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 00:00:21,590 --> 00:00:26,570 ROBERT FIELD: Now the main topic of this lecture 9 00:00:26,570 --> 00:00:31,460 is so important and so beautiful that I 10 00:00:31,460 --> 00:00:35,810 don't want to spend any time reviewing what I did last time. 11 00:00:35,810 --> 00:00:40,280 At the beginning when we talked about the rigid rotor, 12 00:00:40,280 --> 00:00:47,120 I said that this is not just a simple, exactly solved problem 13 00:00:47,120 --> 00:00:49,790 but it tells you about the angular part 14 00:00:49,790 --> 00:00:52,980 of every central force problem. 15 00:00:52,980 --> 00:00:55,640 And it's even more than that. 16 00:00:55,640 --> 00:00:59,370 It enables you to do a certain kind of algebra 17 00:00:59,370 --> 00:01:05,610 with operators which enables you to minimize 18 00:01:05,610 --> 00:01:10,710 the effort of calculating matrix elements 19 00:01:10,710 --> 00:01:14,010 and predicting selection rules simply 20 00:01:14,010 --> 00:01:18,120 by the basis of commutation rules of the operators 21 00:01:18,120 --> 00:01:21,510 without ever looking at wave functions, 22 00:01:21,510 --> 00:01:25,530 without ever looking at differential operators. 23 00:01:25,530 --> 00:01:33,030 This is a really beautiful thing about angular momentum 24 00:01:33,030 --> 00:01:38,160 that if we define the angular momentum in this abstract way-- 25 00:01:38,160 --> 00:01:42,000 and I'll describe what I mean by this epsilon ijk. 26 00:01:42,000 --> 00:01:48,300 If we say we have an operator which obeys this commutation 27 00:01:48,300 --> 00:01:53,240 rule, we will call it an angular momentum. 28 00:01:53,240 --> 00:01:57,110 And we go through some arguments and we 29 00:01:57,110 --> 00:02:02,940 discover the properties of all "angular momentum," in quotes. 30 00:02:02,940 --> 00:02:05,340 Now we define an angular momentum classically 31 00:02:05,340 --> 00:02:06,440 as r cross p. 32 00:02:09,267 --> 00:02:09,850 It's a vector. 33 00:02:16,620 --> 00:02:23,210 Now there are things, operators, where there is no r or p 34 00:02:23,210 --> 00:02:26,860 but we would like to describe them as angular momentum. 35 00:02:26,860 --> 00:02:30,040 One of them is electron spin, something that we sort of take 36 00:02:30,040 --> 00:02:33,730 for granted, and nuclear spin. 37 00:02:33,730 --> 00:02:38,260 NMR is based on these things we call angular momentum 38 00:02:38,260 --> 00:02:39,850 because they obey some rules. 39 00:02:42,620 --> 00:02:45,400 And so what I'm going to do is show the rules 40 00:02:45,400 --> 00:02:48,060 and show where all of this comes from 41 00:02:48,060 --> 00:02:55,530 and that this is an abstract and so kind of dry derivation, 42 00:02:55,530 --> 00:03:00,486 but it has astonishing consequences. 43 00:03:00,486 --> 00:03:03,910 And basically it means that if you've 44 00:03:03,910 --> 00:03:09,065 got angular momenta, if you know these rules, 45 00:03:09,065 --> 00:03:10,940 you're never going to evaluate another matrix 46 00:03:10,940 --> 00:03:13,760 element in your life. 47 00:03:13,760 --> 00:03:16,820 Now, it has another level of complexity. 48 00:03:16,820 --> 00:03:18,830 Sometimes you have operators that 49 00:03:18,830 --> 00:03:23,440 are made out of combinations of angular momenta, 50 00:03:23,440 --> 00:03:25,840 and you can use these sorts of arguments 51 00:03:25,840 --> 00:03:28,720 to derive the matrix elements of them. 52 00:03:28,720 --> 00:03:31,270 That's called the Wigner-Eckart theorem, 53 00:03:31,270 --> 00:03:35,470 and it means that the angular part of every operator 54 00:03:35,470 --> 00:03:39,550 is in your hands without ever looking at a wave function 55 00:03:39,550 --> 00:03:42,270 or a differential operator. 56 00:03:42,270 --> 00:03:44,290 Now we're not going to go there, but this 57 00:03:44,290 --> 00:03:47,470 is a very important area of quantum mechanics. 58 00:03:47,470 --> 00:03:51,780 And you've heard of three j symbols and Racah coefficients. 59 00:03:51,780 --> 00:03:52,580 Maybe you haven't. 60 00:03:52,580 --> 00:03:54,970 But there is just a rich literature 61 00:03:54,970 --> 00:03:57,550 of this sort of stuff. 62 00:03:57,550 --> 00:04:02,100 So today I'm going to talk briefly 63 00:04:02,100 --> 00:04:06,360 about rotational spectra because I'm a spectroscopist. 64 00:04:06,360 --> 00:04:13,040 And from rotational spectra we learn about molecular geometry. 65 00:04:13,040 --> 00:04:16,089 Now it's really strange because why don't we just look 66 00:04:16,089 --> 00:04:18,025 at a molecule and measure it? 67 00:04:18,025 --> 00:04:20,140 Well, we can't because it's smaller 68 00:04:20,140 --> 00:04:22,570 than the wavelength of light that we 69 00:04:22,570 --> 00:04:25,330 would use to illuminate our ruler, 70 00:04:25,330 --> 00:04:28,150 and so we get the structure of a molecule, 71 00:04:28,150 --> 00:04:31,390 the geometric structure, at least part of it, 72 00:04:31,390 --> 00:04:34,050 from the rotational spectrum. 73 00:04:34,050 --> 00:04:36,520 Now I'm only going to talk about the rotational spectrum 74 00:04:36,520 --> 00:04:38,680 of a diatomic molecule. 75 00:04:38,680 --> 00:04:43,810 You don't want me to go further because polyatomic molecules 76 00:04:43,810 --> 00:04:47,130 have extra complexity which you could understand, 77 00:04:47,130 --> 00:04:49,180 but I don't want to go there because we have 78 00:04:49,180 --> 00:04:52,720 a lot of other stuff to do. 79 00:04:52,720 --> 00:04:56,350 I also had promised to talk about visualization 80 00:04:56,350 --> 00:04:59,200 of wave functions, and I'll leave that 81 00:04:59,200 --> 00:05:02,270 to your previous experience. 82 00:05:02,270 --> 00:05:04,780 But I do want to comment. 83 00:05:04,780 --> 00:05:09,940 We often take a sum of wave functions 84 00:05:09,940 --> 00:05:14,380 for positive and negative projection quantum numbers 85 00:05:14,380 --> 00:05:18,820 to make them real or pure imaginary. 86 00:05:18,820 --> 00:05:22,630 When we do that, they are still eigenfunctions of the angular 87 00:05:22,630 --> 00:05:25,300 momentum of squared, but they're not 88 00:05:25,300 --> 00:05:30,090 eigenfunctions of a projection. 89 00:05:30,090 --> 00:05:33,620 And you could-- in fact, maybe you will on Thursday-- 90 00:05:33,620 --> 00:05:39,740 actually evaluate Lz times a symmetrized function. 