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 Open Courseware 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 Open Courseware 7 00:00:17,250 --> 00:00:18,210 at ocw.mit.edu. 8 00:00:22,110 --> 00:00:24,870 ROBERT FIELD: Today is the first of two lectures 9 00:00:24,870 --> 00:00:26,790 on the rigid rotor. 10 00:00:26,790 --> 00:00:30,300 And the rigid rotor is really our first glimpse 11 00:00:30,300 --> 00:00:32,340 of central force problems. 12 00:00:32,340 --> 00:00:35,940 And so although the rigid rotor is an exactly solved problem, 13 00:00:35,940 --> 00:00:38,820 it's also a problem that we-- 14 00:00:38,820 --> 00:00:42,160 it involves something that is universal. 15 00:00:42,160 --> 00:00:46,500 If we have a spherical object, then the angular part 16 00:00:46,500 --> 00:00:50,220 of the Hamiltonian for that spherical object 17 00:00:50,220 --> 00:00:52,810 is solved by the rigid rotor. 18 00:00:52,810 --> 00:00:57,280 And so if we understand the rigid rotor 19 00:00:57,280 --> 00:01:01,380 and know how to draw pictures, we're 20 00:01:01,380 --> 00:01:04,650 going to be able to understand the universal part 21 00:01:04,650 --> 00:01:07,410 of all central force problems. 22 00:01:07,410 --> 00:01:09,240 So it's not just a curiosity. 23 00:01:09,240 --> 00:01:11,910 It's a really major thing. 24 00:01:11,910 --> 00:01:12,420 OK. 25 00:01:12,420 --> 00:01:16,170 And I'm going to do a lot of unconventional stuff 26 00:01:16,170 --> 00:01:18,360 in this lecture and the next lecture, 27 00:01:18,360 --> 00:01:20,500 as far as the rigid rotor is concerned. 28 00:01:20,500 --> 00:01:24,750 Because it's covered to death, but with lots 29 00:01:24,750 --> 00:01:28,140 of equations in all textbooks. 30 00:01:28,140 --> 00:01:34,260 And there is very little effort to give you 31 00:01:34,260 --> 00:01:36,480 independent insights. 32 00:01:36,480 --> 00:01:40,740 And so my discussion of the vector model 33 00:01:40,740 --> 00:01:43,290 is something that I consider really special. 34 00:01:43,290 --> 00:01:45,900 And what I want you to be able to do 35 00:01:45,900 --> 00:01:49,290 is to draw cartoons that capture essentially 36 00:01:49,290 --> 00:01:51,660 all of the important points. 37 00:01:51,660 --> 00:01:55,305 And that you won't find anywhere except in my notes. 38 00:01:58,650 --> 00:02:01,080 But I think it's really important 39 00:02:01,080 --> 00:02:04,390 that you have your own point of view on these problems. 40 00:02:04,390 --> 00:02:10,229 So let me just do a quick review, with the exam in mind, 41 00:02:10,229 --> 00:02:14,520 of the perturbation theory stuff. 42 00:02:14,520 --> 00:02:17,050 So we had a harmonic oscillator. 43 00:02:17,050 --> 00:02:22,080 And it's perturbed by some anharmonicity terms. 44 00:02:22,080 --> 00:02:28,770 So the potential is going to 1/2 kQ squared plus bQ 45 00:02:28,770 --> 00:02:35,040 cubed plus cQ to the fourth. 46 00:02:35,040 --> 00:02:36,240 And we could go on. 47 00:02:36,240 --> 00:02:37,410 But this is enough. 48 00:02:37,410 --> 00:02:41,010 Because it gives a glimpse of all of the important stuff 49 00:02:41,010 --> 00:02:44,310 that emerges qualitatively as well as quantitatively 50 00:02:44,310 --> 00:02:47,460 from perturbation theory. 51 00:02:47,460 --> 00:02:53,130 So this is part of the H0 problem. 52 00:02:53,130 --> 00:02:56,960 And this is H1. 53 00:02:56,960 --> 00:03:02,390 And so you know now that doing second order perturbation 54 00:03:02,390 --> 00:03:05,060 theory is tedious and ugly. 55 00:03:05,060 --> 00:03:09,560 And it's easy to get overwhelmed by the equations. 56 00:03:09,560 --> 00:03:12,020 But what I want to do is to just review 57 00:03:12,020 --> 00:03:16,950 how you get the crucial structure of the problem. 58 00:03:16,950 --> 00:03:23,590 So the Q cubed' term-- 59 00:03:23,590 --> 00:03:27,680 there are no diagonal elements of the Q cubed term. 60 00:03:27,680 --> 00:03:31,790 And so we're always going to be talking about the second order 61 00:03:31,790 --> 00:03:33,650 corrections to the energy. 62 00:03:33,650 --> 00:03:38,000 The selection rules are delta v equals plus and minus 3 63 00:03:38,000 --> 00:03:42,522 and plus n minus 1. 64 00:03:42,522 --> 00:03:47,200 And so what I told you is, if you're going to do the algebra, 65 00:03:47,200 --> 00:03:50,010 which you probably don't want to ever do, 66 00:03:50,010 --> 00:03:52,980 you're going to want to combine the delta v of plus 3 67 00:03:52,980 --> 00:03:56,010 and minus 3 terms of the perturbation sum. 68 00:03:56,010 --> 00:03:58,650 Because the algebra simplifies. 69 00:03:58,650 --> 00:04:01,290 They have similar expressions. 70 00:04:01,290 --> 00:04:09,810 And you have a denominator, which is 3 Hc omega and minus 3 71 00:04:09,810 --> 00:04:10,590 Hc omega. 72 00:04:10,590 --> 00:04:11,440 That factors out. 73 00:04:11,440 --> 00:04:15,150 And you just have a plus 3 and minus 3 in the denominator. 74 00:04:15,150 --> 00:04:19,110 And similarly here, so when you do this, 75 00:04:19,110 --> 00:04:21,300 you lose the highest order term. 76 00:04:21,300 --> 00:04:22,830 What's the highest order term? 77 00:04:22,830 --> 00:04:29,160 It's v plus 1/2 to the third power. 78 00:04:29,160 --> 00:04:33,510 Because we have delta v of 3. 79 00:04:33,510 --> 00:04:35,940 We have Q cubed. 80 00:04:35,940 --> 00:04:39,150 And the matrix elements of Q cubed 81 00:04:39,150 --> 00:04:43,230 are a plus a-dagger cubed. 82 00:04:43,230 --> 00:04:45,440 And the selection rule for them-- 83 00:04:45,440 --> 00:04:49,650 or the matrix-- must go as the square root of the power 84 00:04:49,650 --> 00:04:51,900 of the a's. 85 00:04:51,900 --> 00:04:54,540 And so all of the matrix elements 86 00:04:54,540 --> 00:04:59,640 have the leading term v plus 1/2 to the 3/2 power. 87 00:04:59,640 --> 00:05:02,470 But we're squaring. 88 00:05:02,470 --> 00:05:06,590 And we lose the highest order because of the subtraction. 89 00:05:06,590 --> 00:05:11,495 So this one gives rise to terms in v plus 1/2 squared. 90 00:05:14,080 --> 00:05:22,100 Q4 has selection rules delta v plus or minus 4, 91 00:05:22,100 --> 00:05:24,820 plus or minus 2, and 0. 92 00:05:24,820 --> 00:05:25,770 This is important. 93 00:05:25,770 --> 00:05:28,950 Because this one says, we have a contribution 94 00:05:28,950 --> 00:05:30,840 from the first order-- 95 00:05:30,840 --> 00:05:33,490 in the first order of energy. 96 00:05:33,490 --> 00:05:35,560 And so in the first order of energy, 97 00:05:35,560 --> 00:05:41,745 we're going to have v plus 1/2 to the fourth power-- 98 00:05:46,140 --> 00:05:49,620 I'm sorry-- to the 4/2 power. 99 00:05:53,460 --> 00:05:56,022 Because we only have the matrix element, 100 00:05:56,022 --> 00:05:57,480 we don't square the matrix element. 