1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT Open Courseware 4 00:00:06,030 --> 00:00:10,120 continue to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,660 To make a donation or to view additional materials 6 00:00:12,660 --> 00:00:15,690 from hundreds of MIT courses, visit MIT 7 00:00:15,690 --> 00:00:17,850 open courseware at ocw.mit.edu. 8 00:00:21,555 --> 00:00:25,410 ROBERT FIELD: Today's lecture is one where-- 9 00:00:25,410 --> 00:00:28,270 it's a lecture I've never given before. 10 00:00:28,270 --> 00:00:33,210 And it's very much related to the experiments 11 00:00:33,210 --> 00:00:36,580 we are doing right now in my research group. 12 00:00:36,580 --> 00:00:45,730 And so basically, we have a chirped pulse 13 00:00:45,730 --> 00:00:53,530 of microwave radiation, which is propagating through a sample. 14 00:00:53,530 --> 00:00:59,770 It causes all of the molecules in the sample 15 00:00:59,770 --> 00:01:03,280 to be prepared in some way. 16 00:01:03,280 --> 00:01:05,590 We call it polarized. 17 00:01:05,590 --> 00:01:09,400 And this polarization relaxes by what 18 00:01:09,400 --> 00:01:11,950 we call free induction decay. 19 00:01:11,950 --> 00:01:14,350 And they produce a signal which-- 20 00:01:14,350 --> 00:01:16,881 so we have a pulse of radiation that 21 00:01:16,881 --> 00:01:18,130 propagates through the sample. 22 00:01:21,130 --> 00:01:24,670 The two-level systems in the sample 23 00:01:24,670 --> 00:01:28,570 all get polarized, which we'll talk about today. 24 00:01:28,570 --> 00:01:31,640 And they radiate that polarization. 25 00:01:31,640 --> 00:01:34,690 And we collect it in a detector here. 26 00:01:34,690 --> 00:01:38,830 And so the two important things are 27 00:01:38,830 --> 00:01:41,610 this is a time independent experiment, 28 00:01:41,610 --> 00:01:46,120 and that we have a whole bunch of molecules. 29 00:01:46,120 --> 00:01:48,460 And they're interacting with the radiation 30 00:01:48,460 --> 00:01:51,790 in a way which is complicated. 31 00:01:51,790 --> 00:01:53,500 Because this is not-- 32 00:01:53,500 --> 00:01:56,650 each one of them has quantum states, 33 00:01:56,650 --> 00:01:59,860 but all of the particles in this sample 34 00:01:59,860 --> 00:02:03,790 are somehow interacting with the radiation field 35 00:02:03,790 --> 00:02:08,639 in a way which is uncorrelated. 36 00:02:08,639 --> 00:02:12,070 So we could say all of these particles 37 00:02:12,070 --> 00:02:13,500 are either bosons or fermions. 38 00:02:13,500 --> 00:02:15,990 But we're not going to symmeterize 39 00:02:15,990 --> 00:02:17,900 or anti-symmeterize. 40 00:02:17,900 --> 00:02:22,070 Each of these particles is independent. 41 00:02:22,070 --> 00:02:25,370 And we need a way of describing the quantum mechanics 42 00:02:25,370 --> 00:02:29,540 for an ensemble of independent particles. 43 00:02:29,540 --> 00:02:35,040 So it's a big step towards useful quantum mechanics. 44 00:02:35,040 --> 00:02:37,830 And I'm not going to be able to finish 45 00:02:37,830 --> 00:02:40,620 the lecture as I planned it. 46 00:02:40,620 --> 00:02:43,620 So you should know where I'm going. 47 00:02:43,620 --> 00:02:49,880 And I'm going to be introducing a lot of interesting concepts. 48 00:02:49,880 --> 00:02:54,210 The first 2/3 of lectures votes are typed. 49 00:02:54,210 --> 00:02:56,010 And you could have seen them. 50 00:02:56,010 --> 00:03:00,810 And the rest of them will be typed later today. 51 00:03:00,810 --> 00:03:04,830 This is based on material in Mike Fayer's book, which 52 00:03:04,830 --> 00:03:06,300 is referenced in your notes. 53 00:03:09,320 --> 00:03:13,590 This book is really accessible. 54 00:03:13,590 --> 00:03:15,390 It's not nearly as elegant as some 55 00:03:15,390 --> 00:03:19,350 of the other treatments of interaction of radiation 56 00:03:19,350 --> 00:03:22,650 with two-level systems. 57 00:03:22,650 --> 00:03:24,720 Now I talked about interaction of radiation 58 00:03:24,720 --> 00:03:27,740 with two-level systems in lecture number 19. 59 00:03:27,740 --> 00:03:30,120 And this is a completely different topic 60 00:03:30,120 --> 00:03:35,400 from that, because in that, we were 61 00:03:35,400 --> 00:03:43,380 interested in many transitions. 62 00:03:43,380 --> 00:03:49,990 Let me just say the radiation field that interact 63 00:03:49,990 --> 00:03:53,390 with the molecule is weak. 64 00:03:53,390 --> 00:03:57,350 It interacts with all the molecules, 65 00:03:57,350 --> 00:04:03,650 and the theory is for a weak pulse-- 66 00:04:06,790 --> 00:04:10,900 and the important point in lecture 19 was resonance. 67 00:04:13,790 --> 00:04:18,860 And so we made the dipole approximation. 68 00:04:18,860 --> 00:04:26,000 And each two-level system is separately resonant 69 00:04:26,000 --> 00:04:30,740 and is weakly interacted with, and does something 70 00:04:30,740 --> 00:04:32,830 to the radiation field. 71 00:04:32,830 --> 00:04:36,140 Now here, we're going to be talking 72 00:04:36,140 --> 00:04:41,480 about a two-level system, only two levels. 73 00:04:41,480 --> 00:04:44,180 And the radiation field is really strong, 74 00:04:44,180 --> 00:04:45,890 or is as strong as you want. 75 00:04:45,890 --> 00:04:50,290 And it does something to the two-level system 76 00:04:50,290 --> 00:04:54,870 which results in a signal. 77 00:04:54,870 --> 00:04:58,180 And because the radiation field is strong, 78 00:04:58,180 --> 00:05:02,110 it's not just a matter of taking two levels and mixing them. 79 00:05:02,110 --> 00:05:04,550 The mixing coefficients are not small. 80 00:05:04,550 --> 00:05:07,350 It's not linear response. 