91 00:05:43,060 --> 00:05:47,510 So by symmetrizing them, you get to see the nodal structure, 92 00:05:47,510 --> 00:05:51,510 which is nice, but you lose the fact 93 00:05:51,510 --> 00:05:56,070 that you have eigenfunctions of a projection of angular 94 00:05:56,070 --> 00:06:00,170 momentum, and that's kind of sad. 95 00:06:00,170 --> 00:06:15,720 OK, so spectra-- so we have a diatomic molecule, mA, 96 00:06:15,720 --> 00:06:19,880 mB, and we have a center mass. 97 00:06:19,880 --> 00:06:23,760 And so we're going to be interested in the energy 98 00:06:23,760 --> 00:06:30,060 levels for a rotation of the molecule around that axis which 99 00:06:30,060 --> 00:06:31,785 is perpendicular to the bond axis. 100 00:06:38,590 --> 00:06:46,680 When we do that, we discover that the energy levels 101 00:06:46,680 --> 00:06:49,260 are given by the rotational Hamiltonian. 102 00:06:49,260 --> 00:06:51,390 And for a rotation-- 103 00:06:51,390 --> 00:06:56,850 it's free rotation, so there's no potential. 104 00:06:56,850 --> 00:07:02,100 And the operator is L squared or J squared. 105 00:07:02,100 --> 00:07:04,020 That's another thing. 106 00:07:04,020 --> 00:07:10,680 You already have experienced my use of L and J and maybe 107 00:07:10,680 --> 00:07:12,242 some other things. 108 00:07:12,242 --> 00:07:13,450 They're all angular momentum. 109 00:07:13,450 --> 00:07:16,470 They're all the same sort of thing, 110 00:07:16,470 --> 00:07:21,060 although L is usually referring to electronic coordinates, 111 00:07:21,060 --> 00:07:24,690 and J is usually referring to nuclear coordinates. 112 00:07:24,690 --> 00:07:26,670 Big deal. 113 00:07:26,670 --> 00:07:28,900 But you get a sense that we're talking 114 00:07:28,900 --> 00:07:34,030 about a very rich idea where it doesn't matter 115 00:07:34,030 --> 00:07:35,710 what you name the things. 116 00:07:35,710 --> 00:07:36,835 They follow the same rules. 117 00:07:40,210 --> 00:07:50,020 So we have a single term in the Hamiltonian, mu r0 squared, 118 00:07:50,020 --> 00:07:54,010 or this might be the equilibrium instead of 119 00:07:54,010 --> 00:07:56,177 just the fixed internuclear distance. 120 00:07:56,177 --> 00:07:57,760 But we're talking about a rigid rotor, 121 00:07:57,760 --> 00:08:02,850 so r0 is the internuclear distance. 122 00:08:02,850 --> 00:08:11,595 And now I want to be able to write-- 123 00:08:18,520 --> 00:08:22,525 OK, so I want to have this quantity. 124 00:08:25,120 --> 00:08:28,720 I want to have this quantity in reciprocal centimeter units 125 00:08:28,720 --> 00:08:31,900 because that's what all spectroscopists do, 126 00:08:31,900 --> 00:08:35,340 or sometimes they use megahertz. 127 00:08:35,340 --> 00:08:38,760 In that case, the speed of light is gone. 128 00:08:38,760 --> 00:08:44,790 And when I evaluate the effect of this operator on a wave 129 00:08:44,790 --> 00:08:48,280 function, we get an h bar squared, which cancels that. 130 00:08:48,280 --> 00:08:54,870 We would like to have an energy level expression 131 00:08:54,870 --> 00:09:09,570 EJM is equal to hcBL L plus 1. 132 00:09:09,570 --> 00:09:13,790 So the units of B just accommodate the fact 133 00:09:13,790 --> 00:09:16,090 that we want it in wave numbers. 134 00:09:16,090 --> 00:09:20,270 But this is energy, So we need the hc. 135 00:09:20,270 --> 00:09:25,520 And when the operator operates, we get an h bar squared, 136 00:09:25,520 --> 00:09:30,980 and that's canceled by this factor here. 137 00:09:30,980 --> 00:09:35,290 And so the handy dandy expression 138 00:09:35,290 --> 00:09:46,300 for the rotational constant is 16.85673 times 139 00:09:46,300 --> 00:09:53,180 the reduced mass in AMU units times the internuclear 140 00:09:53,180 --> 00:10:02,540 distance in angstrom units squared reciprocal. 141 00:10:02,540 --> 00:10:08,110 So if you want to know the energy-- 142 00:10:08,110 --> 00:10:11,950 if you want to know the rotational constant 143 00:10:11,950 --> 00:10:15,490 in wave number units, this is the conversion. 144 00:10:15,490 --> 00:10:16,240 Big deal. 145 00:10:28,690 --> 00:10:32,995 So the energy levels are simply-- 146 00:10:35,830 --> 00:10:39,830 I'm going to stick with LM even though I'm hardwired to call it 147 00:10:39,830 --> 00:10:42,920 J. Now if I go back and forth between J and L, 148 00:10:42,920 --> 00:10:46,970 you'll have to forgive me because I just can't-- 149 00:10:46,970 --> 00:10:47,960 yes. 150 00:10:47,960 --> 00:10:56,210 All right, so we have the energy levels, hcB times L L plus 1. 151 00:10:56,210 --> 00:10:59,804 Now L is an integer, and for the simple 152 00:10:59,804 --> 00:11:02,220 diatomics that you're going to deal with, it's an integer. 153 00:11:02,220 --> 00:11:04,460 You can start at zero. 154 00:11:04,460 --> 00:11:15,650 And so the energy levels, the L L plus 1, the L L plus 1 is 2. 155 00:11:15,650 --> 00:11:17,210 I want to make this look right. 156 00:11:20,550 --> 00:11:24,950 B-- this is 1 times 2. 157 00:11:24,950 --> 00:11:26,650 This is 2 times 3. 158 00:11:26,650 --> 00:11:30,210 And the 3 times 4 is 12. 159 00:11:30,210 --> 00:11:34,360 And the important thing is that this energy differences is 2B. 160 00:11:37,580 --> 00:11:41,160 This energy difference is 4B. 161 00:11:41,160 --> 00:11:42,700 This energy difference is 6B. 162 00:11:45,550 --> 00:11:47,450 And so what happens in the spectrum-- 163 00:11:50,320 --> 00:11:52,390 here's energy. 164 00:11:52,390 --> 00:11:54,370 Here's zero. 165 00:11:54,370 --> 00:12:04,910 We have a line here at 2B, a line here at 4B, 6B, 8B. 166 00:12:04,910 --> 00:12:08,960 So if you were able to look at the rotational spectrum, 167 00:12:08,960 --> 00:12:13,660 the lines in the spectrum would be evenly spaced. 168 00:12:13,660 --> 00:12:16,380 The levels are not. 169 00:12:16,380 --> 00:12:18,510 That's very important, especially 170 00:12:18,510 --> 00:12:20,340 when you start doing perturbation theory 171 00:12:20,340 --> 00:12:23,010 because you're going to have energy denominators which 172 00:12:23,010 --> 00:12:25,480 are multiples of a common factor, 173 00:12:25,480 --> 00:12:29,100 but they're not equal to each other. 174 00:12:29,100 --> 00:12:34,600 But we have a spectrum, and it looks really, really trivial. 175 00:12:34,600 --> 00:12:37,100 And textbooks don't talk about this, 176 00:12:37,100 --> 00:12:40,310 but if you have a relatively light diatomic molecule 177 00:12:40,310 --> 00:12:44,930 and you have a laboratory which is equipped with a microwave 178 00:12:44,930 --> 00:12:49,220 spectrometer which is able to generate data that got you 179 00:12:49,220 --> 00:12:52,520 tenure and whatever, it's probably 180 00:12:52,520 --> 00:12:56,900 a spectrometer where the tuning range of the microwave 181 00:12:56,900 --> 00:12:58,100 oscillator is about 30%. 