34:27,630 --> 00:34:36,880 is a superposition of many, or at least several, states. 435 00:34:45,600 --> 00:34:52,469 Whereas in a two-level system, we 436 00:34:52,469 --> 00:34:54,699 can just do a [INAUDIBLE] pulse. 437 00:34:54,699 --> 00:34:57,600 And to put all the atoms in the excited state 438 00:34:57,600 --> 00:34:59,005 is nothing special. 439 00:34:59,005 --> 00:35:03,270 Whereas, to have a cavity filled with photons 440 00:35:03,270 --> 00:35:06,520 and selectively excite the n equals 1 state, 441 00:35:06,520 --> 00:35:09,630 this is special because it's not easy. 442 00:35:09,630 --> 00:35:12,650 It's not straightforward. 443 00:35:12,650 --> 00:35:25,816 So in cavity QED, you can do it if you 444 00:35:25,816 --> 00:35:27,440 have anharmonicities or nonlinearities. 445 00:35:34,250 --> 00:35:35,450 So let me explain that. 446 00:35:39,610 --> 00:35:49,910 Well, it's an anharmonicity or some form of-- so 447 00:35:49,910 --> 00:35:54,270 if you have a situation where you 448 00:35:54,270 --> 00:36:00,650 have your harmonic oscillator, but the energy levels 449 00:36:00,650 --> 00:36:03,790 are not equidistance. 450 00:36:03,790 --> 00:36:09,370 So the difference between this first and second excited state 451 00:36:09,370 --> 00:36:15,500 are not the same, then you can drive the system. 452 00:36:15,500 --> 00:36:17,670 You can prepare Fock state in n equals 453 00:36:17,670 --> 00:36:21,390 1 like in a two-level system. 454 00:36:21,390 --> 00:36:26,090 And you're out of resonance to drive it to higher states. 455 00:36:26,090 --> 00:36:31,885 So here, what you utilize is a sort of two-level system. 456 00:36:31,885 --> 00:36:36,080 And that allows you to create those special state which 457 00:36:36,080 --> 00:36:38,660 are regarded as non-classical, very 458 00:36:38,660 --> 00:36:41,840 special states of the harmonic oscillator. 459 00:36:41,840 --> 00:36:45,400 And one way how you can create it-- well, 460 00:36:45,400 --> 00:36:47,870 if you have an empty cavity, each photon 461 00:36:47,870 --> 00:36:49,340 has the same energy. 462 00:36:49,340 --> 00:36:53,190 Then you have an equidistant harmonic oscillator. 463 00:36:53,190 --> 00:37:11,290 But if you put, for instance-- you add an atom to the cavity, 464 00:37:11,290 --> 00:37:14,600 and the radiation is interacting with the atom, 465 00:37:14,600 --> 00:37:18,580 then you'll get-- we'll talk about it later. 466 00:37:18,580 --> 00:37:20,770 The atom and the photons interact with the Rabi 467 00:37:20,770 --> 00:37:23,410 frequency, and then you get a splitting, 468 00:37:23,410 --> 00:37:24,850 called the normal mode splitting. 469 00:37:29,160 --> 00:37:33,670 And this level splitting is proportional to the Rabi 470 00:37:33,670 --> 00:37:34,800 frequency. 471 00:37:34,800 --> 00:37:36,800 And we'll discuss it later, but many of you 472 00:37:36,800 --> 00:37:39,410 know that the Rabi frequency scales 473 00:37:39,410 --> 00:37:41,840 with the square root of the photon number. 474 00:37:41,840 --> 00:37:44,130 So therefore, you have a splitting 475 00:37:44,130 --> 00:37:47,060 which is proportional to square root 1, square root 476 00:37:47,060 --> 00:37:48,380 2, square root 3. 477 00:37:48,380 --> 00:37:51,040 And you have a spectrum which is no longer 478 00:37:51,040 --> 00:37:52,700 an equidistant system. 479 00:37:52,700 --> 00:37:55,860 And then you can create non-classical states 480 00:37:55,860 --> 00:37:58,420 of the photon field, non-classical states 481 00:37:58,420 --> 00:37:59,668 of a harmonic oscillator. 482 00:38:02,600 --> 00:38:06,760 So anyway, I thought I wanted to bring it up 483 00:38:06,760 --> 00:38:09,663 at the beginning of the class, because a lot what 484 00:38:09,663 --> 00:38:12,130 we are discussing in this class is 485 00:38:12,130 --> 00:38:14,760 we'll rediscover in many situations-- 486 00:38:14,760 --> 00:38:17,760 in atoms, in the light, in the way how 487 00:38:17,760 --> 00:38:20,070 light and atoms interact, harmonic oscillators 488 00:38:20,070 --> 00:38:21,930 and two-level systems. 489 00:38:21,930 --> 00:38:24,170 Often, I say they are the same. 490 00:38:24,170 --> 00:38:25,830 They behave in the same way. 491 00:38:25,830 --> 00:38:29,627 But I hope this introductory [? mark ?] tells you, when can 492 00:38:29,627 --> 00:38:31,210 you think in one limit and when do you 493 00:38:31,210 --> 00:38:32,518 have to apply the other limit? 494 00:38:42,090 --> 00:38:45,660 Any questions about that? 495 00:38:45,660 --> 00:38:47,052 Yes, Nancy. 496 00:38:47,052 --> 00:38:50,496 AUDIENCE: So are we saying here that without changing 497 00:38:50,496 --> 00:38:52,956 the [INAUDIBLE] of the harmonic oscillator, 498 00:38:52,956 --> 00:38:55,908 we cannot use it as a harmonic oscillator? 499 00:38:55,908 --> 00:38:58,977 Like [INAUDIBLE]? 500 00:38:58,977 --> 00:39:01,351 Because when you put an atom in that [INAUDIBLE] changed, 501 00:39:01,351 --> 00:39:04,280 it was no more a harmonic oscillator. 502 00:39:04,280 --> 00:39:05,624 The levels changed. 503 00:39:08,530 --> 00:39:11,620 PROFESSOR: Maybe all I'm saying is 504 00:39:11,620 --> 00:39:13,820 this is a pure harmonic oscillator. 505 00:39:13,820 --> 00:39:19,550 And in a pure harmonic oscillator, 506 00:39:19,550 --> 00:39:22,130 I think it's-- I don't know a proof of it, 507 00:39:22,130 --> 00:39:25,930 but it seems impossible to prepare a system in the first 508 00:39:25,930 --> 00:39:30,170 excited state because every attempt to put an excitation 509 00:39:30,170 --> 00:39:33,659 into the system would carry it higher up. 510 00:39:33,659 --> 00:39:34,950 You would create a wave packet. 