101 00:06:01,630 --> 00:06:08,320 So this gives rise to a term in v plus 1/2 squared. 102 00:06:08,320 --> 00:06:11,980 But it's sensitive to the sign of the Q 103 00:06:11,980 --> 00:06:14,470 to the fourth term in the Hamiltonian. 104 00:06:14,470 --> 00:06:18,180 It's the only thing that's sensitive to the sine. 105 00:06:18,180 --> 00:06:29,740 We also have terms that are v plus 1/2 to the 4/2 times 2. 106 00:06:29,740 --> 00:06:39,500 So we can have matrix elements to the four-- 107 00:06:39,500 --> 00:06:43,670 because it's a fourth power term, we get 4/2. 108 00:06:43,670 --> 00:06:46,220 And because it's squared, we multiply by 2. 109 00:06:46,220 --> 00:06:47,780 And then we lose the highest order. 110 00:06:47,780 --> 00:06:54,010 So we get, from this, v plus 1/2 to the third power. 111 00:06:54,010 --> 00:06:58,870 So what we're hoping to get from the perturbation theory 112 00:06:58,870 --> 00:07:04,060 is the highest order terms in the energy expression. 113 00:07:04,060 --> 00:07:09,730 And so Q to the fourth gives a signed v plus 1/2 squared term. 114 00:07:09,730 --> 00:07:14,950 And it gives an unsigned always positive v plus 1/2 cubed term. 115 00:07:14,950 --> 00:07:21,790 And this one gives v plus 1/2 squared term. 116 00:07:24,380 --> 00:07:26,800 OK. 117 00:07:26,800 --> 00:07:28,390 So this would tell you, when you're 118 00:07:28,390 --> 00:07:32,290 organizing your work, what to expect and how to organize it. 119 00:07:32,290 --> 00:07:35,020 And I'm really not expecting you to do 120 00:07:35,020 --> 00:07:39,890 very much with this level of complexity on an exam problem. 121 00:07:39,890 --> 00:07:42,940 But on homework, all bets are off. 122 00:07:42,940 --> 00:07:44,680 OK. 123 00:07:44,680 --> 00:07:48,010 So that's all I want to do as far as the review is concerned. 124 00:07:50,890 --> 00:07:54,010 So the rigid rotor-- 125 00:07:54,010 --> 00:07:57,070 we have a molecule. 126 00:08:11,060 --> 00:08:14,900 So we have a bond of length r sub 0. 127 00:08:14,900 --> 00:08:16,680 And we have two masses. 128 00:08:16,680 --> 00:08:18,850 And if the masses aren't equal, the center of mass 129 00:08:18,850 --> 00:08:21,789 isn't in the middle. 130 00:08:21,789 --> 00:08:23,330 The center of mass-- you need to know 131 00:08:23,330 --> 00:08:25,670 how to calculate where the center of mass is. 132 00:08:25,670 --> 00:08:28,700 There are all sorts of simple algebra-- 133 00:08:28,700 --> 00:08:30,800 but what is happening is this guy 134 00:08:30,800 --> 00:08:34,914 is rotating without stretching about the center of mass. 135 00:08:43,549 --> 00:08:47,870 Now, what we want to do is think about this problem 136 00:08:47,870 --> 00:08:48,860 as if it were-- 137 00:08:53,710 --> 00:08:55,850 here's our 0. 138 00:08:55,850 --> 00:08:58,740 And this is the reduced mass. 139 00:08:58,740 --> 00:09:04,740 This is a motion of a fictitious particle of mass mu. 140 00:09:04,740 --> 00:09:16,360 Mu is m1 m2 over m1 plus m2 on the surface of a sphere. 141 00:09:16,360 --> 00:09:18,220 They're all the same problem. 142 00:09:18,220 --> 00:09:20,680 But the question is, how do we interpret 143 00:09:20,680 --> 00:09:23,500 what we get from the solution to this problem? 144 00:09:27,120 --> 00:09:32,210 So what we care about is a description of, 145 00:09:32,210 --> 00:09:34,910 where is the molecular axis? 146 00:09:34,910 --> 00:09:37,190 The molecule is rotating. 147 00:09:37,190 --> 00:09:41,090 And we're solving the Schrodinger equation. 148 00:09:41,090 --> 00:09:46,760 And we get things like the expectation value of J squared 149 00:09:46,760 --> 00:09:52,390 and Jz and maybe some other things where 150 00:09:52,390 --> 00:09:55,360 these are the angular momenta. 151 00:09:55,360 --> 00:10:04,270 And how do the eigenfunctions for these states-- 152 00:10:04,270 --> 00:10:05,620 we have a state. 153 00:10:05,620 --> 00:10:12,470 And we have quantum numbers Jm and their probability 154 00:10:12,470 --> 00:10:14,480 amplitudes and theta and phi. 155 00:10:14,480 --> 00:10:16,085 That's a whole lot of stuff to digest. 156 00:10:19,270 --> 00:10:21,580 We want to go as quickly as we can 157 00:10:21,580 --> 00:10:27,650 to how this is related to the thing we really care about, 158 00:10:27,650 --> 00:10:29,080 which is, where is the bond axis? 159 00:10:33,100 --> 00:10:34,970 So the molecule is rotating. 160 00:10:34,970 --> 00:10:40,220 And so the bond axis is moving in laboratory frame. 161 00:10:40,220 --> 00:10:43,450 And we want to be able to take, from that, 162 00:10:43,450 --> 00:10:46,525 the minimal amount of information 163 00:10:46,525 --> 00:10:49,730 that we memorize or remember-- 164 00:10:49,730 --> 00:10:50,990 I don't like "memorize." 165 00:10:50,990 --> 00:10:53,810 Because "memorize" doesn't involve understanding. 166 00:10:53,810 --> 00:10:56,060 But "remembering" does. 167 00:10:56,060 --> 00:11:01,120 We want to be able to, at a drop of a hat, 168 00:11:01,120 --> 00:11:05,240 be able to say, yes, if we pick these two quantum 169 00:11:05,240 --> 00:11:08,180 numbers, which are related to the eigenvalues of these two 170 00:11:08,180 --> 00:11:12,320 operators, we can describe the spatial distribution 171 00:11:12,320 --> 00:11:13,325 of the molecular axis. 172 00:11:17,020 --> 00:11:18,520 There is an extra complication. 173 00:11:25,700 --> 00:11:29,100 So we live in the laboratory. 174 00:11:29,100 --> 00:11:31,520 And we have a coordinate system that we 175 00:11:31,520 --> 00:11:34,850 care about, the lab-fixed coordinates. 176 00:11:34,850 --> 00:11:36,860 And they're always going to be represented 177 00:11:36,860 --> 00:11:39,020 with capital letters. 178 00:11:39,020 --> 00:11:42,350 And there's also the body frame. 179 00:11:42,350 --> 00:11:45,150 And things are in the body frame. 180 00:11:45,150 --> 00:11:47,690 That's what the molecule cares about. 181 00:11:47,690 --> 00:11:50,180 And how do things in the body frame 182 00:11:50,180 --> 00:11:51,830 relate to the laboratory frame? 183 00:11:51,830 --> 00:11:54,890 It's not trivial. 184 00:11:54,890 --> 00:11:58,970 And that's where the real effort at understanding comes. 185 00:11:58,970 --> 00:12:04,150 And it's kind of trivial for a diatomic molecule. 186 00:12:04,150 --> 00:12:10,860 But it's far from trivial when you have a nonlinear molecule, 187 00:12:10,860 --> 00:12:15,760 a molecule with many atoms and properties of each of the atoms 188 00:12:15,760 --> 00:12:19,840 that somehow combine to give things that you observe. 189 00:12:19,840 --> 00:12:25,990 And so these two coordinate systems are very different. 190 00:12:25,990 --> 00:12:30,010 And actually, the solutions of the rigid rotor 191 00:12:30,010 --> 00:12:37,030 Hamiltonian gives you the relationship of the body 192 00:12:37,030 --> 00:12:40,540 coordinates to the laboratory coordinates. 