81 00:05:07,350 --> 00:05:10,500 The mixing is sinusoidal. 82 00:05:10,500 --> 00:05:13,220 The stronger the radiation field, 83 00:05:13,220 --> 00:05:16,680 the mixing changes, and all sorts of interesting things 84 00:05:16,680 --> 00:05:18,000 happen. 85 00:05:18,000 --> 00:05:21,930 So this is a much harder problem than what was 86 00:05:21,930 --> 00:05:26,200 discussed in lecture number 19. 87 00:05:26,200 --> 00:05:28,670 And in order to discuss it, I'm going 88 00:05:28,670 --> 00:05:36,320 to use some important tricks and refer to something 89 00:05:36,320 --> 00:05:40,190 called the density matrix. 90 00:05:40,190 --> 00:05:44,840 The first trick is we have this equation which 91 00:05:44,840 --> 00:05:48,080 can easily be derived. 92 00:05:48,080 --> 00:05:49,820 And most of this lecture, I'm going 93 00:05:49,820 --> 00:05:53,360 to be skipping derivations. 94 00:05:53,360 --> 00:05:56,000 Some of the derivations are going to be in the notes. 95 00:05:56,000 --> 00:05:58,550 So we have some operator. 96 00:05:58,550 --> 00:06:01,640 And we want to know the time dependence of the expectation 97 00:06:01,640 --> 00:06:02,810 value of that operator. 98 00:06:02,810 --> 00:06:17,140 And it's possible to show that the expectation 99 00:06:17,140 --> 00:06:22,570 value of the operator a is given by the expectation 100 00:06:22,570 --> 00:06:28,050 value of the computator of a with the Hamiltonian 101 00:06:28,050 --> 00:06:33,870 plus the expectation value of the partial derivative 102 00:06:33,870 --> 00:06:36,410 of the operator a. 103 00:06:36,410 --> 00:06:42,980 So this is a general and useful equation for the time dependent 104 00:06:42,980 --> 00:06:45,260 of anything. 105 00:06:45,260 --> 00:06:50,510 And it's derived simply by taking the-- 106 00:06:50,510 --> 00:06:55,970 applying the chain rule to this sort of thing. 107 00:07:03,040 --> 00:07:06,900 So we have three terms. 108 00:07:06,900 --> 00:07:12,070 And when you do that, you end up getting this equation. 109 00:07:12,070 --> 00:07:16,090 So this is just this ordinary equation. 110 00:07:16,090 --> 00:07:19,540 And anyway, so this is what happens. 111 00:07:29,150 --> 00:07:33,340 So we're going to have some notation here. 112 00:07:33,340 --> 00:07:34,340 We have a wave function. 113 00:07:37,550 --> 00:07:39,390 And this is a capital psi, so this 114 00:07:39,390 --> 00:07:42,300 is a wave function, a time dependent wave 115 00:07:42,300 --> 00:07:45,260 function that satisfies the time dependent shorter equation. 116 00:07:45,260 --> 00:07:48,420 And we're going to replace that by just something called 117 00:07:48,420 --> 00:07:51,500 little t. 118 00:07:51,500 --> 00:07:59,660 And we can write this thing psi of x and t as the sum over n 119 00:07:59,660 --> 00:08:07,340 Cn psi n of x. 120 00:08:07,340 --> 00:08:18,520 And this becomes in a bracket notation, becomes C Cn n. 121 00:08:21,240 --> 00:08:27,270 So we have a complete ortho normal set of functions. 122 00:08:29,820 --> 00:08:34,250 And this thing is normalized to one. 123 00:08:42,610 --> 00:08:45,220 And now I'm going to introduce this thing called the density 124 00:08:45,220 --> 00:08:45,720 matrix. 125 00:08:54,540 --> 00:09:02,030 This is a very useful quantum mechanical quantity which 126 00:09:02,030 --> 00:09:06,160 replaces the wave function. 127 00:09:06,160 --> 00:09:10,540 It repackages everything we know from the time dependent shorter 128 00:09:10,540 --> 00:09:15,290 equation and the short in your picture of a wave function. 129 00:09:15,290 --> 00:09:16,330 It's equivalent. 130 00:09:16,330 --> 00:09:19,940 It's just arranging it in a different way. 131 00:09:19,940 --> 00:09:25,360 And this different way is extremely powerful, 132 00:09:25,360 --> 00:09:31,000 because what it does is it gets rid of a lot of complexity. 133 00:09:31,000 --> 00:09:33,790 I mean, when you have the time dependent wave functions, 134 00:09:33,790 --> 00:09:39,460 you have this e to the minus i E t over h-bar always kicking 135 00:09:39,460 --> 00:09:40,780 around. 136 00:09:40,780 --> 00:09:47,950 And we get rid of that for most everything. 137 00:09:47,950 --> 00:09:52,820 And it also enables us to do really, really beautiful, 138 00:09:52,820 --> 00:09:56,290 simple calculations of the time dependence of expectation 139 00:09:56,290 --> 00:09:57,560 values. 140 00:09:57,560 --> 00:10:00,280 It's also a quantity where, if you 141 00:10:00,280 --> 00:10:05,890 have a whole bunch of different molecules in the system, 142 00:10:05,890 --> 00:10:08,395 each one of them has a density matrix. 143 00:10:11,110 --> 00:10:13,700 And those density matrices add. 144 00:10:13,700 --> 00:10:17,860 And you have the density matrix for an ensemble. 145 00:10:17,860 --> 00:10:22,320 And so if the populations of different levels are different, 146 00:10:22,320 --> 00:10:29,680 the weights for each of the levels, or each of the systems, 147 00:10:29,680 --> 00:10:30,550 is taken care of. 148 00:10:30,550 --> 00:10:35,320 But we don't worry about coherences between particles 149 00:10:35,320 --> 00:10:39,160 unless we create coherence between particles. 150 00:10:39,160 --> 00:10:41,620 So this is a really powerful thing. 151 00:10:41,620 --> 00:10:46,170 And it's unlike the wave function, 152 00:10:46,170 --> 00:10:53,320 it's observable, because the diagonal elements 153 00:10:53,320 --> 00:10:58,480 of this matrix are populations. 154 00:10:58,480 --> 00:11:04,300 And the off diagonals elements, which we call coherences, 155 00:11:04,300 --> 00:11:06,430 are also observable. 