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 OpenCourseWare 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 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,850 at ocw.mit.edu. 8 00:00:21,560 --> 00:00:26,900 ROBERT FIELD: Today, I'm going to go over a lot of material 9 00:00:26,900 --> 00:00:31,850 that you already know and work an example 10 00:00:31,850 --> 00:00:37,390 with non-degenerate perturbation theory. 11 00:00:37,390 --> 00:00:41,920 This will give me an excuse to use interspersed matrix 12 00:00:41,920 --> 00:00:47,500 and operator notation, and one of the things 13 00:00:47,500 --> 00:00:53,850 that you will be doing is dealing with infinite matrices. 14 00:00:53,850 --> 00:00:58,980 And we don't diagonalize infinite matrices, 15 00:00:58,980 --> 00:01:02,130 but the picture of an infinite matrix 16 00:01:02,130 --> 00:01:05,620 is useful in guiding intuition. 17 00:01:05,620 --> 00:01:10,800 And so we have to know about what those pictures mean 18 00:01:10,800 --> 00:01:15,640 and how to construct them from things that seem to be obvious. 19 00:01:15,640 --> 00:01:21,090 OK, the problem I want to spend most of my time on 20 00:01:21,090 --> 00:01:26,550 is an example of a real molecular potential which 21 00:01:26,550 --> 00:01:30,420 is quadratic but with a cubic and a quartic perturbation 22 00:01:30,420 --> 00:01:39,020 term, and this is a simpler problem than what 23 00:01:39,020 --> 00:01:41,870 I talked about last time. 24 00:01:41,870 --> 00:01:43,370 The last time I really wanted to set 25 00:01:43,370 --> 00:01:47,450 the stage for what you can do with non-degenerate 26 00:01:47,450 --> 00:01:49,520 perturbation theory, and this is going 27 00:01:49,520 --> 00:01:57,860 to be more of dealing with a specific problem. 28 00:01:57,860 --> 00:02:01,850 Now, the algebra does not get simpler, 29 00:02:01,850 --> 00:02:04,770 and so it's easy to get lost in the algebra. 30 00:02:04,770 --> 00:02:08,210 And so I will give you some rules 31 00:02:08,210 --> 00:02:12,320 about what to expect, when you have a potential like this, 32 00:02:12,320 --> 00:02:15,770 and how does that come out of non-degenerate perturbation 33 00:02:15,770 --> 00:02:17,300 theory? 34 00:02:17,300 --> 00:02:22,060 Now, everything that we know about molecules 35 00:02:22,060 --> 00:02:24,560 comes from spectroscopy. 36 00:02:24,560 --> 00:02:31,590 That means transitions between energy levels, and so well, 37 00:02:31,590 --> 00:02:33,450 how do they happen? 38 00:02:33,450 --> 00:02:36,870 And we can lead into that by talking 39 00:02:36,870 --> 00:02:39,720 about the stark effect which is the interaction 40 00:02:39,720 --> 00:02:44,145 of an electric field with a dipole in the molecule. 41 00:02:46,910 --> 00:02:52,770 And so most transitions are caused 42 00:02:52,770 --> 00:02:57,390 by a time-dependent electric field which somehow interacts 43 00:02:57,390 --> 00:03:01,540 with the molecule through its dipole moment, 44 00:03:01,540 --> 00:03:05,700 and so you'll begin to see that today. 45 00:03:11,310 --> 00:03:14,330 Last time, I derived the formulas 46 00:03:14,330 --> 00:03:16,470 for non-degenerate perturbation theory. 47 00:03:29,180 --> 00:03:33,990 Now, I'm a little bit crazy, and I really 48 00:03:33,990 --> 00:03:36,030 believe in perturbation theory. 49 00:03:36,030 --> 00:03:38,730 Most of the textbooks say, well, let's just 50 00:03:38,730 --> 00:03:41,190 stop here at the first-order correction 51 00:03:41,190 --> 00:03:44,980 to the energy levels, and you get nothing 52 00:03:44,980 --> 00:03:48,220 from that except the Zeeman Effect 53 00:03:48,220 --> 00:03:50,380 which is kind of important for NMR. 54 00:03:50,380 --> 00:03:57,850 But in terms of what molecules do, you need this too, 55 00:03:57,850 --> 00:04:03,520 and this is equal to the zero-order energy. 56 00:04:06,730 --> 00:04:09,490 I'm sorry, the zero-order energy which 57 00:04:09,490 --> 00:04:14,530 comes from an exactly solved problem 58 00:04:14,530 --> 00:04:23,230 and then the diagonal matrix element of the everything 59 00:04:23,230 --> 00:04:27,070 that's bad in the world, the first-order Hamiltonian. 60 00:04:27,070 --> 00:04:32,070 And then, we have this complicated-looking thing, 61 00:04:32,070 --> 00:04:43,228 m not equal to n of H n m squared, E n 0 minus E m 0. 62 00:04:46,480 --> 00:04:48,400 Now, the only thing that you might 63 00:04:48,400 --> 00:04:51,670 forget about in this kind of a formula 64 00:04:51,670 --> 00:04:57,280 is is it n before m or the other way around? 65 00:04:57,280 --> 00:05:01,470 And what we know is, OK, here is the nth level, 66 00:05:01,470 --> 00:05:04,290 and here is the mth level, and the interaction causes 67 00:05:04,290 --> 00:05:06,790 this level to be pushed down. 68 00:05:06,790 --> 00:05:13,480 So if the mth level is above the nth level, 69 00:05:13,480 --> 00:05:16,100 this denominator will be negative, 70 00:05:16,100 --> 00:05:18,440 and that causes the pushing down. 71 00:05:18,440 --> 00:05:21,730 And so if you remember the idea of perturbation theory-- 72 00:05:21,730 --> 00:05:26,350 that things interact, and one level gets pushed down, 73 00:05:26,350 --> 00:05:30,040 the other level gets pushed up, equal and opposite-- 74 00:05:30,040 --> 00:05:36,580 you can correct your flaws of memory on which comes first. 75 00:05:36,580 --> 00:05:42,790 And we can write the wave function 76 00:05:42,790 --> 00:05:47,680 as an eigenfunction of the exactly solved problem 77 00:05:47,680 --> 00:06:05,460 plus a term, m not equal to n of H n m 1 E n 0 minus E m 0 times 78 00:06:05,460 --> 00:06:09,082 psi m 0. 79 00:06:09,082 --> 00:06:14,260 Now, the reason this sum does not include the nth level 80 00:06:14,260 --> 00:06:18,370 is, well, if you did, this denominator would blow up, 81 00:06:18,370 --> 00:06:21,370 and you don't need the nth level, because you've already 82 00:06:21,370 --> 00:06:21,940 got it here. 83 00:06:25,570 --> 00:06:30,450 So these are the formulas, and we had said, 84 00:06:30,450 --> 00:06:38,880 OK, we're going to write the Hamiltonian as H0 plus H1, 85 00:06:38,880 --> 00:06:42,360 and this is an exactly solved problem. 86 00:06:42,360 --> 00:06:44,940 This is everything else. 87 00:06:44,940 --> 00:06:49,020 Now, some people will sort this out into small everything else 88 00:06:49,020 --> 00:06:51,010 and big everything else. 89 00:06:51,010 --> 00:06:53,400 But it's foolish, because all that does 90 00:06:53,400 --> 00:06:57,180 is multiply the algebra in a horrible way, 91 00:06:57,180 --> 00:06:58,740 and you don't care, anyway. 92 00:06:58,740 --> 00:07:02,550 You just know that this stuff that's not in this exactly 93 00:07:02,550 --> 00:07:05,070 solved problem gets treated here, 94 00:07:05,070 --> 00:07:09,060 and you're going to know, in principle, how to deal with it, 95 00:07:09,060 --> 00:07:10,830 and there'll be some small problems 96 00:07:10,830 --> 00:07:14,000 that you know how to deal with. 97 00:07:14,000 --> 00:07:24,630 OK, and there is a rule that H n m 1 over E n 0 minus E m 98 00:07:24,630 --> 00:07:29,677 0 absolute value is much less than 1. 