193 00:12:40,540 --> 00:12:42,920 And that's kind of important. 194 00:12:42,920 --> 00:12:47,070 So we'll see how this develops as I go on. 195 00:12:47,070 --> 00:12:48,970 And I feel quite passionate about this. 196 00:12:48,970 --> 00:12:54,820 Because one of the things that I do as an experimentalist is 197 00:12:54,820 --> 00:12:58,000 I observe fully resolved spectra. 198 00:12:58,000 --> 00:13:01,000 And the big-- the most information-rich feature 199 00:13:01,000 --> 00:13:04,460 of the spectrum is the rotational spectrum. 200 00:13:04,460 --> 00:13:09,930 And so it contains a lot of really good stuff. 201 00:13:09,930 --> 00:13:12,320 OK. 202 00:13:12,320 --> 00:13:20,630 So first of all, the Hamiltonian is just the kinetic energy. 203 00:13:20,630 --> 00:13:23,160 Because it's a rigid rotor, it's free. 204 00:13:23,160 --> 00:13:24,642 There's no potential. 205 00:13:27,740 --> 00:13:30,110 But that's the last nice thing about it. 206 00:13:30,110 --> 00:13:36,980 Because we have to go from Cartesian to spherical polar 207 00:13:36,980 --> 00:13:38,390 coordinates. 208 00:13:38,390 --> 00:13:40,850 And there are a lot of unfamiliar things 209 00:13:40,850 --> 00:13:44,740 in this kinetic energy expression. 210 00:13:44,740 --> 00:13:47,099 And when I write it down, you're not going to like it. 211 00:13:47,099 --> 00:13:48,890 Unless you're a mathematician, and you say, 212 00:13:48,890 --> 00:13:52,240 oh, yeah, I want to be able to solve these kinds of equations. 213 00:13:52,240 --> 00:13:56,320 Because the Schrodinger equation is, 214 00:13:56,320 --> 00:14:01,820 considering the simplicity of the problem, terrible. 215 00:14:01,820 --> 00:14:04,370 And except for mathematicians who just love it 216 00:14:04,370 --> 00:14:05,420 because they know-- 217 00:14:05,420 --> 00:14:06,620 oh, I know that equation. 218 00:14:06,620 --> 00:14:09,572 I know how to write down everything that I care about. 219 00:14:09,572 --> 00:14:11,030 But we care about different things. 220 00:14:14,400 --> 00:14:21,250 So we want to know the energy levels of the rigid rotor. 221 00:14:21,250 --> 00:14:25,255 And we also want pictures. 222 00:14:28,530 --> 00:14:33,670 And remember, a picture is a reduced form of all the details 223 00:14:33,670 --> 00:14:35,990 that you have at your disposal. 224 00:14:35,990 --> 00:14:38,140 And you have to really sweat to make 225 00:14:38,140 --> 00:14:41,980 sure you understand every detail of the pictures and the stuff 226 00:14:41,980 --> 00:14:44,560 that you've averaged over, or you've concealed, 227 00:14:44,560 --> 00:14:47,800 because you don't need it, OK? 228 00:14:51,631 --> 00:14:52,130 All right. 229 00:14:52,130 --> 00:14:55,900 So I'm going to be generating a lot of pictures. 230 00:14:55,900 --> 00:14:59,400 And one of the things we want to understand is-- 231 00:14:59,400 --> 00:15:04,460 so we have this vector, J. The J is the angular momentum 232 00:15:04,460 --> 00:15:07,340 of the molecule. 233 00:15:07,340 --> 00:15:09,760 And it's a vector as opposed to a scalar. 234 00:15:09,760 --> 00:15:12,850 That means it has three components. 235 00:15:12,850 --> 00:15:16,140 So for this part of the problem that I've just hidden over 236 00:15:16,140 --> 00:15:19,230 here, that's a one dimensional problem. 237 00:15:19,230 --> 00:15:22,740 We have a momentum and a conjugate coordinate 238 00:15:22,740 --> 00:15:25,230 and it's 1D. 239 00:15:25,230 --> 00:15:26,040 This is 3D. 240 00:15:26,040 --> 00:15:28,260 And there's all sorts of subtle stuff that goes on. 241 00:15:30,950 --> 00:15:31,670 OK. 242 00:15:31,670 --> 00:15:37,760 So we would like to understand how this vector moves. 243 00:15:37,760 --> 00:15:39,680 Well, there is a question. 244 00:15:39,680 --> 00:15:40,994 I said moves. 245 00:15:40,994 --> 00:15:43,160 We're talking about the time independent Schrodinger 246 00:15:43,160 --> 00:15:44,690 equation. 247 00:15:44,690 --> 00:15:45,560 Nothing moves. 248 00:15:49,750 --> 00:15:54,280 We can use our concept of motion from classical mechanics 249 00:15:54,280 --> 00:15:57,490 to describe certain features of the average 250 00:15:57,490 --> 00:16:00,470 of some quantum mechanical system. 251 00:16:00,470 --> 00:16:02,180 And so there is kind of a motion. 252 00:16:02,180 --> 00:16:04,401 But we'll see what that is. 253 00:16:04,401 --> 00:16:04,900 OK. 254 00:16:04,900 --> 00:16:07,000 We have some operators that we like. 255 00:16:20,580 --> 00:16:21,360 OK. 256 00:16:21,360 --> 00:16:24,900 Total angular momentum, projection of the angular 257 00:16:24,900 --> 00:16:28,350 momentum on a laboratory z-axis-- 258 00:16:28,350 --> 00:16:30,990 it's kind of hard to draw capital Z and small z 259 00:16:30,990 --> 00:16:32,610 if you haven't got anybody nearby. 260 00:16:32,610 --> 00:16:36,390 But anyway, this is a capital Z. And this 261 00:16:36,390 --> 00:16:40,230 is the thing that's analogous to the creation and annihilation 262 00:16:40,230 --> 00:16:40,830 operators. 263 00:16:43,350 --> 00:16:44,970 We need them. 264 00:16:44,970 --> 00:16:46,980 And it gives us a lot of insight. 265 00:16:46,980 --> 00:16:49,500 And that will be the subject of Monday's lecture. 266 00:16:52,610 --> 00:16:57,000 And so we're going to have solutions, which are described 267 00:16:57,000 --> 00:16:59,370 by J and m quantum numbers. 268 00:16:59,370 --> 00:17:06,530 And we want to understand what these things look like. 269 00:17:06,530 --> 00:17:07,885 Where are the nodal surfaces? 270 00:17:10,650 --> 00:17:14,030 The nodal surfaces give us basically everything 271 00:17:14,030 --> 00:17:15,199 we want to know. 272 00:17:18,349 --> 00:17:21,280 There is the direct correlation to where the nodes are 273 00:17:21,280 --> 00:17:26,230 and how many of them there are with these quantum numbers. 274 00:17:26,230 --> 00:17:27,790 That's one of the important things 275 00:17:27,790 --> 00:17:28,920 you have to come away with. 276 00:17:34,461 --> 00:17:34,960 OK. 277 00:17:34,960 --> 00:17:38,830 In my next lecture on angular momentum, 278 00:17:38,830 --> 00:17:42,010 I'm going to introduce a commutation rule, 279 00:17:42,010 --> 00:17:55,330 Ji Jj equals I H bar sum over K epsilon I Jk Jk. 280 00:17:55,330 --> 00:17:56,990 Isn't that a strange looking thing? 281 00:17:56,990 --> 00:18:01,252 Because I haven't told you what this epsilon is. 282 00:18:01,252 --> 00:18:02,460 There's lots of names for it. 283 00:18:02,460 --> 00:18:03,000 It won't help. 284 00:18:03,000 --> 00:18:04,833 Because I'm not going to talk about it here. 285 00:18:04,833 --> 00:18:08,430 But this is actually the fundamental definition 286 00:18:08,430 --> 00:18:10,970 of an angular momentum. 