156 00:11:06,430 --> 00:11:10,810 And if you look at the Fourier transform of the emission 157 00:11:10,810 --> 00:11:15,890 from this system, it will consist of several frequencies. 158 00:11:15,890 --> 00:11:20,380 And those frequencies are the off diagonal elements 159 00:11:20,380 --> 00:11:24,320 with the amplitude, the relative weights of those frequencies. 160 00:11:24,320 --> 00:11:27,850 And so one can determine everything in the density 161 00:11:27,850 --> 00:11:29,860 matrix experimentally. 162 00:11:29,860 --> 00:11:32,950 Now it's really-- it's still indirect 163 00:11:32,950 --> 00:11:35,160 because you're making experimental measurements. 164 00:11:35,160 --> 00:11:37,570 But we think about this thing in a way we don't 165 00:11:37,570 --> 00:11:39,800 think about the wave function. 166 00:11:39,800 --> 00:11:41,520 It's really important. 167 00:11:41,520 --> 00:11:44,270 And this is the gateway to almost all 168 00:11:44,270 --> 00:11:49,050 of modern quantum mechanics and statistical mechanics-- 169 00:11:49,050 --> 00:11:50,780 quantum statistical mechanics. 170 00:11:50,780 --> 00:11:53,060 And so this is a really important concept. 171 00:11:53,060 --> 00:11:56,600 And we've protected you from it until now. 172 00:11:56,600 --> 00:12:00,170 And since this is the last lecture both, in this course 173 00:12:00,170 --> 00:12:03,260 and in my teaching of this course forever, 174 00:12:03,260 --> 00:12:07,940 I want to talk about this gateway phenomena. 175 00:12:10,880 --> 00:12:11,855 So what is this? 176 00:12:15,950 --> 00:12:20,210 Well, we denote it by this, this strange notation. 177 00:12:20,210 --> 00:12:24,860 I mean, you're used to this kind of notation where 178 00:12:24,860 --> 00:12:28,890 we have the overlap of bra with a ket 179 00:12:28,890 --> 00:12:32,250 and or abroad with itself. 180 00:12:32,250 --> 00:12:34,590 But this is different. 181 00:12:34,590 --> 00:12:38,420 You know, this is a number and this is a matrix. 182 00:12:46,650 --> 00:12:54,290 And if we have a two-level system, 183 00:12:54,290 --> 00:13:07,800 then we can say that t is equal to C1 of t plus C2 of t. 184 00:13:07,800 --> 00:13:12,530 So state 1, state 2, and we have time dependence. 185 00:13:12,530 --> 00:13:14,640 Now those could be-- 186 00:13:14,640 --> 00:13:16,800 there's lots of stuff that could be in here. 187 00:13:16,800 --> 00:13:18,745 And this is going to be a solution of the time 188 00:13:18,745 --> 00:13:19,870 dependent shorter equation. 189 00:13:22,730 --> 00:13:27,060 So since it's an unfamiliar topic, 190 00:13:27,060 --> 00:13:30,840 I'm going to spend more time talking about the mechanics 191 00:13:30,840 --> 00:13:35,140 than how you use that to solve this problem. 192 00:13:35,140 --> 00:13:37,560 But let's just look at it. 193 00:13:37,560 --> 00:13:40,380 So we have for a two-level level system, 194 00:13:40,380 --> 00:13:43,530 we have-- it's a matrix of 1, 1; a 1, 2; a 2, 1; 195 00:13:43,530 --> 00:13:45,480 and a 2, 2 element. 196 00:13:45,480 --> 00:13:52,520 And so we want the rho 1, 1 matrix on there. 197 00:13:52,520 --> 00:13:54,890 And so it's going to be a 1 here, 198 00:13:54,890 --> 00:13:57,290 then we're going to have a 1 here. 199 00:13:57,290 --> 00:14:12,190 And then we have the C1, 1. 200 00:14:16,900 --> 00:14:22,990 Plus C2, 2. 201 00:14:28,700 --> 00:14:43,780 And then we have C1 star 1 plus C2 star 2. 202 00:14:52,290 --> 00:14:55,490 And so have I got-- am I doing it right now? 203 00:14:55,490 --> 00:15:04,410 So the first thing we do is we look at this inside part. 204 00:15:04,410 --> 00:15:10,250 And we have C1, C1 star. 205 00:15:10,250 --> 00:15:14,710 And we have 1, 1. 206 00:15:14,710 --> 00:15:20,225 And we have C2, C2 star. 207 00:15:24,154 --> 00:15:31,330 Now I'm getting in trouble, because I want this to come out 208 00:15:31,330 --> 00:15:35,900 to be only C1, C1 star. 209 00:15:35,900 --> 00:15:38,191 So what am I doing wrong? 210 00:15:38,191 --> 00:15:40,860 AUDIENCE: [INAUDIBLE] both have C2 halves 211 00:15:40,860 --> 00:15:44,008 the left hand and the right hand half are both [INAUDIBLE].. 212 00:15:50,924 --> 00:15:54,500 See, on the left side, you have 1 on 1 [INAUDIBLE].. 213 00:15:56,842 --> 00:15:57,800 ROBERT FIELD: So here-- 214 00:16:12,820 --> 00:16:25,120 And that's C2 C1 star, but that's 0. 215 00:16:25,120 --> 00:16:29,730 And so anyway, I'm not going to say more. 216 00:16:29,730 --> 00:16:37,340 But this combination is 1, this combination is 0, 217 00:16:37,340 --> 00:16:41,790 this combination is 0, this combination is 1. 218 00:16:41,790 --> 00:16:44,120 And we end up getting-- 219 00:16:44,120 --> 00:16:46,300 and then we end up just getting this. 220 00:16:55,380 --> 00:17:02,756 Rho 1, 2 is equal to C1 C2 star. 221 00:17:02,756 --> 00:17:08,609 Rho 2, 1 is equal to C2 C1 star. 222 00:17:08,609 --> 00:17:15,119 And rho 2, 2 is equal to C2, C2 star. 223 00:17:15,119 --> 00:17:18,760 So we have the elements of this matrix. 224 00:17:18,760 --> 00:17:24,730 And they are expressed in terms of these mixing coefficients 225 00:17:24,730 --> 00:17:25,810 for the states 1 and 2. 226 00:17:32,850 --> 00:17:56,470 Now, if we look at this, we can see that rho 1, 1 plus rho 2, 2 227 00:17:56,470 --> 00:18:04,277 is equal to C1, C1 star plus C2, C2 star. 228 00:18:04,277 --> 00:18:05,860 And that's the normalization integral. 229 00:18:05,860 --> 00:18:06,360 That's 1. 230 00:18:08,950 --> 00:18:16,650 And we have 1, 1; 1, 2 is equal to 2, 1 star. 231 00:18:16,650 --> 00:18:25,650 And so each rho is formation. 