182 00:13:00,989 --> 00:13:01,530 That's a lot. 183 00:13:01,530 --> 00:13:06,470 If you think about NMR, the tuning range is-- 184 00:13:06,470 --> 00:13:09,740 30% is huge. 185 00:13:09,740 --> 00:13:12,560 So, you see this and you say, oh yeah. 186 00:13:12,560 --> 00:13:16,710 I could assign that spectrum because an obvious pattern. 187 00:13:16,710 --> 00:13:19,580 But what happens in the spectrum is you get one line. 188 00:13:23,310 --> 00:13:27,030 And so you say, well, I need to know the internuclear 189 00:13:27,030 --> 00:13:33,230 distance of this molecule to 6 or 8 or 10 digits, 190 00:13:33,230 --> 00:13:34,810 but I get one line. 191 00:13:34,810 --> 00:13:35,870 There's no pattern. 192 00:13:38,790 --> 00:13:40,810 The textbooks are so full of formulas 193 00:13:40,810 --> 00:13:44,880 that they don't indicate that, in reality, you've 194 00:13:44,880 --> 00:13:47,150 got a problem. 195 00:13:47,150 --> 00:13:48,730 And, in fact, in reality you've got 196 00:13:48,730 --> 00:13:50,110 something that's also a gift. 197 00:13:53,250 --> 00:14:07,850 So there's two things that happen, isotopes and vibration. 198 00:14:11,020 --> 00:14:12,470 So we have this one line. 199 00:14:12,470 --> 00:14:16,570 We have a very, very strong, very narrow-- 200 00:14:16,570 --> 00:14:19,180 you can measure the daylights out of it if you wanted to. 201 00:14:22,170 --> 00:14:26,280 And then down here, there's going to be-- 202 00:14:26,280 --> 00:14:33,150 well, actually, sometimes like in chlorine and bromine, 203 00:14:33,150 --> 00:14:36,720 there's a heavy isotope and a light isotope, 204 00:14:36,720 --> 00:14:38,790 and they have similar abundances. 205 00:14:38,790 --> 00:14:40,470 And so you get isotope splittings, 206 00:14:40,470 --> 00:14:46,380 and that's expressed in the reduced mass mA 207 00:14:46,380 --> 00:14:51,420 mB over mA plus mB. 208 00:14:51,420 --> 00:14:55,980 Now the isotope splittings can be really, really small, 209 00:14:55,980 --> 00:15:02,000 but these lines have a width of a part in a million, 210 00:15:02,000 --> 00:15:05,210 maybe even narrower. 211 00:15:05,210 --> 00:15:07,565 And so you can see isotope stuff. 212 00:15:10,640 --> 00:15:12,710 That doesn't tell you anything at all that 213 00:15:12,710 --> 00:15:17,270 you didn't know except maybe that you were confused 214 00:15:17,270 --> 00:15:20,390 about what molecule it was because if you have 215 00:15:20,390 --> 00:15:23,510 a particular atom, it's always born 216 00:15:23,510 --> 00:15:27,950 with the normal isotope ratios. 217 00:15:27,950 --> 00:15:30,230 Except here we have a little problem 218 00:15:30,230 --> 00:15:34,520 where, in sulfur, if you look in minerals, 219 00:15:34,520 --> 00:15:38,990 the isotope ratios are not the naturally abundant 220 00:15:38,990 --> 00:15:41,000 of sulfur isotope. 221 00:15:41,000 --> 00:15:43,580 And this has to do with something really important that 222 00:15:43,580 --> 00:15:46,550 happened 2 and 1/2 billion years ago. 223 00:15:46,550 --> 00:15:48,350 Oxygen happened. 224 00:15:48,350 --> 00:15:56,450 And so isotope ratios are of some geological chemical 225 00:15:56,450 --> 00:16:01,100 significance, but here, if you know what the molecule is, 226 00:16:01,100 --> 00:16:03,980 there will be isotope lines. 227 00:16:03,980 --> 00:16:07,310 And they can be pretty strong depending 228 00:16:07,310 --> 00:16:10,130 on the relative abundance of the different isotopes, 229 00:16:10,130 --> 00:16:12,440 or they can be extremely weak. 230 00:16:12,440 --> 00:16:18,200 So there's stuff, so some grass to be mowed on the baseline. 231 00:16:18,200 --> 00:16:20,270 In addition-- and this is something 232 00:16:20,270 --> 00:16:22,970 that really surprises people. 233 00:16:22,970 --> 00:16:29,300 So here is v equals 0, and way up high is v equals 1. 234 00:16:29,300 --> 00:16:32,780 Typically, the vibrational intervals 235 00:16:32,780 --> 00:16:35,060 are on the order of a thousand times bigger 236 00:16:35,060 --> 00:16:38,580 than the rotational intervals. 237 00:16:38,580 --> 00:16:43,050 And typically, the rotational constant 238 00:16:43,050 --> 00:16:45,180 decreases in steps of about a tenth 239 00:16:45,180 --> 00:16:49,040 of a percent per vibration. 240 00:16:49,040 --> 00:16:52,940 Now we do care about how much it decreases because that allows 241 00:16:52,940 --> 00:16:54,800 us to know a whole bunch of stuff 242 00:16:54,800 --> 00:17:00,490 about how rotation and vibration interact. 243 00:17:00,490 --> 00:17:04,824 And I'm not probably going to do the lecture on the rotation 244 00:17:04,824 --> 00:17:07,030 and vibration interaction unless I 245 00:17:07,030 --> 00:17:09,280 have to give a lecture on something that I can't do 246 00:17:09,280 --> 00:17:12,310 and I'll slip in that one. 247 00:17:12,310 --> 00:17:17,829 So what happens is there are vibrational satellites. 248 00:17:17,829 --> 00:17:19,150 So here's v equals 0. 249 00:17:19,150 --> 00:17:21,290 It has rotational structure. 250 00:17:21,290 --> 00:17:22,420 And here is v equals 1. 251 00:17:22,420 --> 00:17:25,390 It has rotational structure. 252 00:17:25,390 --> 00:17:29,020 The v equals 1 stuff is typically a hundred 253 00:17:29,020 --> 00:17:34,590 to a thousand times weaker than the v equals 0 stuff. 254 00:17:34,590 --> 00:17:36,090 And that's basically telling you, 255 00:17:36,090 --> 00:17:40,800 how does a molecule changes its average 1 256 00:17:40,800 --> 00:17:44,880 over r squared as it vibrates, and that's a useful thing. 257 00:17:44,880 --> 00:17:46,710 It may even be useful on Thursday. 258 00:17:50,120 --> 00:17:54,140 So in addition to hyperfine, there's other small stuff 259 00:17:54,140 --> 00:17:56,000 having to do with vibrations. 260 00:17:56,000 --> 00:17:59,120 And in some experiments that I do, 261 00:17:59,120 --> 00:18:03,140 we use UV light to break a molecule. 262 00:18:03,140 --> 00:18:07,910 And the fragments that we make are born vibrationally excited. 263 00:18:07,910 --> 00:18:14,060 And so by looking at the stuff near the v equals 0 frequency, 264 00:18:14,060 --> 00:18:15,890 you see a whole bunch of stuff which 265 00:18:15,890 --> 00:18:20,090 tells you the populations of the different vibrational levels. 266 00:18:20,090 --> 00:18:22,560 And that's strange because vibration is not part 267 00:18:22,560 --> 00:18:24,180 of the rotational spectrum. 268 00:18:24,180 --> 00:18:27,330 Vibration is big, but we get vibrational information 269 00:18:27,330 --> 00:18:29,220 from the rotational spectrum. 