99 00:07:29,677 --> 00:07:30,510 That's what we mean. 100 00:07:30,510 --> 00:07:31,227 Yes? 101 00:07:31,227 --> 00:07:33,060 AUDIENCE: Are you making a deliberate choice 102 00:07:33,060 --> 00:07:36,971 to put all the badness in the first-order perturbation? 103 00:07:36,971 --> 00:07:37,720 ROBERT FIELD: Yes. 104 00:07:37,720 --> 00:07:41,650 AUDIENCE: Then, what goes into higher order, 105 00:07:41,650 --> 00:07:43,210 if we make that deliberate? 106 00:07:43,210 --> 00:07:44,960 ROBERT FIELD: OK, you can say, well, we've 107 00:07:44,960 --> 00:07:54,890 got things that are important, that obviously 108 00:07:54,890 --> 00:07:56,480 affect the energy levels, and there's 109 00:07:56,480 --> 00:07:59,940 things that are smaller, like hyperfine structure. 110 00:07:59,940 --> 00:08:03,490 And when you use the word hyper, you generally mean it's small, 111 00:08:03,490 --> 00:08:07,610 and so what ends up happening is that you do the big picture, 112 00:08:07,610 --> 00:08:10,820 and then you see some more details. 113 00:08:10,820 --> 00:08:14,840 But it's only a very small number 114 00:08:14,840 --> 00:08:18,140 of people who actually segregate the badness 115 00:08:18,140 --> 00:08:20,720 into real bad and not so bad. 116 00:08:24,470 --> 00:08:31,470 OK, so this works only when this off-diagonal matrix element 117 00:08:31,470 --> 00:08:34,530 is smaller than the energy difference, 118 00:08:34,530 --> 00:08:36,090 and that's going to be true. 119 00:08:39,549 --> 00:08:42,850 Now, here is an infinite matrix, and we're 120 00:08:42,850 --> 00:08:48,670 interested in a certain space, a state space, which 121 00:08:48,670 --> 00:08:51,300 our experiments are designed to measure, 122 00:08:51,300 --> 00:08:53,860 and it depends on what experiment you do. 123 00:08:53,860 --> 00:08:55,870 This might be here, it might be somewhere 124 00:08:55,870 --> 00:09:02,440 in the middle, and the rest, we're not interested, 125 00:09:02,440 --> 00:09:03,790 because it's far away. 126 00:09:03,790 --> 00:09:07,300 Because the energy denominator is so large 127 00:09:07,300 --> 00:09:11,140 that the effect on the energy levels can be ignored, 128 00:09:11,140 --> 00:09:14,890 or you can say, well, we're not really ignoring it. 129 00:09:14,890 --> 00:09:17,560 We're folding the effect of all these levels 130 00:09:17,560 --> 00:09:19,960 through the matrix elements in here 131 00:09:19,960 --> 00:09:24,150 into here by second-order perturbation theory. 132 00:09:24,150 --> 00:09:26,270 We could in principle do that. 133 00:09:26,270 --> 00:09:28,310 We certainly are allowed to think 134 00:09:28,310 --> 00:09:32,630 that we are getting rid of this state space by, in principle, 135 00:09:32,630 --> 00:09:35,480 folding it into this block. 136 00:09:35,480 --> 00:09:40,580 Now, in this block of levels that we care about-- 137 00:09:40,580 --> 00:09:44,480 now I always like to draw these infinite matrices, 138 00:09:44,480 --> 00:09:46,707 or these matrices, by just saying, OK, 139 00:09:46,707 --> 00:09:47,540 here's the diagonal. 140 00:09:50,220 --> 00:09:58,380 And there might be a couple of levels within this state space 141 00:09:58,380 --> 00:10:01,770 we're interested in which, because of accident 142 00:10:01,770 --> 00:10:06,840 or because of something evil, their energy denominator 143 00:10:06,840 --> 00:10:12,900 is small compared to the zero-order energy differences. 144 00:10:12,900 --> 00:10:17,980 But that's usually true for only a few accidentally 145 00:10:17,980 --> 00:10:21,300 degenerate states, and we deal with them 146 00:10:21,300 --> 00:10:24,600 by using the machinery we obtained 147 00:10:24,600 --> 00:10:29,060 from the two-level problem, and so there, we 148 00:10:29,060 --> 00:10:33,280 are actually diagonalizing a small dimension matrix. 149 00:10:33,280 --> 00:10:36,820 We're not diagonalizing it, our computer is diagonalizing it. 150 00:10:36,820 --> 00:10:39,550 You really don't care how it's done, 151 00:10:39,550 --> 00:10:43,030 because you have a machine that will solve 152 00:10:43,030 --> 00:10:46,990 this like difficulty, and so it's OK for this rule 153 00:10:46,990 --> 00:10:49,030 to be violated. 154 00:10:49,030 --> 00:10:58,660 But so what happens, suppose we have two levels, and-- 155 00:10:58,660 --> 00:11:01,080 I just want to make sure I use the same notation 156 00:11:01,080 --> 00:11:03,910 as in the notes. 157 00:11:03,910 --> 00:11:07,690 So here, we have the two zero-order levels, 158 00:11:07,690 --> 00:11:12,940 and they interact and repel each other equal and opposite 159 00:11:12,940 --> 00:11:14,470 amounts. 160 00:11:14,470 --> 00:11:25,430 And these states, which are E2 and E1, are not pure state 2. 161 00:11:25,430 --> 00:11:28,230 They have a mixture of state 1 in them. 162 00:11:28,230 --> 00:11:33,720 Now, in spectroscopy, we always observe these levels 163 00:11:33,720 --> 00:11:35,290 by transitions. 164 00:11:35,290 --> 00:11:39,300 And so suppose we have state 0 down here, 165 00:11:39,300 --> 00:11:48,390 and let us say that this transition between the zero 166 00:11:48,390 --> 00:11:51,105 level and the zero-order level 1 is allowed. 167 00:11:54,510 --> 00:12:02,010 I'll symbolize that by mu 1,0 is not equal to 0. 168 00:12:02,010 --> 00:12:08,550 And the transition between this level and this zero-order level 169 00:12:08,550 --> 00:12:14,160 is forbidden, mu 2,0 is equal to 0. 170 00:12:14,160 --> 00:12:18,270 So we call this a dark state and a bright state, 171 00:12:18,270 --> 00:12:20,360 and now these two states interact, 172 00:12:20,360 --> 00:12:22,920 and we have mixed states. 173 00:12:22,920 --> 00:12:26,100 So the real eigenstates, you wouldn't 174 00:12:26,100 --> 00:12:30,960 have expected to see this level in the spectrum. 175 00:12:30,960 --> 00:12:34,950 But you do, because it's borrowed some bright character 176 00:12:34,950 --> 00:12:39,360 through the perturbation interaction, 177 00:12:39,360 --> 00:12:42,190 and so there's two surprises. 178 00:12:42,190 --> 00:12:46,710 One is everything is sort of describable by a simple set 179 00:12:46,710 --> 00:12:51,760 of equations, and then there's a couple of deviations, 180 00:12:51,760 --> 00:12:54,220 and there's some extra levels that appear. 181 00:12:57,330 --> 00:13:00,090 That's information very rich, because if it's dark, 182 00:13:00,090 --> 00:13:01,410 you can't see it. 183 00:13:01,410 --> 00:13:02,160 It's not a ghost. 184 00:13:02,160 --> 00:13:04,080 It's there, but you can't see it. 185 00:13:04,080 --> 00:13:06,780 Well, the perturbation somehow pulls-- 186 00:13:06,780 --> 00:13:09,300 sometimes-- pulls the curtain back 187 00:13:09,300 --> 00:13:13,160 and enables you to see stuff you need to know about. 