232 00:18:28,870 --> 00:18:34,330 So the density matrix is normalized to 1. 233 00:18:34,330 --> 00:18:36,100 And it's Hermitian matrix. 234 00:18:36,100 --> 00:18:41,080 And we can use all sorts of tricks for Hermitian matrices. 235 00:18:41,080 --> 00:18:45,170 Now we're interested in the time dependence of rho. 236 00:18:45,170 --> 00:18:48,160 And so we're going to use this wonderful equation up 237 00:18:48,160 --> 00:18:50,800 here in order to get the time dependence of rho 238 00:18:50,800 --> 00:18:54,410 because rho like a, is Hermitian operator. 239 00:18:54,410 --> 00:18:56,330 And so we could do that. 240 00:18:56,330 --> 00:19:01,360 And so the time dependence of rho 241 00:19:01,360 --> 00:19:16,272 is going to be equal to the time dependence of t, t. 242 00:19:16,272 --> 00:19:18,680 Where we operating first here. 243 00:19:18,680 --> 00:19:22,470 And then t time dependent. 244 00:19:35,220 --> 00:19:38,810 And when we do this, what we end up getting-- 245 00:19:38,810 --> 00:19:44,260 well, so we have a time dependence of a wave function. 246 00:19:44,260 --> 00:19:46,770 So we use the time dependence shorter equation. 247 00:19:46,770 --> 00:19:48,300 And we insert that. 248 00:19:48,300 --> 00:19:51,940 And using the time dependence shorter equation we have things 249 00:19:51,940 --> 00:19:52,440 like-- 250 00:20:15,390 --> 00:20:18,790 so every time we take the wave function-- 251 00:20:18,790 --> 00:20:25,420 the derivative of a function, we get a Hamiltonian and so on. 252 00:20:25,420 --> 00:20:31,290 And so what we can express this time dependence of the density 253 00:20:31,290 --> 00:20:36,410 matrix by just using the time-- inserting the time dependent 254 00:20:36,410 --> 00:20:38,310 shorter equation repeatedly. 255 00:20:38,310 --> 00:20:43,860 This is why I say this is repackaging the Schrodinger 256 00:20:43,860 --> 00:20:46,740 picture, repackaging the wave function 257 00:20:46,740 --> 00:20:54,120 and writing everything in terms of these matrices. 258 00:21:02,010 --> 00:21:04,390 So that's the first-- 259 00:21:04,390 --> 00:21:05,620 that's what happens here. 260 00:21:09,241 --> 00:21:09,740 Sorry. 261 00:21:09,740 --> 00:21:12,020 That's what happens here. 262 00:21:12,020 --> 00:21:18,470 And then we write this one, and we get plus 1 263 00:21:18,470 --> 00:21:29,330 over minus i h-bar t, t H of t. 264 00:21:33,360 --> 00:21:42,210 And we recognize that that is just one over i h-bar times H 265 00:21:42,210 --> 00:21:42,710 rho. 266 00:21:46,170 --> 00:21:48,900 So the time dependence of the density matrix 267 00:21:48,900 --> 00:21:50,595 is given by this computator. 268 00:21:53,210 --> 00:21:56,150 And the computators are kind of neat because usually 269 00:21:56,150 --> 00:22:00,050 what happens is these two-level things have 270 00:22:00,050 --> 00:22:02,390 very different structures, and you 271 00:22:02,390 --> 00:22:06,470 get rid of something you don't want to deal with anymore. 272 00:22:06,470 --> 00:22:10,790 And so now we actually evaluate these things. 273 00:22:10,790 --> 00:22:19,430 And we do a lot of algebra. 274 00:22:19,430 --> 00:22:23,690 And we get these equations of motion for the elements 275 00:22:23,690 --> 00:22:25,740 of the density matrix. 276 00:22:25,740 --> 00:22:30,260 And so we find the time dependence 277 00:22:30,260 --> 00:22:34,070 of the diagonal element for state 1 278 00:22:34,070 --> 00:22:36,110 is opposite that for state 2. 279 00:22:36,110 --> 00:22:40,720 In other words, population from state 1 280 00:22:40,720 --> 00:22:42,757 is being transferred into state 2. 281 00:22:45,440 --> 00:22:50,900 And that is equal to minus i over h-bar times 282 00:22:50,900 --> 00:23:01,250 H 1, 2 rho 2, 1 minus h 2, 1 rho 1, 2. 283 00:23:05,930 --> 00:23:11,590 And we have rho 1, 2 time dependence 284 00:23:11,590 --> 00:23:19,380 is equal to rho 2, 1 time dependent star. 285 00:23:19,380 --> 00:23:26,530 And that comes out to be minus i h-bar minus i over h-bar, 286 00:23:26,530 --> 00:23:40,800 H 1, 1 minus H 2, 2 rho 1, 2 rho 2, 2 minus rho 1, 1 287 00:23:40,800 --> 00:23:42,030 times H 1, 2. 288 00:23:44,760 --> 00:23:46,680 This is very interesting, but now we 289 00:23:46,680 --> 00:23:48,930 have a couple differential equations 290 00:23:48,930 --> 00:23:50,940 and we can solve them. 291 00:23:50,940 --> 00:23:54,700 But we want to do a trick where we write the Hamiltonian 292 00:23:54,700 --> 00:23:57,885 as a sum of two terms. 293 00:24:00,660 --> 00:24:03,480 This is the time-- the independent part, 294 00:24:03,480 --> 00:24:05,460 and this is the time dependent part. 295 00:24:05,460 --> 00:24:07,240 This is the part that gives us trouble. 296 00:24:07,240 --> 00:24:09,930 This is the part that takes us into territory 297 00:24:09,930 --> 00:24:13,490 that I haven't talked about in time independent. 298 00:24:13,490 --> 00:24:15,720 But it's still-- it's perturbation theory. 299 00:24:15,720 --> 00:24:20,130 This is supposed to be something that 300 00:24:20,130 --> 00:24:23,880 is different from and usually smaller than H 0. 301 00:24:23,880 --> 00:24:26,130 And so we do this. 302 00:24:26,130 --> 00:24:39,960 So H 0, operating on any function gives En times n. 303 00:24:39,960 --> 00:24:42,030 And so we could call these E zeroes, 304 00:24:42,030 --> 00:24:44,154 but we don't need to do that anymore. 305 00:24:49,850 --> 00:24:52,565 And now we do a lot of algebra. 306 00:25:01,400 --> 00:25:04,700 We discover that the time dependence of the density 307 00:25:04,700 --> 00:25:13,160 matrix is given by minus i over h-bar times 308 00:25:13,160 --> 00:25:20,490 H 1 of t times the density matrix. 