1 00:00:00,135 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseware 4 00:00:06,330 --> 00:00:10,690 continue to offer high-quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,250 from hundreds of MIT courses, visit MIT OpenCourseware 7 00:00:17,250 --> 00:00:18,210 at ocw.mit.edu. 8 00:00:21,880 --> 00:00:24,550 ROBERT FIELD: This lecture is not 9 00:00:24,550 --> 00:00:28,060 relevant to this exam or any exam. 10 00:00:28,060 --> 00:00:33,670 It's time-dependent quantum mechanics, which you probably 11 00:00:33,670 --> 00:00:37,060 want to know about, but it's a lot 12 00:00:37,060 --> 00:00:39,700 to digest at the level of this course. 13 00:00:39,700 --> 00:00:47,530 So I'm going to introduce a lot of the tricks and terminology, 14 00:00:47,530 --> 00:00:50,410 and I hope that some of you will care about that 15 00:00:50,410 --> 00:00:52,870 and will go on to use this. 16 00:00:52,870 --> 00:00:57,020 But mostly, this is a first exposure, 17 00:00:57,020 --> 00:00:59,520 and there's a lot of derivation. 18 00:00:59,520 --> 00:01:02,510 And it's hard to see the forest for the trees. 19 00:01:02,510 --> 00:01:05,379 OK, so these are the important things 20 00:01:05,379 --> 00:01:07,390 that I'm going to cover in the lecture. 21 00:01:07,390 --> 00:01:10,540 First, the dipole approximation-- 22 00:01:10,540 --> 00:01:14,950 how can we simplify the interaction between molecules 23 00:01:14,950 --> 00:01:17,090 and electromagnetic radiation? 24 00:01:17,090 --> 00:01:20,050 This is the main simplification, and I'll 25 00:01:20,050 --> 00:01:21,970 explain where it comes from. 26 00:01:21,970 --> 00:01:24,490 Then we have transitions that occur. 27 00:01:24,490 --> 00:01:28,000 And they're caused by a time-dependent perturbation 28 00:01:28,000 --> 00:01:36,190 where the zero-order Hamiltonian is time-independent, 29 00:01:36,190 --> 00:01:39,169 but the perturbation term is time-dependent. 30 00:01:39,169 --> 00:01:40,210 And what does that cause? 31 00:01:40,210 --> 00:01:43,120 It causes transitions. 32 00:01:43,120 --> 00:01:45,700 We're going to express the problem in terms 33 00:01:45,700 --> 00:01:49,510 of the eigenstates of the time-independent Hamiltonian, 34 00:01:49,510 --> 00:01:52,750 the zero-order Hamiltonian, and we know that these always 35 00:01:52,750 --> 00:01:55,600 have this time-dependent factor if we're 36 00:01:55,600 --> 00:01:59,100 doing time-dependent quantum mechanics. 37 00:01:59,100 --> 00:02:00,810 The two crucial approximations are 38 00:02:00,810 --> 00:02:06,030 going to be the electromagnetic field is weak and continuous. 39 00:02:06,030 --> 00:02:09,389 Now many experiments involve short pulses and very intense 40 00:02:09,389 --> 00:02:13,260 pulses, and the time-dependent quantum mechanics 41 00:02:13,260 --> 00:02:16,170 for those problems is completely different, 42 00:02:16,170 --> 00:02:18,480 but you need to understand this in order 43 00:02:18,480 --> 00:02:22,440 to understand what's different about it. 44 00:02:22,440 --> 00:02:24,270 We also assume that we're starting 45 00:02:24,270 --> 00:02:26,250 the system in a single eigenstate, 46 00:02:26,250 --> 00:02:28,680 and that's pretty normal. 47 00:02:28,680 --> 00:02:33,030 But often, you're starting the system in many eigenstates 48 00:02:33,030 --> 00:02:34,080 that are uncorrelated. 49 00:02:34,080 --> 00:02:35,350 We don't talk about that. 50 00:02:35,350 --> 00:02:38,100 That's something that has to do with the density matrix, which 51 00:02:38,100 --> 00:02:41,560 is beyond the level of this course. 52 00:02:41,560 --> 00:02:43,350 And one of the things that happens 53 00:02:43,350 --> 00:02:47,700 is we get this thing called linear response. 54 00:02:47,700 --> 00:02:51,540 Now I went for years hearing the reverence 55 00:02:51,540 --> 00:02:55,150 that people apply to linear response, 56 00:02:55,150 --> 00:02:58,020 but I hadn't a clue what it was. 57 00:02:58,020 --> 00:03:00,270 So you can start out knowing something 58 00:03:00,270 --> 00:03:02,220 about linear response. 59 00:03:02,220 --> 00:03:05,770 Now this all leads up to Fermi's golden rule, 60 00:03:05,770 --> 00:03:10,050 which explains the rate at which transitions 61 00:03:10,050 --> 00:03:13,690 occur between some initial state and some final state. 62 00:03:13,690 --> 00:03:15,420 And there is a lot more complexity 63 00:03:15,420 --> 00:03:18,130 in Fermi's golden rule than what I'm going to present, 64 00:03:18,130 --> 00:03:23,670 but this is the first step in understanding it. 65 00:03:23,670 --> 00:03:26,920 Then I'm going to talk about where 66 00:03:26,920 --> 00:03:30,460 do pure rotation transitions come from and vibrational 67 00:03:30,460 --> 00:03:31,190 transitions. 68 00:03:31,190 --> 00:03:33,460 Then at the end, I'll show a movie 69 00:03:33,460 --> 00:03:37,420 which gives you a sense of what goes on 70 00:03:37,420 --> 00:03:42,220 in making a transition be strong and sharp. 71 00:03:45,020 --> 00:03:49,520 OK, I'm a spectroscopist, and I use spectroscopy 72 00:03:49,520 --> 00:03:53,840 to learn all sorts of secrets that molecules keep. 73 00:03:53,840 --> 00:03:56,690 And in order to do that, I need to record 74 00:03:56,690 --> 00:04:01,190 a spectrum, which basically is you have some radiation source. 75 00:04:01,190 --> 00:04:05,450 And you tune its frequency, and things happen. 76 00:04:05,450 --> 00:04:08,610 And why do the things happen? 77 00:04:08,610 --> 00:04:12,860 How do we understand the interaction 78 00:04:12,860 --> 00:04:16,990 of electromagnetic radiation and a molecule? 79 00:04:16,990 --> 00:04:19,010 And there's really two ways to understand it. 80 00:04:26,150 --> 00:04:42,020 We have one-way molecules as targets, photons as bullets, 81 00:04:42,020 --> 00:04:44,580 and it's a simple geometric picture. 82 00:04:44,580 --> 00:04:47,600 And the size of the target is related to the transition 83 00:04:47,600 --> 00:04:50,820 moments, and it works. 84 00:04:50,820 --> 00:04:52,170 It's very, very simple. 85 00:04:52,170 --> 00:04:54,410 There's no time-dependent quantum mechanics. 86 00:04:54,410 --> 00:04:56,270 It's probabilistic. 87 00:04:56,270 --> 00:05:01,200 And for the first 45 years of my career, 88 00:05:01,200 --> 00:05:07,130 this is the way I handled an understanding of transitions 89 00:05:07,130 --> 00:05:09,597 caused by electromagnetic radiation. 90 00:05:09,597 --> 00:05:10,180 This is wrong. 91 00:05:12,910 --> 00:05:15,170 It has a wide applicability. 