270 00:18:29,220 --> 00:18:32,220 And because the rotational spectrum 271 00:18:32,220 --> 00:18:36,960 is at such high resolution, it's trivial to resolve 272 00:18:36,960 --> 00:18:40,170 and to detect these weak other features. 273 00:18:40,170 --> 00:18:43,410 So as much as I'm going to talk about spectroscopy, 274 00:18:43,410 --> 00:18:47,520 it's a little bit more than I had originally planned. 275 00:18:47,520 --> 00:18:51,180 And now we're going to move to this topic which is dear 276 00:18:51,180 --> 00:18:56,160 to my heart, and it's an example of an abstract algebra 277 00:18:56,160 --> 00:18:58,980 that you use in quantum mechanics. 278 00:18:58,980 --> 00:19:02,350 And there are people who only do this kind of thing 279 00:19:02,350 --> 00:19:05,430 as opposed to solving Schrodinger equation 280 00:19:05,430 --> 00:19:11,220 or even just doing perturbation theory on matrices. 281 00:19:11,220 --> 00:19:14,010 So the rest of today's lecture is 282 00:19:14,010 --> 00:19:19,330 going to be an excursion through here as much as I can do. 283 00:19:19,330 --> 00:19:22,320 It's all clear in the notes, but I think 284 00:19:22,320 --> 00:19:24,450 it's a little bit strange. 285 00:19:24,450 --> 00:19:25,950 Oh, I want to say one more thing. 286 00:19:25,950 --> 00:19:27,960 How do we make assignments? 287 00:19:27,960 --> 00:19:33,252 You all took 5.111 or 5.112 or 3.091, 288 00:19:33,252 --> 00:19:39,080 and there are things that you learn about how big atoms are. 289 00:19:39,080 --> 00:19:41,650 And so you can sort of estimate what 290 00:19:41,650 --> 00:19:44,500 the internuclear distance is-- 291 00:19:44,500 --> 00:19:46,750 maybe to 10% or 20%. 292 00:19:46,750 --> 00:19:49,660 That's not of any chemical use, but it's 293 00:19:49,660 --> 00:19:52,370 enough to assign the spectrum. 294 00:19:52,370 --> 00:19:57,000 So what you do is you say OK, I guess the internuclear distance 295 00:19:57,000 --> 00:19:57,730 is this. 296 00:19:57,730 --> 00:20:01,900 That determines what rotational transition you were observing. 297 00:20:01,900 --> 00:20:04,110 And that has consequences of suppose 298 00:20:04,110 --> 00:20:06,970 you're observing L to L plus 1. 299 00:20:06,970 --> 00:20:11,060 Well, what about L plus to L plus 2 or L minus 1 to L? 300 00:20:11,060 --> 00:20:13,190 So if you make an assignment, you 301 00:20:13,190 --> 00:20:16,370 can predict where the other guys are. 302 00:20:16,370 --> 00:20:19,460 And that would require going to one of your friends 303 00:20:19,460 --> 00:20:21,680 who has a different spectrometer and getting 304 00:20:21,680 --> 00:20:23,420 him to record a spectrum for you, 305 00:20:23,420 --> 00:20:26,000 and that's good for human relations. 306 00:20:26,000 --> 00:20:30,050 And that then enables you to make assignments and know 307 00:20:30,050 --> 00:20:32,570 the rotational constants to as many digits 308 00:20:32,570 --> 00:20:37,480 as you possibly could want, includ-- all the way up to 10. 309 00:20:37,480 --> 00:20:38,500 It's just crazy. 310 00:20:38,500 --> 00:20:42,160 You really don't care about internuclear distances 311 00:20:42,160 --> 00:20:44,350 beyond about a thousandth of an angstrom, 312 00:20:44,350 --> 00:20:45,400 but you can have them. 313 00:20:49,810 --> 00:20:54,100 So first of all, you know we can define an angular 314 00:20:54,100 --> 00:21:00,985 momentum as r cross p, and we can write that as a matrix. 315 00:21:12,150 --> 00:21:14,340 Now I suspect you've all seen this. 316 00:21:14,340 --> 00:21:17,980 These are unit vectors along the x, y, and z directions. 317 00:21:17,980 --> 00:21:23,040 And this is a vector, so there are three components, 318 00:21:23,040 --> 00:21:25,540 and we get three components here. 319 00:21:25,540 --> 00:21:28,142 Now you do want to make sure you know this notation 320 00:21:28,142 --> 00:21:29,100 and know how to use it. 321 00:21:34,360 --> 00:21:39,020 So here is the magic equation. 322 00:21:39,020 --> 00:21:51,606 Li, Lj is equal to ih bar sum over k epsilon ijk Lk. 323 00:21:51,606 --> 00:21:57,540 Well, what is epsilon ijk? 324 00:21:57,540 --> 00:21:59,580 Well, it's got many names, but it's 325 00:21:59,580 --> 00:22:03,900 a really neat tool which is very wonderful in enabling 326 00:22:03,900 --> 00:22:07,900 you to derive new equations. 327 00:22:07,900 --> 00:22:13,660 So if i, j, and k correspond to xyz in cyclic order-- 328 00:22:13,660 --> 00:22:17,790 in other words, xyz, yzx, et cetera-- 329 00:22:17,790 --> 00:22:20,790 then this is plus 1. 330 00:22:20,790 --> 00:22:24,340 If it's in anticyclic order, it's minus 1. 331 00:22:24,340 --> 00:22:27,720 And if any index is repeated, it's 0. 332 00:22:27,720 --> 00:22:31,050 So it packs a real punch, but it enables 333 00:22:31,050 --> 00:22:33,300 you to do fantastic things. 334 00:22:33,300 --> 00:22:40,170 So if we have Lx, Ly, it's equal to ih bar plus 1 times Lz. 335 00:22:48,400 --> 00:22:52,510 And the point of this lecture is with this, 336 00:22:52,510 --> 00:22:56,800 you can derive all of the matrix elements of an angular 337 00:22:56,800 --> 00:22:57,400 momentum-- 338 00:22:57,400 --> 00:23:04,610 L squared, Lz, L plus minus, and anything else. 339 00:23:04,610 --> 00:23:06,214 But these are the important ones, 340 00:23:06,214 --> 00:23:07,630 and this is what we want to derive 341 00:23:07,630 --> 00:23:12,120 from our excursion in matrix element land. 342 00:23:15,160 --> 00:23:18,220 So the first thing we do is we extract 343 00:23:18,220 --> 00:23:22,900 some fundamental equations from this commutator. 344 00:23:31,020 --> 00:23:37,990 So the first equation is that L squared 345 00:23:37,990 --> 00:23:55,140 Lz is equal to Lx squared Lz plus Ly squared Lz plus Lz 346 00:23:55,140 --> 00:23:56,730 squared Lz. 347 00:24:01,240 --> 00:24:04,910 And we know this one is 0, right? 348 00:24:04,910 --> 00:24:06,890 This one, you have to do a little practice, 349 00:24:06,890 --> 00:24:13,320 but you can write this commutation rule as Lx times Lx 350 00:24:13,320 --> 00:24:22,590 comma Lz plus Lx comma Lz Lx. 351 00:24:22,590 --> 00:24:26,610 So if you have a square, you take it out the front side 352 00:24:26,610 --> 00:24:27,810 then the back side. 353 00:24:27,810 --> 00:24:29,565 And now we know what this. 354 00:24:29,565 --> 00:24:31,630 This is minus ih bar Ly. 355 00:24:34,390 --> 00:24:40,810 And this is minus ih bar Ly. 356 00:24:45,130 --> 00:24:48,310 And we do this one, and we discover we have the same thing 357 00:24:48,310 --> 00:24:52,030 except with the opposite sign. 358 00:24:52,030 --> 00:24:56,820 And so what we end up getting is that this is 0. 359 00:24:56,820 --> 00:24:58,480 Now I skipped some steps. 