287 00:18:10,970 --> 00:18:12,770 And from this commutator rule, we 288 00:18:12,770 --> 00:18:18,140 can generate all the matrix elements of angular momentum. 289 00:18:18,140 --> 00:18:23,170 And there's another family of commutation rules 290 00:18:23,170 --> 00:18:25,880 where we have a component of angular momentum 291 00:18:25,880 --> 00:18:29,710 and a component of what we call a spherical tensor. 292 00:18:29,710 --> 00:18:33,310 Any operator can be classified according to spherical tensor 293 00:18:33,310 --> 00:18:34,570 rank. 294 00:18:34,570 --> 00:18:37,300 And that then determines all the matrix elements 295 00:18:37,300 --> 00:18:40,070 of that operator. 296 00:18:40,070 --> 00:18:42,860 And so you can imagine that this is a really powerful way 297 00:18:42,860 --> 00:18:44,510 of approaching stuff. 298 00:18:44,510 --> 00:18:51,470 And what I said before about the commutation rule of x and P sub 299 00:18:51,470 --> 00:18:54,800 x is that many people regard that as the foundation 300 00:18:54,800 --> 00:18:56,697 of quantum mechanics. 301 00:18:56,697 --> 00:18:58,280 And this is just an extension of that. 302 00:19:01,160 --> 00:19:03,580 So there is a way of getting all quantum mechanics 303 00:19:03,580 --> 00:19:07,092 from a few well-chosen commutation rules. 304 00:19:07,092 --> 00:19:08,050 And that's really neat. 305 00:19:11,251 --> 00:19:11,750 OK. 306 00:19:11,750 --> 00:19:12,875 Let's get down to business. 307 00:19:15,920 --> 00:19:16,920 So I already drew this. 308 00:19:16,920 --> 00:19:18,930 But I'll draw it again. 309 00:19:18,930 --> 00:19:21,790 Because I'm going to drop a companion. 310 00:19:21,790 --> 00:19:22,290 OK. 311 00:19:22,290 --> 00:19:23,730 So we have here-- 312 00:19:35,560 --> 00:19:36,060 OK. 313 00:19:36,060 --> 00:19:37,800 So we have an angular momentum. 314 00:19:37,800 --> 00:19:40,260 And we have the bond axis. 315 00:19:40,260 --> 00:19:46,270 And this angular momentum is perpendicular to the bond axis. 316 00:19:46,270 --> 00:19:49,150 So if we know something about how the angular momentum is 317 00:19:49,150 --> 00:19:52,570 distributed in space, we know something about how the bond 318 00:19:52,570 --> 00:19:54,460 axis is distributed in space. 319 00:19:54,460 --> 00:19:57,800 But it's not trivial. 320 00:19:57,800 --> 00:20:00,250 But once you've learned how to make those connections, 321 00:20:00,250 --> 00:20:02,031 it's fine. 322 00:20:02,031 --> 00:20:02,530 OK. 323 00:20:02,530 --> 00:20:03,730 So now, let's draw-- 324 00:20:12,250 --> 00:20:16,250 So we have a right-handed coordinate system. 325 00:20:16,250 --> 00:20:20,110 And we have, say, the vector J. Now, 326 00:20:20,110 --> 00:20:22,000 this is the laboratory frame. 327 00:20:22,000 --> 00:20:34,150 We have J. And we have a molecule, which 328 00:20:34,150 --> 00:20:40,490 is perpendicular to J. And we have 329 00:20:40,490 --> 00:20:45,950 the projection of J on the body axis, on the z-axis, 330 00:20:45,950 --> 00:20:49,470 and that's M. 331 00:20:49,470 --> 00:20:49,970 OK. 332 00:20:49,970 --> 00:20:53,900 So when we talked about the quantum number 333 00:20:53,900 --> 00:20:58,310 J, which is the length of this vector, and m, which 334 00:20:58,310 --> 00:21:00,860 is the projection of that vector on this axis, 335 00:21:00,860 --> 00:21:04,790 that's beginning to be how we understand how this works. 336 00:21:04,790 --> 00:21:06,620 Because this is perpendicular to that. 337 00:21:09,480 --> 00:21:15,135 Now, here we have one of the sins. 338 00:21:25,150 --> 00:21:28,330 So we begin by saying, well, this is a picture. 339 00:21:28,330 --> 00:21:36,780 And J precesses, or moves, on a cone around z. 340 00:21:36,780 --> 00:21:39,840 How do we have motion? 341 00:21:39,840 --> 00:21:43,990 This is a time independent Schrodinger equation. 342 00:21:43,990 --> 00:21:49,630 It's a way of saying, well, it doesn't matter where J is. 343 00:21:49,630 --> 00:21:52,660 It's more or less equally distributed 344 00:21:52,660 --> 00:21:57,170 in probability on this cone. 345 00:21:57,170 --> 00:21:58,734 But not an amplitude. 346 00:22:01,460 --> 00:22:03,130 Remember, when we have something that's 347 00:22:03,130 --> 00:22:06,700 moving in the time independent Schrodinger equation, 348 00:22:06,700 --> 00:22:09,520 we get oscillations. 349 00:22:09,520 --> 00:22:11,830 But we need a complex function in order 350 00:22:11,830 --> 00:22:13,750 to have those oscillations. 351 00:22:13,750 --> 00:22:18,430 If we're going to take psi star psi, there is no complex part. 352 00:22:18,430 --> 00:22:22,900 And you do have uniform amplitude of J. 353 00:22:22,900 --> 00:22:25,720 And so it's sensible to say, well, it got that way 354 00:22:25,720 --> 00:22:27,910 because J precesses. 355 00:22:27,910 --> 00:22:30,450 Maybe. 356 00:22:30,450 --> 00:22:32,970 There is something if you say, well, 357 00:22:32,970 --> 00:22:37,020 maybe if I created a system at t equal 0 358 00:22:37,020 --> 00:22:42,040 where J was at a particular position, 359 00:22:42,040 --> 00:22:45,380 and then I let that thing evolve-- 360 00:22:45,380 --> 00:22:46,670 this is not an eigenstate. 361 00:22:46,670 --> 00:22:50,330 It would be a complicated superposition. 362 00:22:50,330 --> 00:22:52,610 That thing would precess. 363 00:22:52,610 --> 00:22:55,955 And you would observe what we call polarization quantum 364 00:22:55,955 --> 00:22:56,780 beats. 365 00:22:56,780 --> 00:22:59,660 Now, that's getting way ahead of things. 366 00:22:59,660 --> 00:23:07,360 But it is helpful to think about J precessing on this cone. 367 00:23:07,360 --> 00:23:12,840 Because that gives you a sense that the length of J is 368 00:23:12,840 --> 00:23:14,140 conserved. 369 00:23:14,140 --> 00:23:17,800 And the orientation about the z-axis is not. 370 00:23:17,800 --> 00:23:21,610 And it's more or less uniform probability, but not 371 00:23:21,610 --> 00:23:23,560 uniform probability amplitude. 372 00:23:23,560 --> 00:23:27,340 But you don't need that for a lot of the things. 373 00:23:27,340 --> 00:23:29,040 If you're looking at the wave function, 374 00:23:29,040 --> 00:23:31,030 yeah, there's going to be some oscillation. 375 00:23:31,030 --> 00:23:32,740 The real part and the imaginary part 376 00:23:32,740 --> 00:23:36,890 will oscillate in such a way that the probability 377 00:23:36,890 --> 00:23:38,920 is constant about that. 378 00:23:38,920 --> 00:23:40,810 I finally realized that this morning. 379 00:23:40,810 --> 00:23:43,540 So it's not as much of a lie as you think. 380 00:23:43,540 --> 00:23:45,160 OK. 381 00:23:45,160 --> 00:23:49,750 So this is the picture. 382 00:23:49,750 --> 00:23:53,950 And this is what it looks like in the laboratory. 383 00:23:53,950 --> 00:24:02,650 And now, I'm going to prepare you for-- 384 00:24:02,650 --> 00:24:06,040 so suppose you had a molecule. 