92 00:05:15,170 --> 00:05:18,170 But if you try to take it too seriously, 93 00:05:18,170 --> 00:05:20,320 you will miss a lot of good stuff. 94 00:05:24,030 --> 00:05:28,970 The other way is to use the time-dependent 95 00:05:28,970 --> 00:05:34,530 in your equation, and it looks complicated 96 00:05:34,530 --> 00:05:37,196 because we're going to be combining 97 00:05:37,196 --> 00:05:38,820 the time-dependent Schrodinger equation 98 00:05:38,820 --> 00:05:41,217 and the time-independent Schrodinger equation. 99 00:05:41,217 --> 00:05:43,800 We're going to be thinking about the electromagnetic radiation 100 00:05:43,800 --> 00:05:47,730 as waves rather than photons, and that 101 00:05:47,730 --> 00:05:50,250 means there is constructive and destructive interference. 102 00:05:50,250 --> 00:05:52,680 There's phase information, which is not 103 00:05:52,680 --> 00:05:55,140 present in the molecules-as-targets, 104 00:05:55,140 --> 00:05:57,610 photons-as-bullets picture. 105 00:05:57,610 --> 00:05:59,960 Now I don't want you to say, well, I'm 106 00:05:59,960 --> 00:06:01,710 never going to think this way because it's 107 00:06:01,710 --> 00:06:04,430 so easy to think about trends. 108 00:06:04,430 --> 00:06:07,080 And, you know, the Beer-Lambert law, all these things that you 109 00:06:07,080 --> 00:06:12,930 use to describe the probability of an absorption or emission 110 00:06:12,930 --> 00:06:18,190 transition, this sort of thing is really useful. 111 00:06:18,190 --> 00:06:31,540 OK, so this is the right way, and the crucial step 112 00:06:31,540 --> 00:06:33,430 is the dipole approximation. 113 00:06:36,370 --> 00:06:47,140 So we have electromagnetic radiation 114 00:06:47,140 --> 00:06:51,920 being a combination of electric field and magnetic field, 115 00:06:51,920 --> 00:06:57,250 and we can describe the electric field in terms of-- 116 00:07:11,571 --> 00:07:12,070 OK. 117 00:07:14,710 --> 00:07:21,230 So this is a vector, and it's a function of a vector in time. 118 00:07:21,230 --> 00:07:24,760 And there is some magnitude, which is a vector. 119 00:07:24,760 --> 00:07:27,430 And its cosine of this thing, this 120 00:07:27,430 --> 00:07:34,900 is the wave vector, which is 2 pi over the wavelength, 121 00:07:34,900 --> 00:07:37,960 but it also has a direction. 122 00:07:37,960 --> 00:07:40,030 And it points in the direction that the radiation 123 00:07:40,030 --> 00:07:42,360 is propagating. 124 00:07:42,360 --> 00:07:46,390 And this is the position coordinate, 125 00:07:46,390 --> 00:07:47,720 and this is the frequency. 126 00:07:47,720 --> 00:07:54,150 So there's a similar expression for the magnetic part-- 127 00:07:59,130 --> 00:08:00,126 same thing. 128 00:08:07,120 --> 00:08:11,090 I should leave this exposed. 129 00:08:11,090 --> 00:08:13,910 So we know several things. 130 00:08:13,910 --> 00:08:17,060 We know for electromagnetic radiation 131 00:08:17,060 --> 00:08:21,500 that the electric field is always 132 00:08:21,500 --> 00:08:25,010 perpendicular to the magnetic field. 133 00:08:25,010 --> 00:08:33,580 We know a relationship between the constant term, 134 00:08:33,580 --> 00:08:38,090 and we have this k vector, which points in the propagation 135 00:08:38,090 --> 00:08:40,659 direction. 136 00:08:40,659 --> 00:08:41,799 Now the question is-- 137 00:08:50,710 --> 00:08:54,250 because we have-- we have a molecule 138 00:08:54,250 --> 00:08:56,530 and we have the electromagnetic radiation. 139 00:08:59,910 --> 00:09:04,700 And so the question is, what's a typical size for a molecule 140 00:09:04,700 --> 00:09:05,510 in the gas field? 141 00:09:05,510 --> 00:09:06,820 Well, anywhere? 142 00:09:06,820 --> 00:09:08,040 Pick a number. 143 00:09:08,040 --> 00:09:11,250 How big is a molecule? 144 00:09:11,250 --> 00:09:13,664 STUDENT: A couple angstroms? 145 00:09:13,664 --> 00:09:15,330 ROBERT FIELD: I like a couple angstroms. 146 00:09:15,330 --> 00:09:17,010 That's a diatomic molecule. 147 00:09:17,010 --> 00:09:19,080 They're going to be people who like proteins, 148 00:09:19,080 --> 00:09:22,560 and they're going to talk about 10 or 100 nanometers. 149 00:09:22,560 --> 00:09:28,140 But typically, you can say 1 nanometer or 2 angstroms 150 00:09:28,140 --> 00:09:30,960 or something like that. 151 00:09:30,960 --> 00:09:33,940 Now we're going to shine light at a molecule. 152 00:09:33,940 --> 00:09:37,550 What's the typical wavelength of light that we 153 00:09:37,550 --> 00:09:41,740 use to record a spectrum? 154 00:09:41,740 --> 00:09:45,020 Visible wavelength. 155 00:09:45,020 --> 00:09:46,950 What is that? 156 00:09:46,950 --> 00:09:48,950 STUDENT: 400 to 700 nanometers? 157 00:09:48,950 --> 00:09:52,340 ROBERT FIELD: Yeah, so the wavelength of light 158 00:09:52,340 --> 00:09:55,850 is on the order of, say, 500 nanometers. 159 00:09:55,850 --> 00:10:04,130 And if it's in the infrared, it might be 10,000 nanometers. 160 00:10:04,130 --> 00:10:07,100 If it were in the visible, it might be 100-- 161 00:10:07,100 --> 00:10:10,550 in the ultraviolet, it might be as short as 100 nanometers. 162 00:10:10,550 --> 00:10:15,170 But the point is that this wavelength is much, much larger 163 00:10:15,170 --> 00:10:18,470 than the size of a molecule, so this picture here 164 00:10:18,470 --> 00:10:19,400 is complete garbage. 165 00:10:22,290 --> 00:10:27,550 The picture for the ratio-- 166 00:10:27,550 --> 00:10:30,330 so even this is garbage. 167 00:10:30,330 --> 00:10:32,250 The electric field or the magnetic field 168 00:10:32,250 --> 00:10:36,090 that the molecule sees is constant over the length 169 00:10:36,090 --> 00:10:43,750 of the molecule to a very good approximation. 170 00:10:43,750 --> 00:10:52,580 So now we have this expression for the field, 171 00:10:52,580 --> 00:10:57,640 and this is a number which is a very, very small number 172 00:10:57,640 --> 00:11:01,480 times a still pretty small number. 173 00:11:01,480 --> 00:11:05,500 This k dot r is very small. 174 00:11:05,500 --> 00:11:11,750 It says we can expand this in a power series 175 00:11:11,750 --> 00:11:16,520 and throw away everything except the omega t. 176 00:11:16,520 --> 00:11:19,840 That's the dipole approximation. 177 00:11:19,840 --> 00:11:28,000 So all of a sudden, we have as our electric field 178 00:11:28,000 --> 00:11:32,290 just E0 cosine omega t. 179 00:11:34,970 --> 00:11:37,430 That's fantastic. 