188 00:13:13,160 --> 00:13:18,680 Now, you can imagine exciting this two-level system 189 00:13:18,680 --> 00:13:20,390 with a short pulse of light, where 190 00:13:20,390 --> 00:13:23,660 the uncertainty broadening of the short pulse of light 191 00:13:23,660 --> 00:13:27,360 covers these two eigenstates. 192 00:13:27,360 --> 00:13:29,982 What happens then is you get quantum beats, 193 00:13:29,982 --> 00:13:31,440 and that's going to be on the exam. 194 00:13:36,310 --> 00:13:38,750 OK. 195 00:13:38,750 --> 00:13:43,210 So this is a local perturbation, and it's just something in here 196 00:13:43,210 --> 00:13:49,630 that spoils the general rule but it's very information-rich, 197 00:13:49,630 --> 00:13:54,160 and it also is pedagogically fun. 198 00:13:54,160 --> 00:13:55,810 OK. 199 00:13:55,810 --> 00:13:58,750 So now, I'm going to talk about stuff that I've 200 00:13:58,750 --> 00:14:00,610 talked about before, but I'm going 201 00:14:00,610 --> 00:14:03,430 to go a little deeper and slower, 202 00:14:03,430 --> 00:14:08,680 and so let's do this problem. 203 00:14:08,680 --> 00:14:12,160 We have a potential, and for molecules, we 204 00:14:12,160 --> 00:14:16,810 tend to use Q rather than R or X as the displacement 205 00:14:16,810 --> 00:14:17,670 from equilibrium. 206 00:14:17,670 --> 00:14:20,170 The harmonic oscillator coordinate, 207 00:14:20,170 --> 00:14:24,070 and it's the same thing and so we have-- 208 00:14:34,320 --> 00:14:38,030 OK, so this is an exactly solved problem, 209 00:14:38,030 --> 00:14:39,920 and this is something extra. 210 00:14:39,920 --> 00:14:42,370 And so this is a cubic anharmonicity, 211 00:14:42,370 --> 00:14:44,650 and this is a quartic anharmonicity. 212 00:14:44,650 --> 00:14:46,480 In the previous lecture, I talked 213 00:14:46,480 --> 00:14:50,560 about anharmonic couplings between different modes 214 00:14:50,560 --> 00:14:52,850 of the same polyatomic molecule. 215 00:14:52,850 --> 00:14:55,000 Here, this is really a simpler problem, 216 00:14:55,000 --> 00:14:57,430 and I probably should have talked about it first. 217 00:14:57,430 --> 00:14:59,290 But I didn't, because I want to go deeper 218 00:14:59,290 --> 00:15:05,360 here than the big picture, which I described last time. 219 00:15:05,360 --> 00:15:09,380 So we're going to have two terms that we 220 00:15:09,380 --> 00:15:13,247 are going to treat by perturbation theory, 221 00:15:13,247 --> 00:15:15,080 and there are several important things here. 222 00:15:17,980 --> 00:15:23,660 Molecular potential looks like this, not like that. 223 00:15:23,660 --> 00:15:24,160 Right? 224 00:15:24,160 --> 00:15:29,310 This is bond breaking, and it breaks at large displacement. 225 00:15:29,310 --> 00:15:31,400 There is no such thing as a bond breaking as you 226 00:15:31,400 --> 00:15:33,800 squeeze the molecules together. 227 00:15:33,800 --> 00:15:37,310 So this is a crazy idea, and how do you 228 00:15:37,310 --> 00:15:38,770 get an asymmetric potential? 229 00:15:38,770 --> 00:15:47,340 Well, it's cubic, and which sign of b is going to lead to this? 230 00:15:47,340 --> 00:15:47,840 Yes. 231 00:15:47,840 --> 00:15:48,715 AUDIENCE: [INAUDIBLE] 232 00:15:48,715 --> 00:15:49,920 ROBERT FIELD: Right. 233 00:15:49,920 --> 00:15:51,120 OK. 234 00:15:51,120 --> 00:15:59,230 Now, if we're really naive, we say, yeah, negative is good, 235 00:15:59,230 --> 00:16:06,130 but boom, we ignore that. 236 00:16:06,130 --> 00:16:11,290 We don't worry that, if we took this seriously 237 00:16:11,290 --> 00:16:14,000 at large enough displacement, the potential 238 00:16:14,000 --> 00:16:16,280 will go to minus infinity. 239 00:16:16,280 --> 00:16:20,830 We don't worry about tunneling through this barrier. 240 00:16:20,830 --> 00:16:24,280 We just use this to give us something 241 00:16:24,280 --> 00:16:26,350 that has the right shape, and that we can 242 00:16:26,350 --> 00:16:28,480 apply perturbation theory to. 243 00:16:28,480 --> 00:16:31,960 And we're not going to worry about using perturbation theory 244 00:16:31,960 --> 00:16:37,650 to capture this tunneling, at least not now and not 245 00:16:37,650 --> 00:16:40,205 in this course. 246 00:16:40,205 --> 00:16:42,480 OK. 247 00:16:42,480 --> 00:16:47,250 So one thing is we have this term, 248 00:16:47,250 --> 00:16:53,500 and there is something you know immediately about odd powers. 249 00:16:53,500 --> 00:16:57,850 They never have delta v equals 0 matrix elements, 250 00:16:57,850 --> 00:16:59,950 they never have diagonal elements, 251 00:16:59,950 --> 00:17:04,720 and so they do not contribute to the energy in first order. 252 00:17:08,319 --> 00:17:13,420 The only way you know the sign of this perturbation 253 00:17:13,420 --> 00:17:18,460 is from a non-zero first-order contribution, 254 00:17:18,460 --> 00:17:22,030 because when you do second-order perturbation theory, 255 00:17:22,030 --> 00:17:24,650 the matrix element gets squared, and the sign information 256 00:17:24,650 --> 00:17:26,950 is lost. 257 00:17:26,950 --> 00:17:31,540 So there's nothing so far that tells you 258 00:17:31,540 --> 00:17:37,910 what the sign of b is, and the energy level pattern that you 259 00:17:37,910 --> 00:17:43,610 would obtain from either sign of b would be the same. 260 00:17:43,610 --> 00:17:48,170 It's just one of the signs is completely ridiculous. 261 00:17:48,170 --> 00:17:56,320 Now Q to the 4th has selection rules delta v of 4, 2, 0, 262 00:17:56,320 --> 00:17:59,370 minus 2, and minus 4. 263 00:17:59,370 --> 00:18:02,940 So that delta v of 0 term does enter 264 00:18:02,940 --> 00:18:05,430 in first-order perturbation theory, 265 00:18:05,430 --> 00:18:13,600 and so you can determine the sign of any even perturbation 266 00:18:13,600 --> 00:18:15,695 which is useful. 267 00:18:15,695 --> 00:18:18,840 Now, what does this do? 268 00:18:18,840 --> 00:18:30,380 Well, one thing that Q does is, if c is positive, 269 00:18:30,380 --> 00:18:33,080 instead of having a harmonic oscillator, 270 00:18:33,080 --> 00:18:33,995 it makes it steeper. 271 00:18:37,800 --> 00:18:42,520 And if Q is negative, it makes it flatter. 272 00:18:42,520 --> 00:18:47,060 This is typical of bending vibrations, 273 00:18:47,060 --> 00:18:50,967 and so bending vibrations tend to have flat bottom potentials, 274 00:18:50,967 --> 00:18:52,550 but there there's also something else. 275 00:18:55,950 --> 00:18:58,440 Suppose we have two electronic states. 276 00:18:58,440 --> 00:19:01,490 Now, I am cheating, because I'm assuming 277 00:19:01,490 --> 00:19:04,080 you'll accept the idea that there is something 278 00:19:04,080 --> 00:19:06,320 we haven't talked about yet. 279 00:19:06,320 --> 00:19:09,040 But there are different potential curves, 280 00:19:09,040 --> 00:19:14,660 and it's mostly true for atomic molecules. 281 00:19:14,660 --> 00:19:17,230 These two states can't talk to each other 282 00:19:17,230 --> 00:19:19,330 at equilibrium, because a symmetry 283 00:19:19,330 --> 00:19:20,950 exists that prevents that. 284 00:19:20,950 --> 00:19:23,400 And as you move away from equilibrium, 285 00:19:23,400 --> 00:19:25,870 there's a perturbation that gets larger and larger. 