360 00:24:58,480 --> 00:25:04,030 I said them, but I want you to just go through that and see. 361 00:25:04,030 --> 00:25:06,770 So you know what this is. 362 00:25:06,770 --> 00:25:09,130 It's going to be Ly, and it's going 363 00:25:09,130 --> 00:25:12,490 to be minus Ly times ih bar. 364 00:25:12,490 --> 00:25:14,140 And you get the same thing here. 365 00:25:14,140 --> 00:25:17,340 But then you have an LxLy, and you have an LxLy. 366 00:25:24,080 --> 00:25:26,990 And when you do the same trick with this, 367 00:25:26,990 --> 00:25:30,110 you're going to get an Ly and an Lx again, 368 00:25:30,110 --> 00:25:32,360 and they'll be the opposite sign. 369 00:25:32,360 --> 00:25:39,450 So this one is really important because what it says 370 00:25:39,450 --> 00:25:47,790 is that you can take any projection quantum number 371 00:25:47,790 --> 00:25:50,770 and it will commute with the magnitude squared. 372 00:25:50,770 --> 00:25:57,740 The same argument works for Ly and Lz and Lx. 373 00:25:57,740 --> 00:26:01,130 So we have one really powerful commutator 374 00:26:01,130 --> 00:26:13,940 which is that L squared Li equals 0 for x, y, and z, 375 00:26:13,940 --> 00:26:19,070 which means since we like L squared and Lz-- 376 00:26:19,070 --> 00:26:21,890 we could add like Lx instead of Lz, 377 00:26:21,890 --> 00:26:24,530 but we tend to favor these-- 378 00:26:24,530 --> 00:26:31,070 that L squared and Lz are operators 379 00:26:31,070 --> 00:26:35,430 that can have a common set of eigenfunctions. 380 00:26:35,430 --> 00:26:38,090 If we have two operators that commute, 381 00:26:38,090 --> 00:26:40,790 the eigenfunctions of one can be the eigenfunctions 382 00:26:40,790 --> 00:26:41,360 of the other. 383 00:26:41,360 --> 00:26:42,440 Very convenient. 384 00:26:46,760 --> 00:26:49,580 Then there's another operator that we can derive, 385 00:26:49,580 --> 00:26:50,270 and that is-- 386 00:26:55,430 --> 00:26:59,680 let's define this thing, a step up 387 00:26:59,680 --> 00:27:04,450 or step down or our raising or lowering operator-- 388 00:27:04,450 --> 00:27:06,340 we don't know that yet-- 389 00:27:06,340 --> 00:27:07,960 Lx plus or minus iLy. 390 00:27:17,620 --> 00:27:22,360 So we might want to know the commutation rule of Lz 391 00:27:22,360 --> 00:27:23,350 with L plus minus. 392 00:27:27,430 --> 00:27:34,010 We know how to write this out because we have Lz with Lx, 393 00:27:34,010 --> 00:27:37,270 and we know that's going to be a minus ih bar Ly. 394 00:27:40,340 --> 00:27:49,390 And we have Lz with iLy, and that's going to be a minus Lx. 395 00:27:49,390 --> 00:27:51,885 Anyway, I'm going to just write down the final result, 396 00:27:51,885 --> 00:28:00,290 that this is equal to plus or minus h bar times L plus minus. 397 00:28:04,100 --> 00:28:09,050 The algebra of this operator enables 398 00:28:09,050 --> 00:28:11,870 you to slice through any derivation 399 00:28:11,870 --> 00:28:14,960 as fast as you can write once you've loaded this 400 00:28:14,960 --> 00:28:16,190 into your head. 401 00:28:16,190 --> 00:28:16,925 Yes? 402 00:28:16,925 --> 00:28:19,730 AUDIENCE: So for the epsilon, how do you [INAUDIBLE]?? 403 00:28:19,730 --> 00:28:21,740 Is it like xy becomes-- 404 00:28:21,740 --> 00:28:23,380 if it's cyclical it's positive? 405 00:28:23,380 --> 00:28:24,380 ROBERT FIELD: I'm sorry? 406 00:28:24,380 --> 00:28:28,590 AUDIENCE: When you say the epsilon thing, epsilon ijk, 407 00:28:28,590 --> 00:28:32,164 so you're saying that if it's in order, it's 1? 408 00:28:32,164 --> 00:28:34,080 ROBERT FIELD: Let's just do this a little bit. 409 00:28:34,080 --> 00:28:38,720 Let's say we have Lx and Ly. 410 00:28:38,720 --> 00:28:42,680 Well, we know that that's going to give Lz. 411 00:28:42,680 --> 00:28:49,520 And xyz, ijk, that's cyclic order. 412 00:28:49,520 --> 00:28:52,340 We say that's the home base. 413 00:28:52,340 --> 00:28:59,004 And if we have yxz, that would be anticyclic, 414 00:28:59,004 --> 00:29:00,420 and so that would be a minus sign. 415 00:29:00,420 --> 00:29:02,070 You know that just by looking at this, 416 00:29:02,070 --> 00:29:04,860 and you say if we switch this, the sign of the commutator 417 00:29:04,860 --> 00:29:07,140 has to switch. 418 00:29:07,140 --> 00:29:09,850 There's a lot of stuff loaded in there. 419 00:29:09,850 --> 00:29:12,820 And once you've sort of processed it, 420 00:29:12,820 --> 00:29:14,550 it becomes automatic. 421 00:29:14,550 --> 00:29:15,850 You forget the beauty of it. 422 00:29:18,560 --> 00:29:21,800 So are you satisfied? 423 00:29:21,800 --> 00:29:23,580 Everybody else? 424 00:29:23,580 --> 00:29:26,120 All right, so now let's do another one. 425 00:29:30,120 --> 00:29:33,160 Let's look at L squared L plus minus. 426 00:29:36,760 --> 00:29:40,210 Well, this one is super easy because we already 427 00:29:40,210 --> 00:29:44,440 know that L squared commutes with Lx, Ly, and Lz. 428 00:29:44,440 --> 00:29:46,810 So I just need to just write 0 here 429 00:29:46,810 --> 00:29:50,710 because this is Lx plus or minus iLy, 430 00:29:50,710 --> 00:29:53,270 and we know L squared commutes with both of them. 431 00:30:02,930 --> 00:30:05,915 Now comes the abstract and weird stuff. 432 00:30:10,970 --> 00:30:13,550 We're starting to use the commutators to derive 433 00:30:13,550 --> 00:30:15,470 the matrix elements and selection rules. 434 00:30:18,180 --> 00:30:23,760 So let us say that we have some function which 435 00:30:23,760 --> 00:30:28,440 is an eigenfunction of L squared and Lz. 436 00:30:28,440 --> 00:30:33,330 And so we're entitled to say that L squared operating 437 00:30:33,330 --> 00:30:40,520 on this function gives an eigenvalue we call lambda. 438 00:30:40,520 --> 00:30:44,920 And we can also say that Lz operating on the function 439 00:30:44,920 --> 00:30:49,020 gives a different concept mu. 440 00:30:49,020 --> 00:30:52,190 Now this lambda and mu have no significance. 441 00:30:52,190 --> 00:30:53,940 They're just numbers. 442 00:30:53,940 --> 00:30:56,440 There's not something that's going to pop up here that says, 443 00:30:56,440 --> 00:30:59,114 oh yeah, this means something. 444 00:31:02,290 --> 00:31:06,580 So now we're going to use the fact that this function, which 445 00:31:06,580 --> 00:31:10,000 we're allowed to have as a simultaneous eigenfunction of L 446 00:31:10,000 --> 00:31:15,780 squared and Lz with its own set of eigenvalues, this function, 447 00:31:15,780 --> 00:31:20,260 we are going to operate on it and derive some useful results 448 00:31:20,260 --> 00:31:23,590 that all are based on the commutation rules. 