180 00:11:37,430 --> 00:11:41,630 So we've gotten rid of the spatial degree of freedom, 181 00:11:41,630 --> 00:11:44,430 and that enables us to do all sorts of things 182 00:11:44,430 --> 00:11:47,700 that would have required a lot more justification. 183 00:11:47,700 --> 00:11:54,320 Now sometimes we need to keep higher order terms 184 00:11:54,320 --> 00:11:55,070 in this expansion. 185 00:11:55,070 --> 00:11:57,920 We've kept none of them, just the zero order term. 186 00:11:57,920 --> 00:12:03,350 And so if we do, that's called quadrupole or octopole or 187 00:12:03,350 --> 00:12:06,380 hexadecapole, and there are transitions 188 00:12:06,380 --> 00:12:10,520 that are not dipole allowed but are quadrupole allowed. 189 00:12:10,520 --> 00:12:14,120 And they're incredibly weak because k 190 00:12:14,120 --> 00:12:16,100 dot r is really, really small. 191 00:12:19,520 --> 00:12:25,140 Now the intensity of quadrupole-allowed transitions 192 00:12:25,140 --> 00:12:30,230 is on the order of a million times smaller than dipole. 193 00:12:30,230 --> 00:12:31,570 So why go there? 194 00:12:31,570 --> 00:12:35,890 Well, sometimes the dipole transitions are forbidden. 195 00:12:35,890 --> 00:12:38,550 And so if you're going to get the molecule to talk to you, 196 00:12:38,550 --> 00:12:41,120 you're going to have to somehow make use 197 00:12:41,120 --> 00:12:43,010 of the quadrupole transitions. 198 00:12:43,010 --> 00:12:45,770 But it's a completely different kind of experiment 199 00:12:45,770 --> 00:12:48,290 because you have to have an incredibly long path 200 00:12:48,290 --> 00:12:51,720 length and a relatively high number density. 201 00:12:51,720 --> 00:12:54,170 And so you don't want to go there, 202 00:12:54,170 --> 00:12:56,270 and that's something that's beside-- 203 00:12:56,270 --> 00:12:58,800 aside from what we care about. 204 00:12:58,800 --> 00:13:05,420 So now many of you are going to be doing experiments involving 205 00:13:05,420 --> 00:13:13,280 light, and that will involve the electric field. 206 00:13:13,280 --> 00:13:16,830 Some of you will be doing magnetic resonance, 207 00:13:16,830 --> 00:13:19,680 and they will be thinking entirely 208 00:13:19,680 --> 00:13:21,450 about the magnetic field. 209 00:13:24,690 --> 00:13:26,560 The theory is the same. 210 00:13:26,560 --> 00:13:32,330 It's just the main actor is a little bit different. 211 00:13:32,330 --> 00:13:36,740 Now if we're dealing with an electric field, 212 00:13:36,740 --> 00:13:40,670 we are interested in the symmetry 213 00:13:40,670 --> 00:13:45,410 of this operator, which is the electric field dotted 214 00:13:45,410 --> 00:13:51,390 into the molecular dipole moment, 215 00:13:51,390 --> 00:13:56,610 and that operator has odd parity. 216 00:13:56,610 --> 00:14:00,000 And so now I'm not going to tell you what parity is. 217 00:14:00,000 --> 00:14:03,150 But because this has odd parity, there 218 00:14:03,150 --> 00:14:06,840 are only transitions between states of opposite parity, 219 00:14:06,840 --> 00:14:12,180 whereas this, the magnetic operator, has even parity. 220 00:14:12,180 --> 00:14:15,000 And so they only have transitions between states 221 00:14:15,000 --> 00:14:16,410 of the same parity. 222 00:14:16,410 --> 00:14:18,710 Now you want to be curious about what parity is, 223 00:14:18,710 --> 00:14:20,250 and I'm not going to tell you. 224 00:14:20,250 --> 00:14:26,160 OK, so the problem is tremendously 225 00:14:26,160 --> 00:14:29,280 simplified by the fact that now we just 226 00:14:29,280 --> 00:14:33,900 have a time-dependent field, which 227 00:14:33,900 --> 00:14:36,490 is constant over the molecule. 228 00:14:36,490 --> 00:14:40,110 So the molecule is seeing an oscillatory field, 229 00:14:40,110 --> 00:14:43,140 but the whole molecule is feeling that same field. 230 00:14:46,870 --> 00:14:50,010 OK, now we're ready to start doing quantum mechanics. 231 00:14:57,540 --> 00:15:02,040 So the interaction term, the thing that causes 232 00:15:02,040 --> 00:15:03,330 transitions to occur-- 233 00:15:05,970 --> 00:15:11,570 the electric interaction term, which we're going to call 234 00:15:11,570 --> 00:15:15,720 H1 because it's a perturbation. 235 00:15:15,720 --> 00:15:18,480 We're going to be doing something in perturbation 236 00:15:18,480 --> 00:15:21,550 theory, but it's time-dependent perturbation theory, 237 00:15:21,550 --> 00:15:24,300 which is a whole lot more complicated and rich 238 00:15:24,300 --> 00:15:26,040 than ordinary time-independent. 239 00:15:26,040 --> 00:15:31,500 Now many of you have found time-independent perturbation 240 00:15:31,500 --> 00:15:36,270 theory tedious and algebraically complicated. 241 00:15:36,270 --> 00:15:37,770 Time-dependent perturbation theory 242 00:15:37,770 --> 00:15:41,460 for these kinds of operators is not tedious. 243 00:15:41,460 --> 00:15:43,050 It's really beautiful. 244 00:15:43,050 --> 00:15:45,510 And there are many, many cases. 245 00:15:45,510 --> 00:15:47,580 It's not just having another variable. 246 00:15:47,580 --> 00:15:50,550 There's a lot of really neat stuff. 247 00:15:50,550 --> 00:15:55,290 And what I'm going to present today or I am presenting today 248 00:15:55,290 --> 00:16:00,180 is the theory for CW radiation-- that's continuous radiation-- 249 00:16:00,180 --> 00:16:05,430 really weak, interacting with a molecule or a system 250 00:16:05,430 --> 00:16:09,190 in a single quantum state initially. 251 00:16:09,190 --> 00:16:11,560 And it's important. 252 00:16:11,560 --> 00:16:16,850 The really weak and the CW are two really important features. 253 00:16:16,850 --> 00:16:20,030 And the single quantum state is just a convenience. 254 00:16:20,030 --> 00:16:20,990 We can deal with that. 255 00:16:20,990 --> 00:16:22,930 That's not a big deal, but it does 256 00:16:22,930 --> 00:16:29,050 involve using a different, more physical, or a more correct 257 00:16:29,050 --> 00:16:32,050 definition of what we mean by an average measurement 258 00:16:32,050 --> 00:16:35,240 on a system of many particles. 259 00:16:35,240 --> 00:16:38,170 And you'll hear the word "density matrix" if you go on 260 00:16:38,170 --> 00:16:39,421 in physical chemistry. 261 00:16:39,421 --> 00:16:41,170 But I'm not going to do anything about it, 262 00:16:41,170 --> 00:16:43,400 but that's how we deal with it. 263 00:16:43,400 --> 00:16:53,230 OK, so this is going to be minus mu-- 264 00:16:53,230 --> 00:16:59,720 it's a vector-- dot E of t, which is also a vector. 