286 00:19:25,870 --> 00:19:32,020 And as a result, what happens is that this potential 287 00:19:32,020 --> 00:19:33,395 does something like that. 288 00:19:38,330 --> 00:19:43,040 So you get either a flattening or an actual extra pair 289 00:19:43,040 --> 00:19:46,940 of minima, and this one gets sharper. 290 00:19:46,940 --> 00:19:48,880 That's called a vibronic interaction, 291 00:19:48,880 --> 00:19:53,690 and that's fairly important in polyatomic molecules. 292 00:19:53,690 --> 00:19:56,990 But so we can begin to understand these things 293 00:19:56,990 --> 00:20:04,510 just by dealing with these terms in the potential. 294 00:20:04,510 --> 00:20:06,720 OK. 295 00:20:06,720 --> 00:20:12,570 So now, we're off to the races, and my goal here 296 00:20:12,570 --> 00:20:16,830 is to make you comfortable with either the operator 297 00:20:16,830 --> 00:20:20,140 or the matrix notation, and so I'm 298 00:20:20,140 --> 00:20:23,440 going to go back and forth between them in what might 299 00:20:23,440 --> 00:20:26,190 seem to be a random manner. 300 00:20:26,190 --> 00:20:26,950 OK. 301 00:20:26,950 --> 00:20:31,420 Now, you know that a dagger. 302 00:20:31,420 --> 00:20:33,720 Now, we can call it a dagger with a hat, 303 00:20:33,720 --> 00:20:36,370 or we can call it a dagger double underline-- 304 00:20:36,370 --> 00:20:40,240 bold, hat, matrix, operator. 305 00:20:40,240 --> 00:20:43,970 And you know that this operates on v 306 00:20:43,970 --> 00:20:52,710 to give v plus 1 square root psi v plus 1, 307 00:20:52,710 --> 00:20:57,420 or in bracket notation, we could write this as v 308 00:20:57,420 --> 00:21:03,400 plus 1 a dagger v. 309 00:21:03,400 --> 00:21:05,260 Now this is a shorthand. 310 00:21:05,260 --> 00:21:07,240 It's a wonderful shorthand. 311 00:21:07,240 --> 00:21:11,130 It's easier to draw this than wave functions. 312 00:21:11,130 --> 00:21:12,630 But we have to know, OK, what does 313 00:21:12,630 --> 00:21:19,440 this look like in the matrix for a dagger? 314 00:21:19,440 --> 00:21:29,380 So here is a dagger, and it's a matrix, and what goes in here? 315 00:21:29,380 --> 00:21:33,350 Well, the first thing you do, this is infinite, 316 00:21:33,350 --> 00:21:35,900 so you need some sort of a way of drawing something 317 00:21:35,900 --> 00:21:38,930 that's infinite so that you understand what it is. 318 00:21:38,930 --> 00:21:42,860 And so the first thing you do is you know that there is no-- 319 00:21:42,860 --> 00:21:47,280 all of the diagonal elements are 0. 320 00:21:47,280 --> 00:21:51,120 Now, most of the elements in this matrix are 0, 321 00:21:51,120 --> 00:21:54,690 and you don't want to draw them, because you'll just be spending 322 00:21:54,690 --> 00:21:57,390 all your time writing 0's. 323 00:21:57,390 --> 00:22:03,410 So we want a shorthand, and so now this is a matrix element. 324 00:22:03,410 --> 00:22:09,860 This is the row, and this is the column, and so where 325 00:22:09,860 --> 00:22:15,870 do I put this square root of E plus 1? 326 00:22:15,870 --> 00:22:18,310 AUDIENCE: [INAUDIBLE] 327 00:22:21,740 --> 00:22:25,720 ROBERT FIELD: So here, we have the row. 328 00:22:29,300 --> 00:22:31,880 I get confused about this, so before I 329 00:22:31,880 --> 00:22:35,780 accept your answer, which I want to reject, 330 00:22:35,780 --> 00:22:39,810 I have to think carefully about it. 331 00:22:39,810 --> 00:22:46,385 So let us say this is 0, and this is 1, so we have the 1-- 332 00:22:50,880 --> 00:22:51,580 you're right. 333 00:22:54,260 --> 00:22:58,735 OK, and so we have the square root of 1, 334 00:22:58,735 --> 00:23:06,010 the square root of 2, square root of n. 335 00:23:06,010 --> 00:23:08,620 And everything else is 0, so we can write big 0's. 336 00:23:11,267 --> 00:23:12,850 So that's what this matrix looks like. 337 00:23:15,910 --> 00:23:21,280 Now, if we're using computers, instead of multiplying 338 00:23:21,280 --> 00:23:24,850 these matrices to have say Q to the 13th, 339 00:23:24,850 --> 00:23:30,080 or a dagger to the 13th, you can just multiply these matrices. 340 00:23:30,080 --> 00:23:33,660 That's an easy request for the computer. 341 00:23:33,660 --> 00:23:36,450 It's not such an easy request for you, 342 00:23:36,450 --> 00:23:38,700 but you end up getting matrices. 343 00:23:38,700 --> 00:23:41,610 When you have integer powers of these, 344 00:23:41,610 --> 00:23:47,100 you get a matrix with a diagonal and then 345 00:23:47,100 --> 00:23:53,154 another diagonal separated by a diagonal of 0's, and so you 346 00:23:53,154 --> 00:23:55,070 know what these things are going to look like. 347 00:24:00,240 --> 00:24:06,580 So you might ask, OK, what does a look like? 348 00:24:06,580 --> 00:24:08,260 And there are several ways to say 349 00:24:08,260 --> 00:24:10,870 what a is going to look like, because it's 350 00:24:10,870 --> 00:24:13,975 the conjugate transpose of a dagger. 351 00:24:17,010 --> 00:24:19,650 Well, these are all real numbers, 352 00:24:19,650 --> 00:24:21,790 and so all you do is flip this on its diagonal, 353 00:24:21,790 --> 00:24:22,590 and now you get a. 354 00:24:27,590 --> 00:24:29,450 Then there's another actor in this game, 355 00:24:29,450 --> 00:24:33,160 and that's n, the number operator, 356 00:24:33,160 --> 00:24:41,310 and the number operator is going to be a, a dagger 357 00:24:41,310 --> 00:24:43,470 or is it going to be a dagger a? 358 00:24:49,900 --> 00:24:51,992 So which is it? 359 00:24:51,992 --> 00:24:53,896 AUDIENCE: [INAUDIBLE] 360 00:24:55,324 --> 00:24:56,276 ROBERT FIELD: This. 361 00:24:56,276 --> 00:24:57,228 AUDIENCE: [INAUDIBLE] 362 00:24:57,228 --> 00:25:00,100 ROBERT FIELD: Yes. 363 00:25:00,100 --> 00:25:05,650 OK, one way to remember this is when 364 00:25:05,650 --> 00:25:09,040 you operate with either a or a dagger, 365 00:25:09,040 --> 00:25:11,484 it connects two vibrational levels, 366 00:25:11,484 --> 00:25:13,150 and the thing you put in the square root 367 00:25:13,150 --> 00:25:15,010 is the larger of the two quantum numbers. 368 00:25:18,740 --> 00:25:21,300 OK. 369 00:25:21,300 --> 00:25:25,860 So what would the number operator matrix look like? 370 00:25:36,060 --> 00:25:37,790 So what do I put here? 371 00:25:40,664 --> 00:25:42,101 Yes. 372 00:25:42,101 --> 00:25:44,496 AUDIENCE: [INAUDIBLE] 373 00:25:47,219 --> 00:25:49,510 ROBERT FIELD: But what's the lowest vibrational quantum 374 00:25:49,510 --> 00:25:49,870 number? 375 00:25:49,870 --> 00:25:50,730 AUDIENCE: [INAUDIBLE] 376 00:25:50,730 --> 00:25:51,563 ROBERT FIELD: Right. 377 00:25:56,920 --> 00:26:01,490 OK, and so once you've practiced a little bit, 378 00:26:01,490 --> 00:26:04,610 you can write these things. 379 00:26:04,610 --> 00:26:06,720 And it's not just an arbitrary thing, 380 00:26:06,720 --> 00:26:10,420 because you want to be able to visualize what you're doing. 