309 00:25:20,490 --> 00:25:24,190 So this is very much like what we did before, but now we have 310 00:25:24,190 --> 00:25:27,080 that the time dependence is entirely due to the time 311 00:25:27,080 --> 00:25:28,370 independent Hamiltonian. 312 00:25:33,250 --> 00:25:35,650 So everything associated with H 0 313 00:25:35,650 --> 00:25:38,050 is gone from this equation of motion. 314 00:25:43,030 --> 00:25:47,320 So now let's just be specific. 315 00:25:47,320 --> 00:25:49,210 So here is a two-level level system. 316 00:25:49,210 --> 00:25:51,330 This is state 1. 317 00:25:51,330 --> 00:25:53,750 This is state 2. 318 00:25:53,750 --> 00:25:59,280 This difference is delta E. 319 00:25:59,280 --> 00:26:03,950 And we're going to call that h-bar omega 0. 320 00:26:03,950 --> 00:26:08,690 So this is the frequency difference 321 00:26:08,690 --> 00:26:10,420 between levels 1 and 2. 322 00:26:13,300 --> 00:26:25,478 H 0 is equal to minus h h-omega over 2 h-bar omega over 2 0, 0. 323 00:26:25,478 --> 00:26:26,702 We like that, right? 324 00:26:26,702 --> 00:26:27,285 It's diagonal. 325 00:26:31,940 --> 00:26:36,000 h1 is where all the trouble comes. 326 00:26:36,000 --> 00:26:50,370 And we're going to call that h-bar times e x 1, 2 E0 cosine 327 00:26:50,370 --> 00:26:52,620 omega t. 328 00:26:52,620 --> 00:26:55,430 This is not an energy. 329 00:26:55,430 --> 00:26:58,440 This is an electric field. 330 00:26:58,440 --> 00:27:01,940 So this is the strength of the perturbation. 331 00:27:01,940 --> 00:27:08,000 And this is the dipole matrix element between levels 1 and 2. 332 00:27:08,000 --> 00:27:14,610 So we have a dipole moment times an electric field multiplied 333 00:27:14,610 --> 00:27:16,410 by h-bar. 334 00:27:16,410 --> 00:27:24,780 So this quantity here, has units of angular frequency. 335 00:27:24,780 --> 00:27:30,580 And we call it omega 1, which is the Rabi frequency. 336 00:27:30,580 --> 00:27:33,545 It gets a special name because Rabi was special. 337 00:27:36,290 --> 00:27:38,860 And so we're going to be-- 338 00:27:38,860 --> 00:27:42,530 and this is-- expresses the strength of the interaction. 339 00:27:42,530 --> 00:27:46,230 So we have a molecular antenna mu 1, 2. 340 00:27:46,230 --> 00:27:47,830 And we have the external field. 341 00:27:47,830 --> 00:27:50,180 And they're interacting with each other. 342 00:27:50,180 --> 00:27:53,200 And so this is the strength of the badness, 343 00:27:53,200 --> 00:27:54,220 except its goodness. 344 00:27:54,220 --> 00:27:55,750 Because we want to see transitions. 345 00:28:00,710 --> 00:28:04,340 So now we do a little bit of playing with notation 346 00:28:04,340 --> 00:28:07,815 because there's just a lot of stuff that's going on. 347 00:28:07,815 --> 00:28:10,130 and we have to understand it. 348 00:28:10,130 --> 00:28:13,760 So we're going to call the state-- 349 00:28:13,760 --> 00:28:18,140 we're going to separate the time dependent-- 350 00:28:18,140 --> 00:28:21,290 the time independent part of the wave functions from the time 351 00:28:21,290 --> 00:28:22,520 dependent. 352 00:28:22,520 --> 00:28:25,610 And so state 1-- 353 00:28:25,610 --> 00:28:28,970 this is the full time dependent wave function. 354 00:28:28,970 --> 00:28:41,561 And it's going to be minus i omega 0 t over 2. 355 00:28:41,561 --> 00:28:48,160 in Other words, we should have had zeros here-- 356 00:28:52,160 --> 00:28:58,780 times 1, right. 357 00:28:58,780 --> 00:29:00,710 So this is the time independent part, 358 00:29:00,710 --> 00:29:03,300 and this is the time dependent part. 359 00:29:03,300 --> 00:29:15,420 And 2 is e to the minus i omega 0 t over 2, 2 prime. 360 00:29:15,420 --> 00:29:19,680 Notice these two guys have the same sign. 361 00:29:19,680 --> 00:29:23,190 This bothers me a lot. 362 00:29:23,190 --> 00:29:28,930 But it's true, because we have opposite signs here, 363 00:29:28,930 --> 00:29:31,990 and we have a bra and a ket. 364 00:29:31,990 --> 00:29:34,480 And they end up having the same signs. 365 00:29:37,870 --> 00:29:44,700 So that means that h 1 looks like this, 0, 366 00:29:44,700 --> 00:30:00,810 0 omega 1 cosine omega t e to the minus i omega 0 t. 367 00:30:00,810 --> 00:30:11,770 And here we have omega 1 cosine omega t e to the pi omega 0 t. 368 00:30:11,770 --> 00:30:13,640 So this is a 2 by 2 matrix. 369 00:30:13,640 --> 00:30:16,390 Diagonal elements are 0. 370 00:30:16,390 --> 00:30:19,960 Off diagonal elements are this omega. 371 00:30:19,960 --> 00:30:23,170 The strength of the interaction times the frequency 372 00:30:23,170 --> 00:30:29,650 of the applied radiation times the oscillating factor. 373 00:30:39,960 --> 00:30:42,600 So now we go back and we calculate 374 00:30:42,600 --> 00:30:48,600 the equation of motion, bringing in this h 1 term. 375 00:30:48,600 --> 00:30:59,330 And so we have minus i over h-bar h 1 rho. 376 00:30:59,330 --> 00:31:01,850 And we get some complicated equations of motions. 377 00:31:05,430 --> 00:31:09,540 And I don't really want to write them out, 378 00:31:09,540 --> 00:31:16,370 because it takes a while, and they're in your notes. 379 00:31:16,370 --> 00:31:22,080 And I'm going to make the crucial approximation, 380 00:31:22,080 --> 00:31:26,350 the rotating wave approximation. 381 00:31:26,350 --> 00:31:31,860 Notice we have a cosine omega t. 382 00:31:31,860 --> 00:31:35,620 We can write that as e to the i omega t plus e 383 00:31:35,620 --> 00:31:37,560 to the minus i omega t. 384 00:31:37,560 --> 00:31:40,320 And so basically what we're doing 385 00:31:40,320 --> 00:31:43,574 is we're going to do a trick. 