511 00:39:34,950 --> 00:39:36,792 You would create a superposition. 512 00:39:36,792 --> 00:39:38,000 So you have to do some thing. 513 00:39:38,000 --> 00:39:40,540 You have to break the degeneracy of the spectrum 514 00:39:40,540 --> 00:39:42,310 of the harmonic oscillator. 515 00:39:42,310 --> 00:39:44,990 Of course, what you can do is you can put in an atom. 516 00:39:44,990 --> 00:39:49,350 You can use the atom as an aid to just put in exactly one 517 00:39:49,350 --> 00:39:52,750 photon into your cavity, and then you can remove the atom. 518 00:39:52,750 --> 00:39:55,750 Then you are back to an ideal harmonic oscillator, 519 00:39:55,750 --> 00:39:58,730 but you have overcome the limitation 520 00:39:58,730 --> 00:40:02,020 of the harmonic oscillator in preparing certain states. 521 00:40:08,940 --> 00:40:13,980 Another take-home message you may take from this discussion 522 00:40:13,980 --> 00:40:16,650 is harmonic oscillators-- yes, we 523 00:40:16,650 --> 00:40:18,200 have quantum harmonic oscillators. 524 00:40:18,200 --> 00:40:22,540 But even the quantum harmonic oscillator 525 00:40:22,540 --> 00:40:24,160 follows a classical description. 526 00:40:24,160 --> 00:40:28,150 So the real quantumness-- what makes quantum optics quantum 527 00:40:28,150 --> 00:40:33,330 optics and cavity QED a wonderful example of quantum 528 00:40:33,330 --> 00:40:38,120 physics is the physics embedded in a two-level system. 529 00:40:38,120 --> 00:40:41,650 That we can put one quantum excitation into something, 530 00:40:41,650 --> 00:40:45,650 exactly one is as much quantum as you can get. 531 00:40:45,650 --> 00:40:49,050 This is realized in a two-level system 532 00:40:49,050 --> 00:40:52,220 and this is related to the phenomenon of saturation. 533 00:40:52,220 --> 00:40:54,120 You can saturate a two-level system, 534 00:40:54,120 --> 00:40:59,010 but you cannot saturate a harmonic oscillator. 535 00:40:59,010 --> 00:41:02,440 So with that, let me make the transition 536 00:41:02,440 --> 00:41:08,020 to another simple system. 537 00:41:08,020 --> 00:41:09,585 And we want to spend some time on it. 538 00:41:12,230 --> 00:41:16,150 And these are rotating systems. 539 00:41:23,120 --> 00:41:27,300 So a system which rotates. 540 00:41:27,300 --> 00:41:29,741 Well, what do you think? 541 00:41:29,741 --> 00:41:33,330 Will it behave, using the discussion we just 542 00:41:33,330 --> 00:41:35,940 had, more like a classical system 543 00:41:35,940 --> 00:41:37,870 or more like a quantum-mechanical system? 544 00:41:53,300 --> 00:41:56,170 Of course, I gave you a very special definition. 545 00:41:56,170 --> 00:41:59,340 What brings out quantum mechanics in a system? 546 00:42:04,150 --> 00:42:07,240 The harmonic oscillator is always linear. 547 00:42:07,240 --> 00:42:09,920 You can drive it as hard as you want. 548 00:42:09,920 --> 00:42:12,520 You drive it hundred times stronger 549 00:42:12,520 --> 00:42:14,940 and the reaction is hundred times more. 550 00:42:14,940 --> 00:42:17,230 Everything is linear. 551 00:42:17,230 --> 00:42:19,935 The quantumness of a two-level system comes from saturation. 552 00:42:22,830 --> 00:42:24,405 What about a rotating system? 553 00:42:27,170 --> 00:42:28,645 Something which can go in a circle. 554 00:42:35,936 --> 00:42:36,852 AUDIENCE: [INAUDIBLE]. 555 00:42:41,772 --> 00:42:44,724 So you're not going to have this degeneracy [INAUDIBLE]. 556 00:42:48,629 --> 00:42:49,670 PROFESSOR: OK, very good. 557 00:42:49,670 --> 00:42:53,920 You're immediately applying what is the spectrum. 558 00:42:53,920 --> 00:42:56,057 The spectrum is not equidistant, so it 559 00:42:56,057 --> 00:42:57,265 should bring in a difference. 560 00:42:59,920 --> 00:43:02,700 It's sometimes hard to ask a simple question 561 00:43:02,700 --> 00:43:04,700 without giving the answer away. 562 00:43:04,700 --> 00:43:08,180 But what I had in mind was a gyroscope, 563 00:43:08,180 --> 00:43:11,350 a gyroscope which is precessing. 564 00:43:11,350 --> 00:43:13,550 And what I wanted to sort of lead you 565 00:43:13,550 --> 00:43:16,900 with the question is, if you have something which rotates, 566 00:43:16,900 --> 00:43:18,230 the amplitude is limited. 567 00:43:20,820 --> 00:43:24,530 A rotating object, let's assume a magnetic, classical magnetic 568 00:43:24,530 --> 00:43:25,380 moment. 569 00:43:25,380 --> 00:43:29,940 It can have a precession angle which is 180 degrees, 570 00:43:29,940 --> 00:43:31,550 but that's a maximum. 571 00:43:31,550 --> 00:43:35,770 In other words, the rotating system when you excite it 572 00:43:35,770 --> 00:43:40,070 has a maximum amplitude, exactly as a two-level system. 573 00:43:40,070 --> 00:43:46,930 So that's what a rotating system and a two-level system-- 574 00:43:46,930 --> 00:43:48,780 actually then, I'm now specializing 575 00:43:48,780 --> 00:43:52,380 on more rotating gyroscope. 576 00:43:52,380 --> 00:43:55,430 If you have a free rotator, this, of course, 577 00:43:55,430 --> 00:43:58,110 can rotate with [INAUDIBLE] angular momentum 578 00:43:58,110 --> 00:44:01,380 and the excitation spectrum would not be bound. 579 00:44:01,380 --> 00:44:03,850 So I think I have to rephrase the question the next time 580 00:44:03,850 --> 00:44:05,970 I teach the class. 