381 00:26:10,420 --> 00:26:13,930 Because you're dealing with multiple infinities of objects, 382 00:26:13,930 --> 00:26:16,600 and you want to focus only on the ones you care about. 383 00:26:16,600 --> 00:26:20,500 And with a little bit of guidance from these pictures, 384 00:26:20,500 --> 00:26:22,480 you can do what you need to do. 385 00:26:27,824 --> 00:26:28,324 OK. 386 00:26:36,430 --> 00:26:39,850 Well, we know what the operator Q is 387 00:26:39,850 --> 00:26:43,660 in terms of a dimensionalist version of Q, 388 00:26:43,660 --> 00:26:49,570 and that's h bar over-- now, I'm going to be using this. 389 00:26:56,900 --> 00:26:58,710 OK, this is the dimensionalist Q. 390 00:26:58,710 --> 00:27:01,980 That's what the twiddle means, and these are constants. 391 00:27:01,980 --> 00:27:11,000 And now I'm using omega twiddle, because spectroscopists always 392 00:27:11,000 --> 00:27:16,140 use omega in wave number units, reciprocal centimeter units. 393 00:27:16,140 --> 00:27:19,340 Which is a terrible thing, because first of all, wave 394 00:27:19,340 --> 00:27:20,730 number doesn't have a unit. 395 00:27:20,730 --> 00:27:22,130 It's a quantity. 396 00:27:22,130 --> 00:27:24,680 And centimeters are things that we're not supposed to use, 397 00:27:24,680 --> 00:27:26,600 because we use MKS. 398 00:27:26,600 --> 00:27:29,120 But spectroscopists are stubborn, 399 00:27:29,120 --> 00:27:33,050 and when we observe transitions, we always 400 00:27:33,050 --> 00:27:37,620 talk about wave numbers not energy. 401 00:27:37,620 --> 00:27:41,790 And so the difference between wave numbers and energies 402 00:27:41,790 --> 00:27:46,350 is the factor of hc, not h bar c. 403 00:27:46,350 --> 00:27:49,630 So anyway, this is the conversion factor, when 404 00:27:49,630 --> 00:27:52,030 omega is in wave number units. 405 00:27:56,020 --> 00:28:07,480 And we can go further and relate Q twiddle well, 406 00:28:07,480 --> 00:28:11,070 let's just not do that. 407 00:28:11,070 --> 00:28:20,280 We can relate this to h bar over 4 pi 408 00:28:20,280 --> 00:28:26,680 c u omega square root times a plus a dagger. 409 00:28:26,680 --> 00:28:28,540 OK? 410 00:28:28,540 --> 00:28:32,680 So this is something we did before. 411 00:28:32,680 --> 00:28:46,940 Omega twiddle is k over mu square root 1 over 2 pi c-- 412 00:28:46,940 --> 00:28:49,400 hc. 413 00:28:49,400 --> 00:28:52,280 So this what you're used to, and this 414 00:28:52,280 --> 00:28:55,910 is the extra stuff that we have to carry along in order 415 00:28:55,910 --> 00:28:59,675 to work in wave number units. 416 00:28:59,675 --> 00:29:00,175 OK. 417 00:29:06,200 --> 00:29:09,170 So we have operators that can be expressed 418 00:29:09,170 --> 00:29:20,050 like a, a, a dagger, a or a dagger, a, a, a or anywhere you 419 00:29:20,050 --> 00:29:21,790 put the dagger anywhere. 420 00:29:21,790 --> 00:29:27,140 You look at this, and you say immediately I know two things. 421 00:29:27,140 --> 00:29:29,810 I know the selection rule. 422 00:29:29,810 --> 00:29:35,100 The selection rule is count up the a's and count up 423 00:29:35,100 --> 00:29:44,330 the a daggers, and so this is delta v of minus 2. 424 00:29:44,330 --> 00:29:46,100 So is this. 425 00:29:46,100 --> 00:29:50,730 All of the positions of the a dagger are delta v of minus 2, 426 00:29:50,730 --> 00:29:53,870 but the numbers, the matrix elements, are different. 427 00:29:53,870 --> 00:29:55,490 You know this from the last exam. 428 00:29:58,090 --> 00:30:07,300 So we can have v minus 2 v and whatever comes 429 00:30:07,300 --> 00:30:10,370 in here in one of those forms. 430 00:30:10,370 --> 00:30:12,460 So those are the only non-zero elements, 431 00:30:12,460 --> 00:30:16,210 and you know how to mechanically figure out 432 00:30:16,210 --> 00:30:17,930 what is the value of the matrix element. 433 00:30:21,130 --> 00:30:25,150 Now, in order to simplify the algebra, which is not 434 00:30:25,150 --> 00:30:30,400 essential to the physics, you want to take all of the terms 435 00:30:30,400 --> 00:30:37,120 that results say from Q to the 4th 436 00:30:37,120 --> 00:30:40,190 and arrange them according to selection rule. 437 00:30:40,190 --> 00:30:43,610 And then take all of the terms that have the same selection 438 00:30:43,610 --> 00:30:47,350 rule and combining them to a single number. 439 00:30:53,060 --> 00:30:56,610 And you use the computation rule to be 440 00:30:56,610 --> 00:30:59,745 able to reverse the order of terms. 441 00:31:03,371 --> 00:31:04,620 Well, I don't want to do that. 442 00:31:08,240 --> 00:31:13,780 So suppose we have an a, a dagger, and we can write 443 00:31:13,780 --> 00:31:19,072 that as a, a dagger plus a dagger, a. 444 00:31:19,072 --> 00:31:22,350 And so if we want to convert something 445 00:31:22,350 --> 00:31:25,590 like this to something like that, 446 00:31:25,590 --> 00:31:32,930 we know that this has a value of plus 1, so we can do that. 447 00:31:32,930 --> 00:31:36,080 And that's tedious, and you have to do it. 448 00:31:36,080 --> 00:31:39,290 But one of the things that is kind of nice 449 00:31:39,290 --> 00:31:41,750 is when you have a problem that's 450 00:31:41,750 --> 00:31:47,360 cubic or quartic or quintic, you mess around with this operator 451 00:31:47,360 --> 00:31:51,470 algebra once in your life, and you put it on a sheet of paper, 452 00:31:51,470 --> 00:31:52,530 and you refer to it. 453 00:31:52,530 --> 00:31:54,230 It doesn't matter what the molecule 454 00:31:54,230 --> 00:31:58,580 is, what the constant in front of Q to the 3rd or 4th or 13th 455 00:31:58,580 --> 00:31:59,540 is. 456 00:31:59,540 --> 00:32:02,777 If you've done the operator algebra, you're fine. 457 00:32:02,777 --> 00:32:04,860 Now, you might say, well, I don't want to do that. 458 00:32:04,860 --> 00:32:07,410 I'm going to have the computer do that. 459 00:32:07,410 --> 00:32:09,360 Well, fine, you can have the computer do that, 460 00:32:09,360 --> 00:32:13,020 and then a computer will tell you what the matrix looks like, 461 00:32:13,020 --> 00:32:15,260 and you can do what you need to do. 462 00:32:15,260 --> 00:32:15,930 OK. 463 00:32:15,930 --> 00:32:21,330 So spectroscopists call the vibrational energy formula 464 00:32:21,330 --> 00:32:26,360 G of v. I don't know why the letter G is always use, 465 00:32:26,360 --> 00:32:30,530 but it is, and so this is the same thing 466 00:32:30,530 --> 00:32:32,870 as vibrational energy. 467 00:32:32,870 --> 00:32:35,750 And the vibrational energy, I'm going to put the tilde on it. 468 00:32:35,750 --> 00:32:41,030 You will never find a tilde in any spectroscopy note paper. 469 00:32:41,030 --> 00:32:45,260 We assume that you understand that the only units 470 00:32:45,260 --> 00:32:50,230 for spectroscopic quantities are reciprocal centimeters, 471 00:32:50,230 --> 00:32:54,720 but for this purposes, I have finally caved, 472 00:32:54,720 --> 00:32:57,350 and I said, OK, I'm going to put the tildes on. 473 00:32:57,350 --> 00:33:10,730 So the energy levels, now you might ask, why this? 474 00:33:10,730 --> 00:33:13,140 This is the first anharmonicity constant. 