386 00:31:43,574 --> 00:31:44,990 We have the Hamiltonian, and we're 387 00:31:44,990 --> 00:31:48,040 going to go to a rotating coordinate system. 388 00:31:48,040 --> 00:31:52,950 And if we choose the rotational coordinate the rotation 389 00:31:52,950 --> 00:32:01,440 frequency right, we can almost exactly cancel omega 0 terms. 390 00:32:01,440 --> 00:32:05,430 And so we have two terms, one rotating like this, 391 00:32:05,430 --> 00:32:09,720 which is canceling or trying to cancel omega 0, 392 00:32:09,720 --> 00:32:15,870 and one rotating like this, which is adding to omega 0. 393 00:32:15,870 --> 00:32:19,100 And so what we end up getting is a slowly oscillating term, 394 00:32:19,100 --> 00:32:22,640 which we like, and a rapidly oscillating term, 395 00:32:22,640 --> 00:32:25,040 which we can throw away. 396 00:32:25,040 --> 00:32:27,620 That's the approximation. 397 00:32:27,620 --> 00:32:30,170 And this is commonly used. 398 00:32:30,170 --> 00:32:33,830 And I can write this in terms of transformations. 399 00:32:33,830 --> 00:32:37,550 And although we think about going to a rotating coordinate 400 00:32:37,550 --> 00:32:45,060 system, for each two-level system, 401 00:32:45,060 --> 00:32:47,580 we can rotate at a different frequency 402 00:32:47,580 --> 00:32:52,380 to cancel or make nearly canceling the off diagonal 403 00:32:52,380 --> 00:32:53,100 elements. 404 00:32:53,100 --> 00:32:57,390 So although the molecule doesn't rotate 405 00:32:57,390 --> 00:33:00,180 at different frequencies, our transformation 406 00:33:00,180 --> 00:33:03,900 attacks the coupling between states individually. 407 00:33:03,900 --> 00:33:06,870 And you can imply as many rotating wave core 408 00:33:06,870 --> 00:33:08,550 transformations as you want. 409 00:33:08,550 --> 00:33:09,997 But we have a two-level system. 410 00:33:09,997 --> 00:33:10,830 So we only have one. 411 00:33:17,790 --> 00:33:28,320 And so we do this. 412 00:33:28,320 --> 00:33:35,940 And we skip a lot of steps, because it's complicated 413 00:33:35,940 --> 00:33:39,600 and because we don't have a lot of time. 414 00:33:39,600 --> 00:33:47,100 We now have the time dependence of the 1, 1 element. 415 00:33:47,100 --> 00:33:48,930 And it's expressed as omega 1. 416 00:33:48,930 --> 00:33:50,760 I've skipped a lot of steps. 417 00:33:50,760 --> 00:33:53,010 But you can do those steps. 418 00:33:53,010 --> 00:33:55,500 The important thing is what we're going to see here. 419 00:33:55,500 --> 00:34:06,760 We have e to the i omega 0 minus omega t rho 1, 2. 420 00:34:06,760 --> 00:34:12,000 And we have a minus e to the minus i omega 421 00:34:12,000 --> 00:34:17,679 0 minus omega t times rho 2, 1. 422 00:34:20,659 --> 00:34:26,610 And we have 2, 2 dot is equal to minus rho 1, 1 dot. 423 00:34:26,610 --> 00:34:30,090 And we have rho 1, 2 dot-- 424 00:34:30,090 --> 00:34:32,380 this is the important guy-- 425 00:34:32,380 --> 00:34:39,520 is equal to i omega 1 over 2 e to the minus i omega 426 00:34:39,520 --> 00:34:46,872 0 minus omega t rho 1, 1 minus rho 2, 2. 427 00:34:46,872 --> 00:34:49,330 So we have a whole bunch of coupled differential equations, 428 00:34:49,330 --> 00:34:53,230 but each of them have these factors here where 429 00:34:53,230 --> 00:34:57,220 you have omega 0 minus omega. 430 00:34:57,220 --> 00:34:59,830 I've thrown away the omega 0 plus omega terms. 431 00:35:02,610 --> 00:35:04,940 And now it really starts to look good, 432 00:35:04,940 --> 00:35:08,300 because we can make these-- 433 00:35:08,300 --> 00:35:14,960 so when we make omega equal to omega 0, well, this is just 1. 434 00:35:14,960 --> 00:35:16,290 Everything is simple. 435 00:35:16,290 --> 00:35:17,990 We're on resonance. 436 00:35:17,990 --> 00:35:25,650 And so what we do is we create another symbol, delta omega, 437 00:35:25,650 --> 00:35:28,670 which is omega 0 minus omega. 438 00:35:28,670 --> 00:35:30,900 So this is the oscillating frequency applied. 439 00:35:30,900 --> 00:35:33,772 This is the intrinsic level spacing in the molecule. 440 00:35:37,840 --> 00:35:42,640 And so we can now write the solution 441 00:35:42,640 --> 00:35:47,447 to this differential equation for each 442 00:35:47,447 --> 00:35:49,030 of the elements of the density matrix. 443 00:35:55,460 --> 00:35:57,680 And we're going to actually define another symbol. 444 00:36:00,350 --> 00:36:04,750 We going to have the symbol omega sub e. 445 00:36:04,750 --> 00:36:07,360 This is not the vibrational frequency. 446 00:36:07,360 --> 00:36:10,690 This is just a symbol that is used a lot in literature, 447 00:36:10,690 --> 00:36:17,710 and that it comes out to be delta omega squared 448 00:36:17,710 --> 00:36:20,830 plus omega 1 squared. 449 00:36:24,120 --> 00:36:27,950 So in solving the density matrix equation, 450 00:36:27,950 --> 00:36:34,490 it turns out we care about this extra frequency. 451 00:36:34,490 --> 00:36:37,850 If delta omega is 0, well, then there's nothing surprising. 452 00:36:37,850 --> 00:36:39,680 Well, maybe e is just omega 1. 453 00:36:42,320 --> 00:36:48,500 But this allows for there to be an effect of the detuning. 454 00:36:48,500 --> 00:36:52,790 So basically what you're doing is when you go to the rotating 455 00:36:52,790 --> 00:37:02,420 coordinate system, you have an intrinsic frequency separation. 456 00:37:02,420 --> 00:37:07,320 And so in the rotating coordinate system, 457 00:37:07,320 --> 00:37:09,320 you have two levels that are different. 