581 00:44:05,970 --> 00:44:09,540 I wanted to ask you here about a special rotating system, which 582 00:44:09,540 --> 00:44:12,330 is a precessing gyroscope. 583 00:44:12,330 --> 00:44:14,310 So rotating system. 584 00:44:14,310 --> 00:44:25,430 If you think about precessing gyroscope, 585 00:44:25,430 --> 00:44:32,460 it has a bound on the amplitude it can be excited. 586 00:44:37,650 --> 00:44:42,640 So what I want to show you, today and in the next lecture, 587 00:44:42,640 --> 00:44:49,895 is the motion of classical magnetic moments. 588 00:44:58,900 --> 00:45:02,300 When you think about the motion of classical magnetic moments, 589 00:45:02,300 --> 00:45:05,900 think about a compass needle, a magnetized needle 590 00:45:05,900 --> 00:45:07,025 which has angular momentum. 591 00:45:11,240 --> 00:45:14,780 And then, the system is acted upon with a magnetic field. 592 00:45:14,780 --> 00:45:16,820 So this is our system. 593 00:45:16,820 --> 00:45:20,090 And if angular momentum, magnetic moments, and torque 594 00:45:20,090 --> 00:45:24,590 come into play, we have the physics of classical rotation. 595 00:45:24,590 --> 00:45:29,070 But the excitation spectrum here is limited 596 00:45:29,070 --> 00:45:31,390 because you can flip a compass needle 597 00:45:31,390 --> 00:45:33,010 and this is a maximum excitation. 598 00:45:33,010 --> 00:45:35,710 When the North Pole points in the opposite direction, that's 599 00:45:35,710 --> 00:45:38,190 the maximum excitation you can give it. 600 00:45:38,190 --> 00:45:42,210 So therefore, it has a limited amplitude of its excitation, 601 00:45:42,210 --> 00:45:45,090 unlike a harmonic oscillator. 602 00:45:45,090 --> 00:45:50,180 And at this point you may say, but maybe somewhat 603 00:45:50,180 --> 00:45:53,825 similar or analogous to a two-level system. 604 00:45:57,980 --> 00:46:01,190 But the surprising result is-- at least 605 00:46:01,190 --> 00:46:04,090 it was surprising when I first learned about it. 606 00:46:04,090 --> 00:46:06,450 That it's not just somewhat analogous 607 00:46:06,450 --> 00:46:08,580 to a two-level system, it actually 608 00:46:08,580 --> 00:46:13,050 captures exactly a lot of the properties 609 00:46:13,050 --> 00:46:15,480 of the dynamics of a two-level system. 610 00:46:15,480 --> 00:46:16,710 So let me write that down. 611 00:46:16,710 --> 00:46:25,820 The motion of classical magnetic moments provides a model. 612 00:46:25,820 --> 00:46:33,740 It's actually an exact model which 613 00:46:33,740 --> 00:46:47,266 captures essentially all features 614 00:46:47,266 --> 00:46:49,840 of the quantum mechanical two-level system. 615 00:46:55,430 --> 00:46:59,570 I want to show you today and the next lecture the concepts 616 00:46:59,570 --> 00:47:04,750 or Rabi frequency, of generalized of resonant Rabi 617 00:47:04,750 --> 00:47:10,190 frequency, all of that you find in the classical motion 618 00:47:10,190 --> 00:47:13,510 of a magnetic moment. 619 00:47:13,510 --> 00:47:16,070 Or, for instance, the physics of rapid [INAUDIBLE] 620 00:47:16,070 --> 00:47:18,850 following, [INAUDIBLE]. 621 00:47:18,850 --> 00:47:21,010 A lot of physics we would usually 622 00:47:21,010 --> 00:47:23,180 associate with the quantum system, 623 00:47:23,180 --> 00:47:25,244 we find it here in a purely classical system. 624 00:47:32,030 --> 00:47:37,040 What aspects of the two-level system will we not find? 625 00:47:37,040 --> 00:47:37,540 Any ideas? 626 00:47:45,224 --> 00:47:45,724 Will? 627 00:47:45,724 --> 00:47:47,700 AUDIENCE: Spontaneous emission? 628 00:47:47,700 --> 00:47:50,050 PROFESSOR: Spontaneous emission, definitely. 629 00:47:50,050 --> 00:47:50,950 Yes. 630 00:47:50,950 --> 00:47:53,150 But actually, in a two-level system, 631 00:47:53,150 --> 00:47:57,110 in a quantum-mechanical two-level system, which 632 00:47:57,110 --> 00:47:59,460 we drive with a single frequency, 633 00:47:59,460 --> 00:48:01,610 spontaneous emission is also missing. 634 00:48:01,610 --> 00:48:04,380 Spontaneous emission, as we will discuss later, 635 00:48:04,380 --> 00:48:08,690 only comes into play when we say the excited state of the system 636 00:48:08,690 --> 00:48:11,860 interacts with many, many modes and not just 637 00:48:11,860 --> 00:48:13,920 the one mode we apply. 638 00:48:13,920 --> 00:48:16,045 And typically, if you go to high frequency, 639 00:48:16,045 --> 00:48:19,470 we have an optical oscillator, we cannot avoid spontaneous 640 00:48:19,470 --> 00:48:20,330 emission. 641 00:48:20,330 --> 00:48:24,610 Whereas, for a quantum-mechanical spin 1/2 642 00:48:24,610 --> 00:48:27,290 interacting with microwaves, we can completely 643 00:48:27,290 --> 00:48:30,100 eliminate spontaneous emission. 644 00:48:30,100 --> 00:48:33,480 So spontaneous emission, I would say, 645 00:48:33,480 --> 00:48:36,570 comes into play at high frequency. 646 00:48:36,570 --> 00:48:37,830 So that's correct. 647 00:48:37,830 --> 00:48:41,070 But there is one aspect even at low frequency, 648 00:48:41,070 --> 00:48:44,075 one aspect of quantum mechanics which we cannot capture. 649 00:48:55,460 --> 00:48:58,930 AUDIENCE: Having different G in a magnetic field? 650 00:48:58,930 --> 00:49:02,490 PROFESSOR: Different G factors, yes. 651 00:49:02,490 --> 00:49:05,560 That's, as we see, a more quantitative aspect. 652 00:49:05,560 --> 00:49:09,830 But there is one very important feature about quantum mechanics 653 00:49:09,830 --> 00:49:12,866 you will never get in a classical system. 