385 00:24:06,040 --> 00:24:14,670 And you have a little person standing on the molecule frame. 386 00:24:14,670 --> 00:24:17,875 And now, you have somebody out here observing as the molecule 387 00:24:17,875 --> 00:24:18,375 rotates. 388 00:24:22,982 --> 00:24:26,790 Well, and here's J. All information that you're 389 00:24:26,790 --> 00:24:30,800 allowed to know from outside the molecule 390 00:24:30,800 --> 00:24:33,620 comes from the projection of whatever 391 00:24:33,620 --> 00:24:41,650 is in the molecule frame on J. And that's 392 00:24:41,650 --> 00:24:42,950 what the vector model does. 393 00:24:42,950 --> 00:24:44,920 It tells you how to take stuff you 394 00:24:44,920 --> 00:24:48,160 know about the individual atoms and project it 395 00:24:48,160 --> 00:24:51,974 on the thing that communicates with the outside world. 396 00:24:51,974 --> 00:24:53,390 There are a couple of other things 397 00:24:53,390 --> 00:24:54,765 that I'm going to say about this. 398 00:24:54,765 --> 00:24:57,190 But now, let's get down to the business 399 00:24:57,190 --> 00:25:00,430 of actually doing a little bit of the solution 400 00:25:00,430 --> 00:25:01,718 of the Schrodinger equation. 401 00:25:14,180 --> 00:25:18,170 So for our free rotor, the potential 402 00:25:18,170 --> 00:25:25,220 is r equals r0 theta phi. 403 00:25:25,220 --> 00:25:29,630 And it's equal to 0 if r equals r0. 404 00:25:29,630 --> 00:25:32,060 And it's equal to infinity if it's not. 405 00:25:32,060 --> 00:25:35,810 So this is very much like a particle in an infinite box. 406 00:25:35,810 --> 00:25:40,810 In fact, you can do a really cheap solution 407 00:25:40,810 --> 00:25:44,740 if you say, well, let's consider a particle 408 00:25:44,740 --> 00:25:48,020 in an infinite circular box. 409 00:25:48,020 --> 00:25:51,620 Well, you can solve this problem just using the de Broglie 410 00:25:51,620 --> 00:25:53,380 wavelength. 411 00:25:53,380 --> 00:25:57,250 And you get that the energy levels for this circular box 412 00:25:57,250 --> 00:26:01,800 problem go as the quantum number squared. 413 00:26:01,800 --> 00:26:04,870 And that's almost exactly like the solutions 414 00:26:04,870 --> 00:26:09,980 to the free rotor. 415 00:26:09,980 --> 00:26:14,390 The difference is the energies for this go 416 00:26:14,390 --> 00:26:16,040 as the quantum number squared. 417 00:26:16,040 --> 00:26:18,530 And the energies for this go as the quantum number 418 00:26:18,530 --> 00:26:20,476 times the quantum number plus 1. 419 00:26:20,476 --> 00:26:21,350 It's almost the same. 420 00:26:24,250 --> 00:26:25,870 OK. 421 00:26:25,870 --> 00:26:31,930 So because the potential is constant as long as r is fixed, 422 00:26:31,930 --> 00:26:35,800 the Hamiltonian is just theta-- 423 00:26:42,540 --> 00:26:43,040 OK. 424 00:26:43,040 --> 00:26:47,520 And so we have to understand this kinetic energy. 425 00:26:54,080 --> 00:26:57,960 For 1D problems, the kinetic energy 426 00:26:57,960 --> 00:27:04,618 is P squared over 2 mu, the linear momentum. 427 00:27:04,618 --> 00:27:08,260 I should put a hat on this. 428 00:27:08,260 --> 00:27:12,430 Well, we have motion in three dimensions, 429 00:27:12,430 --> 00:27:14,150 or two dimensions, theta and phi. 430 00:27:16,810 --> 00:27:21,860 And so we want to do something. 431 00:27:21,860 --> 00:27:24,880 We want to generate a kinetic energy Hamiltonian, which 432 00:27:24,880 --> 00:27:28,480 is analogous to this, some kind of momentum squared 433 00:27:28,480 --> 00:27:30,520 over some kind of a mass. 434 00:27:30,520 --> 00:27:32,050 And so we can go by analogy. 435 00:27:32,050 --> 00:27:34,310 Or we can go back to classical mechanics. 436 00:27:34,310 --> 00:27:40,190 We know that the orbital angular momentum is r cross P. 437 00:27:40,190 --> 00:27:42,340 So these are two vectors. 438 00:27:42,340 --> 00:27:44,320 Cross product is a vector. 439 00:27:44,320 --> 00:27:45,550 We know all that stuff. 440 00:27:45,550 --> 00:27:48,280 We know how to write the cross product 441 00:27:48,280 --> 00:27:51,017 in terms of a matrix involving unit vectors 442 00:27:51,017 --> 00:27:51,850 and stuff like that. 443 00:27:51,850 --> 00:27:55,390 You've done that before. 444 00:27:55,390 --> 00:27:56,580 OK. 445 00:27:56,580 --> 00:28:09,020 So now, we have an angular motion, omega. 446 00:28:09,020 --> 00:28:17,210 And we would like to know what the velocity of the mass points 447 00:28:17,210 --> 00:28:19,130 are on a sphere. 448 00:28:19,130 --> 00:28:26,750 Or if we think of this as just a particle of mass mu rotating, 449 00:28:26,750 --> 00:28:28,550 then we want to know its velocity. 450 00:28:28,550 --> 00:28:31,670 And to get from the angular velocity 451 00:28:31,670 --> 00:28:36,800 to the linear velocity, the linear velocity is r omega. 452 00:28:44,790 --> 00:28:49,810 So we can say, all right, knowing-- 453 00:28:49,810 --> 00:28:55,180 well, for a vector cross product, if the two things-- 454 00:28:55,180 --> 00:28:57,700 the r and the P-- 455 00:28:57,700 --> 00:29:07,010 are always orthogonal, well, then we just do r times P. 456 00:29:07,010 --> 00:29:11,970 And in fact, for motion on this sphere, here's r. 457 00:29:11,970 --> 00:29:15,750 And the motion is always orthogonal to r. 458 00:29:15,750 --> 00:29:24,900 And so we can write that the angular momentum is m1 r1 459 00:29:24,900 --> 00:29:33,560 squared omega plus m2 r2 squared times omega. 460 00:29:33,560 --> 00:29:36,980 Or it's just I omega. 461 00:29:43,250 --> 00:29:44,840 OK. 462 00:29:44,840 --> 00:29:52,970 And so we can write the kinetic energy term just 463 00:29:52,970 --> 00:29:58,840 goes as L squared over 2I where I 464 00:29:58,840 --> 00:30:05,530 is the sum of the individual masses M sub r I squared. 465 00:30:09,190 --> 00:30:10,490 OK. 466 00:30:10,490 --> 00:30:15,900 So we have an angular momentum operator, 467 00:30:15,900 --> 00:30:18,560 which is the guts of the kinetic energy. 468 00:30:22,340 --> 00:30:24,230 OK. 469 00:30:24,230 --> 00:30:28,060 Now, we're thinking Cartesian. 470 00:30:28,060 --> 00:30:30,680 And this is a spherical problem. 471 00:30:30,680 --> 00:30:33,070 And so we want to go from Cartesian 472 00:30:33,070 --> 00:30:37,338 coordinates to spherical polar coordinates. 473 00:30:37,338 --> 00:30:40,290 And that's a non-trivial problem. 474 00:30:40,290 --> 00:30:43,620 And it's also an extremely boring problem. 475 00:30:43,620 --> 00:30:45,300 And what you get is also something 476 00:30:45,300 --> 00:30:48,600 that's not terribly rewarding, except it's the differential 477 00:30:48,600 --> 00:30:50,170 equation you have solve. 