1 00:00:00,090 --> 00:00:01,631 FEMALE SPEAKER: The following content 2 00:00:01,631 --> 00:00:03,820 is provided under a Creative 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 00:00:23,180 --> 00:00:25,700 PROFESSOR: So we're now starting to get 9 00:00:25,700 --> 00:00:28,310 into a really familiar territory, which 10 00:00:28,310 --> 00:00:29,451 is the hydrogen atom. 11 00:00:32,159 --> 00:00:40,770 And it's a short step from a hydrogen atom to molecules. 12 00:00:40,770 --> 00:00:41,790 And we're chemists. 13 00:00:41,790 --> 00:00:45,280 We make molecules. 14 00:00:45,280 --> 00:00:49,530 But one question that would be a legitimate question to ask 15 00:00:49,530 --> 00:00:52,810 is, what does a hydrogen atom have to do with molecules, 16 00:00:52,810 --> 00:00:55,070 because it's an atom and it's the simplest atom. 17 00:00:55,070 --> 00:01:00,390 And what I hope to show is that what 18 00:01:00,390 --> 00:01:03,090 we learn from looking at the hydrogen atom 19 00:01:03,090 --> 00:01:06,870 has all sorts of good non textbook stuff 20 00:01:06,870 --> 00:01:09,660 that prepares you for understanding stuff 21 00:01:09,660 --> 00:01:11,580 in molecules. 22 00:01:11,580 --> 00:01:15,960 And I am going to stress that because you're not 23 00:01:15,960 --> 00:01:19,350 going to see it in McQuarrie and you've probably never seen 24 00:01:19,350 --> 00:01:24,120 it elsewhere except in freshman chemistry when they expect 25 00:01:24,120 --> 00:01:26,790 that you understand the periodic table based 26 00:01:26,790 --> 00:01:28,830 on some simple concepts. 27 00:01:28,830 --> 00:01:31,480 And they come from the hydrogen atom. 28 00:01:31,480 --> 00:01:39,600 And so I'm going to attempt to make those connections. 29 00:01:39,600 --> 00:01:42,990 My last lecture was I told you something you're not 30 00:01:42,990 --> 00:01:44,110 going to be tested on. 31 00:01:44,110 --> 00:01:47,520 But it's about spectroscopy. 32 00:01:47,520 --> 00:01:49,770 That's how we learned almost everything 33 00:01:49,770 --> 00:01:53,730 we know about small molecules and a lot of stuff 34 00:01:53,730 --> 00:01:55,860 about big molecules too. 35 00:01:55,860 --> 00:01:59,540 The crucial approximations in that 36 00:01:59,540 --> 00:02:05,820 were the dipole approximation where the radiation field 37 00:02:05,820 --> 00:02:08,250 is such that the molecule field's 38 00:02:08,250 --> 00:02:11,580 a uniform but oscillating electromagnetic field. 39 00:02:14,620 --> 00:02:18,110 In developing the theory that I presented, 40 00:02:18,110 --> 00:02:24,140 we assume that only one of the initial eigenstates 41 00:02:24,140 --> 00:02:26,780 is populated at t equal zero. 42 00:02:26,780 --> 00:02:32,210 And only one final eigenstate gets tickled so 43 00:02:32,210 --> 00:02:34,100 that it gets populated. 44 00:02:34,100 --> 00:02:38,060 And so even though there is an infinity of states, 45 00:02:38,060 --> 00:02:43,280 the theory specifies we start with one definite one and only 46 00:02:43,280 --> 00:02:49,960 one final state gets selected because of resonance. 47 00:02:49,960 --> 00:02:52,030 And this is what we do all the time. 48 00:02:52,030 --> 00:02:55,120 We have an infinite dimension problem, 49 00:02:55,120 --> 00:02:58,780 and we discover that we only really need 50 00:02:58,780 --> 00:03:02,490 to worry about a very small number of states. 51 00:03:02,490 --> 00:03:06,420 And then we deal with that. 52 00:03:06,420 --> 00:03:07,920 There are a couple other assumptions 53 00:03:07,920 --> 00:03:11,280 that were essential for this first step 54 00:03:11,280 --> 00:03:14,360 into a time-dependent Hamiltonian. 55 00:03:14,360 --> 00:03:18,080 And that is the field is weak and so you get linear response. 56 00:03:18,080 --> 00:03:22,550 That means that the increase in the mixing coefficient 57 00:03:22,550 --> 00:03:26,600 for the final state is proportional to the coupling 58 00:03:26,600 --> 00:03:30,450 matrix element times time. 59 00:03:30,450 --> 00:03:34,930 Now mixing coefficients can't get larger than 1, 60 00:03:34,930 --> 00:03:40,900 and so clearly we can't use a simple theory without saying, 61 00:03:40,900 --> 00:03:44,200 OK, we don't care about the mixing coefficients. 62 00:03:44,200 --> 00:03:47,170 We care about the rate of increase. 63 00:03:47,170 --> 00:03:53,260 And so by going from amplitude to rates, 64 00:03:53,260 --> 00:03:57,940 we are able to have a theory that's generally applicable. 65 00:03:57,940 --> 00:04:02,830 Now what I did was to talk about a CW radiation field, 66 00:04:02,830 --> 00:04:05,400 continuous. 67 00:04:05,400 --> 00:04:10,350 Many experiments use pulsed fields. 68 00:04:10,350 --> 00:04:15,040 Many experiments use an extremely strong pulsed field. 69 00:04:15,040 --> 00:04:18,320 That I hope to revisit later in the course. 70 00:04:18,320 --> 00:04:23,770 But the CW probably is the best way to begin. 71 00:04:23,770 --> 00:04:25,540 So now let's talk about the hydrogen atom. 72 00:04:30,180 --> 00:04:41,780 We did the rigid rotor, and that led to some angular momenta. 73 00:04:41,780 --> 00:04:44,160 Well, let's just picture. 74 00:04:44,160 --> 00:04:48,940 So we have this rigid rotor, and it's rotating 75 00:04:48,940 --> 00:04:50,980 about the center of mass. 76 00:04:50,980 --> 00:04:53,980 And the Schrodinger equation tells us 77 00:04:53,980 --> 00:04:58,030 the probability amplitudes of the rotor 78 00:04:58,030 --> 00:05:03,790 axis relative to the laboratory frame. 79 00:05:03,790 --> 00:05:05,980 And we got angular momenta, which 80 00:05:05,980 --> 00:05:12,680 we can denote by a little l, big L, J, and many other things. 81 00:05:12,680 --> 00:05:16,990 And this is important because, if it's an angular momentum, 82 00:05:16,990 --> 00:05:18,340 you know it. 83 00:05:18,340 --> 00:05:21,680 You know everything about it. 84 00:05:21,680 --> 00:05:25,760 And you don't know yet that, if you have two angular momenta, 85 00:05:25,760 --> 00:05:29,194 you could know everything about the interactions between them. 86 00:05:29,194 --> 00:05:30,860 That's called the Wigner-Eckart theorem. 87 00:05:30,860 --> 00:05:32,820 That's not in this course. 