654 00:49:12,866 --> 00:49:14,660 AUDIENCE: Spin? 655 00:49:14,660 --> 00:49:15,913 PROFESSOR: Spin. 656 00:49:15,913 --> 00:49:16,829 AUDIENCE: [INAUDIBLE]. 657 00:49:21,110 --> 00:49:23,037 AUDIENCE: [INAUDIBLE]. 658 00:49:23,037 --> 00:49:23,620 PROFESSOR: OK. 659 00:49:23,620 --> 00:49:25,320 I think you're skirting around. 660 00:49:25,320 --> 00:49:28,140 It's a quantum measurement process and projection. 661 00:49:28,140 --> 00:49:30,930 If you perform a measurement on a compass needle, 662 00:49:30,930 --> 00:49:32,520 it can be at any angle. 663 00:49:32,520 --> 00:49:34,680 But if you do a measurement on a quantum system, 664 00:49:34,680 --> 00:49:36,360 you do a projection. 665 00:49:36,360 --> 00:49:39,090 After the measurement projects a system 666 00:49:39,090 --> 00:49:41,910 in either spin up or spin down. 667 00:49:41,910 --> 00:49:44,910 So the probabilistic nature, the projection 668 00:49:44,910 --> 00:49:46,990 occurring in a quantum measurement 669 00:49:46,990 --> 00:49:49,070 is, of course, absent in a classical system. 670 00:49:55,930 --> 00:49:58,770 But when you say spin 1/2 and quantum levels, 671 00:49:58,770 --> 00:50:00,790 this is sort of implied in it. 672 00:50:00,790 --> 00:50:03,330 If there are only two levels, there's only up and down 673 00:50:03,330 --> 00:50:05,840 and not an infinite number of angles. 674 00:50:05,840 --> 00:50:09,560 So what we will actually see is that we find an exact analogy 675 00:50:09,560 --> 00:50:13,340 between the classic system and the quantum mechanical system 676 00:50:13,340 --> 00:50:16,130 when we compare expectation values. 677 00:50:16,130 --> 00:50:18,450 But the individual measurement, the individual quantum 678 00:50:18,450 --> 00:50:22,070 measurement because it is projective is different. 679 00:50:45,400 --> 00:50:50,990 OK, with that motivation, we are now 680 00:50:50,990 --> 00:50:53,300 talking about magnetic resonance. 681 00:51:02,450 --> 00:51:09,400 And we will later do a fully quantum mechanical description, 682 00:51:09,400 --> 00:51:15,460 but to get the concepts and also understand the analogies 683 00:51:15,460 --> 00:51:18,450 to a classical system, we want to understand 684 00:51:18,450 --> 00:51:27,090 what happens when we have a classic magnetic moment 685 00:51:27,090 --> 00:51:30,280 in magnetic fields. 686 00:51:30,280 --> 00:51:34,220 And that includes static fields, but then we 687 00:51:34,220 --> 00:51:36,280 want to excite the system. 688 00:51:36,280 --> 00:51:41,505 We want to drive the system, and this is time-varying fields. 689 00:51:44,630 --> 00:51:50,360 And we will assume that the fields are spatially uniform. 690 00:51:50,360 --> 00:51:59,540 So let me just remind you of the obvious equations of motion 691 00:51:59,540 --> 00:52:04,351 that also allows me to introduce the nomenclature. 692 00:52:04,351 --> 00:52:13,280 The interaction energy between a classical magnetic moment 693 00:52:13,280 --> 00:52:21,310 mu and the magnetic field is mu dot B. 694 00:52:21,310 --> 00:52:31,560 The force is the gradient of the interaction energy, 695 00:52:31,560 --> 00:52:36,340 but it is 0 for uniform fields. 696 00:52:36,340 --> 00:52:40,690 So therefore, we don't need to look at the force. 697 00:52:40,690 --> 00:52:48,240 But the next thing which we then have to consider is the torque. 698 00:52:57,150 --> 00:53:01,350 And when we think about the classical magnetic moment, 699 00:53:01,350 --> 00:53:03,020 you can think about a compass needle. 700 00:53:03,020 --> 00:53:06,310 But magnetic materials are complicated. 701 00:53:06,310 --> 00:53:09,460 If I think about the simplest magnetic moment, 702 00:53:09,460 --> 00:53:19,550 I think about a loop of current I and area A. 703 00:53:19,550 --> 00:53:24,780 And that's sort of the classical model for magnetic moment. 704 00:53:35,110 --> 00:53:39,180 So we have a magnetic moment mu. 705 00:53:39,180 --> 00:53:46,180 And if we now add a magnetic field, which is at an angle, 706 00:53:46,180 --> 00:53:48,380 we have a torque. 707 00:53:48,380 --> 00:53:51,290 But just to make sure the torque is 708 00:53:51,290 --> 00:53:53,760 something which is nothing else than the Lewin's 709 00:53:53,760 --> 00:53:57,760 force on the electrons. 710 00:53:57,760 --> 00:54:01,310 But since electron is forced to go in a circle, 711 00:54:01,310 --> 00:54:04,460 we don't have to look at the Lewin's force microscopically. 712 00:54:04,460 --> 00:54:06,980 We just immediately jump to the torque. 713 00:54:06,980 --> 00:54:10,010 And the torque is what describes the dynamics of the system. 714 00:54:18,755 --> 00:54:23,620 So we have torque. 715 00:54:23,620 --> 00:54:28,230 When we have torque, we want to formulate the problem 716 00:54:28,230 --> 00:54:29,800 in terms of angular momentum. 717 00:54:35,830 --> 00:54:41,550 And our equation of motion is the classical equation 718 00:54:41,550 --> 00:54:45,570 of motion that the derivative of angular momentum 719 00:54:45,570 --> 00:54:50,420 is given by that. 720 00:54:50,420 --> 00:54:56,590 Now, what makes those equations immediately solvable-- 721 00:54:56,590 --> 00:54:59,130 and to find the very easy limit is 722 00:54:59,130 --> 00:55:04,840 that the magnetic moment of the system we assume 723 00:55:04,840 --> 00:55:07,486 is proportional to its angular momentum. 