478 00:30:50,170 --> 00:30:54,990 So T, when you go to spherical polar coordinates, 479 00:30:54,990 --> 00:31:06,150 is minus h squared over 2I 1 over sine theta 480 00:31:06,150 --> 00:31:10,380 partial with respect to theta times 481 00:31:10,380 --> 00:31:19,200 sine theta partial with respect to theta plus 1 over sine 482 00:31:19,200 --> 00:31:25,990 squared theta second partial with respect to phi. 483 00:31:25,990 --> 00:31:30,360 So that's the kinetic energy operator. 484 00:31:30,360 --> 00:31:35,760 So first of all, you say, what am I going to do? 485 00:31:35,760 --> 00:31:39,410 And the first thing you say is, separate the variables. 486 00:31:39,410 --> 00:31:41,080 So you do that. 487 00:31:41,080 --> 00:31:43,960 And you do the standard trick for separating variables. 488 00:31:43,960 --> 00:31:46,630 And you get two differential equations-- the theta 489 00:31:46,630 --> 00:31:48,250 equation and the phi equation. 490 00:31:51,230 --> 00:31:51,950 I'm sorry? 491 00:31:51,950 --> 00:31:53,140 AUDIENCE: H bar! 492 00:31:53,140 --> 00:31:56,380 ROBERT FIELD: When I wrote it, I said, why is that not H bar? 493 00:31:56,380 --> 00:31:58,790 OK. 494 00:31:58,790 --> 00:32:05,370 All right, so when you separate variables, 495 00:32:05,370 --> 00:32:06,820 we have this result-- 496 00:32:06,820 --> 00:32:10,450 1 over phi of phi. 497 00:32:10,450 --> 00:32:14,430 This is the phi part of the equation. 498 00:32:22,185 --> 00:32:23,740 Whoops, yeah, that's right. 499 00:32:30,470 --> 00:32:33,330 m squared is the separation constant. 500 00:32:33,330 --> 00:32:37,280 So we arrange to have a differential equation that 501 00:32:37,280 --> 00:32:43,130 depends only on phi and another that depends only on theta. 502 00:32:43,130 --> 00:32:45,360 And they're equal to each other. 503 00:32:45,360 --> 00:32:48,660 And so we call the thing that they're equal to a constant. 504 00:32:48,660 --> 00:32:50,090 And we call it m squared. 505 00:32:50,090 --> 00:32:53,370 We like that because m can be positive or negative. 506 00:32:53,370 --> 00:32:57,200 And the sign of m determines whether you 507 00:32:57,200 --> 00:33:01,820 have an oscillating function or an exponential function. 508 00:33:01,820 --> 00:33:05,490 You're all familiar with that. 509 00:33:05,490 --> 00:33:07,320 And so this is the phi equation. 510 00:33:09,880 --> 00:33:12,340 And this one, you can solve easily. 511 00:33:12,340 --> 00:33:14,530 You know the solution to this. 512 00:33:14,530 --> 00:33:17,500 You could, with a little bit of thought, 513 00:33:17,500 --> 00:33:23,450 write down the solution and have normalized functions. 514 00:33:23,450 --> 00:33:24,400 So I'll just do that. 515 00:33:29,580 --> 00:33:36,290 So the phi part of the solution is-- 516 00:33:42,210 --> 00:33:46,870 it's not the order in which I wrote my notes-- 517 00:33:46,870 --> 00:33:52,050 1 over the square root of 2 pi times e to the i m phi. 518 00:33:52,050 --> 00:33:56,560 And m is equal to 0 plus and minus 1 plus and minus 519 00:33:56,560 --> 00:33:58,450 2 et cetera. 520 00:33:58,450 --> 00:34:02,490 Quantization comes from imposition 521 00:34:02,490 --> 00:34:06,410 of what we call periodic boundary conditions. 522 00:34:06,410 --> 00:34:11,590 The wave function has to be the same for phi and phi 523 00:34:11,590 --> 00:34:14,600 plus 2 pi and phi plus 4 pi. 524 00:34:14,600 --> 00:34:17,590 And so that gives quantization. 525 00:34:17,590 --> 00:34:18,820 So this is the phi part. 526 00:34:18,820 --> 00:34:19,530 That's simple. 527 00:34:19,530 --> 00:34:20,139 That's simple. 528 00:34:20,139 --> 00:34:20,764 It's wonderful. 529 00:34:20,764 --> 00:34:23,949 We understand that perfectly. 530 00:34:23,949 --> 00:34:26,110 And one of the nice things about the vector model 531 00:34:26,110 --> 00:34:28,480 is mostly that's what you're focused on. 532 00:34:37,060 --> 00:34:41,550 So immediately, we can draw some pictures. 533 00:34:41,550 --> 00:34:47,610 And we can look at the nodes in the xy plane. 534 00:34:51,080 --> 00:35:00,760 So m equals 0, there are no nodes. 535 00:35:03,930 --> 00:35:12,910 m equals 1, first of all, let's draw where phi is 0. 536 00:35:12,910 --> 00:35:14,200 So this is phi. 537 00:35:14,200 --> 00:35:19,969 And so there can be a nodal plane like that. 538 00:35:19,969 --> 00:35:21,760 And there could be a nodal plane like that. 539 00:35:29,450 --> 00:35:34,320 And already, you know this looks like an S orbital. 540 00:35:34,320 --> 00:35:37,810 And this looks like a Px orbital and a Py orbital. 541 00:35:40,830 --> 00:35:45,660 And then we could also have-- 542 00:35:45,660 --> 00:35:56,150 yeah, m equals 2, we could have this, and this, and that-- 543 00:35:56,150 --> 00:35:59,550 no node, one node, two nodes. 544 00:35:59,550 --> 00:36:00,650 I'm sorry, yeah. 545 00:36:05,040 --> 00:36:06,630 What am I doing here? 546 00:36:06,630 --> 00:36:08,160 This is one node. 547 00:36:12,890 --> 00:36:16,790 I have to be a little bit more careful here. 548 00:36:16,790 --> 00:36:20,687 So what I wrote down is not true here. 549 00:36:20,687 --> 00:36:22,020 There are going to be two nodes. 550 00:36:22,020 --> 00:36:23,853 So they're probably like this and like that. 551 00:36:27,000 --> 00:36:32,550 But it's clear that what I have in my handwritten notes-- 552 00:36:32,550 --> 00:36:34,640 I'm not sure what the printed notes say. 553 00:36:34,640 --> 00:36:36,290 But there are going to be two nodes. 554 00:36:36,290 --> 00:36:40,850 And so immediately, you know something about-- 555 00:36:40,850 --> 00:36:42,560 if you draw a reduced picture and you 556 00:36:42,560 --> 00:36:47,990 show the nodal structure in the xy plane, 557 00:36:47,990 --> 00:36:54,500 you can tell what the projection of the angular momentum quantum 558 00:36:54,500 --> 00:36:58,049 number is, which is really important. 559 00:37:11,530 --> 00:37:13,900 There's the other differential equation. 560 00:37:13,900 --> 00:37:17,145 And that's the theta differential equation. 561 00:37:20,460 --> 00:37:25,880 And that is complicated. 562 00:37:25,880 --> 00:37:30,170 It is an exactly solved differential equation. 563 00:37:30,170 --> 00:37:32,160 But one of the things it gives you, 564 00:37:32,160 --> 00:37:35,150 which is easily memorized-- remembered, I'm sorry-- 565 00:37:35,150 --> 00:37:39,170 is that the energy levels have-- 566 00:37:46,260 --> 00:37:49,780 where this is the overlying momentum. 567 00:37:49,780 --> 00:37:51,370 So you can solve this. 568 00:37:51,370 --> 00:37:53,950 And you can get the eigenenergies. 569 00:37:53,950 --> 00:37:58,690 And the eigenenergies do not depend on m. 570 00:37:58,690 --> 00:38:01,519 They only depend on the L quantum number. 