449 00:31:29,060 --> 00:31:35,390 So let us take L squared operating on L plus minus 450 00:31:35,390 --> 00:31:36,330 times f. 451 00:31:39,520 --> 00:31:50,510 And we know that L plus minus commutes with L squared. 452 00:31:50,510 --> 00:31:55,420 So we can write L plus minus times L squared f. 453 00:31:58,200 --> 00:32:02,720 But L squared operating on f gives lambda. 454 00:32:02,720 --> 00:32:05,620 We have L plus minus lambda f. 455 00:32:13,990 --> 00:32:16,110 Oh, isn't that interesting? 456 00:32:16,110 --> 00:32:24,256 We have-- I'll just write it-- lambda times L plus minus f-- 457 00:32:24,256 --> 00:32:27,220 L plus minus f. 458 00:32:27,220 --> 00:32:34,150 So it's saying that this thing is an eigenfunction of L 459 00:32:34,150 --> 00:32:38,219 squared with eigenvalue lambda. 460 00:32:38,219 --> 00:32:39,010 Well, we knew that. 461 00:32:48,040 --> 00:32:58,340 So L plus minus operating on f does not 462 00:32:58,340 --> 00:33:05,040 change lambda, the eigenvalue of L squared. 463 00:33:21,640 --> 00:33:23,850 Now let's use another one. 464 00:33:23,850 --> 00:33:28,732 Let's use Lz L plus minus. 465 00:33:28,732 --> 00:33:29,990 Well, I derived it. 466 00:33:29,990 --> 00:33:36,626 It's plus or minus h bar times L plus minus. 467 00:33:36,626 --> 00:33:38,470 And if I didn't derive it, I should have, 468 00:33:38,470 --> 00:33:40,190 but I'm pretty sure I did. 469 00:33:40,190 --> 00:33:42,850 And so now what we can do is write 470 00:33:42,850 --> 00:33:48,880 Lz L plus minus minus L plus minus Lz 471 00:33:48,880 --> 00:33:53,550 is equal to h plus or minus h bar L plus minus. 472 00:34:10,400 --> 00:34:16,514 Let's stick in a function on the right, f, f, f. 473 00:34:19,820 --> 00:34:24,260 So now we have these operators operating on the same function. 474 00:34:27,900 --> 00:34:32,100 Well, we don't yet know what L plus minus does to f, 475 00:34:32,100 --> 00:34:35,260 but we know what Lz does to it. 476 00:34:35,260 --> 00:34:38,130 And so what we can write immediately 477 00:34:38,130 --> 00:34:43,620 is that Lz operating on L plus minus 478 00:34:43,620 --> 00:34:51,679 f is equal to plus or minus h bar 479 00:34:51,679 --> 00:35:01,850 L plus minus f plus mu L plus minus f. 480 00:35:01,850 --> 00:35:04,560 Well, that's interesting. 481 00:35:04,560 --> 00:35:09,000 So we see that we can rearrange this, 482 00:35:09,000 --> 00:35:15,060 and we could write plus minus h bar L plus minus f is 483 00:35:15,060 --> 00:35:25,520 equal to mu L plus minus f plus Lz L plus minus f-- 484 00:35:35,747 --> 00:35:37,610 that's h bar, OK. 485 00:35:40,190 --> 00:35:42,910 Oh, I'm sorry, L plus minus f there. 486 00:35:46,476 --> 00:35:47,970 So what's this telling us? 487 00:35:56,540 --> 00:35:58,430 So we can simply combine these terms. 488 00:35:58,430 --> 00:36:00,020 We have the L plus minus f here. 489 00:36:05,030 --> 00:36:12,650 And so we can write mu plus h bar times L plus minus f. 490 00:36:12,650 --> 00:36:15,680 That's the point. 491 00:36:15,680 --> 00:36:18,624 So we have this operator operating-- 492 00:36:22,206 --> 00:36:24,330 AUDIENCE: I don't think you want the whole second-- 493 00:36:24,330 --> 00:36:24,840 ROBERT FIELD: I'm sorry? 494 00:36:24,840 --> 00:36:26,839 AUDIENCE: The first line goes straight to there. 495 00:36:26,839 --> 00:36:28,722 I think your second line's [INAUDIBLE].. 496 00:36:34,720 --> 00:36:37,780 ROBERT FIELD: I took this thing over to here. 497 00:36:37,780 --> 00:36:39,730 So let's just rewrite that again. 498 00:36:39,730 --> 00:36:47,030 We have Lz L plus minus f is equal to this. 499 00:36:47,030 --> 00:36:52,300 And so here we have mu, the eigenvalue, 500 00:36:52,300 --> 00:36:55,550 and it's been increased by h bar. 501 00:36:55,550 --> 00:36:59,390 And so what that tells us is that we 502 00:36:59,390 --> 00:37:03,120 have a manifold of levels-- 503 00:37:03,120 --> 00:37:10,770 mu, et cetera. 504 00:37:10,770 --> 00:37:13,710 So we get a manifold of levels that are equally spaced, 505 00:37:13,710 --> 00:37:16,620 spaced by h bar. 506 00:37:19,590 --> 00:37:23,190 AUDIENCE: I think it also should be plus or minus h bar, right? 507 00:37:26,190 --> 00:37:28,730 ROBERT FIELD: Plus minus h bar-- 508 00:37:28,730 --> 00:37:29,230 yeah. 509 00:37:32,670 --> 00:37:36,160 So we have this manifold of levels, 510 00:37:36,160 --> 00:37:40,110 and so what we can say is, well, this isn't going to go forever. 511 00:37:43,290 --> 00:37:47,240 This is a ladder of equally spaced levels, 512 00:37:47,240 --> 00:37:52,020 and it will have a highest and a lowest member. 513 00:37:52,020 --> 00:38:03,380 And so we can say, all right, well, suppose we have f max mu, 514 00:38:03,380 --> 00:38:06,400 and we have L plus operating on it. 515 00:38:06,400 --> 00:38:07,410 That's going to give 0. 516 00:38:10,280 --> 00:38:15,186 And at the same time we can say we have L minus min mu 517 00:38:15,186 --> 00:38:16,310 and that's going to give 0. 518 00:38:16,310 --> 00:38:17,730 We're going to use both of these. 519 00:38:23,030 --> 00:38:26,606 Now I'm just going to leave that there. 520 00:38:26,606 --> 00:38:27,670 Oh, I'm not. 521 00:38:27,670 --> 00:38:34,130 I'm going to say, all right, so since we 522 00:38:34,130 --> 00:38:35,420 have this arrangement-- 523 00:38:41,570 --> 00:38:43,050 all right, I am skipping something, 524 00:38:43,050 --> 00:38:44,216 and I don't want to skip it. 525 00:38:47,650 --> 00:38:57,790 So if we have L plus operating on the maximum value of mu, 526 00:38:57,790 --> 00:38:59,080 we get 0. 527 00:38:59,080 --> 00:39:04,690 And the next one down is down by an integer number of L, 528 00:39:04,690 --> 00:39:15,630 and so we can say that Lz operating on f max mu 529 00:39:15,630 --> 00:39:19,260 is equal to h bar L, some integer. 530 00:39:19,260 --> 00:39:24,010 Now this L is chosen with some prejudice. 531 00:39:24,010 --> 00:39:24,550 Yes? 532 00:39:24,550 --> 00:39:26,480 AUDIENCE: Why is there an f of x? 533 00:39:31,610 --> 00:39:33,990 ROBERT FIELD: Now I have to cheat. 534 00:39:33,990 --> 00:39:39,740 I'm going to apply an argument which is not 535 00:39:39,740 --> 00:39:42,564 based on just abstract vectors. 536 00:39:42,564 --> 00:39:43,730 We have an angular momentum. 537 00:39:43,730 --> 00:39:46,930 It has a certain length. 538 00:39:46,930 --> 00:39:50,920 We know the projection of that angular momentum on some axis 539 00:39:50,920 --> 00:39:52,540 cannot be longer than its length. 