724 00:55:11,110 --> 00:55:13,330 Well, if you have a mechanical object which 725 00:55:13,330 --> 00:55:18,860 goes in a-- if you have a charged object which circles 726 00:55:18,860 --> 00:55:22,020 around a central potential, then you, of course, 727 00:55:22,020 --> 00:55:24,780 find immediately that if it moves faster, 728 00:55:24,780 --> 00:55:26,260 it has more angular momentum. 729 00:55:26,260 --> 00:55:28,600 It has a larger magnetic moment. 730 00:55:28,600 --> 00:55:34,770 So we use that as the defining equation for what 731 00:55:34,770 --> 00:55:37,070 is called the gyromagnetic ratio. 732 00:55:42,400 --> 00:55:45,130 Which, of course, is very closely related 733 00:55:45,130 --> 00:55:49,160 to G factors, which we define later on for atoms. 734 00:55:49,160 --> 00:55:54,750 The gyromagnetic ratio is the ratio between magnetic moment 735 00:55:54,750 --> 00:55:55,670 and angular momentum. 736 00:55:59,990 --> 00:56:08,270 And then, we find that the derivative of angular momentum 737 00:56:08,270 --> 00:56:10,920 is given by this equation. 738 00:56:10,920 --> 00:56:16,820 And this is now an equation which 739 00:56:16,820 --> 00:56:18,460 you have seen in classical mechanics 740 00:56:18,460 --> 00:56:20,090 and in many situations. 741 00:56:20,090 --> 00:56:23,700 The solution of that is a pure precession. 742 00:56:23,700 --> 00:56:28,120 The motion is pure precession of the angular momentum 743 00:56:28,120 --> 00:56:33,280 around the axis of the magnetic field. 744 00:56:39,530 --> 00:56:43,830 So in other words, we have the axis of the magnetic field. 745 00:56:47,380 --> 00:56:50,830 We have the angular momentum. 746 00:56:50,830 --> 00:57:05,930 And at a constant tipping angle, we 747 00:57:05,930 --> 00:57:10,000 have the tip of the angular momentum 748 00:57:10,000 --> 00:57:14,430 precesses around the magnetic field. 749 00:57:14,430 --> 00:57:24,320 And the precession happens with an angular frequency 750 00:57:24,320 --> 00:57:26,155 which is called the Larmor frequency. 751 00:57:33,840 --> 00:57:38,490 The Larmor frequency, the frequency of precession, 752 00:57:38,490 --> 00:57:40,910 is proportional to the magnetic field 753 00:57:40,910 --> 00:57:42,841 and the gyromagnetic ratio. 754 00:57:48,500 --> 00:57:53,290 So let me give you an example for an electron. 755 00:57:53,290 --> 00:58:03,680 The gyromagnetic ratio is 2 pi times 2.8 megahertz per Gauss. 756 00:58:07,060 --> 00:58:09,080 And we've discussed last class what 757 00:58:09,080 --> 00:58:11,510 it means when I take out 2 pi. 758 00:58:11,510 --> 00:58:15,290 Because the Larmor frequency is an angular frequency. 759 00:58:15,290 --> 00:58:18,300 And angular frequency is not measured in Hertz 760 00:58:18,300 --> 00:58:20,210 because there is a 2 pi factor. 761 00:58:20,210 --> 00:58:24,321 And I just make it obvious where the 2 pi factor is hidden. 762 00:58:24,321 --> 00:58:25,570 Now, this is for the electron. 763 00:58:28,270 --> 00:58:33,870 But if you have an ensemble of classical charges, 764 00:58:33,870 --> 00:58:37,530 an [INAUDIBLE] distribution of classical charges-- 765 00:58:37,530 --> 00:58:40,510 well, with the same charge to mass ratio, 766 00:58:40,510 --> 00:58:50,840 you find that the gyromagnetic ratio is 1/2 of that. 767 00:58:53,430 --> 00:58:55,860 And this here is the Bohr magneton, 768 00:58:55,860 --> 00:58:58,546 which we will use quite often in this course. 769 00:59:06,930 --> 00:59:10,380 The third example is the proton. 770 00:59:10,380 --> 00:59:14,400 The proton is heavier, has a heavier mass. 771 00:59:14,400 --> 00:59:18,030 About 1,000 times heavier than the electron. 772 00:59:18,030 --> 00:59:20,650 And therefore, the Larmor frequency is not 773 00:59:20,650 --> 00:59:24,310 megahertz per Gauss, it is kilohertz per Gauss. 774 00:59:35,970 --> 00:59:37,130 Any questions? 775 00:59:37,130 --> 00:59:42,190 These are more definitions and setting the stage. 776 00:59:50,060 --> 00:59:53,110 Let me make a note. 777 00:59:53,110 --> 00:59:55,110 It's one of the many notes I will 778 00:59:55,110 --> 00:59:58,330 make in this course about factors of 2. 779 00:59:58,330 --> 01:00:00,260 There are factors of 2. 780 01:00:00,260 --> 01:00:03,600 If you miss it, you qualitatively miss the physics. 781 01:00:03,600 --> 01:00:05,520 And let me in that context by talking 782 01:00:05,520 --> 01:00:08,390 about precession frequencies and magnetic moment, 783 01:00:08,390 --> 01:00:11,460 explain a factor of 2 which is related 784 01:00:11,460 --> 01:00:12,910 to the G factor of the electron. 785 01:00:16,460 --> 01:00:25,730 So if you have the electron which has spin of 1/2, 786 01:00:25,730 --> 01:00:27,560 in units of the Bohr magneton, what 787 01:00:27,560 --> 01:00:31,010 is the magnetic moment of the electron? 788 01:00:31,010 --> 01:00:34,040 1/2, 1, or 2? 789 01:00:34,040 --> 01:00:37,020 What is the magnetic moment of the electron? 790 01:00:37,020 --> 01:00:38,618 1/2, 1, or 2? 791 01:00:41,522 --> 01:00:43,942 AUDIENCE: 2. 792 01:00:43,942 --> 01:00:44,920 AUDIENCE: [INAUDIBLE]. 793 01:00:44,920 --> 01:00:47,250 PROFESSOR: I should have a clicker question on there. 794 01:00:47,250 --> 01:00:47,940 No. 795 01:00:47,940 --> 01:00:49,520 It's 1. 796 01:00:49,520 --> 01:00:52,290 1 Bohr magneton. 797 01:00:52,290 --> 01:00:54,570 And let me sort of show the level structure of it. 798 01:00:58,470 --> 01:01:00,510 This is [INAUDIBLE] energy. 799 01:01:00,510 --> 01:01:05,170 You have spin up and you have spin down. 800 01:01:05,170 --> 01:01:13,450 The difference is 2.8 megahertz per Gauss. 