265 00:16:59,720 --> 00:17:02,720 Now a dot product, that looks really neat. 266 00:17:02,720 --> 00:17:06,500 However, this is a vector in the molecular frame, 267 00:17:06,500 --> 00:17:09,150 and this is a vector in the laboratory frame. 268 00:17:09,150 --> 00:17:12,530 So this dot product is a whole bunch more complicated 269 00:17:12,530 --> 00:17:15,720 than you would think. 270 00:17:15,720 --> 00:17:17,910 Now I do want to mention that when 271 00:17:17,910 --> 00:17:22,680 we talk about the rigid rotor, the rigid rotor is telling 272 00:17:22,680 --> 00:17:26,550 us what is the probability amplitude of the orientation 273 00:17:26,550 --> 00:17:30,420 of the molecular frame relative to the laboratory frame. 274 00:17:30,420 --> 00:17:34,710 So that is where all this information about these two 275 00:17:34,710 --> 00:17:38,980 different coordinate systems reside, 276 00:17:38,980 --> 00:17:42,330 and we'll see a little bit of that. 277 00:17:42,330 --> 00:17:49,012 OK, there's a similar expression for the magnetic term. 278 00:17:49,012 --> 00:17:50,470 I'm just not going to write it down 279 00:17:50,470 --> 00:17:53,290 because it's just too much stuff to write down. 280 00:17:53,290 --> 00:17:57,580 So the Hamiltonian, the time-independent Hamiltonian, 281 00:17:57,580 --> 00:18:04,200 can be expressed as H0 plus H1 of t. 282 00:18:07,190 --> 00:18:10,460 This looks exactly like time-independent perturbation 283 00:18:10,460 --> 00:18:14,420 theory, except this guy, which makes all the complications is 284 00:18:14,420 --> 00:18:17,990 time dependent. 285 00:18:17,990 --> 00:18:21,690 But this says, OK, we can find a whole set, 286 00:18:21,690 --> 00:18:26,280 a complete set of eignenenergies and eigenfunctions. 287 00:18:26,280 --> 00:18:29,892 And we know how to write the time-dependent Schrodinger-- 288 00:18:29,892 --> 00:18:31,850 the solutions of the time-dependent Schrodinger 289 00:18:31,850 --> 00:18:34,790 equation if this is the whole game. 290 00:18:34,790 --> 00:18:37,250 So we're going to use these as basis functions 291 00:18:37,250 --> 00:18:40,340 just as we did in ordinary perturbation theory. 292 00:18:45,697 --> 00:18:55,220 So H0 times some eigenfunction, which now I'm 293 00:18:55,220 --> 00:19:02,493 writing as explicitly time-dependent is En phi n t 294 00:19:02,493 --> 00:19:08,180 equals 0 e to the minus i En t over h-bar. 295 00:19:08,180 --> 00:19:09,890 So this is a solution. 296 00:19:09,890 --> 00:19:13,201 This thing is a solution to the time-dependent Schrodinger 297 00:19:13,201 --> 00:19:13,700 equation. 298 00:19:18,870 --> 00:19:23,930 And so when the external field is off, 299 00:19:23,930 --> 00:19:28,450 then the only states that we consider 300 00:19:28,450 --> 00:19:31,840 are eigenstates of the zero-order Hamiltonian, 301 00:19:31,840 --> 00:19:35,260 and they can be time dependent. 302 00:19:35,260 --> 00:19:49,570 But if we write psi n star of t times psi n of t, 303 00:19:49,570 --> 00:19:54,380 well, that's not time dependent if this is an eigenstate. 304 00:19:54,380 --> 00:19:58,220 So the only way we get time dependence 305 00:19:58,220 --> 00:20:01,760 is by having this time-dependent perturbation term. 306 00:20:06,050 --> 00:20:13,820 OK, so let's take some initial state. 307 00:20:13,820 --> 00:20:20,870 And let us call that initial state some arbitrary state. 308 00:20:20,870 --> 00:20:24,830 And we can always write this as a superposition 309 00:20:24,830 --> 00:20:30,455 of zero-order states. 310 00:20:37,270 --> 00:20:45,260 OK, and now, unfortunately, both the coefficients 311 00:20:45,260 --> 00:20:48,020 in this linear combination and the functions 312 00:20:48,020 --> 00:20:49,261 are time dependent. 313 00:20:51,910 --> 00:20:54,480 So this means when we're going to be applying 314 00:20:54,480 --> 00:20:56,400 the time-dependent Schrodinger equation, 315 00:20:56,400 --> 00:20:59,500 we take a partial derivative with respect to t, 316 00:20:59,500 --> 00:21:03,310 we get derivatives with this and this. 317 00:21:03,310 --> 00:21:05,260 So it's an extra level of complexity, 318 00:21:05,260 --> 00:21:08,130 but we can deal with it, because one 319 00:21:08,130 --> 00:21:10,560 of the things that we keep coming back to 320 00:21:10,560 --> 00:21:14,920 is that everything we talk about is expressed 321 00:21:14,920 --> 00:21:19,290 as a linear combination of t equals zero eigenstates 322 00:21:19,290 --> 00:21:21,489 of the zero-order Hamiltonian. 323 00:21:26,540 --> 00:21:28,590 OK, so the time-dependent Schrodinger equation-- 324 00:21:28,590 --> 00:21:34,120 i h-bar partial with respect to t of the-- 325 00:21:38,800 --> 00:21:42,120 yeah, of the wave function is equal to-- 326 00:21:56,600 --> 00:22:00,570 OK, that's our friend or our new friend 327 00:22:00,570 --> 00:22:04,600 because the old friend was too simple. 328 00:22:04,600 --> 00:22:10,130 And so, well, we can represent this partial derivative 329 00:22:10,130 --> 00:22:15,210 just using dots because the equations I'm 330 00:22:15,210 --> 00:22:18,000 going to be putting on the board are hideous, 331 00:22:18,000 --> 00:22:23,940 and so we want to use every abbreviation we can. 332 00:22:23,940 --> 00:22:27,960 This is written as a product of time-dependent coefficients 333 00:22:27,960 --> 00:22:29,670 and time-dependent functions. 334 00:22:29,670 --> 00:22:32,580 When we apply the derivative to it, 335 00:22:32,580 --> 00:22:36,580 we're going to get derivatives of each. 336 00:22:36,580 --> 00:23:03,205 And so that's the left-hand side. 337 00:23:07,630 --> 00:23:10,600 OK, and let's look at this left-hand side for a minute. 338 00:23:15,760 --> 00:23:18,230 OK, so we've got something that we don't really 339 00:23:18,230 --> 00:23:22,671 know what to do with, but this guy, we know that this is-- 340 00:23:22,671 --> 00:23:26,750 this time-dependent wave function 341 00:23:26,750 --> 00:23:30,860 is something that we can use the time-dependent Schrodinger 342 00:23:30,860 --> 00:23:33,860 equation on and get a simplification. 343 00:23:33,860 --> 00:23:34,819 So the left-hand side-- 344 00:23:34,819 --> 00:23:36,401 I haven't written the right-hand side. 345 00:23:36,401 --> 00:23:38,090 I'm just working on the left-hand side 346 00:23:38,090 --> 00:23:40,890 of what we get when we start to write this equation. 