88 00:05:32,820 --> 00:05:35,450 But the important thing is, if you've got angular momenta, 89 00:05:35,450 --> 00:05:41,280 you're on solid ground and you can do stuff that doesn't ever 90 00:05:41,280 --> 00:05:44,140 need to be repeated. 91 00:05:44,140 --> 00:05:47,650 And so one of the things that we learned about the angular 92 00:05:47,650 --> 00:05:53,350 momentum was that we have-- 93 00:05:53,350 --> 00:05:58,180 even though it's going to be L as the angular momentum mostly 94 00:05:58,180 --> 00:06:02,050 in the hydrogen atom, I'm going to switch to J, 95 00:06:02,050 --> 00:06:03,850 which is my favorite notation. 96 00:06:03,850 --> 00:06:08,740 So J squared operating on a state-- 97 00:06:08,740 --> 00:06:13,600 now I can denote it like this or I can denote it like this. 98 00:06:13,600 --> 00:06:16,180 It's the same general idea. 99 00:06:16,180 --> 00:06:18,080 These are not quite equivalent things. 100 00:06:22,620 --> 00:06:30,450 And you get H bar squared J A plus one JM. 101 00:06:30,450 --> 00:06:36,440 And we have JZ operating on JM. 102 00:06:36,440 --> 00:06:41,100 And you get H bar M JM. 103 00:06:41,100 --> 00:06:43,930 And we can have J plus minus operating on JM. 104 00:06:46,600 --> 00:06:49,620 And that's H bar times this more complicated 105 00:06:49,620 --> 00:06:57,327 looking thing, J plus 1 minus M, M plus or minus 1. 106 00:06:57,327 --> 00:06:59,160 So that's the only bit you have to remember. 107 00:07:02,440 --> 00:07:04,900 And you might say, well, is it plus or minus 1 108 00:07:04,900 --> 00:07:07,480 or minus or plus 1? 109 00:07:07,480 --> 00:07:09,360 And so if you choose-- 110 00:07:09,360 --> 00:07:17,160 M is equal to J. Then you know that if you 111 00:07:17,160 --> 00:07:20,010 have M is equal to J, this is going 112 00:07:20,010 --> 00:07:25,030 to be 0 if we're doing a raising operator. 113 00:07:25,030 --> 00:07:29,260 And so there's nothing to remember 114 00:07:29,260 --> 00:07:32,530 if you are willing to go to an extreme situation 115 00:07:32,530 --> 00:07:35,560 and decide whether it's plus minus or minus or plus. 116 00:07:43,990 --> 00:07:49,160 Now the rigid rotor is a universal problem, 117 00:07:49,160 --> 00:07:50,020 which is solved. 118 00:07:50,020 --> 00:07:53,510 It deals with all central force problems. 119 00:07:53,510 --> 00:07:56,800 Everything that's round, the rigid rotor 120 00:07:56,800 --> 00:07:59,360 is a fantastic starting point. 121 00:07:59,360 --> 00:08:03,250 So one-electron atoms for sure-- 122 00:08:03,250 --> 00:08:05,620 many-electron atoms, maybe. 123 00:08:05,620 --> 00:08:07,860 We'll see. 124 00:08:07,860 --> 00:08:11,220 But anything that's round, the rigid rotor 125 00:08:11,220 --> 00:08:14,910 is a good zero point for dealing with the angular 126 00:08:14,910 --> 00:08:17,840 part of the problem. 127 00:08:17,840 --> 00:08:21,580 So for the hydrogen atom, we have a potential. 128 00:08:24,390 --> 00:08:25,990 It looks like this. 129 00:08:28,850 --> 00:08:32,679 And so the radial potential is new. 130 00:08:32,679 --> 00:08:36,440 For that rigid rotor, the radial potential was-- 131 00:08:36,440 --> 00:08:42,970 it's just 0 for the R is equal to R zero. 132 00:08:42,970 --> 00:08:47,270 And not zero-- then the potential is infinite. 133 00:08:47,270 --> 00:08:49,840 Now here we have potential, which 134 00:08:49,840 --> 00:08:53,200 doesn't go to infinity here and it does something strange here 135 00:08:53,200 --> 00:08:56,700 because you can't get to negative R 136 00:08:56,700 --> 00:08:59,270 in spherical polar coordinates. 137 00:08:59,270 --> 00:09:02,200 So is it a boundary condition or is it 138 00:09:02,200 --> 00:09:09,430 just an accident of the way we use coordinates? 139 00:09:09,430 --> 00:09:13,510 So we're going to apply what we know about angular momenta 140 00:09:13,510 --> 00:09:15,280 to the hydrogen atom. 141 00:09:15,280 --> 00:09:18,310 So our potential for the hydrogen atom 142 00:09:18,310 --> 00:09:23,860 is going to be expressed in terms 143 00:09:23,860 --> 00:09:27,230 of the distance of the electron from the nucleus 144 00:09:27,230 --> 00:09:30,220 and then the theta phi coordinates, which you already 145 00:09:30,220 --> 00:09:30,970 know. 146 00:09:30,970 --> 00:09:33,480 And the theta phi part is universal, 147 00:09:33,480 --> 00:09:37,900 and the R part is special to each problem that 148 00:09:37,900 --> 00:09:39,970 has physical symmetry. 149 00:09:39,970 --> 00:09:43,960 And there are different kinds of approximations you use 150 00:09:43,960 --> 00:09:45,790 to be able to deal with them. 151 00:09:45,790 --> 00:09:49,540 The nice thing about it is it's one dimensional, 152 00:09:49,540 --> 00:09:53,830 and we're very good at thinking about one dimensional problems. 153 00:09:53,830 --> 00:09:56,950 And even if the problem isn't exactly one dimensional 154 00:09:56,950 --> 00:09:59,470 or it has some hidden stuff to it, 155 00:09:59,470 --> 00:10:04,960 we can extend what we know from one dimensional problems 156 00:10:04,960 --> 00:10:06,700 and get a great deal of insight. 157 00:10:06,700 --> 00:10:09,710 And if you have a one dimensional problem, 158 00:10:09,710 --> 00:10:14,380 it's very easy to describe the potential in one dimension 159 00:10:14,380 --> 00:10:18,430 and the eigen functions in one dimension. 160 00:10:18,430 --> 00:10:22,698 And so we can begin to really understand everything. 161 00:10:26,180 --> 00:10:31,410 So we're going to have a wave function, which is 162 00:10:31,410 --> 00:10:32,660 going to have quantum numbers. 163 00:10:38,690 --> 00:10:40,770 And we're going to be able to write it 164 00:10:40,770 --> 00:10:43,185 as a product of two parts. 165 00:10:51,270 --> 00:10:55,950 Well, this is the same thing we had before for the rigid rotor. 166 00:10:55,950 --> 00:10:59,580 So we're going to be able to take this rotary equation 167 00:10:59,580 --> 00:11:04,610 and separate the wave function into two parts, a radial part 168 00:11:04,610 --> 00:11:06,030 and an angular part. 169 00:11:06,030 --> 00:11:07,620 And this is old and this is new. 170 00:11:13,800 --> 00:11:19,350 Because we've got this, we can use all of that stuff 171 00:11:19,350 --> 00:11:21,150 without taking a breath, except maybe 172 00:11:21,150 --> 00:11:25,575 getting the right letter L, J, S, whatever. 