801 01:01:13,450 --> 01:01:20,870 And if you ask, what is the precession frequency 802 01:01:20,870 --> 01:01:22,980 of an electron in a magnetic field? 803 01:01:22,980 --> 01:01:26,410 It's 2.8 megahertz if the field is 1 Gauss. 804 01:01:26,410 --> 01:01:28,560 And if you want to drive the rotation, 805 01:01:28,560 --> 01:01:30,705 if you want to change the precession angle-- we'll 806 01:01:30,705 --> 01:01:32,080 talk about that in great detail-- 807 01:01:32,080 --> 01:01:36,340 you better drive the system at 2.8 megahertz. 808 01:01:36,340 --> 01:01:40,770 But 2.8 megahertz is the difference of plus 1.4 809 01:01:40,770 --> 01:01:43,160 and minus 1.4. 810 01:01:43,160 --> 01:01:46,470 And therefore, the energy of the electron in a magnetic field 811 01:01:46,470 --> 01:01:50,420 is either plus or minus 1.4 megahertz per Gauss, 812 01:01:50,420 --> 01:01:52,970 and 1.4 is 1 Bohr magneton. 813 01:01:52,970 --> 01:01:56,590 The magnetic moment of the electron is 1 Bohr magneton. 814 01:02:00,780 --> 01:02:07,130 So precesses at 2.8. 815 01:02:07,130 --> 01:02:15,230 OK, but let us contrast this with a classical current 816 01:02:15,230 --> 01:02:26,000 distribution, which has 1 unit of h bar, which 817 01:02:26,000 --> 01:02:35,040 means the magnetic moment is 1 Bohr magneton exactly 818 01:02:35,040 --> 01:02:38,300 as the electron has. 819 01:02:38,300 --> 01:02:46,330 Well, quantum mechanically means it 820 01:02:46,330 --> 01:02:48,330 has three different level-- minus 1, 821 01:02:48,330 --> 01:02:52,380 [INAUDIBLE] 0, and 1 because it has 1 unit of angular momentum. 822 01:02:52,380 --> 01:02:56,440 Since the system has 1 Bohr magneton, when 823 01:02:56,440 --> 01:02:58,940 the system stands up or stands down, 824 01:02:58,940 --> 01:03:03,150 the difference between spin up and spin down 825 01:03:03,150 --> 01:03:06,485 is 2.8 megahertz per Gauss. 826 01:03:09,650 --> 01:03:13,920 OK, my question now is, what is the precession frequency 827 01:03:13,920 --> 01:03:18,730 of this classical charge distribution which 828 01:03:18,730 --> 01:03:21,850 has 1 unit h bar of angular momentum? 829 01:03:21,850 --> 01:03:24,640 And I've just shown you the level structure. 830 01:03:24,640 --> 01:03:28,020 If you will now create a wave packet of those three 831 01:03:28,020 --> 01:03:31,430 levels, which means-- a wave packet of the three levels 832 01:03:31,430 --> 01:03:35,420 means you have a spin which points in one direction. 833 01:03:35,420 --> 01:03:37,998 What is the precession frequency of that system? 834 01:03:42,690 --> 01:03:45,630 Let's have a clicker question. 835 01:03:45,630 --> 01:03:47,580 So what is the precession frequency? 836 01:03:47,580 --> 01:03:49,041 Oops, what happened? 837 01:03:59,755 --> 01:04:04,980 So let me give you three choices. 838 01:04:08,860 --> 01:04:15,360 2.4, 1.4, or 0.0 megahertz per Gauss. 839 01:04:15,360 --> 01:04:41,570 So please vote for A, B, or C. 840 01:04:41,570 --> 01:04:49,510 Yes, it's 1.4. 841 01:04:49,510 --> 01:04:52,220 So they have the same magnetic moment. 842 01:04:52,220 --> 01:04:55,870 Quantum mechanically, you would see here is a G factor of 2. 843 01:04:55,870 --> 01:04:58,110 Here's a G factor of 1. 844 01:04:58,110 --> 01:05:03,050 But the easiest explanation is it precesses. 845 01:05:03,050 --> 01:05:06,400 The precession is a beat note between two energy levels. 846 01:05:06,400 --> 01:05:10,040 Here, you have the beat note between those two energy 847 01:05:10,040 --> 01:05:13,350 levels happens at 1.4 megahertz. 848 01:05:13,350 --> 01:05:15,036 Therefore, when you want to drive it 849 01:05:15,036 --> 01:05:16,536 with an external radiation, you have 850 01:05:16,536 --> 01:05:17,960 to drive it at 1.4 megahertz. 851 01:05:17,960 --> 01:05:19,880 You want to drive it level by level. 852 01:05:19,880 --> 01:05:23,380 Whereas, this system has a beat note between two levels, 853 01:05:23,380 --> 01:05:25,510 and the difference is 2.8 megahertz. 854 01:05:25,510 --> 01:05:28,940 Anyway, whenever you get confused about factors of 2 855 01:05:28,940 --> 01:05:31,840 with magnetic moment and precessing system, 856 01:05:31,840 --> 01:05:34,530 just think about those two simple examples. 857 01:05:34,530 --> 01:05:39,920 They have all the factors of 2 hidden in the simplest example 858 01:05:39,920 --> 01:05:40,420 possible. 859 01:05:45,040 --> 01:05:45,735 All right. 860 01:05:50,040 --> 01:05:51,800 We have a rotating system. 861 01:05:51,800 --> 01:05:53,490 have a system which precesses. 862 01:05:57,040 --> 01:06:01,360 So we want to learn about rotations in general. 863 01:06:01,360 --> 01:06:12,610 And what I want to show you is that 864 01:06:12,610 --> 01:06:16,920 under very general circumstances, 865 01:06:16,920 --> 01:06:23,950 we can solve the equation, the dynamics of the system, 866 01:06:23,950 --> 01:06:27,000 by going into rotating frame. 867 01:06:27,000 --> 01:06:30,690 You all know about rotating wave [INAUDIBLE] 868 01:06:30,690 --> 01:06:33,360 rotating wave approximations that's in quantum physics. 869 01:06:33,360 --> 01:06:36,050 I'm simply talking about a classical system 870 01:06:36,050 --> 01:06:37,750 and I want to solve the equations 871 01:06:37,750 --> 01:06:40,630 for the classical system by going to rotating frame. 872 01:06:40,630 --> 01:06:43,244 And I want to show you where this is exact and where not. 873 01:06:47,990 --> 01:06:50,190 OK, so this is actually something 874 01:06:50,190 --> 01:06:58,530 which we do in undergraduate, in 8.01-- definitely, in 8.012. 