571 00:38:01,519 --> 00:38:02,727 And they have this nice form. 572 00:38:06,000 --> 00:38:06,540 OK. 573 00:38:06,540 --> 00:38:09,690 Now, in my notes and in all the textbooks, 574 00:38:09,690 --> 00:38:15,750 there are these horrendously beautiful detailed expressions 575 00:38:15,750 --> 00:38:20,790 of the solution, the mathematical form 576 00:38:20,790 --> 00:38:24,240 to the solution of the theta equation. 577 00:38:24,240 --> 00:38:26,910 And it's the Legendre equation. 578 00:38:26,910 --> 00:38:29,070 And there's Legendre polynomials. 579 00:38:29,070 --> 00:38:31,500 And remember, with the harmonic oscillator, I told you, 580 00:38:31,500 --> 00:38:34,180 you don't ever want to look at these things. 581 00:38:34,180 --> 00:38:35,560 The computer will look at them. 582 00:38:35,560 --> 00:38:37,185 And if you have integrals, the computer 583 00:38:37,185 --> 00:38:38,790 will know how to do those integrals. 584 00:38:38,790 --> 00:38:40,530 Because you told it how. 585 00:38:40,530 --> 00:38:42,630 And you don't have to keep that in your head. 586 00:38:42,630 --> 00:38:46,870 You've got better things to do with your limited attention. 587 00:38:46,870 --> 00:38:49,390 So there is a solution. 588 00:38:49,390 --> 00:38:51,810 And it has a form, which I have decided 589 00:38:51,810 --> 00:38:55,050 that I won't talk about. 590 00:38:55,050 --> 00:38:59,280 Because you get much more insight into real problems 591 00:38:59,280 --> 00:39:01,530 from the vector model. 592 00:39:01,530 --> 00:39:07,360 Now, the vector model is really only 593 00:39:07,360 --> 00:39:12,150 marginally useful for a diatomic molecule 594 00:39:12,150 --> 00:39:17,250 in an electronic state, which you don't know about yet, which 595 00:39:17,250 --> 00:39:22,100 is called sigma sigma plus, where there is no angular 596 00:39:22,100 --> 00:39:25,120 momentum associated with the electron 597 00:39:25,120 --> 00:39:27,160 and when there is no angular momentum associated 598 00:39:27,160 --> 00:39:30,760 with the nuclei. 599 00:39:30,760 --> 00:39:34,680 So this is just practice for real life 600 00:39:34,680 --> 00:39:39,780 when you have to understand other things about what is 601 00:39:39,780 --> 00:39:42,310 living in the molecular frame. 602 00:39:42,310 --> 00:39:48,450 But this is hard enough to understand completely that it's 603 00:39:48,450 --> 00:39:50,340 a worthwhile investment. 604 00:39:50,340 --> 00:39:52,680 And you will be doing most of the understanding 605 00:39:52,680 --> 00:39:55,710 of more complicated problems on your own 606 00:39:55,710 --> 00:39:58,500 if you ever do anything with diatomic molecules 607 00:39:58,500 --> 00:40:00,520 or polyatomic molecules. 608 00:40:00,520 --> 00:40:03,040 And so, OK. 609 00:40:08,300 --> 00:40:09,910 So what is the vector model? 610 00:40:16,970 --> 00:40:20,540 Well, the things we want to know about the vector model 611 00:40:20,540 --> 00:40:23,870 is, how long is J? 612 00:40:23,870 --> 00:40:28,640 So if we have a state with J and m quantum numbers, 613 00:40:28,640 --> 00:40:32,320 how long is the vector associated with J? 614 00:40:32,320 --> 00:40:36,190 And what is the angle-- 615 00:40:36,190 --> 00:40:42,910 what is the projection of J on the laboratory z-axis? 616 00:40:42,910 --> 00:40:45,670 So we want to know the length of J. 617 00:40:45,670 --> 00:40:55,200 And we want to know the projection of J on z. 618 00:40:55,200 --> 00:40:56,280 OK. 619 00:40:56,280 --> 00:41:03,260 And we want to know the angle between the z-axis and J. 620 00:41:03,260 --> 00:41:04,700 That's the beginning of a picture. 621 00:41:07,830 --> 00:41:12,700 And again, we stick with this idea that J-- 622 00:41:12,700 --> 00:41:14,750 so here's the z-axis. 623 00:41:14,750 --> 00:41:19,850 And J precesses about z. 624 00:41:23,970 --> 00:41:27,860 And here is an angle we want to know. 625 00:41:27,860 --> 00:41:31,950 This theta is a different theta from the theta in-- 626 00:41:31,950 --> 00:41:35,860 in fact, I should call it alpha. 627 00:41:35,860 --> 00:41:36,360 OK. 628 00:41:40,000 --> 00:41:42,370 So some things we know-- 629 00:41:42,370 --> 00:41:49,570 we know that the eigenvalues of the rigid rotor equation 630 00:41:49,570 --> 00:41:51,430 are this. 631 00:41:51,430 --> 00:42:00,010 And we know that Jz operator operating on y Lm 632 00:42:00,010 --> 00:42:07,584 gives H bar m y Lm. 633 00:42:11,900 --> 00:42:21,590 And J squared operating on y Lm gives H bar squared J J plus 1. 634 00:42:25,070 --> 00:42:27,180 So what do we do with these two things? 635 00:42:27,180 --> 00:42:31,640 Well, one is to say, all right, the length of J 636 00:42:31,640 --> 00:42:34,640 is going to be the square root-- 637 00:42:34,640 --> 00:42:37,700 times y Lm. 638 00:42:37,700 --> 00:42:43,790 So the length of J is going to be the square root 639 00:42:43,790 --> 00:42:47,390 of H bar squared J J plus 1. 640 00:42:51,980 --> 00:42:58,109 So that becomes H bar and approximately J plus 1/2. 641 00:42:58,109 --> 00:42:59,650 Because the square root of J J plus 1 642 00:42:59,650 --> 00:43:01,390 is almost exactly J plus 1/2. 643 00:43:06,810 --> 00:43:09,180 When you get really high J, it's exactly. 644 00:43:09,180 --> 00:43:12,420 At low J, It's a little bit less than exact. 645 00:43:12,420 --> 00:43:15,470 So we know the length of J is this. 646 00:43:15,470 --> 00:43:19,670 And we know the length of m is H bar m. 647 00:43:19,670 --> 00:43:21,790 The length of J z is H bar m. 648 00:43:34,930 --> 00:43:43,280 So now, we know that this is H bar m. 649 00:43:43,280 --> 00:43:48,120 And this is H bar times J plus 1/2. 650 00:43:51,750 --> 00:43:57,300 And so this is alpha. 651 00:43:57,300 --> 00:44:03,210 So we know how to calculate the cosine of alpha. 652 00:44:03,210 --> 00:44:07,230 So the cosine of alpha is equal to H 653 00:44:07,230 --> 00:44:20,150 bar m over H bar J plus 1/2, or m over J-- 654 00:44:20,150 --> 00:44:24,170 oh, right, plus 1/2, really small. 655 00:44:24,170 --> 00:44:27,320 So the angle, the cosine of the angle 656 00:44:27,320 --> 00:44:35,180 that j makes with the z-axis is m over J. 657 00:44:35,180 --> 00:44:37,820 And all of a sudden, now, we have lengths 658 00:44:37,820 --> 00:44:40,250 of things and angles of things. 659 00:44:40,250 --> 00:44:41,810 And we can do all sorts of stuff. 660 00:44:41,810 --> 00:44:46,730 Remember, what we care about is, if this is J, 661 00:44:46,730 --> 00:44:50,210 the body axis is like that. 662 00:44:50,210 --> 00:44:52,190 So we now know where the body axis 663 00:44:52,190 --> 00:44:55,460 is relative to the laboratory. 664 00:44:55,460 --> 00:45:00,320 So we have the body axis is perpendicular to J. 665 00:45:00,320 --> 00:45:03,670 And so we can then calculate where everything is. 666 00:45:12,560 --> 00:45:15,870 All right, so special cases-- 667 00:45:15,870 --> 00:45:25,250 if m is equal to plus or minus J, then 668 00:45:25,250 --> 00:45:27,590 what does that mean for this picture? 