173 00:11:29,330 --> 00:11:33,910 So really the hydrogen atom is just one thing 174 00:11:33,910 --> 00:11:37,585 with some curve balls thrown at you in the latter stages. 175 00:11:45,350 --> 00:11:48,620 One of the things you also want to be able to do-- 176 00:11:48,620 --> 00:11:52,310 because nodes are so important in determining 177 00:11:52,310 --> 00:11:57,290 both the names of the states and how they behave 178 00:11:57,290 --> 00:12:05,240 in various situations, including external fields and excitation 179 00:12:05,240 --> 00:12:07,830 by electromagnetic radiation-- 180 00:12:07,830 --> 00:12:10,730 you want to be able to understand the nodal surfaces. 181 00:12:10,730 --> 00:12:14,230 And you already know this one. 182 00:12:14,230 --> 00:12:16,770 So how many nodal surfaces are there 183 00:12:16,770 --> 00:12:18,630 if you have a particular value of L? 184 00:12:21,440 --> 00:12:21,950 Yes. 185 00:12:21,950 --> 00:12:22,640 AUDIENCE: L. 186 00:12:22,640 --> 00:12:27,505 PROFESSOR: Right, and if you have a particular value of M, 187 00:12:27,505 --> 00:12:29,770 how many nodes are there in the xy plane? 188 00:12:37,790 --> 00:12:39,223 You're hot-- 189 00:12:39,223 --> 00:12:42,980 AUDIENCE: I mean, I know if M is 0, 190 00:12:42,980 --> 00:12:47,150 the entire angular momentum is tipped into the xy plane. 191 00:12:47,150 --> 00:12:52,760 So that means that the axis of the rotor 192 00:12:52,760 --> 00:12:56,655 has to be orthogonal to the angular momentum. 193 00:12:56,655 --> 00:12:58,670 So that means that the probability density is 194 00:12:58,670 --> 00:13:02,252 oriented along Z, I think. 195 00:13:02,252 --> 00:13:03,710 PROFESSOR: Now you're saying things 196 00:13:03,710 --> 00:13:05,168 that I have to stop and think about 197 00:13:05,168 --> 00:13:07,760 because you're not telling me what I expected to know. 198 00:13:07,760 --> 00:13:13,670 So if M is equal to 0, L is perpendicular 199 00:13:13,670 --> 00:13:19,685 to the quantization axis, and there are no nodes. 200 00:13:19,685 --> 00:13:21,310 AUDIENCE: Well, it depends on if you're 201 00:13:21,310 --> 00:13:25,000 talking about the nodes of the probability 202 00:13:25,000 --> 00:13:28,750 density of the rotor, where the axis is. 203 00:13:28,750 --> 00:13:32,410 PROFESSOR: That's true, but where the axis 204 00:13:32,410 --> 00:13:36,670 is determined by theta. 205 00:13:36,670 --> 00:13:42,100 And when theta is equal to pi over 2, you're in the xy plane. 206 00:13:42,100 --> 00:13:48,430 And the number of nodes in the xy plane 207 00:13:48,430 --> 00:13:58,370 is M, or absolute value of M. The phi part of the rigid rotor 208 00:13:58,370 --> 00:13:59,420 is simple. 209 00:13:59,420 --> 00:14:03,300 It's a differential equation that everybody can solve. 210 00:14:03,300 --> 00:14:05,570 We already know that one, and we know what 211 00:14:05,570 --> 00:14:06,810 the wave functions look like. 212 00:14:15,240 --> 00:14:26,550 We can write the hydrogen atom Schrodinger equation, 213 00:14:26,550 --> 00:14:32,500 and it's very quick to show separation of variables. 214 00:14:32,500 --> 00:14:34,480 And I'll do that in a minute. 215 00:14:34,480 --> 00:14:43,230 And then we have the pictures of the separated parts-- 216 00:14:43,230 --> 00:14:48,140 RNL of R and YLM of theta phi. 217 00:14:50,900 --> 00:14:55,670 And essential in these pictures is the number of nodes. 218 00:14:55,670 --> 00:14:57,920 In the spacing between nodes, remember 219 00:14:57,920 --> 00:15:01,210 the semiclassical approximation. 220 00:15:01,210 --> 00:15:04,620 We know that Mr. DeBroglie really hit it out of the park 221 00:15:04,620 --> 00:15:12,070 by saying that the wavelength is H over P. 222 00:15:12,070 --> 00:15:13,600 For every one dimensional problem, 223 00:15:13,600 --> 00:15:16,250 we know what to do with that. 224 00:15:16,250 --> 00:15:19,970 And we're going to discover that, for the radial problem, 225 00:15:19,970 --> 00:15:22,640 we have a very simple way of determining 226 00:15:22,640 --> 00:15:25,770 what the classical momentum is. 227 00:15:25,770 --> 00:15:29,790 And so we know everything about the nodes and the node spacing 228 00:15:29,790 --> 00:15:32,250 and the amplitudes between nodes and how 229 00:15:32,250 --> 00:15:39,220 to evaluate every integral of some power of R or Z. 230 00:15:39,220 --> 00:15:41,580 And so there's just an enormous amount 231 00:15:41,580 --> 00:15:45,270 gotten from the semiclassical picture 232 00:15:45,270 --> 00:15:48,450 once you are familiar with this. 233 00:15:48,450 --> 00:15:55,560 And so we can say that we have the classical wavelength, 234 00:15:55,560 --> 00:15:57,480 or the semiclassical wavelength. 235 00:15:57,480 --> 00:16:12,460 It has an index R. So we have a momentum, a linear momentum 236 00:16:12,460 --> 00:16:14,596 with a quantum number on it for L. See, 237 00:16:14,596 --> 00:16:15,595 that's a little strange. 238 00:16:18,500 --> 00:16:20,960 But that tells us what the potential is going to be 239 00:16:20,960 --> 00:16:22,860 and that tells us what to use in order 240 00:16:22,860 --> 00:16:25,500 to determine the wavelength. 241 00:16:25,500 --> 00:16:27,890 And this is really the core of how 242 00:16:27,890 --> 00:16:30,230 we can go way beyond textbooks. 243 00:16:30,230 --> 00:16:34,105 We can do-- in our heads or on a simple piece of paper, 244 00:16:34,105 --> 00:16:34,730 we can do this. 245 00:16:34,730 --> 00:16:41,645 And we can draw pictures, and we can evaluate matrix elements. 246 00:16:45,040 --> 00:16:49,210 Without any complicated integral tables, 247 00:16:49,210 --> 00:16:54,310 you can make estimates that are incredibly important. 248 00:16:54,310 --> 00:17:03,960 And from that you can get expectation values and also 249 00:17:03,960 --> 00:17:05,910 off-diagonal matrix elements of integer 250 00:17:05,910 --> 00:17:07,871 powers of the coordinate. 251 00:17:10,460 --> 00:17:13,079 That's an enormous amount of stuff that you can do. 252 00:17:13,079 --> 00:17:15,700 Now you can't do it yet. 253 00:17:15,700 --> 00:17:19,240 But after Wednesday's lecture, you will. 254 00:17:19,240 --> 00:17:23,869 And so this will be Wednesday. 255 00:17:23,869 --> 00:17:31,655 And then we'll have evidence of electron spin. 256 00:17:34,940 --> 00:17:39,130 And I will get to that today. 