347 00:23:40,890 --> 00:23:53,230 And what we get is we know that the time dependence of this 348 00:23:53,230 --> 00:24:01,570 is equal to 1 over i h-bar times the Hamiltonian operating 349 00:24:01,570 --> 00:24:04,790 on phi n. 350 00:24:07,736 --> 00:24:10,620 Is that what I want? 351 00:24:10,620 --> 00:24:12,310 I can't read my notes so I have to-- 352 00:24:12,310 --> 00:24:14,350 I have to be-- 353 00:24:14,350 --> 00:24:16,930 yeah, so we've just taken that 1 over i h-bar. 354 00:24:20,740 --> 00:24:25,150 This is going to be the time-independent Hamiltonian, 355 00:24:25,150 --> 00:24:26,370 the zero-order Hamiltonian. 356 00:24:26,370 --> 00:24:27,970 And we know what we get here. 357 00:24:33,000 --> 00:24:33,500 Yes? 358 00:24:33,500 --> 00:24:35,060 STUDENT: So all your phi n's, those 359 00:24:35,060 --> 00:24:36,642 are the zero-order solutions? 360 00:24:36,642 --> 00:24:37,850 ROBERT FIELD: That's correct. 361 00:24:37,850 --> 00:24:39,475 STUDENT: So they're unperturbed states? 362 00:24:39,475 --> 00:24:43,370 ROBERT FIELD: They're unperturbed eigenstates of H0. 363 00:24:43,370 --> 00:24:50,210 And if it's psi n of t, it has the e to the i En of t-- 364 00:24:52,940 --> 00:24:56,510 En t over h-bar factor implicit, and we're 365 00:24:56,510 --> 00:24:58,700 going to be using that. 366 00:24:58,700 --> 00:25:06,910 All right, so what we get when we 367 00:25:06,910 --> 00:25:09,400 take that partial derivative, we get a simplification. 368 00:25:19,510 --> 00:25:22,120 OK, let me just write the right-hand side 369 00:25:22,120 --> 00:25:23,140 of this equation too. 370 00:25:23,140 --> 00:25:30,140 So we have the simplified left-hand side, 371 00:25:30,140 --> 00:25:34,210 which is psi n c-- 372 00:25:34,210 --> 00:25:37,270 I've never lectured on time-dependent perturbation 373 00:25:37,270 --> 00:25:38,650 theory before. 374 00:25:38,650 --> 00:25:41,740 And so although I think I understand it, 375 00:25:41,740 --> 00:25:47,140 it's not as available in core as it ought to be. 376 00:25:47,140 --> 00:25:52,030 OK, so we have this minus-- 377 00:25:56,829 --> 00:25:58,120 where did the wave function go? 378 00:26:04,630 --> 00:26:07,930 Well, there's got to be a phi in here 379 00:26:07,930 --> 00:26:21,380 and then minus i over h-bar En cn t over the times phi n of t. 380 00:26:21,380 --> 00:26:30,010 That's the left-hand side in the bracket here. 381 00:26:30,010 --> 00:26:33,050 OK, and the right-hand side of the original equation, 382 00:26:33,050 --> 00:26:56,940 that is just some n cn t En plus H1 of t phi n t. 383 00:26:56,940 --> 00:27:01,420 OK, it takes a little imagination, 384 00:27:01,420 --> 00:27:05,260 but this and the terms associated 385 00:27:05,260 --> 00:27:07,360 with that are the same. 386 00:27:07,360 --> 00:27:10,330 This happened when we did non-degenerate perturbation 387 00:27:10,330 --> 00:27:11,110 theory. 388 00:27:11,110 --> 00:27:14,320 We looked at the lambdas of one equation. 389 00:27:14,320 --> 00:27:18,910 There was a cancellation of two ugly terms. 390 00:27:18,910 --> 00:27:21,820 And so what ends up happening is we 391 00:27:21,820 --> 00:27:26,720 get a tremendous simplification of the problem. 392 00:27:26,720 --> 00:27:49,530 And so the left-hand side of the equation has the form, 393 00:27:49,530 --> 00:27:53,640 and the right-hand side has the form over here without 394 00:27:53,640 --> 00:27:54,680 the extra term-- 395 00:27:54,680 --> 00:28:09,820 sum over n, cn of t H1 of t psi n of t. 396 00:28:18,640 --> 00:28:23,337 OK, and now we have this equation. 397 00:28:23,337 --> 00:28:24,920 We have this simple thing here, and we 398 00:28:24,920 --> 00:28:27,110 have this ugly thing here. 399 00:28:27,110 --> 00:28:40,160 And we want to simplify this by multiplying on the left 400 00:28:40,160 --> 00:28:43,230 by psi F of t so-- 401 00:28:46,520 --> 00:28:48,206 and integrating with respect to tau. 402 00:28:51,860 --> 00:28:54,750 F is for final. 403 00:28:54,750 --> 00:28:56,820 So we're interested in the transition 404 00:28:56,820 --> 00:28:59,560 from some initial state to some final state. 405 00:28:59,560 --> 00:29:02,270 So we're going to massage this. 406 00:29:02,270 --> 00:29:05,340 And when we do that, we get-- 407 00:29:09,690 --> 00:29:12,510 I've clearly skipped a step, but it doesn't matter-- 408 00:29:12,510 --> 00:29:26,790 i h-bar cf dot of t is equal to this integral sum c n of t 409 00:29:26,790 --> 00:29:30,060 integral cf of-- 410 00:29:30,060 --> 00:29:42,820 phi f of t H1 f of t phi n of t, e tau. 411 00:29:46,170 --> 00:29:50,940 This is a very important equation 412 00:29:50,940 --> 00:29:54,240 because we have a simple derivative of the coefficient 413 00:29:54,240 --> 00:29:57,220 that we want, and it's expressed as an integral. 414 00:29:57,220 --> 00:30:00,940 And we have an integral between an eigenstate 415 00:30:00,940 --> 00:30:04,440 of the zero-order Hamiltonian and another eigenstate. 416 00:30:04,440 --> 00:30:17,430 And this is just H1 f n of t. 417 00:30:17,430 --> 00:30:21,715 OK, so we have these guys. 418 00:30:27,810 --> 00:30:29,830 So what we want to know is, all right, 419 00:30:29,830 --> 00:30:33,590 this is the thing that's making stuff happen. 420 00:30:33,590 --> 00:30:38,010 This is a matrix element of this term. 421 00:30:38,010 --> 00:30:48,600 Well, H1 of t, which is equal to v cosine omega t 422 00:30:48,600 --> 00:30:57,810 can be written as v times 1/2 e to the i omega t plus e 423 00:30:57,810 --> 00:31:00,930 to the minus i omega t. 424 00:31:00,930 --> 00:31:04,530 This is really neat because you notice 425 00:31:04,530 --> 00:31:08,520 we have these complex oscillating field terms, 426 00:31:08,520 --> 00:31:11,850 and we have on each of these wave functions 427 00:31:11,850 --> 00:31:14,970 a complex oscillating term. 428 00:31:14,970 --> 00:31:20,700 And what ends up happening is that we get this equation. 429 00:31:20,700 --> 00:31:27,340 i h-bar cf dot of t is equal to-- 430 00:31:27,340 --> 00:31:29,390 and this is-- you know, it's ugly. 431 00:31:29,390 --> 00:31:30,060 It gets big. 432 00:31:30,060 --> 00:31:32,040 A lot of stuff has to be written, 433 00:31:32,040 --> 00:31:34,750 and I have to transfer from my notes to here. 434 00:31:34,750 --> 00:31:36,720 And then you have to transfer to your paper. 435 00:31:36,720 --> 00:31:38,670 And there is going to be-- 436 00:31:38,670 --> 00:31:40,290 there will be printed lecture notes. 437 00:31:40,290 --> 00:31:42,840 And in fact, there may actually be printed lecture notes 438 00:31:42,840 --> 00:31:44,470 for this lecture. 