458 00:37:09,320 --> 00:37:12,251 And there's a stark effect between them. 459 00:37:12,251 --> 00:37:14,150 And you diagonalize this stark effect 460 00:37:14,150 --> 00:37:17,600 using second order perturbation theory or just 461 00:37:17,600 --> 00:37:20,800 the diagonizing the matrix. 462 00:37:20,800 --> 00:37:24,200 And so that gives rise to this extra term here, 463 00:37:24,200 --> 00:37:32,330 because you have the oscillation frequency and the Rabi 464 00:37:32,330 --> 00:37:33,500 frequency. 465 00:37:33,500 --> 00:37:36,390 And anyway, when you do the transformation, 466 00:37:36,390 --> 00:37:38,100 you get these terms. 467 00:37:38,100 --> 00:37:41,640 And so here is now the solution in the rotating wave 468 00:37:41,640 --> 00:37:43,262 approximation. 469 00:37:43,262 --> 00:37:51,380 Rho 1, 1 is equal to 1 minus omega 1 squared over omega 470 00:37:51,380 --> 00:37:58,760 e squared sine squared omega 0 t over 2. 471 00:38:02,240 --> 00:38:08,520 We have rho 2, 2 is equal to just 472 00:38:08,520 --> 00:38:13,140 omega 1 squared over the e squared sine 473 00:38:13,140 --> 00:38:20,050 squared omega e t over 2. 474 00:38:20,050 --> 00:38:22,320 We have omega 1, 2-- 475 00:38:22,320 --> 00:38:26,030 rho 1, 2, which is equal to something more complicated 476 00:38:26,030 --> 00:38:27,140 looking. 477 00:38:27,140 --> 00:38:37,260 Omega 1 over omega e squared times i omega 0-- 478 00:38:37,260 --> 00:38:50,530 omega e, sorry, or 2 sine omega e t minus delta omega sine 479 00:38:50,530 --> 00:38:56,950 squared omega e t over 2 times now 480 00:38:56,950 --> 00:39:03,790 e to the minus i delta omega t. 481 00:39:03,790 --> 00:39:05,260 It looks complicated. 482 00:39:05,260 --> 00:39:08,890 And we get a similar term for who 2, 1. 483 00:39:08,890 --> 00:39:13,740 It's just equal to rho 1, 2 complex conjugate. 484 00:39:13,740 --> 00:39:20,310 And so now what we see is these populations 485 00:39:20,310 --> 00:39:24,170 are oscillating not at-- 486 00:39:27,180 --> 00:39:29,130 It's a e, not a 0. 487 00:39:32,680 --> 00:39:35,140 They're oscillating at a slightly shifted frequency. 488 00:39:37,840 --> 00:39:40,710 But they're oscillating sinusoidally. 489 00:39:40,710 --> 00:39:42,930 And we have an amplitude term, which 490 00:39:42,930 --> 00:39:47,250 is omega 1 over omega e quantity squared. 491 00:39:47,250 --> 00:39:50,850 Omega e is a little bigger than omega 1. 492 00:39:50,850 --> 00:39:54,450 So this is less than 1. 493 00:39:54,450 --> 00:39:59,600 So it's just like the situation in the absence 494 00:39:59,600 --> 00:40:03,290 of this oscillating field, that you just 495 00:40:03,290 --> 00:40:07,820 get a slightly, slightly shifted oscillation 496 00:40:07,820 --> 00:40:11,510 frequency, and a slightly reduced co-factor. 497 00:40:14,690 --> 00:40:18,320 The coherence terms-- so these are populations, populations 498 00:40:18,320 --> 00:40:20,460 going back and forth between 1 and 2 499 00:40:20,460 --> 00:40:22,830 at a slightly shifted frequency. 500 00:40:22,830 --> 00:40:26,570 And then we have this, which looks horrible. 501 00:40:29,510 --> 00:40:33,585 And now, for some more insights. 502 00:40:37,870 --> 00:40:45,970 If we make omega 1 much larger than delta omega-- 503 00:40:45,970 --> 00:40:48,340 in other words, the Rabi frequency much larger 504 00:40:48,340 --> 00:40:52,330 than the tuning, it might as well not be detuned. 505 00:40:52,330 --> 00:40:55,690 We get back the simple picture. 506 00:40:55,690 --> 00:41:03,670 We get rho 1 is equal to cosine squared omega 1 of t 507 00:41:03,670 --> 00:41:05,430 over 2, et cetera. 508 00:41:12,350 --> 00:41:18,977 So we have what we call free precession. 509 00:41:25,950 --> 00:41:29,270 Each of the elements of the density-- 510 00:41:29,270 --> 00:41:31,370 the density matrix is telling you 511 00:41:31,370 --> 00:41:35,600 that the system is going back and forth sinusoidally 512 00:41:35,600 --> 00:41:39,030 or co-sinusoidally cosine squared. 513 00:41:39,030 --> 00:41:45,080 And what happens to level 1 is the opposite 514 00:41:45,080 --> 00:41:46,790 of what happens at level 2. 515 00:41:46,790 --> 00:41:50,540 And everything is simple and the system just oscillates. 516 00:42:01,860 --> 00:42:17,130 Suppose we apply radiation or delta t a short time. 517 00:42:17,130 --> 00:42:20,970 And so what we're interested in-- here is t equals 0. 518 00:42:20,970 --> 00:42:22,940 This is time. 519 00:42:22,940 --> 00:42:25,080 And this is t equals 0. 520 00:42:25,080 --> 00:42:27,690 And before t equals 0, we do something. 521 00:42:27,690 --> 00:42:28,830 We apply the radiation. 522 00:42:31,960 --> 00:42:37,340 And we apply the radiation for a time, which gives rise 523 00:42:37,340 --> 00:42:39,015 to a certain flipping. 524 00:42:48,960 --> 00:42:51,600 And so what we choose. 525 00:42:51,600 --> 00:42:59,560 We have delta t is equal to theta over omega 1, 526 00:42:59,560 --> 00:43:06,860 or theta is equal to delta t omega 1, the Rabi frequency. 527 00:43:06,860 --> 00:43:11,280 And so if we choose a flip angle, which we call 528 00:43:11,280 --> 00:43:17,200 say a pi pulse, theta is a pi. 529 00:43:17,200 --> 00:43:22,360 And what ends up happening is that we transfer population 530 00:43:22,360 --> 00:43:25,390 entirely from level 1 to level 2. 531 00:43:36,250 --> 00:43:42,780 When we do that, we get no off diagonal elements 532 00:43:42,780 --> 00:43:43,970 of the density matrix. 533 00:43:43,970 --> 00:43:45,480 They are zero. 534 00:43:45,480 --> 00:43:55,330 So if at t equals 0, we have everything in level 1, 535 00:43:55,330 --> 00:44:02,560 and we have applied this 0 pulse or a pi pulse, 536 00:44:02,560 --> 00:44:03,760 we have no coherence. 