257 00:17:39,130 --> 00:17:40,430 So this is the menu. 258 00:17:40,430 --> 00:17:43,130 Now let's start delivering some of this stuff. 259 00:17:48,360 --> 00:17:51,510 I have to write some big equations. 260 00:17:51,510 --> 00:17:57,920 So the Hamiltonian is kinetic energy plus potential energy, 261 00:17:57,920 --> 00:18:00,510 and kinetic energy was-- 262 00:18:00,510 --> 00:18:01,790 I'm sorry. 263 00:18:01,790 --> 00:18:06,710 For the rigid rotor, V was zero. 264 00:18:06,710 --> 00:18:08,650 And everything was in the kinetic energy. 265 00:18:08,650 --> 00:18:10,660 Well, it's not quite true. 266 00:18:10,660 --> 00:18:15,420 And so, for hydrogen-- so we know 267 00:18:15,420 --> 00:18:21,720 this is P squared over 2 mu, and mu for the hydrogen atom, 268 00:18:21,720 --> 00:18:23,130 reduced mass. 269 00:18:23,130 --> 00:18:32,750 And we know this is just Coulomb's Law, minus e 270 00:18:32,750 --> 00:18:42,240 squared over 4 pi epsilon zero R, H bar squared. 271 00:18:45,750 --> 00:18:51,570 I'm sorry-- it was supposed to be e squared. 272 00:18:51,570 --> 00:18:52,800 So this is the classical-- 273 00:18:55,610 --> 00:18:57,820 So these are the parts. 274 00:18:57,820 --> 00:19:00,470 But the since it's spherical, we're 275 00:19:00,470 --> 00:19:04,840 not going to work in Cartesian coordinates. 276 00:19:04,840 --> 00:19:09,560 And in fact, that's a very strong statement. 277 00:19:09,560 --> 00:19:12,270 If you have a spherical problem, don't 278 00:19:12,270 --> 00:19:14,190 start in Cartesian coordinates because it's 279 00:19:14,190 --> 00:19:18,720 a horrible mess transforming to spherical polar coordinates. 280 00:19:18,720 --> 00:19:22,820 Just remember the spherical polar coordinates. 281 00:19:22,820 --> 00:19:24,990 So you know how spherical polar coordinates work. 282 00:19:24,990 --> 00:19:26,031 I'm not going to draw it. 283 00:19:28,260 --> 00:19:36,850 So the kinetic energy term is-- 284 00:19:44,690 --> 00:19:51,590 and this is the Laplacian, and that's a terrible thing. 285 00:19:51,590 --> 00:19:58,070 And Del squared-- it looks like it's 286 00:19:58,070 --> 00:20:02,428 going to be a real nightmare partial with respect to R, 287 00:20:02,428 --> 00:20:08,550 R squared partial with respect to R. And we have 1 over R 288 00:20:08,550 --> 00:20:13,460 squared sine squared theta partial with respect 289 00:20:13,460 --> 00:20:21,130 to theta, sine theta partial with respect to theta. 290 00:20:21,130 --> 00:20:23,560 And then a third term-- 291 00:20:23,560 --> 00:20:28,640 1 over R squared sine squared. 292 00:20:28,640 --> 00:20:30,100 I got a square here. 293 00:20:30,100 --> 00:20:32,170 That's wrong. 294 00:20:32,170 --> 00:20:34,570 And this is sine squared theta. 295 00:20:34,570 --> 00:20:36,340 I heard a mumble over there. 296 00:20:43,420 --> 00:20:48,005 Yeah, but I'm just doing Del squared. 297 00:20:48,005 --> 00:20:50,130 AUDIENCE: With your first Del, it's H plus squared. 298 00:20:50,130 --> 00:20:50,921 PROFESSOR: My what? 299 00:20:50,921 --> 00:20:53,410 AUDIENCE: Your first time you used the Del operator, it's H 300 00:20:53,410 --> 00:20:55,280 plus squared. 301 00:20:55,280 --> 00:20:56,662 PROFESSOR: Oh, yes, yes. 302 00:20:56,662 --> 00:20:58,078 AUDIENCE: And then the second one, 303 00:20:58,078 --> 00:21:00,165 if you could clarify that that's a Del-- 304 00:21:00,165 --> 00:21:01,040 PROFESSOR: I'm sorry. 305 00:21:01,040 --> 00:21:01,920 AUDIENCE: The second Del squared, 306 00:21:01,920 --> 00:21:03,764 if you could clarify that's a Del squared. 307 00:21:03,764 --> 00:21:05,120 It looks like a-- 308 00:21:05,120 --> 00:21:06,870 PROFESSOR: It looks like a terrible thing. 309 00:21:11,640 --> 00:21:18,690 And the last part is a second derivative with respect to phi. 310 00:21:18,690 --> 00:21:22,710 This looks like a terrible thing to build on. 311 00:21:22,710 --> 00:21:27,670 But with a little bit of trickery-- 312 00:21:27,670 --> 00:21:32,900 and that is, suppose we multiply this equation by R squared-- 313 00:21:32,900 --> 00:21:41,340 then we have killed the R squared terms here and here. 314 00:21:41,340 --> 00:21:44,020 And we're going to be able to separate it. 315 00:21:44,020 --> 00:21:50,980 And so we are able to write an equation which 316 00:21:50,980 --> 00:21:53,230 has the separability built in-- 317 00:21:53,230 --> 00:21:56,784 and so partial with respect to R, 318 00:21:56,784 --> 00:22:05,300 R squared partial with respect to R plus L 319 00:22:05,300 --> 00:22:23,980 squared plus 2 mu H R squared V of R minus E psi. 320 00:22:23,980 --> 00:22:26,150 That's a Schrodinger equation. 321 00:22:26,150 --> 00:22:32,020 So we have an R dependent term and another R dependent term 322 00:22:32,020 --> 00:22:36,510 and a theta phi dependent term all in this one, nice operator 323 00:22:36,510 --> 00:22:37,570 that we've understood. 324 00:22:41,380 --> 00:22:44,780 So now there's one more trick. 325 00:22:44,780 --> 00:22:47,330 In order for the shorter equation 326 00:22:47,330 --> 00:22:51,860 to be separated into a theta phi part and an R part 327 00:22:51,860 --> 00:22:54,230 is we need a commutator. 328 00:22:54,230 --> 00:23:00,170 And so that commutator is this. 329 00:23:00,170 --> 00:23:05,000 What is a commutator between L squared and any function of R? 330 00:23:08,330 --> 00:23:08,910 Yes. 331 00:23:08,910 --> 00:23:11,243 AUDIENCE: Their operators depend on different variables? 332 00:23:11,243 --> 00:23:13,250 PROFESSOR: Absolutely, that's really important. 333 00:23:13,250 --> 00:23:16,310 We often encounter operators that 334 00:23:16,310 --> 00:23:18,780 depend on different variables. 335 00:23:18,780 --> 00:23:21,420 And when they do, they commute with each other, 336 00:23:21,420 --> 00:23:24,200 which is an incredibly convenient thing because that 337 00:23:24,200 --> 00:23:26,690 means we can set up the problem as a product 338 00:23:26,690 --> 00:23:31,238 of the eigenfunctions of the different operators. 339 00:23:31,238 --> 00:23:38,880 So this means that we can write a theta phi term, which 340 00:23:38,880 --> 00:23:44,190 we completely know, and an R term, which we don't know 341 00:23:44,190 --> 00:23:46,200 and contains all of the interesting stuff. 