439 00:31:44,470 --> 00:31:47,430 But if they're not, they will be soon. 440 00:31:47,430 --> 00:31:51,600 OK, and so we get this differential equation, which 441 00:31:51,600 --> 00:32:03,540 is the sum over n c n of t integral psi f star of t times 442 00:32:03,540 --> 00:32:10,710 1/2 v, v to the i omega t plus e to the minus i omega 443 00:32:10,710 --> 00:32:18,530 t times psi n of t, e tau. 444 00:32:21,510 --> 00:32:24,510 Well, these guys have time dependence, 445 00:32:24,510 --> 00:32:26,780 and so we can put that in. 446 00:32:26,780 --> 00:32:39,210 And now this integral has the form psi f star 0 1/2 v, 447 00:32:39,210 --> 00:32:54,520 and we have e to the minus I omega and f minus omega t. 448 00:32:54,520 --> 00:32:58,070 Omega nf, the difference in-- 449 00:32:58,070 --> 00:33:05,210 so omega nf is En minus Ef over h-bar. 450 00:33:08,970 --> 00:33:14,670 And so we have minus this oscillating term, minus omega 451 00:33:14,670 --> 00:33:24,930 t, and then we have e to the minus i nft plus omega t. 452 00:33:30,080 --> 00:33:34,620 So here this isn't well, it's so ugly because of my stupidity 453 00:33:34,620 --> 00:33:35,120 here. 454 00:33:35,120 --> 00:33:38,360 But what we have here is a resonance integral. 455 00:33:38,360 --> 00:33:42,740 We have something that's oscillating fast 456 00:33:42,740 --> 00:33:45,880 minus something that's oscillating fast. 457 00:33:45,880 --> 00:33:48,140 And we have the same thing plus something 458 00:33:48,140 --> 00:33:50,930 that's oscillating fast. 459 00:33:50,930 --> 00:33:55,550 So those terms are zero because we have an integral that 460 00:33:55,550 --> 00:33:57,740 is oscillating. 461 00:33:57,740 --> 00:33:58,392 I'm sorry. 462 00:33:58,392 --> 00:34:00,350 It's oscillating between positive and negative, 463 00:34:00,350 --> 00:34:01,670 positive, negative. 464 00:34:04,880 --> 00:34:10,489 And as long as omega is different from omega nf, 465 00:34:10,489 --> 00:34:16,040 those integrals are zero because this integrand, as we integrate 466 00:34:16,040 --> 00:34:20,420 to t equals infinity or to any time, 467 00:34:20,420 --> 00:34:24,300 is oscillating about zero, and it's small. 468 00:34:24,300 --> 00:34:30,650 However, if omega is the same as minus omega nf 469 00:34:30,650 --> 00:34:35,270 or plus omega nf, well, then this thing is 1 times t. 470 00:34:38,120 --> 00:34:40,400 It gets really big. 471 00:34:40,400 --> 00:34:42,920 Now we're talking about coefficients, which 472 00:34:42,920 --> 00:34:45,139 are related to probabilities. 473 00:34:45,139 --> 00:34:47,060 And so these coefficients had better not 474 00:34:47,060 --> 00:34:51,989 go get really big because probability is always 475 00:34:51,989 --> 00:34:54,980 going to be less than 1. 476 00:34:54,980 --> 00:34:57,200 OK, so what we're going to do now 477 00:34:57,200 --> 00:35:00,410 is collect the rubble in a form that it turns out 478 00:35:00,410 --> 00:35:01,380 to be really useful. 479 00:35:13,510 --> 00:35:17,310 So we have an equation for the time 480 00:35:17,310 --> 00:35:23,580 dependence of a final state, and it's expressed as a sum over n. 481 00:35:23,580 --> 00:35:27,570 But if we say, oh, let's make our initial state 482 00:35:27,570 --> 00:35:30,850 just one of those. 483 00:35:30,850 --> 00:35:33,690 So our initial state is-- 484 00:35:33,690 --> 00:35:37,990 let's call it ci. 485 00:35:37,990 --> 00:35:41,380 And we say, well, the system is not 486 00:35:41,380 --> 00:35:47,540 in any other state other than the i state, and this is weak. 487 00:35:47,540 --> 00:35:51,040 So we can neglect all of the other states 488 00:35:51,040 --> 00:35:54,370 where n is not equal to i. 489 00:35:54,370 --> 00:36:01,870 And if they're not there, cn has to be 1, 490 00:36:01,870 --> 00:36:04,240 so we can forget about it. 491 00:36:04,240 --> 00:36:07,150 So we end up with this incredibly wonderful 492 00:36:07,150 --> 00:36:09,170 simple equation. 493 00:36:09,170 --> 00:36:10,630 So we make the two approximations. 494 00:36:10,630 --> 00:36:13,690 Single state, the perturbation is really weak, 495 00:36:13,690 --> 00:36:20,720 and we get cf of t is equal to the vfi-- 496 00:36:20,720 --> 00:36:22,510 the off-diagonal matrix element-- 497 00:36:22,510 --> 00:36:30,578 over 2i h-bar times the integral from 0 to t e 498 00:36:30,578 --> 00:36:44,720 to the minus i omega i f t minus omega times 499 00:36:44,720 --> 00:36:52,930 e to the i omega i f plus omega dt. 500 00:36:57,350 --> 00:37:00,260 Well, all complexity is gone. 501 00:37:00,260 --> 00:37:03,690 We have the amount of the final state, 502 00:37:03,690 --> 00:37:07,650 and it's expressed by a matrix element and some time 503 00:37:07,650 --> 00:37:09,500 dependence. 504 00:37:09,500 --> 00:37:18,580 And this is a resonant situation where if omega t, omega t-- 505 00:37:18,580 --> 00:37:24,570 if omega is equal to omega i f, fine. 506 00:37:24,570 --> 00:37:29,490 Then this is zero, the exponent is zero, 507 00:37:29,490 --> 00:37:32,770 we get t here from that. 508 00:37:32,770 --> 00:37:35,150 And we get zero from that one because that's oscillating 509 00:37:35,150 --> 00:37:36,441 so fast it doesn't do anything. 510 00:37:39,330 --> 00:37:45,430 But that's a problem because this c is a probability. 511 00:37:45,430 --> 00:37:50,580 And so the square of c had better not be larger than one, 512 00:37:50,580 --> 00:37:54,240 and this is cruising to be larger than 1. 513 00:37:54,240 --> 00:37:56,940 But we don't care about cw. 514 00:37:56,940 --> 00:37:58,530 What we really care about-- well, 515 00:37:58,530 --> 00:38:03,000 what is the rate as opposed to the probability? 516 00:38:03,000 --> 00:38:11,130 OK, because the rate of increase of state f 517 00:38:11,130 --> 00:38:16,080 is something that we can calculate from this integral 518 00:38:16,080 --> 00:38:19,880 simply by taking the-- 519 00:38:19,880 --> 00:38:23,695 we multiply the integral by 1 over T 520 00:38:23,695 --> 00:38:29,850 if the limit T goes to infinity. 521 00:38:32,820 --> 00:38:34,550 And now we get a new equation, which 522 00:38:34,550 --> 00:38:40,210 is called Fermi's golden rule. 523 00:38:40,210 --> 00:38:43,100 OK, so I'm skipping some steps, and I'm 524 00:38:43,100 --> 00:38:45,260 doing things in the wrong order. 525 00:38:45,260 --> 00:38:48,920 But so first of all, the probability 526 00:38:48,920 --> 00:38:52,250 of the transition from the i state to the f state 527 00:38:52,250 --> 00:38:53,820 as a function of time. 528 00:38:53,820 --> 00:38:56,330 So the probability is going to keep growing. 