537 00:44:10,130 --> 00:44:17,910 If we have a pi over 2 pulse, well then, 538 00:44:17,910 --> 00:44:22,890 we've equalized the two-level populations, 539 00:44:22,890 --> 00:44:24,870 and we created a maximum coherence. 540 00:44:27,990 --> 00:44:30,600 And this guy radiates. 541 00:44:30,600 --> 00:44:33,035 So now we have an oscillating dipole. 542 00:44:33,035 --> 00:44:36,980 And it's broadcasting radiation. 543 00:44:36,980 --> 00:44:40,280 And so all of the two-level systems, 544 00:44:40,280 --> 00:44:44,820 if you use a flip angle of pi over 2, 545 00:44:44,820 --> 00:44:49,010 you get a maximum polarization, they're 546 00:44:49,010 --> 00:44:52,680 radiating to my detector, which is up there. 547 00:44:52,680 --> 00:44:53,850 And I'm happy. 548 00:44:53,850 --> 00:44:58,480 I detect their resonance frequency. 549 00:44:58,480 --> 00:45:01,950 And so the experiments work. 550 00:45:01,950 --> 00:45:08,580 So we're pretty much done. 551 00:45:08,580 --> 00:45:11,900 So I mean, what we are doing is we're 552 00:45:11,900 --> 00:45:18,020 creating a time dependent dipole. 553 00:45:18,020 --> 00:45:21,380 And that dipole radiates something which we call-- 554 00:45:27,280 --> 00:45:31,810 if we have a sample like this, that sample-- 555 00:45:31,810 --> 00:45:34,000 all of the molecules in the sample 556 00:45:34,000 --> 00:45:40,150 are contributing to the radiation of this dipole. 557 00:45:40,150 --> 00:45:44,410 But they all have slightly different frequencies, 558 00:45:44,410 --> 00:45:49,120 because the field that polarized them wasn't uniform. 559 00:45:49,120 --> 00:45:52,230 In a perfect experiment it would be. 560 00:45:52,230 --> 00:45:54,972 And so they have different frequencies 561 00:45:54,972 --> 00:45:56,055 and they get out of phase. 562 00:45:59,530 --> 00:46:04,300 Or conservation of energy as the two-level system 563 00:46:04,300 --> 00:46:06,550 radiates from the situation where 564 00:46:06,550 --> 00:46:10,330 you have equal populations to everybody in the lowest state, 565 00:46:10,330 --> 00:46:11,290 there is a decay. 566 00:46:11,290 --> 00:46:18,910 So there's decays that causes the signal, which 567 00:46:18,910 --> 00:46:25,070 we call free induction decay, to dephase or decay. 568 00:46:25,070 --> 00:46:27,917 But the important thing is, you observe the signal 569 00:46:27,917 --> 00:46:29,500 and it tells you what you want to know 570 00:46:29,500 --> 00:46:32,770 about the level system, the two-level system, or the end 571 00:46:32,770 --> 00:46:35,000 level system. 572 00:46:35,000 --> 00:46:39,760 And it's a very powerful way of understanding the interaction 573 00:46:39,760 --> 00:46:42,100 of radiation with matter, because it 574 00:46:42,100 --> 00:46:45,670 focuses on near resonance. 575 00:46:45,670 --> 00:46:49,170 And near resonance for one two-level system 576 00:46:49,170 --> 00:46:51,100 is not near resonance. 577 00:46:51,100 --> 00:46:54,430 For another-- and so you're picking out one, 578 00:46:54,430 --> 00:46:56,650 and you get really good signals. 579 00:46:56,650 --> 00:46:58,540 And you can actually do-- 580 00:46:58,540 --> 00:47:00,460 by chirping the pulse-- 581 00:47:00,460 --> 00:47:03,909 you can have one two-level system, and a little bit later, 582 00:47:03,909 --> 00:47:05,200 another two-level-level system. 583 00:47:05,200 --> 00:47:06,130 They all radiate. 584 00:47:06,130 --> 00:47:07,660 They all get polarized. 585 00:47:07,660 --> 00:47:10,270 They all radiate at their own frequency. 586 00:47:10,270 --> 00:47:13,540 And you can detect the signal in the time domain 587 00:47:13,540 --> 00:47:16,120 and get everything you want in a simple experiment. 588 00:47:16,120 --> 00:47:19,630 This experiment has enabled us to do spectroscopy 589 00:47:19,630 --> 00:47:22,860 a million times faster than was possible before. 590 00:47:22,860 --> 00:47:25,690 A million is a big number. 591 00:47:25,690 --> 00:47:29,160 And so I think it's important. 592 00:47:29,160 --> 00:47:34,310 And I think that this sort of theory is germane, 593 00:47:34,310 --> 00:47:38,480 not just for high resolution frequency domain 594 00:47:38,480 --> 00:47:41,660 experiments, in fact, it's basically 595 00:47:41,660 --> 00:47:42,830 a time domain experiment. 596 00:47:42,830 --> 00:47:45,320 You're detecting something in the time domain, 597 00:47:45,320 --> 00:47:49,440 and Fourier transferring back to the frequency domain. 598 00:47:49,440 --> 00:47:51,050 So there are ultra fast experiments 599 00:47:51,050 --> 00:47:54,440 where you create polarizations and they-- 600 00:47:54,440 --> 00:47:57,170 it is what is modern experimental physical 601 00:47:57,170 --> 00:47:59,430 chemistry. 602 00:47:59,430 --> 00:48:02,880 And the notes that I will produce 603 00:48:02,880 --> 00:48:06,300 will be far clearer than these lectures, this lecture. 604 00:48:06,300 --> 00:48:09,190 But it really is a gateway. 605 00:48:09,190 --> 00:48:12,530 And I hope that some of you will walk through that gateway. 606 00:48:12,530 --> 00:48:16,170 And it's been a pleasure for me lecturing to you 607 00:48:16,170 --> 00:48:18,840 for the last time in 5.61. 608 00:48:18,840 --> 00:48:22,020 I really enjoyed doing this. 609 00:48:22,020 --> 00:48:24,725 Thanks. 610 00:48:24,725 --> 00:48:27,620 [APPLAUSE] 611 00:48:27,620 --> 00:48:28,916 Thank them. 612 00:48:28,916 --> 00:48:32,740 [APPLAUSE] 613 00:48:32,740 --> 00:48:34,340 Well, I got to take the hydrogen atom. 614 00:48:34,340 --> 00:48:38,240 [LAUGHTER] 615 00:48:38,240 --> 00:48:39,756 Thank you.