342 00:23:50,450 --> 00:23:52,890 So let's forget about theta and phi. 343 00:23:52,890 --> 00:23:54,760 Let's just look at the R part. 344 00:24:02,710 --> 00:24:04,750 Well, the way we separated variables, 345 00:24:04,750 --> 00:24:06,670 we had the Schrodinger equation and we 346 00:24:06,670 --> 00:24:10,360 divided by the wave function, which would be 347 00:24:10,360 --> 00:24:17,210 of R of R, Y L M of theta phi. 348 00:24:17,210 --> 00:24:22,790 And on one side of the equality, we have only the angle part. 349 00:24:22,790 --> 00:24:25,400 And so when we do that, when we divide by R, 350 00:24:25,400 --> 00:24:27,680 we kill the R part on this side. 351 00:24:27,680 --> 00:24:30,900 And on the other side, it's the opposite. 352 00:24:30,900 --> 00:24:33,710 And so we get two pieces-- 353 00:24:33,710 --> 00:24:36,230 one is only dependent on theta and phi 354 00:24:36,230 --> 00:24:38,330 and one is only dependent on R. So they both 355 00:24:38,330 --> 00:24:40,040 have to be a constant. 356 00:24:40,040 --> 00:24:42,680 So we get two separate differential equations. 357 00:24:42,680 --> 00:24:45,570 And we've already dealt with one of them. 358 00:24:45,570 --> 00:24:57,430 So what you end up getting is 1 over R of R times stuff 359 00:24:57,430 --> 00:25:09,131 times R of R is equal to 1 over YLM L squared YLM-- 360 00:25:09,131 --> 00:25:09,630 sorry. 361 00:25:13,700 --> 00:25:15,710 So this is the separation. 362 00:25:15,710 --> 00:25:17,900 This is all R stuff. 363 00:25:17,900 --> 00:25:21,960 This is all theta stuff and separation constant. 364 00:25:21,960 --> 00:25:24,350 Well, this one is inviting a separation concept 365 00:25:24,350 --> 00:25:27,830 because L squared operating on YLM 366 00:25:27,830 --> 00:25:30,679 is H bar squared, LL plus 1. 367 00:25:30,679 --> 00:25:31,970 That's the separation constant. 368 00:25:38,320 --> 00:25:44,140 So now let's look for the only time 369 00:25:44,140 --> 00:25:48,350 at the R part of the differential equation. 370 00:25:48,350 --> 00:25:53,080 And so this writing out all the pieces honestly-- 371 00:26:31,600 --> 00:26:33,070 we have-- there's one more. 372 00:26:40,090 --> 00:26:43,269 So that's the Schrodinger equation for the radial part. 373 00:26:43,269 --> 00:26:44,560 It looks a little bit annoying. 374 00:26:47,680 --> 00:26:52,600 The important trick is that we've 375 00:26:52,600 --> 00:26:58,410 taken the separation constant, and it has an R dependence 376 00:26:58,410 --> 00:27:04,020 but when we divide through by 2 mu HR squared. 377 00:27:04,020 --> 00:27:07,650 And we say, oh, well, let's call these two things together-- 378 00:27:07,650 --> 00:27:15,510 VL of R. This is the effect of potential, 379 00:27:15,510 --> 00:27:18,790 and it depends on the value of L. 380 00:27:18,790 --> 00:27:22,390 So this is just like an ordinary one dimensional problem 381 00:27:22,390 --> 00:27:24,250 except now, for every value of L, 382 00:27:24,250 --> 00:27:27,140 we have a different potential. 383 00:27:27,140 --> 00:27:30,520 And this potential is-- 384 00:27:30,520 --> 00:27:36,470 OK, when L is not equal to zero, this potential 385 00:27:36,470 --> 00:27:40,350 goes to infinity at R equals 0, which 386 00:27:40,350 --> 00:27:44,340 is bad, except it's good because it keeps the particle ever 387 00:27:44,340 --> 00:27:46,425 from getting close to the nucleus. 388 00:27:49,710 --> 00:27:51,870 And that's what Mr. Schrodinger-- 389 00:27:51,870 --> 00:27:53,970 I'm sorry what Mr. Bohr was thinking about, 390 00:27:53,970 --> 00:27:56,150 that we have only circular orbits 391 00:27:56,150 --> 00:28:03,570 or we have only orbits that are away from the place 392 00:28:03,570 --> 00:28:08,740 where the Coulomb interaction would be infinite. 393 00:28:08,740 --> 00:28:11,130 And I mean, there are several things that's 394 00:28:11,130 --> 00:28:12,390 wrong with Bohr's picture. 395 00:28:12,390 --> 00:28:15,580 One is that we have orbits. 396 00:28:15,580 --> 00:28:20,280 And the other is that we don't include L equals 0, 397 00:28:20,280 --> 00:28:28,020 but this is telling you a lot of really important stuff 398 00:28:28,020 --> 00:28:32,490 because the value of L determines 399 00:28:32,490 --> 00:28:36,930 the importance of this thing that goes to infinity at R 400 00:28:36,930 --> 00:28:39,490 equals 0. 401 00:28:39,490 --> 00:28:44,450 And that has very significant consequences as far as 402 00:28:44,450 --> 00:28:47,270 which orbital angular momentum states we're dealing with. 403 00:28:54,420 --> 00:29:01,440 So we're going to get the usual LML quantum numbers, 404 00:29:01,440 --> 00:29:04,200 and we're going to get another one from the radial part. 405 00:29:04,200 --> 00:29:08,100 And since this is a 1D equation, there's 406 00:29:08,100 --> 00:29:10,420 only one quantum number. 407 00:29:10,420 --> 00:29:12,790 And we're going to call it L. And now 408 00:29:12,790 --> 00:29:17,180 there's this word principal, and there's really two words-- 409 00:29:17,180 --> 00:29:19,835 principle with an LE and principal with an AL. 410 00:29:22,880 --> 00:29:27,200 Which one do you think is appropriate? 411 00:29:27,200 --> 00:29:28,680 I'm sorry. 412 00:29:28,680 --> 00:29:30,229 I can't hear. 413 00:29:30,229 --> 00:29:30,770 AUDIENCE: AL. 414 00:29:30,770 --> 00:29:32,710 PROFESSOR: AL is the appropriate one. 415 00:29:32,710 --> 00:29:36,570 Principle has to do with something fundamental. 416 00:29:36,570 --> 00:29:40,980 Principal has to do with something that's important. 417 00:29:40,980 --> 00:29:44,720 And I can't tell you how many times 418 00:29:44,720 --> 00:29:48,498 people who should know better use the wrong principal. 419 00:29:52,470 --> 00:29:55,620 You'll never do that now because it was like, what's nu? 420 00:29:55,620 --> 00:29:56,460 C over lambda. 421 00:29:59,730 --> 00:30:01,590 So this is the right principal. 422 00:30:10,464 --> 00:30:18,110 If we do something clever and we have the radial part 423 00:30:18,110 --> 00:30:20,240 and we say, let us replace it by 1 424 00:30:20,240 --> 00:30:29,900 over R times this new function chi L of R, well, 425 00:30:29,900 --> 00:30:35,380 when we do that, this equation becomes really simple. 