475 00:33:13,140 --> 00:33:16,410 It's not a product of two numbers. 476 00:33:16,410 --> 00:33:19,580 It's just what people wrote originally, 477 00:33:19,580 --> 00:33:23,450 because they sort of thought of it as a product of two numbers, 478 00:33:23,450 --> 00:33:26,870 but it's really only one. 479 00:33:26,870 --> 00:33:33,470 And that's times v plus 1/2 squared, 480 00:33:33,470 --> 00:33:45,760 and then the next term is omega e, ye, e plus 1/2 cubed. 481 00:33:45,760 --> 00:33:48,370 So this is a dumb power series in the vibrational quantum 482 00:33:48,370 --> 00:33:52,860 number, and so in the spectrum, one 483 00:33:52,860 --> 00:33:58,050 is able to fit the spectrum to these sorts of things. 484 00:33:58,050 --> 00:34:00,270 So that's what you get experimentally, 485 00:34:00,270 --> 00:34:07,640 but what you want to know is we want 486 00:34:07,640 --> 00:34:11,210 to know the force constant, the reduced 487 00:34:11,210 --> 00:34:16,850 mass, the cubic anharmonicity constant, the quartic 488 00:34:16,850 --> 00:34:18,870 anharmonicity constant, and whatever. 489 00:34:18,870 --> 00:34:21,730 So these are structural parameters, 490 00:34:21,730 --> 00:34:24,050 and these are molecular parameters. 491 00:34:26,750 --> 00:34:29,540 People like to call them spectroscopic parameters, 492 00:34:29,540 --> 00:34:32,115 but that implies something more fundamental. 493 00:34:32,115 --> 00:34:35,429 These are just what you measure in the spectrum. 494 00:34:35,429 --> 00:34:39,110 And so we want to know the relationship between the things 495 00:34:39,110 --> 00:34:42,674 we measure and the things we want to know, 496 00:34:42,674 --> 00:34:44,590 and so that's what perturbation theory is for. 497 00:34:52,960 --> 00:34:54,070 OK. 498 00:34:54,070 --> 00:34:56,469 Now, to risk boring you, I'm just 499 00:34:56,469 --> 00:35:00,410 going to go over material that we've done before. 500 00:35:00,410 --> 00:35:06,720 So if we have Q to the n, we have 501 00:35:06,720 --> 00:35:12,440 this constant out in front, h bar over 4 pi c 502 00:35:12,440 --> 00:35:23,170 mu omega twiddle to the n over 2, a plus a dagger to the n. 503 00:35:28,090 --> 00:35:31,000 And so we're going to be constantly dealing 504 00:35:31,000 --> 00:35:34,060 with terms like a plus a dagger squared 505 00:35:34,060 --> 00:35:37,745 and plus a dagger cubed and so on. 506 00:35:37,745 --> 00:35:40,120 These are the things I said you're going to work out once 507 00:35:40,120 --> 00:35:42,610 in your life and either remember or just 508 00:35:42,610 --> 00:35:45,520 become so practiced with it, you'll do it faster. 509 00:35:48,470 --> 00:35:52,430 And so these contain the values of the matrix elements 510 00:35:52,430 --> 00:35:53,540 and the selection rules. 511 00:35:57,200 --> 00:35:58,200 OK. 512 00:35:58,200 --> 00:36:06,170 So I'm going to do an example of the cube, 513 00:36:06,170 --> 00:36:10,850 and that's, of course, this thing in constant the 3/2 514 00:36:10,850 --> 00:36:15,870 and then a cubed plus a dagger cubed 515 00:36:15,870 --> 00:36:19,430 plus a whole bunch of terms which have two a's 516 00:36:19,430 --> 00:36:21,770 and one a dagger. 517 00:36:21,770 --> 00:36:27,990 So let's just put it a squared and a dagger. 518 00:36:27,990 --> 00:36:29,810 This isn't a computation rule. 519 00:36:29,810 --> 00:36:32,540 This is just three terms that you have to deal with 520 00:36:32,540 --> 00:36:36,020 and three terms that have a dagger squared and a. 521 00:36:38,570 --> 00:36:41,300 So that's what you do your work on, 522 00:36:41,300 --> 00:36:44,420 because you don't want to be messing around 523 00:36:44,420 --> 00:36:48,700 once you start doing the perturbation sums, 524 00:36:48,700 --> 00:36:50,980 because it's ugly enough. 525 00:36:50,980 --> 00:36:53,800 So you want to simplify this as much as possible, and you do. 526 00:36:57,880 --> 00:37:01,460 And so the purpose is this is delta v of minus 3. 527 00:37:01,460 --> 00:37:03,770 This is delta v of plus 3. 528 00:37:03,770 --> 00:37:05,520 This is delta v of minus 1. 529 00:37:05,520 --> 00:37:08,300 This is delta v of plus 1. 530 00:37:08,300 --> 00:37:14,060 You arrange the terms according to selection rules, 531 00:37:14,060 --> 00:37:16,450 and so what you end up getting is 532 00:37:16,450 --> 00:37:25,230 Q cubed is equal to this thing, to the 3/2 times 533 00:37:25,230 --> 00:37:41,840 a cubed plus 3 a n plus 3 a dagger n plus 1 plus a. 534 00:37:41,840 --> 00:37:46,470 OK, the algebra is tedious, pretty simple. 535 00:37:46,470 --> 00:37:49,080 And what you usually want to do is 536 00:37:49,080 --> 00:37:52,170 have the thing that changes the vibrational quantum 537 00:37:52,170 --> 00:37:55,170 number after the thing that preserves it, 538 00:37:55,170 --> 00:37:58,230 because then it's easy just to write down these matrix 539 00:37:58,230 --> 00:38:00,920 elements just by inspection. 540 00:38:00,920 --> 00:38:01,730 OK? 541 00:38:01,730 --> 00:38:03,330 You could do it the other way around. 542 00:38:03,330 --> 00:38:07,550 It's more complicated if you put the n first and then the a, 543 00:38:07,550 --> 00:38:11,000 but it's up to you. 544 00:38:11,000 --> 00:38:14,760 So these then are what you work on, 545 00:38:14,760 --> 00:38:17,371 and now we start doing non-degenerate perturbation 546 00:38:17,371 --> 00:38:17,870 theory. 547 00:38:17,870 --> 00:38:21,200 The first thing you do is you want 548 00:38:21,200 --> 00:38:24,620 to know, well, what is the first-order correction 549 00:38:24,620 --> 00:38:26,210 to the energy? 550 00:38:26,210 --> 00:38:37,995 And if this is Q to the 3rd power, then this, that's 0. 551 00:38:40,710 --> 00:38:42,870 So you like 0's, but in this case, 552 00:38:42,870 --> 00:38:45,570 you're kind of disappointed, because you 553 00:38:45,570 --> 00:38:49,230 don't know what the sign of the coefficient of Q 554 00:38:49,230 --> 00:38:54,210 to the 3rd power is from any experiment or at least 555 00:38:54,210 --> 00:38:57,240 any experiment at the level we've described. 556 00:38:57,240 --> 00:39:02,760 When we introduce rotation as well as vibration, 557 00:39:02,760 --> 00:39:05,130 there will be something that reports the sign 558 00:39:05,130 --> 00:39:09,300 of the coefficient of Q 3rd. 559 00:39:09,300 --> 00:39:13,470 So now, we're stuck, and we have to start doing 560 00:39:13,470 --> 00:39:15,930 all of this second-order stuff. 561 00:39:24,610 --> 00:39:25,270 OK. 562 00:39:25,270 --> 00:39:31,220 Well, we have this bQ to the 3rd power, 563 00:39:31,220 --> 00:39:35,480 and so we get squares of the matrix element. 564 00:39:35,480 --> 00:39:37,510 So we get a b squared. 