540 00:39:57,144 --> 00:39:59,060 I mean, I'm uncomfortable making that argument 541 00:39:59,060 --> 00:40:04,100 because I should be able to say it in a more abstract way, 542 00:40:04,100 --> 00:40:05,240 but this is, in fact-- 543 00:40:08,810 --> 00:40:12,080 we know there cannot be an infinite number of projection 544 00:40:12,080 --> 00:40:16,640 quantum numbers, values of the projection quantum number that 545 00:40:16,640 --> 00:40:21,660 aren't reached by applying L plus and L minus. 546 00:40:21,660 --> 00:40:23,450 It must be limited. 547 00:40:23,450 --> 00:40:33,260 And so we're going to call the maximum value of mu h bar L 548 00:40:33,260 --> 00:40:36,470 or L. 549 00:40:36,470 --> 00:40:39,260 Now I have to derive a new commutation 550 00:40:39,260 --> 00:40:42,090 rule based on the original one. 551 00:40:42,090 --> 00:40:43,820 No, let's not erase this. 552 00:40:43,820 --> 00:40:45,220 We might want to see it again. 553 00:40:49,200 --> 00:40:54,400 So let's ask, well, what does this combination of operators 554 00:40:54,400 --> 00:40:54,900 do? 555 00:41:07,980 --> 00:41:11,640 Well, this is surely equal to Lx squared, 556 00:41:11,640 --> 00:41:16,140 and we get a plus i and a minus i, 557 00:41:16,140 --> 00:41:20,620 and so it's going to be plus Ly squared. 558 00:41:20,620 --> 00:41:29,180 And then we get i times LyLx, and we get a minus i 559 00:41:29,180 --> 00:41:32,100 times LxLy. 560 00:41:41,710 --> 00:41:45,770 This is L squared minus Lx squared. 561 00:41:45,770 --> 00:41:49,750 We have the square root of 2 components of L squared, 562 00:41:49,750 --> 00:41:53,160 and so this is equal to the difference. 563 00:41:56,100 --> 00:42:04,680 And now we express this as i times LyLx. 564 00:42:09,180 --> 00:42:10,926 And what is this? 565 00:42:10,926 --> 00:42:15,778 This is plus ih bar Lx. 566 00:42:21,030 --> 00:42:23,830 AUDIENCE: I think you wrote an x [INAUDIBLE].. 567 00:42:23,830 --> 00:42:33,460 ROBERT FIELD: OK, this is, yes, x, and that's Lz. 568 00:42:33,460 --> 00:42:35,860 I didn't like what I wrote because I 569 00:42:35,860 --> 00:42:41,980 want to have everything but the z and the L squared 570 00:42:41,980 --> 00:42:45,850 disappearing, and so we get that we have 571 00:42:45,850 --> 00:42:49,600 L squared minus Lz squared. 572 00:42:49,600 --> 00:42:55,020 And then we have plus ih bar Lz. 573 00:42:58,606 --> 00:43:00,550 I lost the plus and minus. 574 00:43:00,550 --> 00:43:01,540 No I didn't. 575 00:43:01,540 --> 00:43:02,590 OK, that's it. 576 00:43:06,310 --> 00:43:10,540 And so we can rearrange this and say L squared 577 00:43:10,540 --> 00:43:22,930 is equal to Lz squared minus or plus h bar Lz plus L 578 00:43:22,930 --> 00:43:25,230 plus minus L minus plus. 579 00:43:28,390 --> 00:43:32,350 So we can use this equation-- 580 00:43:32,350 --> 00:43:34,770 OK, I'd better not-- 581 00:43:34,770 --> 00:43:36,260 to derive some good stuff. 582 00:43:36,260 --> 00:43:39,560 I better erase some stuff or access a board. 583 00:43:42,835 --> 00:43:44,460 We're actually pretty close to the end, 584 00:43:44,460 --> 00:43:47,800 so I might actually finish this. 585 00:43:47,800 --> 00:43:50,610 So we're going to use this equation to find-- 586 00:43:50,610 --> 00:43:57,300 so we want lambda, the value of lambda for the top rung 587 00:43:57,300 --> 00:43:59,890 of the manifold over here. 588 00:43:59,890 --> 00:44:07,740 So we apply L squared to f max mu. 589 00:44:07,740 --> 00:44:12,780 And we know we have an equation here 590 00:44:12,780 --> 00:44:17,100 which enables us to evaluate what the consequences of that 591 00:44:17,100 --> 00:44:19,900 will be, and it will be Lz squared 592 00:44:19,900 --> 00:44:35,460 f max mu minus and plus h bar Lz f max mu plus L 593 00:44:35,460 --> 00:44:41,864 plus L minus f max mu. 594 00:44:45,650 --> 00:44:51,170 So if we take the bottom sign, that L plus on f max 595 00:44:51,170 --> 00:44:53,130 is going to give 0. 596 00:44:53,130 --> 00:44:57,060 So we're looking at the bottom sign, and we have a 0 here, 597 00:44:57,060 --> 00:45:07,960 and so we have L squared f max mu is equal to Lz 598 00:45:07,960 --> 00:45:21,110 squared f max mu minus or plus h bar Lz f max mu plus 0. 599 00:45:21,110 --> 00:45:22,660 Isn't that interesting? 600 00:45:26,570 --> 00:45:31,670 So we know that Lz is going to give-- 601 00:45:31,670 --> 00:45:43,050 so we're going to get an h bar squared and mu max squared. 602 00:45:43,050 --> 00:45:49,140 We're going to get a minus h bar h bar mu max. 603 00:45:56,640 --> 00:46:01,020 So what this is telling us is that L squared operating on f 604 00:46:01,020 --> 00:46:13,630 max mu is given by l because we said that we're 605 00:46:13,630 --> 00:46:20,300 going to take the maximum value of mu to be h bar L. 606 00:46:20,300 --> 00:46:24,590 So I shouldn't have had an extra h bar here. 607 00:46:24,590 --> 00:46:31,280 So we get this result. So lambda-- 608 00:46:34,470 --> 00:46:37,930 so L squared operating on this gives 609 00:46:37,930 --> 00:46:41,890 the-- oh yeah, maximum mu. 610 00:46:41,890 --> 00:46:49,260 So it's telling us that L squared 611 00:46:49,260 --> 00:46:57,020 f max mu is equal to h bar squared l l plus 1. 612 00:46:57,020 --> 00:47:00,290 Now that is why we chose that constant to be l. 613 00:47:06,630 --> 00:47:09,900 And we do a similar argument for the lowest rung of the ladder. 614 00:47:15,190 --> 00:47:19,070 And for the lowest rung of the ladder, 615 00:47:19,070 --> 00:47:21,860 we know there must be a lowest rung, 616 00:47:21,860 --> 00:47:27,090 and so we will simply say, OK, for the lowest 617 00:47:27,090 --> 00:47:37,550 rung of the ladder we're going to get that mu is equal to h 618 00:47:37,550 --> 00:47:43,730 bar lambda bar mu min. 619 00:47:43,730 --> 00:47:47,570 And we do some stuff, and we discover 620 00:47:47,570 --> 00:47:54,260 that lambda has to be equal to h bar squared l l plus 1. 621 00:47:54,260 --> 00:47:58,940 And using this other relationship and the top sign, 622 00:47:58,940 --> 00:48:03,747 we get h bar squared l bar l bar minus 1. 623 00:48:08,220 --> 00:48:10,180 And there's two ways to solve this. 624 00:48:10,180 --> 00:48:13,570 One is that l is equal to minus l bar, 625 00:48:13,570 --> 00:48:18,790 and the other is that l bar is equal to l plus 1. 626 00:48:25,290 --> 00:48:28,700 Well, this is the lowest rung of the ladder. 627 00:48:34,072 --> 00:48:36,590 Wait a minute, let me just make sure I'm 628 00:48:36,590 --> 00:48:39,420 doing the logic correctly. 629 00:48:39,420 --> 00:48:42,580 It's this one, OK, here. 630 00:48:42,580 --> 00:48:44,410 So this is the lowest rung of the ladder, 631 00:48:44,410 --> 00:48:51,400 and l bar is supposedly larger than l. 632 00:48:51,400 --> 00:48:53,960 Can't be, so this is impossible. 