669 00:45:27,590 --> 00:45:30,210 Well, we have the z-axis. 670 00:45:30,210 --> 00:45:37,150 And we have J. And J is almost exactly along the z-axis. 671 00:45:37,150 --> 00:45:39,795 It would be exactly along if there wasn't this plus 1/2. 672 00:45:47,100 --> 00:45:51,690 And so what that means is the bond 673 00:45:51,690 --> 00:45:56,770 axis is almost exactly rotating in the xy plane. 674 00:45:59,930 --> 00:46:03,590 So if we choose m equals plus or minus J, 675 00:46:03,590 --> 00:46:07,420 the molecule is rotating in the xy plane. 676 00:46:07,420 --> 00:46:10,160 That's kind of nice to know. 677 00:46:10,160 --> 00:46:15,010 The other easy case is m equals 0. 678 00:46:15,010 --> 00:46:21,150 If m equals 0, J is perpendicular to the z-axis. 679 00:46:21,150 --> 00:46:28,720 And so the body axis is in the yz plane. 680 00:46:28,720 --> 00:46:31,700 Or if we're coming around here, it's in the xz plane. 681 00:46:31,700 --> 00:46:36,330 So we have xz yz. 682 00:46:36,330 --> 00:46:39,010 And together, what it says is the body 683 00:46:39,010 --> 00:46:44,690 axis is more along z than along anything else. 684 00:46:44,690 --> 00:46:46,760 That's a lot of insight. 685 00:46:46,760 --> 00:46:51,310 And so we can go to our friendly extreme cases, special cases. 686 00:46:51,310 --> 00:46:54,806 And we could say, where's the body axis? 687 00:46:54,806 --> 00:46:56,180 In the laboratory, which is where 688 00:46:56,180 --> 00:46:57,470 we're making the observations. 689 00:47:00,410 --> 00:47:03,580 That should make you happy. 690 00:47:03,580 --> 00:47:06,680 Because there's no equations here. 691 00:47:06,680 --> 00:47:10,940 There's just a lot of pictures that you can understand 692 00:47:10,940 --> 00:47:12,815 and develop your intuition. 693 00:47:18,040 --> 00:47:21,331 Now, I want a blackboard. 694 00:47:21,331 --> 00:47:21,830 OK. 695 00:47:21,830 --> 00:47:25,370 Suppose you have a laboratory in which you 696 00:47:25,370 --> 00:47:26,570 can make a molecular beam. 697 00:47:30,932 --> 00:47:32,140 And that's easy enough to do. 698 00:47:32,140 --> 00:47:34,360 You squirt molecules out of a pinhole. 699 00:47:34,360 --> 00:47:38,350 And you have some kind of an aperture. 700 00:47:38,350 --> 00:47:44,770 And so you have a directed flow of molecules in one direction. 701 00:47:44,770 --> 00:47:46,570 And you could also say, all right, 702 00:47:46,570 --> 00:47:48,760 I'm going to do something with my laser. 703 00:47:48,760 --> 00:47:50,950 And I'm going to make the molecules, which 704 00:47:50,950 --> 00:47:58,390 are like a helicopter, in other words, 705 00:47:58,390 --> 00:48:04,420 the molecular axis is rotating in a circle 706 00:48:04,420 --> 00:48:08,620 about the propagation axis of the beam, so a helicopter. 707 00:48:12,390 --> 00:48:14,620 And we can also do this. 708 00:48:14,620 --> 00:48:21,210 And so these are two ways that you can prepare molecules 709 00:48:21,210 --> 00:48:21,780 in this beam. 710 00:48:24,630 --> 00:48:28,710 And you could say, oh, I've got a whole bunch of molecules 711 00:48:28,710 --> 00:48:30,120 that are not in the beam. 712 00:48:30,120 --> 00:48:32,910 And these molecules are going to collide 713 00:48:32,910 --> 00:48:39,015 with the molecules outside the beam and get scattered. 714 00:48:39,015 --> 00:48:43,650 Well, which one is going to be scattered more? 715 00:48:43,650 --> 00:48:47,480 The one that sweeps out a big volume or the one that 716 00:48:47,480 --> 00:48:54,640 sweeps out something only essentially one dimension? 717 00:48:54,640 --> 00:48:55,265 That's insight. 718 00:48:57,824 --> 00:48:59,240 There's lots of other things, too. 719 00:48:59,240 --> 00:49:01,280 Now, let's do something which is actually 720 00:49:01,280 --> 00:49:02,420 slightly exam relevant. 721 00:49:06,480 --> 00:49:08,670 Suppose we have a diatomic molecule 722 00:49:08,670 --> 00:49:14,450 which is positive on one end and negative on the other, OK? 723 00:49:14,450 --> 00:49:17,500 And we're going to apply an electric field. 724 00:49:17,500 --> 00:49:20,640 And if the electric field is in this direction 725 00:49:20,640 --> 00:49:23,890 and the molecule is doing this, the molecule doesn't care. 726 00:49:27,380 --> 00:49:30,130 But if the electric field is in that direction 727 00:49:30,130 --> 00:49:34,450 and the molecule is doing this, the energy levels-- 728 00:49:34,450 --> 00:49:38,142 the energy goes up and down as the molecule rotates. 729 00:49:38,142 --> 00:49:39,850 Well, how is that? 730 00:49:39,850 --> 00:49:42,580 The molecule was free to precess. 731 00:49:42,580 --> 00:49:45,860 And we knew the amplitude along this axis. 732 00:49:45,860 --> 00:49:50,230 But if there's some angle dependence, what we're doing 733 00:49:50,230 --> 00:49:55,870 is we're mixing in a different J. 734 00:49:55,870 --> 00:50:00,750 And so that's telling you that, if you apply an electric field, 735 00:50:00,750 --> 00:50:03,500 there will be a Stark effect. 736 00:50:03,500 --> 00:50:07,490 And it will affect the m equals 0 levels differently 737 00:50:07,490 --> 00:50:10,440 from the m equals J levels. 738 00:50:10,440 --> 00:50:15,480 Now, I'm not going to tell you which one is more affected. 739 00:50:15,480 --> 00:50:18,330 But it's clear that one is hardly affected. 740 00:50:18,330 --> 00:50:21,780 And the other is profoundly affected. 741 00:50:21,780 --> 00:50:24,870 And that gives rise to a splitting of the energy 742 00:50:24,870 --> 00:50:27,790 levels in an electric field, which 743 00:50:27,790 --> 00:50:31,540 depends on J at m in a way that you would recognize. 744 00:50:35,410 --> 00:50:36,390 Time to quit. 745 00:50:36,390 --> 00:50:39,300 I did what I was hoping I could do. 746 00:50:39,300 --> 00:50:42,720 Now, there is some stuff at the end of the notes 747 00:50:42,720 --> 00:50:45,620 which is standard textbook material about polar plots 748 00:50:45,620 --> 00:50:47,670 of the wave functions. 749 00:50:47,670 --> 00:50:49,620 These polar plots are useful. 750 00:50:49,620 --> 00:50:52,860 But most of the time, when you first encounter them, 751 00:50:52,860 --> 00:50:56,910 you have no idea what they mean, except that they 752 00:50:56,910 --> 00:51:00,109 express where the nodes are. 753 00:51:00,109 --> 00:51:01,650 Now, you want to go back and you want 754 00:51:01,650 --> 00:51:05,280 to actually think about what these polar plots mean 755 00:51:05,280 --> 00:51:10,500 and whether you can use them as an auxiliary 756 00:51:10,500 --> 00:51:12,720 to this vector picture. 757 00:51:12,720 --> 00:51:13,220 OK. 758 00:51:13,220 --> 00:51:17,615 So I will then lecture, and it's exam relevant, 759 00:51:17,615 --> 00:51:21,890 on commutation rules on Monday. 760 00:51:21,890 --> 00:51:27,280 And then have a good weekend.