426 00:30:35,380 --> 00:30:40,530 So what we get when we make that substitution is H bar squared 427 00:30:40,530 --> 00:30:52,040 over 2 mu H second derivative with respect to R plus VL of R 428 00:30:52,040 --> 00:30:57,950 minus E chi L of R is equal to 0. 429 00:30:57,950 --> 00:31:02,250 That looks like a differential equation we've seen before. 430 00:31:02,250 --> 00:31:04,820 It's a simple one dimensional differential equation-- 431 00:31:04,820 --> 00:31:06,710 kinetic energy, potential energy. 432 00:31:06,710 --> 00:31:07,690 But it's not. 433 00:31:07,690 --> 00:31:13,140 There's some kinetic energy hidden in the potential energy. 434 00:31:13,140 --> 00:31:16,580 But this is simple, and we can deal with this a lot. 435 00:31:16,580 --> 00:31:18,080 I'm not going to. 436 00:31:18,080 --> 00:31:20,709 But if you're going to actually do stuff, 437 00:31:20,709 --> 00:31:23,000 you're going to be wanting to look at this differential 438 00:31:23,000 --> 00:31:24,680 equation. 439 00:31:24,680 --> 00:31:26,975 But I'm going to forego that pleasure. 440 00:31:31,100 --> 00:31:34,810 One of the problems with the radial equation 441 00:31:34,810 --> 00:31:39,130 is the fact that R is a special kind of coordinate. 442 00:31:39,130 --> 00:31:41,860 It can't go negative. 443 00:31:41,860 --> 00:31:48,730 And so treating the boundary condition for R is equal to 0 444 00:31:48,730 --> 00:31:49,840 is a little subtle. 445 00:31:53,280 --> 00:31:58,920 And it turns out that when L is equal to 0, then R of 0 446 00:31:58,920 --> 00:31:59,880 is not 0. 447 00:32:02,570 --> 00:32:09,340 But for all other values of L, R of 0 is 0. 448 00:32:09,340 --> 00:32:18,340 And all of NMR depends on L being 449 00:32:18,340 --> 00:32:25,130 0 because the electron feels the nucleus. 450 00:32:25,130 --> 00:32:29,450 It doesn't experience an infinite singularity. 451 00:32:29,450 --> 00:32:30,950 It feels the nucleus. 452 00:32:30,950 --> 00:32:34,910 And when L is not equal to 0, then it's 453 00:32:34,910 --> 00:32:38,030 sensing the nucleus at a distance. 454 00:32:38,030 --> 00:32:41,060 And that leads to some very small splittings 455 00:32:41,060 --> 00:32:43,580 called hyperfine. 456 00:32:43,580 --> 00:32:48,180 And so there's different kinds of hyperfine structure. 457 00:32:48,180 --> 00:32:52,200 But anyway, this is a really subtle and important point. 458 00:32:57,757 --> 00:33:01,230 How much time left? 459 00:33:01,230 --> 00:33:02,350 Yes? 460 00:33:02,350 --> 00:33:04,340 No. 461 00:33:04,340 --> 00:33:07,151 So now pictures of orbitals-- 462 00:33:09,740 --> 00:33:13,970 So the problem with pictures of orbitals 463 00:33:13,970 --> 00:33:18,710 is now we have a function of three variables 464 00:33:18,710 --> 00:33:23,460 and it's equal to some complex number. 465 00:33:23,460 --> 00:33:25,880 So we need two degrees of freedom 466 00:33:25,880 --> 00:33:29,910 to present a complex number and we have three variables. 467 00:33:29,910 --> 00:33:34,940 And so representing that on a two dimensional sheet of paper 468 00:33:34,940 --> 00:33:35,990 is horrible. 469 00:33:35,990 --> 00:33:38,150 But we have this wonderful factorization 470 00:33:38,150 --> 00:33:42,380 where we have RNL of R. And we could draw that easily. 471 00:33:42,380 --> 00:33:44,460 We don't need any special skill. 472 00:33:44,460 --> 00:33:48,380 And we have YLM of theta phi. 473 00:33:48,380 --> 00:33:50,120 And we've got lots of practice with that, 474 00:33:50,120 --> 00:33:51,830 although what we've practiced with 475 00:33:51,830 --> 00:33:54,140 may not be completely understood yet. 476 00:33:58,630 --> 00:34:00,760 So we have ways of representing these. 477 00:34:00,760 --> 00:34:05,770 And so we go through the understanding 478 00:34:05,770 --> 00:34:09,030 of the hydrogen atom by looking at these two things separately. 479 00:34:12,330 --> 00:34:16,365 Now for the radial part, the energy levels-- 480 00:34:27,409 --> 00:34:29,429 where this is the Rydberg constant. 481 00:34:29,429 --> 00:34:32,540 It's a combination or fundamental constants. 482 00:34:32,540 --> 00:34:37,070 And for hydrogen, the Rydberg constant 483 00:34:37,070 --> 00:34:46,440 is equal to 109,737.319-- 484 00:34:46,440 --> 00:34:49,770 there's actually more digits, wave numbers-- 485 00:34:49,770 --> 00:34:56,590 times mu H over mu infinite. 486 00:34:56,590 --> 00:34:59,260 Well, this is actually the Rydberg constant 487 00:34:59,260 --> 00:35:00,355 for an infinite mass. 488 00:35:09,900 --> 00:35:20,610 And so mu H is equal to the mass of the electron 489 00:35:20,610 --> 00:35:25,969 times the mass of the proton over the mass of the electron 490 00:35:25,969 --> 00:35:27,135 plus the mass of the proton. 491 00:35:30,180 --> 00:35:33,230 Now the mass of the proton is much bigger 492 00:35:33,230 --> 00:35:34,790 than the mass of the electron. 493 00:35:34,790 --> 00:35:37,610 And so you can use this as a trick. 494 00:35:37,610 --> 00:35:39,310 You can say, well, what is it? 495 00:35:39,310 --> 00:35:41,430 Well, we know what this is. 496 00:35:41,430 --> 00:35:42,509 It's easy to calculate. 497 00:35:42,509 --> 00:35:43,550 But there are two limits. 498 00:35:43,550 --> 00:35:47,810 What is the smallest possible reduced mass? 499 00:35:47,810 --> 00:35:49,265 And that would be for-- 500 00:35:55,790 --> 00:36:00,240 if the mass of the proton is infinite, 501 00:36:00,240 --> 00:36:02,560 then we just get the mass of the electron. 502 00:36:02,560 --> 00:36:05,790 That's the biggest reduced mass. 503 00:36:05,790 --> 00:36:08,310 And the smallest is, if we have positronium, 504 00:36:08,310 --> 00:36:13,380 where we have an electron bound to a proton-- 505 00:36:13,380 --> 00:36:17,680 and then when we do that, we get half. 506 00:36:17,680 --> 00:36:18,180 I'm sorry. 507 00:36:18,180 --> 00:36:19,520 It's not a proton. 508 00:36:19,520 --> 00:36:24,590 It's a positively charged particle, 509 00:36:24,590 --> 00:36:27,020 which we call a positron. 510 00:36:27,020 --> 00:36:34,010 And so each of the terms here is the mass of the electron. 511 00:36:34,010 --> 00:36:43,820 And so the range is from 1/2 me to me 512 00:36:43,820 --> 00:36:47,490 depending on what particles you're dealing with. 513 00:36:47,490 --> 00:36:50,690 And so that's a useful thing. 