875 01:06:58,530 --> 01:07:05,500 But let me remind you when we have a rotating vector which 876 01:07:05,500 --> 01:07:11,190 rotates with a constant angular frequency, 877 01:07:11,190 --> 01:07:17,564 then the time derivative of the vector is the cross product. 878 01:07:27,380 --> 01:07:30,280 But now we want to allow-- so this 879 01:07:30,280 --> 01:07:34,000 is when the vector is constant and it just rotates. 880 01:07:34,000 --> 01:07:41,950 But now, we want to assume that there is something else. 881 01:07:41,950 --> 01:07:43,770 There is an arbitrary time dependence 882 01:07:43,770 --> 01:07:46,900 of the vector in the rotating frame. 883 01:07:46,900 --> 01:07:49,690 So we have a vector which changes 884 01:07:49,690 --> 01:07:51,680 according to A dot [INAUDIBLE]. 885 01:07:51,680 --> 01:07:53,700 But it also rotates. 886 01:07:53,700 --> 01:07:55,810 And that means-- and this is exactly 887 01:07:55,810 --> 01:08:00,700 shown in classical mechanics that in the inertia frame, 888 01:08:00,700 --> 01:08:04,340 the time derivative is the sum of the two. 889 01:08:04,340 --> 01:08:07,290 It is the change of the vector in the rotating 890 01:08:07,290 --> 01:08:19,399 frame plus omega cross A. So it has this equation. 891 01:08:19,399 --> 01:08:27,760 Has the simple two limiting cases 892 01:08:27,760 --> 01:08:31,859 that if there is no change in the rotating frame, 893 01:08:31,859 --> 01:08:35,580 then we retrieve the kinematics of pure rotation. 894 01:08:38,580 --> 01:08:41,370 When our rotating frame is not rotating 895 01:08:41,370 --> 01:08:45,689 or it rotates at 0 angular frequency, then of course 896 01:08:45,689 --> 01:08:52,220 the two time derivatives are the same. 897 01:08:52,220 --> 01:09:00,069 But anyway, what I derived for you is an operator equation 898 01:09:00,069 --> 01:09:03,720 that the time derivative in the rotating frame 899 01:09:03,720 --> 01:09:07,080 is related to the time derivative in the inertia 900 01:09:07,080 --> 01:09:08,800 frame in this way. 901 01:09:11,930 --> 01:09:18,950 And now we want to apply it to our angular momentum L dot. 902 01:09:26,609 --> 01:09:30,920 So this is just applying the operator equation 903 01:09:30,920 --> 01:09:35,550 to our angular momentum L. And now 904 01:09:35,550 --> 01:09:39,810 we want to specialize that-- we just 905 01:09:39,810 --> 01:09:48,435 discussed that the time [INAUDIBLE]. 906 01:09:56,522 --> 01:09:58,870 I'm just looking for a sign problem, 907 01:09:58,870 --> 01:10:04,710 but it's sometimes hard to fix sign problems at the board. 908 01:10:04,710 --> 01:10:07,710 The creation of motion for the angular momentum 909 01:10:07,710 --> 01:10:16,091 was that it's L cross gamma B. 910 01:10:16,091 --> 01:10:17,090 Oh, I changed the order. 911 01:10:17,090 --> 01:10:19,170 There is no sign problem. 912 01:10:19,170 --> 01:10:22,630 And then, I add this. 913 01:10:22,630 --> 01:10:27,770 So if we now describe our precessing, 914 01:10:27,770 --> 01:10:31,430 classical magnetic moment, which has the equation of motion 915 01:10:31,430 --> 01:10:35,310 that L dot is L cross gamma B, if you describe it 916 01:10:35,310 --> 01:10:40,080 in a rotating frame, then the equation of motion 917 01:10:40,080 --> 01:10:41,530 gets modified as follows. 918 01:10:44,200 --> 01:10:50,750 Now, what happens is we-- let me factor out the gamma. 919 01:10:50,750 --> 01:10:55,260 Gamma L cross B is the real field. 920 01:11:00,270 --> 01:11:04,200 So what we observe is that when we 921 01:11:04,200 --> 01:11:06,530 go into a rotating frame-- and this 922 01:11:06,530 --> 01:11:14,390 is exact-- that the real magnetic field gets replaced 923 01:11:14,390 --> 01:11:19,100 by an effective magnetic field. 924 01:11:19,100 --> 01:11:25,210 Because there is an extra term added to it, 925 01:11:25,210 --> 01:11:27,822 which we can call a fictitious magnetic field. 926 01:11:32,690 --> 01:11:35,970 So this is just an exact transformation 927 01:11:35,970 --> 01:11:38,300 of our equation of motion for a precessing 928 01:11:38,300 --> 01:11:40,980 system into the rotating frame. 929 01:11:40,980 --> 01:11:44,700 And now, of course, we haven't made any assumptions 930 01:11:44,700 --> 01:11:47,660 what the rotating frequency is. 931 01:11:47,660 --> 01:11:52,370 But if you would choose the rotating frequency 932 01:11:52,370 --> 01:12:00,330 to be the Larmor frequency minus gamma times B, 933 01:12:00,330 --> 01:12:05,010 then our effective magnetic field vanishes. 934 01:12:05,010 --> 01:12:09,070 And then we know, because there is no magnetic field, 935 01:12:09,070 --> 01:12:15,310 that the angular momentum is constant in the rotating frame. 936 01:12:19,330 --> 01:12:23,000 In other words, the dynamics of the system 937 01:12:23,000 --> 01:12:26,130 means that L is constant in a rotating frame. 938 01:12:26,130 --> 01:12:28,840 And if you want to know what happens in the original, 939 01:12:28,840 --> 01:12:31,374 in the [? lab ?] frame, in the inertial frame, 940 01:12:31,374 --> 01:12:32,540 we just have to rotate back. 941 01:12:51,030 --> 01:12:55,620 OK, that's something we want to take advantage of. 942 01:12:55,620 --> 01:13:01,590 But we fully apply it in the next class. 943 01:13:01,590 --> 01:13:05,670 I have only a few minutes left today, 944 01:13:05,670 --> 01:13:09,760 and I want to spend those few minutes to talk 945 01:13:09,760 --> 01:13:12,840 about another factor of 2. 