529 00:38:56,330 --> 00:39:00,230 That's why we want to do this trick with dividing by t. 530 00:39:00,230 --> 00:39:01,801 What time is it? 531 00:39:01,801 --> 00:39:02,300 OK. 532 00:39:09,740 --> 00:39:17,150 That's just cf of t squared, and that's just 533 00:39:17,150 --> 00:39:28,760 the fi over 4 h-bar squared times this integral 0 to t e 534 00:39:28,760 --> 00:39:39,200 to the plus and e to the minus term dt squared. 535 00:39:39,200 --> 00:39:45,340 OK, the integrals survive only if omega is equal to omega i f. 536 00:39:49,630 --> 00:39:52,580 And if we convert to a rate so that the rate is 537 00:39:52,580 --> 00:40:04,890 going to be Wfi, which is going to be Vfi over 4 h-bar 538 00:40:04,890 --> 00:40:14,331 squared times the sum of two delta functions Vi 539 00:40:14,331 --> 00:40:27,440 minus Ef minus omega plus sum of Ei minus Ef plus omega. 540 00:40:30,420 --> 00:40:34,530 So the rate is just this simple thing-- 541 00:40:34,530 --> 00:40:36,990 the square matrix element and a delta 542 00:40:36,990 --> 00:40:39,570 function-- saying either it's an absorption or emission 543 00:40:39,570 --> 00:40:42,250 transition on resonance, and we're cooked. 544 00:40:42,250 --> 00:40:46,700 OK, so now I want to show some pictures 545 00:40:46,700 --> 00:40:54,540 of a movie, which will make this whole thing make more sense. 546 00:40:54,540 --> 00:40:57,240 This is for a vibrational transition. 547 00:40:57,240 --> 00:41:00,815 So we have the electric field-- 548 00:41:03,770 --> 00:41:06,530 the dipole interacting with the electric field. 549 00:41:06,530 --> 00:41:10,410 And now let's just turn on the time dependence. 550 00:41:10,410 --> 00:41:14,100 OK, so this is the interaction term. 551 00:41:14,100 --> 00:41:18,050 We add that interaction term to the zero-order Hamiltonian, 552 00:41:18,050 --> 00:41:24,650 and so we end up getting a big effect of the potential. 553 00:41:24,650 --> 00:41:27,450 The potential's going like this, like that. 554 00:41:27,450 --> 00:41:30,660 And so the eigenfunctions of that potential 555 00:41:30,660 --> 00:41:35,260 are going to be profoundly affected, and so let's do that. 556 00:41:35,260 --> 00:41:38,760 Let's go to the next. 557 00:41:38,760 --> 00:41:42,980 All right, so here now we have a realistic small field, 558 00:41:42,980 --> 00:41:44,870 and now this is small. 559 00:41:44,870 --> 00:41:46,640 And you can hardly see this thing moving. 560 00:41:51,520 --> 00:42:00,830 OK, now what we have is the wave function of this. 561 00:42:00,830 --> 00:42:12,610 And what we see is if omega is 1/4 the energy, if omega 562 00:42:12,610 --> 00:42:16,120 is much smaller than the vibrational frequency 563 00:42:16,120 --> 00:42:21,730 or much larger, we get very little effect of the time 564 00:42:21,730 --> 00:42:24,750 dependent. 565 00:42:24,750 --> 00:42:27,330 You can see that the wave function is just 566 00:42:27,330 --> 00:42:28,560 moving a little bit. 567 00:42:28,560 --> 00:42:30,630 The potential is jiggling around, 568 00:42:30,630 --> 00:42:33,990 whether the perturbation is strong or weak. 569 00:42:33,990 --> 00:42:35,590 It's not on resonance. 570 00:42:35,590 --> 00:42:41,810 And now let's go to the resonance. 571 00:42:41,810 --> 00:42:43,940 Now what's happening is the potential 572 00:42:43,940 --> 00:42:46,640 is moving not too much, but the wave function 573 00:42:46,640 --> 00:42:50,050 is diving all over the place. 574 00:42:50,050 --> 00:42:51,810 And if we ask, well, what does that really 575 00:42:51,810 --> 00:42:59,230 look like as a sum of terms, the thing 576 00:42:59,230 --> 00:43:01,810 that's different from the zero-order wave function 577 00:43:01,810 --> 00:43:03,730 is this. 578 00:43:03,730 --> 00:43:08,070 So zero-order wave function is one nodeless thing. 579 00:43:08,070 --> 00:43:14,850 This is the time-dependent term, and it looks like V equals 1 So 580 00:43:14,850 --> 00:43:19,060 what this shows is, yes, there are-- 581 00:43:19,060 --> 00:43:23,890 if we have a time-dependent field, and it's resonant, 582 00:43:23,890 --> 00:43:25,720 then we get a very strong interaction 583 00:43:25,720 --> 00:43:27,700 even though the field is weak. 584 00:43:27,700 --> 00:43:31,570 And it causes the appearance of the other level 585 00:43:31,570 --> 00:43:32,841 but oscillating. 586 00:43:35,610 --> 00:43:39,030 And so resonance is really important, 587 00:43:39,030 --> 00:43:41,450 and selection rule is really important. 588 00:43:41,450 --> 00:43:46,650 The selection rule for the vibrational transitions 589 00:43:46,650 --> 00:43:49,620 has to do with the form. 590 00:43:49,620 --> 00:43:54,790 Oh, I shouldn't be rushing at all. 591 00:43:54,790 --> 00:43:59,915 OK, so let's draw a picture. 592 00:44:02,840 --> 00:44:06,210 And this is the part that has puzzled me for a long time, 593 00:44:06,210 --> 00:44:09,620 but I've got it now. 594 00:44:09,620 --> 00:44:20,170 So here we have a picture of the molecule. 595 00:44:20,170 --> 00:44:22,800 And this end is positive, and this end is negative. 596 00:44:22,800 --> 00:44:25,280 And we have a positive electrode, 597 00:44:25,280 --> 00:44:27,570 and we have a negative electrode. 598 00:44:27,570 --> 00:44:30,050 So that's an electric field. 599 00:44:30,050 --> 00:44:35,800 And so now the positive electrode is saying, 600 00:44:35,800 --> 00:44:38,690 you better go away, and you better come here. 601 00:44:38,690 --> 00:44:42,370 So it's trying to use compressed bond. 602 00:44:42,370 --> 00:44:44,470 And now this field oscillates, and so it's 603 00:44:44,470 --> 00:44:48,120 compressing and expanding. 604 00:44:48,120 --> 00:44:50,960 Now that's what's going on. 605 00:44:50,960 --> 00:44:53,260 But how does quantum mechanics account for it? 606 00:44:53,260 --> 00:44:57,100 Well, quantum mechanics says in order for the bond length 607 00:44:57,100 --> 00:45:00,710 to change, we have to mix in some other state. 608 00:45:00,710 --> 00:45:06,460 So we have the ground state, and we have an excited state 609 00:45:06,460 --> 00:45:08,630 that looks like that. 610 00:45:08,630 --> 00:45:11,560 And so the field is mixing these two. 611 00:45:11,560 --> 00:45:19,600 Now that means that the operator is mu 0 plus derivative of mu 612 00:45:19,600 --> 00:45:25,600 with respect to the electric field times q. 613 00:45:25,600 --> 00:45:31,750 So this is the thing that allows some mixing of an excited 614 00:45:31,750 --> 00:45:33,680 state into the ground state. 