514 00:36:50,690 --> 00:36:55,040 And so the Rydberg constant for hydrogen 515 00:36:55,040 --> 00:36:59,491 is smaller than the reduced mass of the infinite. 516 00:36:59,491 --> 00:36:59,990 I'm sorry. 517 00:36:59,990 --> 00:37:05,270 It's smaller than that for the infinitely mass nucleus. 518 00:37:05,270 --> 00:37:11,810 And it is the value that made Mr. Bohr very happy-- 519 00:37:11,810 --> 00:37:15,890 679. 520 00:37:15,890 --> 00:37:19,430 So the important thing is this mass scaled, 521 00:37:19,430 --> 00:37:25,296 or reduced mass scaled, Rydberg constant explains to a part 522 00:37:25,296 --> 00:37:28,970 in 10 to the 10th all the energy levels of one-electron 523 00:37:28,970 --> 00:37:29,870 systems-- 524 00:37:29,870 --> 00:37:34,010 hydrogen, helium plus, lithium 2 plus. 525 00:37:34,010 --> 00:37:34,540 That's it. 526 00:37:37,725 --> 00:37:39,200 And that's fantastic. 527 00:37:45,560 --> 00:37:48,740 Now we have to talk to become really familiar 528 00:37:48,740 --> 00:37:53,500 with this Rnl of our function. 529 00:38:06,670 --> 00:38:09,200 And one of the questions is, how many radial nodes. 530 00:38:15,710 --> 00:38:21,570 And you know that for 1S there aren't any nodes. 531 00:38:24,990 --> 00:38:28,510 And you know for 2P there aren't any radial nodes. 532 00:38:31,110 --> 00:38:34,230 And so what we need is something that 533 00:38:34,230 --> 00:38:38,870 goes like n minus L minus 1. 534 00:38:38,870 --> 00:38:41,740 The number of radial nodes, which 535 00:38:41,740 --> 00:38:45,100 is all you need to know about the radial A function, 536 00:38:45,100 --> 00:38:50,070 is how many nodes are and how far are they apart 537 00:38:50,070 --> 00:38:53,710 and what's the amplitude of each loop between nodes. 538 00:38:53,710 --> 00:38:57,770 Semi classical theory gives you all of that. 539 00:38:57,770 --> 00:39:01,970 And often when you're calculating an integral, 540 00:39:01,970 --> 00:39:04,550 all you care about is the amplitude in the first loop. 541 00:39:07,190 --> 00:39:11,880 And so instead of having to evaluate an integral, 542 00:39:11,880 --> 00:39:16,770 you just figure out what is the envelope function based 543 00:39:16,770 --> 00:39:18,810 on the classical momentum function. 544 00:39:24,950 --> 00:39:27,410 Now the thing that everybody's been waiting for. 545 00:39:33,280 --> 00:39:33,780 Spin. 546 00:39:39,660 --> 00:39:49,950 So remember, we know an angular momentum is R cross P, 547 00:39:49,950 --> 00:39:53,177 and we know that there isn't any internal structure. 548 00:39:53,177 --> 00:39:55,260 Or at least we don't know about internal structure 549 00:39:55,260 --> 00:39:58,930 of the electron or a proton. 550 00:39:58,930 --> 00:40:04,850 And so we can't somehow say, well, it's R cross P. 551 00:40:04,850 --> 00:40:06,260 So we have the Zeeman effect. 552 00:40:14,530 --> 00:40:17,400 So we can look at the Zeeman effect 553 00:40:17,400 --> 00:40:20,000 for an atom in a magnetic field. 554 00:40:20,000 --> 00:40:23,860 And we have several things that we know. 555 00:40:23,860 --> 00:40:28,950 So we have the magnetic moment of the electron 556 00:40:28,950 --> 00:40:37,470 is equal to minus the charge on the electron times 2ME times 557 00:40:37,470 --> 00:40:40,850 L. Well, that's an angular momentum. 558 00:40:48,980 --> 00:40:57,070 So if we had circulating charge, well, that circulating charge 559 00:40:57,070 --> 00:41:00,680 will produce a magnetic moment, which has a magnitude which 560 00:41:00,680 --> 00:41:02,330 is related to the velocity. 561 00:41:04,890 --> 00:41:09,120 And what is L divided by mass? 562 00:41:09,120 --> 00:41:11,660 It's a velocity. 563 00:41:11,660 --> 00:41:13,530 And we have the charge here. 564 00:41:13,530 --> 00:41:17,440 So this is a perfectly reasonable thing. 565 00:41:17,440 --> 00:41:22,940 And that's good because we know that, if we have electrons 566 00:41:22,940 --> 00:41:25,520 with some kind of internal structure, 567 00:41:25,520 --> 00:41:26,900 we know what this is going to do. 568 00:41:29,630 --> 00:41:36,390 Now the magnetic potential is equal to minus 569 00:41:36,390 --> 00:41:40,080 the magnetic moment times the external magnetic field. 570 00:41:42,840 --> 00:41:51,920 And so what we have is E, BC, LZ over 2 NE. 571 00:41:55,110 --> 00:41:56,970 Well, this is another very good thing 572 00:41:56,970 --> 00:42:02,440 because, not only do we know what this is, 573 00:42:02,440 --> 00:42:06,370 we know that it has only diagonal elements. 574 00:42:06,370 --> 00:42:12,000 And so we can do first order perturbation theory. 575 00:42:15,330 --> 00:42:18,960 If this is so easy, if this is the Hamiltonian, 576 00:42:18,960 --> 00:42:22,410 we can just tack that on and it adds an extra splitting 577 00:42:22,410 --> 00:42:23,820 to our energy levels. 578 00:42:27,230 --> 00:42:32,350 The only tricky thing is, we don't know that it's LZ. 579 00:42:32,350 --> 00:42:35,360 We just know that it has a magnetic moment 580 00:42:35,360 --> 00:42:39,640 or it could have a magnetic moment. 581 00:42:39,640 --> 00:42:41,450 And so this wonderful experiment-- 582 00:42:41,450 --> 00:42:45,920 suppose we start with a 1S state, 583 00:42:45,920 --> 00:42:48,880 and you go to a 2P state. 584 00:42:48,880 --> 00:42:53,800 Now this implies that we know something 585 00:42:53,800 --> 00:42:58,422 about the selection rules for electromagnetic transitions. 586 00:42:58,422 --> 00:43:00,130 And so for an electric dipole transition, 587 00:43:00,130 --> 00:43:02,500 we go from 1S to 2P. 588 00:43:02,500 --> 00:43:05,470 Now this is not how it was done initially 589 00:43:05,470 --> 00:43:09,820 because the frequency of this transition 590 00:43:09,820 --> 00:43:12,370 is well into the vacuum ultraviolet. 591 00:43:12,370 --> 00:43:16,090 And in the days when quantum mechanics was being developed, 592 00:43:16,090 --> 00:43:18,510 that was a hard experiment. 593 00:43:18,510 --> 00:43:24,750 So one used an S to P transition on some other atom, 594 00:43:24,750 --> 00:43:31,580 like mercury, but let's pretend it was hydrogen. 595 00:43:31,580 --> 00:43:35,380 So this is an angular momentum of 0. 596 00:43:35,380 --> 00:43:40,490 So we would expect it would be ML equals 0. 