565 00:39:37,510 --> 00:39:45,615 We get this thing to the 3rd power, 566 00:39:45,615 --> 00:39:47,490 because we're squaring the matrix element, so 567 00:39:47,490 --> 00:39:51,030 this bunch of constants. 568 00:39:51,030 --> 00:39:56,370 And then, you get a matrix element v prime, 569 00:39:56,370 --> 00:40:07,360 some operator v, then we get an energy denominator. 570 00:40:07,360 --> 00:40:10,380 Now, there's a lot of symbols, but we 571 00:40:10,380 --> 00:40:17,940 want to simplify things as much as possible, hc omega times v 572 00:40:17,940 --> 00:40:20,670 minus v prime. 573 00:40:20,670 --> 00:40:23,670 So this is matrix element squared over an energy 574 00:40:23,670 --> 00:40:27,150 denominator, and what you really want to know 575 00:40:27,150 --> 00:40:30,580 is what is the quantum number dependence of everything? 576 00:40:30,580 --> 00:40:34,710 OK, so the operator here is either Q dagger cubed, 577 00:40:34,710 --> 00:40:37,770 Q dagger-- 578 00:40:37,770 --> 00:40:47,940 I'm sorry, a cubed, a times n, a dagger n plus 1, or a dagger 579 00:40:47,940 --> 00:40:49,140 cubed. 580 00:40:49,140 --> 00:40:52,980 And so we know how to write all of these matrix elements, 581 00:40:52,980 --> 00:40:55,590 trivially, no work. 582 00:40:55,590 --> 00:40:59,070 Once we've simplified here, it's really trivial 583 00:40:59,070 --> 00:41:04,910 to write the squared matrix element. 584 00:41:07,620 --> 00:41:18,030 OK, so we do it, and so we arrange things 585 00:41:18,030 --> 00:41:23,985 according to delta v and we have delta v 586 00:41:23,985 --> 00:41:30,600 of plus 3, plus 1, minus 1, and minus 3. 587 00:41:30,600 --> 00:41:36,450 And so what we get from the square of the matrix element 588 00:41:36,450 --> 00:41:47,470 a dagger cubed, we get v plus 1, v plus 2, v plus 3, 589 00:41:47,470 --> 00:41:52,180 and we also have an energy denominator. 590 00:41:52,180 --> 00:41:57,820 And we're going to get for this one 1 over minus 3, 591 00:41:57,820 --> 00:42:05,380 because the initial quantum number is v, 592 00:42:05,380 --> 00:42:10,420 and the second quantum number, v prime, is v plus 3, 593 00:42:10,420 --> 00:42:14,520 and so we get a minus 3 in the energy denominator. 594 00:42:14,520 --> 00:42:17,430 And then the plus 1, that comes out to be-- 595 00:42:23,106 --> 00:42:25,800 I've got it in a different order in my notes-- 596 00:42:25,800 --> 00:42:41,550 that comes out to be 9, v plus 1, v plus 2, v plus 1 squared, 597 00:42:41,550 --> 00:42:45,423 and the energy denominator for this is 1 over minus 1. 598 00:42:48,261 --> 00:42:49,090 OK. 599 00:42:49,090 --> 00:42:50,630 Then, we have the-- 600 00:42:50,630 --> 00:42:54,210 I'm going to skip this one-- we have the matrix element here, 601 00:42:54,210 --> 00:42:59,537 and that's going to be v minus 1, v minus 2. 602 00:42:59,537 --> 00:43:01,120 Remember, we're squaring these things. 603 00:43:01,120 --> 00:43:05,420 So that's why we don't have those square roots anymore, 604 00:43:05,420 --> 00:43:07,560 and we have an energy denominator 1 over 3. 605 00:43:10,430 --> 00:43:14,320 OK, now advice-- you don't like this, 606 00:43:14,320 --> 00:43:17,740 and you want to minimize your effort. 607 00:43:17,740 --> 00:43:20,470 And so it turns out that if you take the terms 608 00:43:20,470 --> 00:43:23,020 with the equal and opposite energy denominators 609 00:43:23,020 --> 00:43:26,975 and combine them, simplifications occur. 610 00:43:26,975 --> 00:43:28,600 One of the things you can see is you're 611 00:43:28,600 --> 00:43:31,060 going to have a v cubed here, and you're 612 00:43:31,060 --> 00:43:33,010 going to have a v cubed here. 613 00:43:33,010 --> 00:43:39,930 They're going to cancel, because we have a 1 over minus 3 614 00:43:39,930 --> 00:43:42,750 and a 1 over plus 3. 615 00:43:42,750 --> 00:43:45,000 So you get an algebraic simplification 616 00:43:45,000 --> 00:43:50,280 when you take these terms pairwise and combine them. 617 00:43:50,280 --> 00:43:54,700 Now, you're never in your life going to do this, 618 00:43:54,700 --> 00:43:58,660 but if you did do it, this is how you end up 619 00:43:58,660 --> 00:44:04,390 with formulas which are simple. 620 00:44:04,390 --> 00:44:06,670 They're horrible getting there, unless you 621 00:44:06,670 --> 00:44:07,910 know how to get there. 622 00:44:07,910 --> 00:44:08,860 So now there's a rule. 623 00:44:15,920 --> 00:44:19,990 So if the perturbation is Q to the n, 624 00:44:19,990 --> 00:44:33,800 then the highest-order term involves v plus 1/2 625 00:44:33,800 --> 00:44:35,133 to the n minus 1. 626 00:44:38,837 --> 00:44:42,240 The reason for that is in the matrix element 627 00:44:42,240 --> 00:44:49,910 the highest-order term is v plus 1/2 to the 3 over 2, 628 00:44:49,910 --> 00:44:54,399 and then you square it, you get to the 3rd power, 629 00:44:54,399 --> 00:44:56,065 and then the highest-order term cancels. 630 00:45:01,640 --> 00:45:09,730 So we know that if we're dealing with Q to the 3rd power, 631 00:45:09,730 --> 00:45:14,750 we're going to get a term v plus 1/2 squared. 632 00:45:14,750 --> 00:45:19,550 If we're dealing with Q to the 4th power, 633 00:45:19,550 --> 00:45:22,270 well, then we get something from second order 634 00:45:22,270 --> 00:45:25,040 from the first-order correction to the energy 635 00:45:25,040 --> 00:45:29,270 a delta v of one matrix element. 636 00:45:29,270 --> 00:45:34,760 What we've done is square it, and we get v plus 1/2 637 00:45:34,760 --> 00:45:35,930 squared also. 638 00:45:39,260 --> 00:45:45,070 OK and the off-diagonal matrix element, 639 00:45:45,070 --> 00:45:49,660 we're going to get from the highest order 640 00:45:49,660 --> 00:45:55,140 from the off-diagonal is v plus 1/2 cubed. 641 00:45:55,140 --> 00:45:58,500 So one of the things that is great about the algebra 642 00:45:58,500 --> 00:46:00,420 is with a little bit of practice, 643 00:46:00,420 --> 00:46:07,470 you know how to organize things, and if you do the algebra, 644 00:46:07,470 --> 00:46:09,660 you collect the highest-order terms. 645 00:46:09,660 --> 00:46:12,060 Then, there's that kind of cascading result, 646 00:46:12,060 --> 00:46:16,410 and you get the lowest-order terms in simplified form. 647 00:46:16,410 --> 00:46:19,500 So this is irrelevant, but this is how you do it, 648 00:46:19,500 --> 00:46:21,822 if you're a professional. 649 00:46:21,822 --> 00:46:27,110 So with these results, we can determine 650 00:46:27,110 --> 00:46:30,920 the relationship between these molecular constants. 651 00:46:30,920 --> 00:46:31,910 Where did I put them? 652 00:46:31,910 --> 00:46:33,110 Oh, probably on this board. 