633 00:48:53,960 --> 00:48:55,180 This is correct. 634 00:48:55,180 --> 00:49:00,590 And what we end up getting is this relationship, 635 00:49:00,590 --> 00:49:03,410 and so mu can be equal to-- 636 00:49:03,410 --> 00:49:09,700 and this is l l minus 1 minus l stepped to 1. 637 00:49:14,670 --> 00:49:17,370 This seems very weird and not very interesting 638 00:49:17,370 --> 00:49:22,200 until you say, well, how do I satisfy this? 639 00:49:22,200 --> 00:49:24,435 Well, if l is an integer, it's obvious. 640 00:49:27,030 --> 00:49:30,200 If l is a half integer, it shouldn't be quite so obvious, 641 00:49:30,200 --> 00:49:31,540 but it's true. 642 00:49:31,540 --> 00:49:37,150 So we can have integer l and half-integer l. 643 00:49:42,780 --> 00:49:44,690 That's weird. 644 00:49:44,690 --> 00:49:47,270 We can show no connection between the integer 645 00:49:47,270 --> 00:49:48,685 l's and the half-integer l's. 646 00:49:51,220 --> 00:49:53,770 They belong to completely different problems, 647 00:49:53,770 --> 00:49:56,980 but this abstract argument says, yeah, 648 00:49:56,980 --> 00:50:00,430 we can have integer l's and half-integer l's. 649 00:50:00,430 --> 00:50:05,160 And if we have electron-- 650 00:50:05,160 --> 00:50:09,280 well, we call it electron spin because we want 651 00:50:09,280 --> 00:50:11,460 it to be an angular momentum. 652 00:50:11,460 --> 00:50:15,580 Spin is sort of an angular momentum or nuclear spin. 653 00:50:15,580 --> 00:50:19,120 And we discover that there are patterns of energy levels 654 00:50:19,120 --> 00:50:22,660 which enable us to count the number of projection 655 00:50:22,660 --> 00:50:23,410 components. 656 00:50:26,210 --> 00:50:29,530 And if you have an integer l, you'll 657 00:50:29,530 --> 00:50:37,580 get 2l plus 1 components, which is an odd number. 658 00:50:37,580 --> 00:50:39,170 And if it's a half integer you get 659 00:50:39,170 --> 00:50:42,290 2l plus 1 components, which is an even number. 660 00:50:45,500 --> 00:50:48,290 And so it turns out that our definition of an angular 661 00:50:48,290 --> 00:50:51,110 momentum is more general than we thought. 662 00:50:51,110 --> 00:50:53,270 It allows there to be both integer 663 00:50:53,270 --> 00:50:56,870 and half-integer angular momentum. 664 00:50:56,870 --> 00:51:00,740 And this means we can have angular momenta where 665 00:51:00,740 --> 00:51:04,550 we can't define it in terms of r cross b. 666 00:51:04,550 --> 00:51:06,500 It's defined by the commutation rule. 667 00:51:06,500 --> 00:51:08,380 It's more general. 668 00:51:08,380 --> 00:51:09,280 It's more abstract. 669 00:51:09,280 --> 00:51:11,670 It's beautiful. 670 00:51:11,670 --> 00:51:18,720 And I don't have time to finish the job, but in the notes 671 00:51:18,720 --> 00:51:20,880 you can see that we can derive the matrix 672 00:51:20,880 --> 00:51:23,370 elements for the raising and lowering operators too. 673 00:51:23,370 --> 00:51:26,430 And the angular momentum matrix elements 674 00:51:26,430 --> 00:51:30,300 are that L plus minus operating on a function 675 00:51:30,300 --> 00:51:33,660 gives this combination square root, 676 00:51:33,660 --> 00:51:36,790 and it raises or lowers m. 677 00:51:36,790 --> 00:51:39,940 So it's sort of like what we have for the a's and a daggers 678 00:51:39,940 --> 00:51:46,630 for the harmonic oscillator, but it's not as good because you 679 00:51:46,630 --> 00:51:48,970 can't generate all the L's. 680 00:51:48,970 --> 00:51:52,450 You can generate the m sub L's. 681 00:51:52,450 --> 00:51:56,870 And that's great, but there's still something that remains 682 00:51:56,870 --> 00:52:00,050 to be done to generate the different L's. 683 00:52:00,050 --> 00:52:02,100 That's not a problem. 684 00:52:02,100 --> 00:52:05,480 It's just there's not a simple way to do it, 685 00:52:05,480 --> 00:52:07,170 at least not simple to me. 686 00:52:07,170 --> 00:52:12,890 And so now anytime we're faced with a problem involving 687 00:52:12,890 --> 00:52:15,080 angular momenta, we have a prayer 688 00:52:15,080 --> 00:52:19,120 of writing down the matrix elements without ever looking 689 00:52:19,120 --> 00:52:23,400 at the wave function, without ever looking at a differential 690 00:52:23,400 --> 00:52:24,422 operator. 691 00:52:27,134 --> 00:52:29,870 And we can also say, well, let's suppose 692 00:52:29,870 --> 00:52:41,100 we had some operator that involves L and S, 693 00:52:41,100 --> 00:52:43,380 now that we know that we have these things, 694 00:52:43,380 --> 00:52:46,470 and L plus S can be called J. So now we 695 00:52:46,470 --> 00:52:49,140 have two different operators, two different angular momenta. 696 00:52:49,140 --> 00:52:54,090 We have S and we have the total of J. Well, 697 00:52:54,090 --> 00:52:55,270 they're all angular momenta. 698 00:52:55,270 --> 00:52:58,620 They're going to satisfy their selection rules and the matrix 699 00:52:58,620 --> 00:53:02,670 elements, and we can calculate all 700 00:53:02,670 --> 00:53:06,870 of these matrix elements, including things like L dot S 701 00:53:06,870 --> 00:53:09,180 and whatever. 702 00:53:09,180 --> 00:53:12,780 So it just opens up a huge area where 703 00:53:12,780 --> 00:53:14,820 before you would say, well, I got 704 00:53:14,820 --> 00:53:16,224 to look at the wave function. 705 00:53:16,224 --> 00:53:17,640 I've got to look at this integral. 706 00:53:17,640 --> 00:53:19,230 No more. 707 00:53:19,230 --> 00:53:24,010 But there is one thing, and that is these arguments 708 00:53:24,010 --> 00:53:25,600 do not determine-- 709 00:53:25,600 --> 00:53:27,790 I mean, when you take the square root of something 710 00:53:27,790 --> 00:53:31,340 you can have a positive value and a negative value. 711 00:53:31,340 --> 00:53:33,820 That corresponds to a phase ambiguity, 712 00:53:33,820 --> 00:53:37,430 and these arguments don't resolve that. 713 00:53:37,430 --> 00:53:41,660 At some point you have to decide on the phase and be consistent. 714 00:53:41,660 --> 00:53:43,730 And since you're never looking at wave functions, 715 00:53:43,730 --> 00:53:48,980 that actually is a frequent source of error. 716 00:53:48,980 --> 00:53:52,340 But that's the only defect in this whole thing. 717 00:53:52,340 --> 00:53:55,610 So that's it for the exam. 718 00:53:55,610 --> 00:53:58,700 I will talk about something that will make you a little bit 719 00:53:58,700 --> 00:54:01,220 more comfortable about some of the exam 720 00:54:01,220 --> 00:54:05,140 questions on Wednesday, but it's not going to be tested.