946 01:13:12,840 --> 01:13:15,933 Now, let me ask you the following. 947 01:13:21,580 --> 01:13:30,010 If you have an electron in a magnetic field, well, 948 01:13:30,010 --> 01:13:37,695 you know that the electron goes in circles. 949 01:13:40,460 --> 01:13:42,380 It's the cyclotron motion of the electron. 950 01:13:45,095 --> 01:13:51,334 Now, just give me one second. 951 01:14:06,290 --> 01:14:11,525 I forgot to mention something for the classical system. 952 01:14:18,350 --> 01:14:27,400 For a classical charge distribution, 953 01:14:27,400 --> 01:14:35,150 the Larmor frequency is the charge of the particle. 954 01:14:35,150 --> 01:14:38,240 In case of the electron, it's e. 955 01:14:38,240 --> 01:14:43,030 Divided by 2m times B. So the Larmor frequency 956 01:14:43,030 --> 01:14:47,630 is e over 2m times B. 957 01:14:47,630 --> 01:14:51,460 So therefore, we know that when we 958 01:14:51,460 --> 01:14:58,140 have an [INAUDIBLE] of positive and negative charges, 959 01:14:58,140 --> 01:15:01,800 and there is an effective magnetic moment, 960 01:15:01,800 --> 01:15:06,740 that this magnetic moment would precess at the Larmor 961 01:15:06,740 --> 01:15:22,810 frequency which is given by this expression. 962 01:15:22,810 --> 01:15:26,800 So who knows what the frequency of the cyclotron motion is? 963 01:15:31,800 --> 01:15:37,920 So when we have a free electron, what 964 01:15:37,920 --> 01:15:40,694 is the frequency at which it revolves? 965 01:15:40,694 --> 01:15:41,860 At which it goes in circles? 966 01:15:41,860 --> 01:15:44,320 AUDIENCE: 2 times the Larmor frequency. 967 01:15:44,320 --> 01:15:46,785 PROFESSOR: It's two times the Larmor frequency. 968 01:15:53,220 --> 01:15:54,740 OK, I just wanted to mention it. 969 01:15:54,740 --> 01:15:58,755 There's an important factor of 2 which you should know about. 970 01:16:02,190 --> 01:16:05,290 In previous classes, I spent 10 or 20 minutes 971 01:16:05,290 --> 01:16:08,160 to teach you about a few of them which 972 01:16:08,160 --> 01:16:10,100 is called Larmor's theorem. 973 01:16:13,580 --> 01:16:17,190 But I summarized the argument on the atomic physics wiki 974 01:16:17,190 --> 01:16:19,290 and I can't say more here in class 975 01:16:19,290 --> 01:16:20,830 than I've written on the wiki. 976 01:16:20,830 --> 01:16:26,350 So please read on our atomic physics wiki 977 01:16:26,350 --> 01:16:28,300 about Larmor's theorem. 978 01:16:28,300 --> 01:16:36,860 Larmor's theorem shows you that under certain assumptions, 979 01:16:36,860 --> 01:16:40,900 you can transform away the effect of a magnetic field 980 01:16:40,900 --> 01:16:42,378 by going to the Larmor frequency. 981 01:16:45,730 --> 01:16:54,170 That looks exactly like what we have discussed here. 982 01:16:54,170 --> 01:16:57,030 But what we discussed here was exact. 983 01:16:57,030 --> 01:16:58,450 There was no approximation. 984 01:16:58,450 --> 01:17:01,380 Whereas, the derivation of Larmor's theorem, 985 01:17:01,380 --> 01:17:04,040 which talks about charge distributions-- 986 01:17:04,040 --> 01:17:07,370 not about magnetic moments, about charge distributions-- 987 01:17:07,370 --> 01:17:09,780 has to make certain approximations. 988 01:17:09,780 --> 01:17:11,480 So just want to point your attention 989 01:17:11,480 --> 01:17:14,930 to that there are two derivations about Larmor 990 01:17:14,930 --> 01:17:15,930 frequency. 991 01:17:15,930 --> 01:17:18,600 One is exact, which I gave to you. 992 01:17:18,600 --> 01:17:21,160 There is another one which is Larmor's theorem, which 993 01:17:21,160 --> 01:17:24,930 applies to isolated charges which is not exact. 994 01:17:24,930 --> 01:17:29,230 But they both conclude that you can transform away 995 01:17:29,230 --> 01:17:31,700 the effect of a magnetic field by going 996 01:17:31,700 --> 01:17:36,000 to rotating frame at the Larmor frequency. 997 01:17:36,000 --> 01:17:40,520 And the fact that Larmor's theorem is not exact 998 01:17:40,520 --> 01:17:45,430 is actually illustrated by this example of a free electron 999 01:17:45,430 --> 01:17:48,590 where you have a factor of 2. 1000 01:17:48,590 --> 01:17:53,840 And this comes because the term which 1001 01:17:53,840 --> 01:17:58,070 you neglect when you derive Larmor's theorem 1002 01:17:58,070 --> 01:18:01,660 is negligible if the situation is 1003 01:18:01,660 --> 01:18:04,520 that you have electrons and charges forming 1004 01:18:04,520 --> 01:18:05,930 magnetic moments. 1005 01:18:05,930 --> 01:18:09,300 But if you have a free electron, the neglected term 1006 01:18:09,300 --> 01:18:12,410 is exactly 1/2 of the dominant term. 1007 01:18:12,410 --> 01:18:15,910 And that is why the cyclotron frequency is twice 1008 01:18:15,910 --> 01:18:18,110 about Larmor frequency. 1009 01:18:18,110 --> 01:18:21,730 So never confuse the cyclotron and the Larmor frequency. 1010 01:18:21,730 --> 01:18:23,930 And the factor of 2 is not related 1011 01:18:23,930 --> 01:18:26,260 to a G factor of the electron or such. 1012 01:18:26,260 --> 01:18:29,280 It's really the difference between the physics 1013 01:18:29,280 --> 01:18:32,115 of a free charge and the physics of a magnetic moment. 1014 01:18:34,810 --> 01:18:35,485 Any questions? 1015 01:18:40,090 --> 01:18:42,955 OK, well, then we are finished for today. 1016 01:18:42,955 --> 01:18:46,240 A reminder, no class on Wednesday, but we 1017 01:18:46,240 --> 01:18:49,710 have class on Friday in the different classroom.