653 00:46:36,120 --> 00:46:38,580 No. 654 00:46:38,580 --> 00:46:39,840 Oh yeah, it's right here. 655 00:46:39,840 --> 00:46:43,130 So we have omega e, omega e xe, omega e ye, 656 00:46:43,130 --> 00:46:46,010 and we have then the relationships 657 00:46:46,010 --> 00:46:48,770 between these things which you measure 658 00:46:48,770 --> 00:46:50,990 and these things that you want to know. 659 00:46:50,990 --> 00:46:53,240 So the stuff you want to know is encoded 660 00:46:53,240 --> 00:46:56,420 in the spectrum in a not particularly complicated way-- 661 00:46:56,420 --> 00:46:58,070 it's just not a very interesting way, 662 00:46:58,070 --> 00:47:02,990 but you have to do it in order to get it. 663 00:47:02,990 --> 00:47:06,960 OK, so I've got just a few minutes left, 664 00:47:06,960 --> 00:47:09,230 but I want to get to the really interesting stuff. 665 00:47:12,330 --> 00:47:24,720 So if you have some expression for the potential, 666 00:47:24,720 --> 00:47:29,910 it's easy to go and get the expression for the energy 667 00:47:29,910 --> 00:47:34,700 levels and the wave functions. 668 00:47:34,700 --> 00:47:37,950 And you can use the vector picture or the wave function 669 00:47:37,950 --> 00:47:38,450 picture. 670 00:47:38,450 --> 00:47:42,860 It doesn't matter, and so this is 671 00:47:42,860 --> 00:47:45,320 enough to do the spectroscopy. 672 00:47:45,320 --> 00:47:49,410 It tells you not only where are the energies, 673 00:47:49,410 --> 00:47:52,670 but because some transitions are supposed 674 00:47:52,670 --> 00:47:56,600 to be weak because of the dipole selection rule 675 00:47:56,600 --> 00:47:58,970 or whatever, it says, well, there 676 00:47:58,970 --> 00:48:01,460 are some transitions that have borrowed intensity. 677 00:48:05,780 --> 00:48:08,930 It also tells you-- 678 00:48:08,930 --> 00:48:12,970 suppose we make a coherent superposition state at T 679 00:48:12,970 --> 00:48:18,410 equals 0 using a short pulse, and suppose 680 00:48:18,410 --> 00:48:23,520 in the linear combination of zero-order states, 681 00:48:23,520 --> 00:48:27,380 there is only one that is bright. 682 00:48:27,380 --> 00:48:30,020 And so then it tells us, if we know 683 00:48:30,020 --> 00:48:34,880 how to go from basis states to eigenstates, 684 00:48:34,880 --> 00:48:39,800 we can go backwards, and we can write the expression 685 00:48:39,800 --> 00:48:44,240 for the T equals 0 superposition in terms 686 00:48:44,240 --> 00:48:48,520 of some sum over psi v0 cv. 687 00:48:54,030 --> 00:48:55,770 And if we have this, as long as we 688 00:48:55,770 --> 00:49:00,730 are writing a superposition in terms of eigenstates, 689 00:49:00,730 --> 00:49:05,156 we know immediately how to get to this, 690 00:49:05,156 --> 00:49:06,655 and then we've got all the dynamics. 691 00:49:10,060 --> 00:49:13,870 So the perturbation theory enables 692 00:49:13,870 --> 00:49:17,590 you to say, if I want to work in the time domain, 693 00:49:17,590 --> 00:49:19,420 I know what to do. 694 00:49:23,920 --> 00:49:25,710 You're going to get-- 695 00:49:25,710 --> 00:49:26,620 in the time domain-- 696 00:49:26,620 --> 00:49:30,670 you're going to get a signal that oscillates, 697 00:49:30,670 --> 00:49:35,080 and it oscillates at frequencies corresponding to energy level 698 00:49:35,080 --> 00:49:39,590 differences divided by h bar. 699 00:49:39,590 --> 00:49:44,640 And so what frequencies will appear in the Fourier transform 700 00:49:44,640 --> 00:49:51,780 of the spectrum, and what are the amplitudes of those Fourier 701 00:49:51,780 --> 00:49:53,190 components? 702 00:49:53,190 --> 00:49:55,860 You can calculate all of those stuff. 703 00:49:55,860 --> 00:49:57,870 It all comes from perturbation theory. 704 00:49:57,870 --> 00:50:02,670 These mixing coefficients you get by perturbation theory. 705 00:50:02,670 --> 00:50:07,340 And remember, if you have a transformation that 706 00:50:07,340 --> 00:50:15,420 diagonalizes the Hamiltonian that the eigenstates correspond 707 00:50:15,420 --> 00:50:21,720 to, the columns of T dagger, and the expression 708 00:50:21,720 --> 00:50:25,320 of the zero-order states, in terms of the eigenstates, 709 00:50:25,320 --> 00:50:28,440 corresponds to either the columns of T 710 00:50:28,440 --> 00:50:30,216 or the rows of T dagger. 711 00:50:33,120 --> 00:50:35,430 So once you do the perturbation theory, 712 00:50:35,430 --> 00:50:42,630 you can go to the frequency domain 713 00:50:42,630 --> 00:50:46,140 spectrum with intensities and frequency 714 00:50:46,140 --> 00:50:51,160 or the time domain spectrum with amplitude and frequencies. 715 00:50:51,160 --> 00:50:52,110 It's all there. 716 00:50:52,110 --> 00:50:53,980 This is a complete tool. 717 00:50:53,980 --> 00:50:56,220 It's the kind of tool that you can 718 00:50:56,220 --> 00:50:59,910 use for an enormous number of problems, 719 00:50:59,910 --> 00:51:02,640 and so you'd better get comfortable with perturbation 720 00:51:02,640 --> 00:51:06,090 theory, because the people who aren't comfortable 721 00:51:06,090 --> 00:51:09,242 can't do anything except talk about it. 722 00:51:09,242 --> 00:51:12,860 But if you want to actually solve problems, especially 723 00:51:12,860 --> 00:51:15,530 problems on an exam, you want to know 724 00:51:15,530 --> 00:51:17,750 how to use perturbation theory. 725 00:51:17,750 --> 00:51:22,130 And you also want to know how to read and construct 726 00:51:22,130 --> 00:51:26,090 the relevant notation in the vector picture, 727 00:51:26,090 --> 00:51:29,090 because a vector picture and the matrix picture 728 00:51:29,090 --> 00:51:31,790 is the one where you see the entire structure 729 00:51:31,790 --> 00:51:33,900 of the problem. 730 00:51:33,900 --> 00:51:38,610 And you can decide on how you're going to organize your time 731 00:51:38,610 --> 00:51:41,110 or what are the important things that I'm going 732 00:51:41,110 --> 00:51:43,460 to get from this analysis. 733 00:51:43,460 --> 00:51:46,780 And so it's much better than the Schrodinger picture, 734 00:51:46,780 --> 00:51:49,690 because with the Schrodinger picture, 735 00:51:49,690 --> 00:51:52,210 you're just solving a differential equation and one 736 00:51:52,210 --> 00:51:55,060 problem at a time, whereas with the matrix picture, 737 00:51:55,060 --> 00:51:58,510 you're solving all problems at once. 738 00:51:58,510 --> 00:52:02,310 This is really a wonderful thing, 739 00:52:02,310 --> 00:52:05,780 and so that's why I'm taking a very different path 740 00:52:05,780 --> 00:52:08,730 from what is in the textbooks. 741 00:52:08,730 --> 00:52:10,980 Your wonderful textbook McQuarrie 742 00:52:10,980 --> 00:52:14,650 does not do second-order perturbation theory. 743 00:52:14,650 --> 00:52:19,190 So nothing you want can be calculated, unless you're 744 00:52:19,190 --> 00:52:24,560 dealing with NMR, and you're dealing with magnetic dipole 745 00:52:24,560 --> 00:52:25,730 transitions. 746 00:52:25,730 --> 00:52:31,400 And then, you can get a lot of good stuff 747 00:52:31,400 --> 00:52:33,170 from first-order perturbation theory, 748 00:52:33,170 --> 00:52:38,370 and you can avoid second order until you grow up. 749 00:52:38,370 --> 00:52:39,700 OK, I'm done. 750 00:52:39,700 --> 00:52:42,390 I'll be talking about rigid rotor next time, 751 00:52:42,390 --> 00:52:46,310 and I will be talking about it also in an unconventional way.