615 00:45:33,680 --> 00:45:37,210 This is our friend the harmonic oscillator-- 616 00:45:37,210 --> 00:45:39,310 operator, displacement operator. 617 00:45:39,310 --> 00:45:42,440 It has selection rules delta v equals plus or minus 1, 618 00:45:42,440 --> 00:45:45,740 and that's all. 619 00:45:45,740 --> 00:45:50,630 So a vibrational transition is caused 620 00:45:50,630 --> 00:45:55,640 by the derivative of the-- 621 00:45:58,839 --> 00:46:05,150 yeah, that's-- no, derivative of the dipole moment with respect 622 00:46:05,150 --> 00:46:07,650 to Q. 623 00:46:07,650 --> 00:46:12,750 So did I have it right? 624 00:46:12,750 --> 00:46:14,100 Yes. 625 00:46:14,100 --> 00:46:16,170 So this is something-- 626 00:46:16,170 --> 00:46:20,040 we can calculate how the dipole moment depends 627 00:46:20,040 --> 00:46:21,810 on the displacement from equilibrium, 628 00:46:21,810 --> 00:46:26,060 but this is the operator that causes the mixing of states. 629 00:46:26,060 --> 00:46:30,690 So one of the things I've loved to do over the years 630 00:46:30,690 --> 00:46:34,670 is to write a cumulative exam in which I ask, well, 631 00:46:34,670 --> 00:46:38,110 what is it that causes a vibrational transition? 632 00:46:38,110 --> 00:46:40,290 What does a molecule have to have in order 633 00:46:40,290 --> 00:46:42,480 to have a vibrational transition? 634 00:46:42,480 --> 00:46:44,250 And also what does a molecule have 635 00:46:44,250 --> 00:46:47,610 to have to have a rotational transition? 636 00:46:47,610 --> 00:46:50,100 Well, this is what causes the rotational transition 637 00:46:50,100 --> 00:46:55,210 because we can think of the dipole moment interacting 638 00:46:55,210 --> 00:46:59,950 with a field, which is going like that or like that. 639 00:46:59,950 --> 00:47:04,720 And so what that does is it causes a torque on the system. 640 00:47:04,720 --> 00:47:07,330 It doesn't change the dipole moment, 641 00:47:07,330 --> 00:47:09,640 doesn't stretch the molecule. 642 00:47:09,640 --> 00:47:17,650 It causes a transition, and this is 643 00:47:17,650 --> 00:47:23,920 expressed in terms of the interaction 644 00:47:23,920 --> 00:47:32,230 mu dot E. This dot product, this cosine theta, 645 00:47:32,230 --> 00:47:34,060 is the operator that causes this. 646 00:47:34,060 --> 00:47:44,830 We call the relationship between the laboratory and the body 647 00:47:44,830 --> 00:47:46,690 fixed coordinate system is determined 648 00:47:46,690 --> 00:47:50,720 by the cosine of some angle, and the cosine of the angle 649 00:47:50,720 --> 00:47:55,970 is what's responsible for a pure rotational transition. 650 00:47:55,970 --> 00:47:57,720 And we have vibrational transitions 651 00:47:57,720 --> 00:48:00,240 where they are derivative of the dipole with respect 652 00:48:00,240 --> 00:48:02,160 to the coordinate. 653 00:48:02,160 --> 00:48:06,980 Now let's say we have nitrogen-- 654 00:48:06,980 --> 00:48:11,830 no dipole moment, no derivative of the dipole moment. 655 00:48:11,830 --> 00:48:13,380 Suppose we have CO. 656 00:48:13,380 --> 00:48:15,850 CO has a very small dipole moment 657 00:48:15,850 --> 00:48:19,570 and a huge derivative of the dipole moment with respect 658 00:48:19,570 --> 00:48:21,100 to displacement. 659 00:48:21,100 --> 00:48:24,670 And so CO has really strong vibrational transitions 660 00:48:24,670 --> 00:48:27,430 and rather weak rotational transitions. 661 00:48:27,430 --> 00:48:34,180 So if it happened that CO had zero permanent dipole moment, 662 00:48:34,180 --> 00:48:36,490 it would have no rotational transition. 663 00:48:36,490 --> 00:48:39,100 But as you go up to higher V's, then it would not be zero. 664 00:48:39,100 --> 00:48:41,840 And you would see rotational transitions. 665 00:48:41,840 --> 00:48:44,800 And so there's all sorts of insights that come from this. 666 00:48:44,800 --> 00:48:48,460 And so now we know what causes transitions. 667 00:48:48,460 --> 00:48:51,140 There is some operator, which causes 668 00:48:51,140 --> 00:48:53,980 mixing of some wave functions. 669 00:48:53,980 --> 00:48:55,810 And the time-dependent perturbation theory 670 00:48:55,810 --> 00:49:00,910 when it's resonant mixes only one state. 671 00:49:00,910 --> 00:49:03,130 We have selection rules which we understand just 672 00:49:03,130 --> 00:49:04,510 by looking at the wave fun-- 673 00:49:04,510 --> 00:49:08,290 looking at the matrix elements, and now we 674 00:49:08,290 --> 00:49:11,770 have a big understanding of what is 675 00:49:11,770 --> 00:49:13,330 going to appear in a spectrum. 676 00:49:13,330 --> 00:49:15,170 What are the intensities in the spectrum? 677 00:49:15,170 --> 00:49:17,140 What are the transitions? 678 00:49:17,140 --> 00:49:18,980 Which transitions are going to be allowed? 679 00:49:18,980 --> 00:49:22,600 Which are going to be forbidden? 680 00:49:22,600 --> 00:49:24,170 And that's kind of useful. 681 00:49:24,170 --> 00:49:28,790 So there is this tremendously tedious algebra, 682 00:49:28,790 --> 00:49:31,540 which I didn't do a very good job displaying, 683 00:49:31,540 --> 00:49:33,760 but you don't need it because, at the end, 684 00:49:33,760 --> 00:49:36,640 you get Fermi's golden rule, which says 685 00:49:36,640 --> 00:49:40,310 transitions occur on resonance. 686 00:49:40,310 --> 00:49:42,400 Now if you're a little bit off resonance, 687 00:49:42,400 --> 00:49:46,780 well, then the stationary phase in the oscillating exponential 688 00:49:46,780 --> 00:49:50,590 persists for a while, and then it goes away. 689 00:49:50,590 --> 00:49:54,100 And so you get a little bit of slightly off-resonance 690 00:49:54,100 --> 00:49:58,180 transition probability, and you get other things too. 691 00:49:58,180 --> 00:50:01,780 But you already now have enough to understand basically 692 00:50:01,780 --> 00:50:06,940 everything you need to begin to make sense 693 00:50:06,940 --> 00:50:13,190 of the interaction of radiation with molecules correctly, 694 00:50:13,190 --> 00:50:15,020 and this isn't bullets and targets. 695 00:50:15,020 --> 00:50:19,370 This is waves with phases, and so there 696 00:50:19,370 --> 00:50:21,980 are all sorts of things you have to do to be honest about it. 697 00:50:21,980 --> 00:50:24,530 But you know what the actors are, 698 00:50:24,530 --> 00:50:26,810 and that's really a useful thing. 699 00:50:26,810 --> 00:50:31,150 And you're never going to be tested on this from me. 700 00:50:31,150 --> 00:50:34,390 OK, good luck on the exam tomorrow night.