597 00:43:40,490 --> 00:43:46,970 And here we have it could split into three components. 598 00:43:46,970 --> 00:43:56,530 And this is an L equals 1, 0 minus 1, because that's minus-- 599 00:43:56,530 --> 00:43:59,350 there with a minus sign somewhere. 600 00:43:59,350 --> 00:44:01,700 That's what we expect. 601 00:44:01,700 --> 00:44:05,920 And so being naive, you might expect transitions like this. 602 00:44:09,070 --> 00:44:12,070 Well, the transitions where you do not 603 00:44:12,070 --> 00:44:14,260 change the projection quantum number 604 00:44:14,260 --> 00:44:18,040 are done with Z polarized radiation. 605 00:44:18,040 --> 00:44:23,360 And so if it's Z polarized, you get only delta ML equals 0. 606 00:44:23,360 --> 00:44:25,890 And if you have x and y, you have delta 607 00:44:25,890 --> 00:44:29,030 ML equals plus and minus 1. 608 00:44:29,030 --> 00:44:32,380 So depending on how the experiment was done, 609 00:44:32,380 --> 00:44:35,680 you would expect to see one, two, 610 00:44:35,680 --> 00:44:38,694 or three Zeeman components. 611 00:44:41,480 --> 00:44:43,610 And they did the experiment, and they 612 00:44:43,610 --> 00:44:45,980 saw more than three components. 613 00:44:45,980 --> 00:44:48,440 They saw five components. 614 00:44:48,440 --> 00:44:51,780 And that's possibly for a number of reasons. 615 00:44:51,780 --> 00:44:53,770 But they saw five components. 616 00:44:53,770 --> 00:44:57,040 So they knew that there was something else going on. 617 00:44:57,040 --> 00:45:09,110 And so you have to try to collect enough information 618 00:45:09,110 --> 00:45:13,490 to have a simple minded picture that will explain it all. 619 00:45:16,300 --> 00:45:18,090 So one thing you might do is say, OK, 620 00:45:18,090 --> 00:45:24,180 suppose there is another quantum number, another thing. 621 00:45:24,180 --> 00:45:26,920 And we're going to call it spin. 622 00:45:26,920 --> 00:45:29,890 And we don't know whether spin is integer or half integer. 623 00:45:29,890 --> 00:45:33,910 We know from our exercise with computation rules 624 00:45:33,910 --> 00:45:37,390 that both half integer and integer angular 625 00:45:37,390 --> 00:45:39,350 momentum are possible. 626 00:45:39,350 --> 00:45:43,820 And so we can say we have a spin. 627 00:45:43,820 --> 00:45:45,430 And it could be 1/2. 628 00:45:45,430 --> 00:45:46,150 It could be 1. 629 00:45:46,150 --> 00:45:47,600 It could be anything. 630 00:45:47,600 --> 00:45:50,290 And then we start looking at the details of what we observe. 631 00:45:53,130 --> 00:45:58,500 And what we find is, all we need is spin 1/2 632 00:45:58,500 --> 00:46:00,660 to account for almost everything. 633 00:46:03,540 --> 00:46:08,590 However, if it were spin 1/2, then you get two here. 634 00:46:08,590 --> 00:46:14,830 And you'd get two here, two, here and two here. 635 00:46:14,830 --> 00:46:15,790 And that's six. 636 00:46:15,790 --> 00:46:18,510 That's bigger than five. 637 00:46:18,510 --> 00:46:21,350 So you need to know something else. 638 00:46:21,350 --> 00:46:23,810 And one thing is quite reasonable-- you can say, 639 00:46:23,810 --> 00:46:28,790 well, the spin thing is mysterious, 640 00:46:28,790 --> 00:46:32,810 and electromagnetic radiation acts on the spatial 641 00:46:32,810 --> 00:46:34,584 coordinates. 642 00:46:34,584 --> 00:46:36,250 And there aren't any spatial coordinates 643 00:46:36,250 --> 00:46:39,100 of the internal structure of the electron. 644 00:46:39,100 --> 00:46:43,390 And so we have a selection rule delta MS equals 0. 645 00:46:47,809 --> 00:46:48,850 That still doesn't do it. 646 00:46:51,500 --> 00:46:53,630 You still need something else, and that 647 00:46:53,630 --> 00:46:58,940 is that the proportionality concept between the angular 648 00:46:58,940 --> 00:47:06,320 momentum and the energy, which is called the G factor. 649 00:47:06,320 --> 00:47:10,940 For orbital angular matter, the G factor is 1, 650 00:47:10,940 --> 00:47:12,620 or the proportionality constant is 1. 651 00:47:12,620 --> 00:47:18,070 So the G factor is L. And the G factor for the electron 652 00:47:18,070 --> 00:47:20,570 turns out to be 2-- 653 00:47:20,570 --> 00:47:23,330 not exactly 2, just a little more than 2-- 654 00:47:23,330 --> 00:47:26,510 Nobel Prize for that. 655 00:47:26,510 --> 00:47:30,360 And so with those extra little things, 656 00:47:30,360 --> 00:47:35,390 then every detail of the Zeeman effect is understood. 657 00:47:35,390 --> 00:47:38,540 And we say, oh, well, the electron has a spin of 1/2, 658 00:47:38,540 --> 00:47:40,480 and it like it acts like an angular momentum. 659 00:47:40,480 --> 00:47:44,580 It obeys the angular momentum computation rules. 660 00:47:44,580 --> 00:47:48,480 It also obeys the computation rules. 661 00:47:48,480 --> 00:47:50,600 Well, I won't say that. 662 00:47:50,600 --> 00:47:52,940 So I should stop-- 663 00:47:52,940 --> 00:47:56,660 there are other things that provide us information 664 00:47:56,660 --> 00:47:58,700 that there is something else. 665 00:47:58,700 --> 00:48:03,980 And one, I'm called by some people the spin orbit kid 666 00:48:03,980 --> 00:48:07,550 because I've made a whole lot of mileage on using spin orbits 667 00:48:07,550 --> 00:48:08,315 splittings. 668 00:48:08,315 --> 00:48:13,690 And the spin orbit Hamiltonian has the form L.S 669 00:48:13,690 --> 00:48:19,700 And this gives rise to splittings of a doublet P 670 00:48:19,700 --> 00:48:21,290 state, for example, as what you would 671 00:48:21,290 --> 00:48:24,620 see in the excited state of hydrogen 672 00:48:24,620 --> 00:48:27,110 into two components at zero field. 673 00:48:29,620 --> 00:48:33,310 And so there's all sorts of really wonderful stuff. 674 00:48:33,310 --> 00:48:36,070 And I'm going to cheat you out of almost all of it 675 00:48:36,070 --> 00:48:39,580 because we're going to go over to the semiclassical picture 676 00:48:39,580 --> 00:48:41,660 next time. 677 00:48:41,660 --> 00:48:45,410 And we'll understand everything about Rydberg states 678 00:48:45,410 --> 00:48:48,560 and about how do we estimate everything 679 00:48:48,560 --> 00:48:52,110 having to do with the radial part of the wave function. 680 00:48:52,110 --> 00:48:54,640 And there are some astonishing things.