1 00:00:00,060 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,810 under a Creative Commons license. 3 00:00:03,810 --> 00:00:06,540 Your support will help MIT OpenCourseWare continue 4 00:00:06,540 --> 00:00:10,120 to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,305 at ocw.mit.edu. 8 00:00:21,740 --> 00:00:24,600 PROFESSOR: All right. 9 00:00:24,600 --> 00:00:29,660 Today we'll be talking a little about angular momentum. 10 00:00:29,660 --> 00:00:33,220 Continuing the discussion of those vector operators 11 00:00:33,220 --> 00:00:37,080 and their identities that we had last time. 12 00:00:37,080 --> 00:00:42,190 So it will allow us to make quite a bit of progress 13 00:00:42,190 --> 00:00:46,190 with those operators, and understand them better. 14 00:00:46,190 --> 00:00:51,000 Then we'll go through the algebraic analysis 15 00:00:51,000 --> 00:00:52,470 of the spectrum. 16 00:00:52,470 --> 00:00:56,100 This is something that probably you've 17 00:00:56,100 --> 00:01:01,170 seen in some way or another, perhaps in not so much detail. 18 00:01:01,170 --> 00:01:04,090 But you're probably somewhat familiar, 19 00:01:04,090 --> 00:01:07,050 but it's good to see it again. 20 00:01:07,050 --> 00:01:12,210 And finally at the end we'll discuss an application 21 00:01:12,210 --> 00:01:16,700 that is related to your last problem in the homework. 22 00:01:16,700 --> 00:01:20,130 And it's a rather mysterious thing 23 00:01:20,130 --> 00:01:22,680 that I think one should appreciate 24 00:01:22,680 --> 00:01:26,990 how unusual the result is, related 25 00:01:26,990 --> 00:01:30,180 to the two dimensional harmonic oscillator. 26 00:01:30,180 --> 00:01:35,950 So I'll begin by reminding you of a few things. 27 00:01:35,950 --> 00:01:40,460 We have L, which is r cross p. 28 00:01:40,460 --> 00:01:43,220 And we managed to prove last time 29 00:01:43,220 --> 00:01:49,430 that that was equal to p cross r, with a minus sign. 30 00:01:49,430 --> 00:01:51,840 And then part of the problem's that you're 31 00:01:51,840 --> 00:01:55,720 solving with angular momentum use 32 00:01:55,720 --> 00:01:59,320 the concept of a vector and the rotations. 33 00:01:59,320 --> 00:02:11,320 So if u is a vector under rotations-- 34 00:02:11,320 --> 00:02:15,490 to say that something is a vector under rotations means 35 00:02:15,490 --> 00:02:25,800 the following, means that if you compute Li commutator with uj, 36 00:02:25,800 --> 00:02:26,810 you can put a hat. 37 00:02:26,810 --> 00:02:29,650 All these things are operators, all these vectors. 38 00:02:29,650 --> 00:02:33,650 So maybe I won't put a hat here on the blackboard. 39 00:02:33,650 --> 00:02:40,010 Then you're supposed to get i, epsilon, ijk, ih bar. 40 00:02:40,010 --> 00:02:44,170 Epsilon, ijk, uk. 41 00:02:44,170 --> 00:02:47,610 So that's a definition if you wish. 42 00:02:47,610 --> 00:02:53,990 Any object that does that is a vector under rotations. 43 00:02:53,990 --> 00:02:59,880 And something that in the homework you can verify 44 00:02:59,880 --> 00:03:07,910 is that r and p are vectors under rotation. 45 00:03:07,910 --> 00:03:13,490 That is, if you put here xj, you get this thing with xk. 46 00:03:13,490 --> 00:03:17,610 If you put here pj, you get this thing with pk. 47 00:03:17,610 --> 00:03:20,230 If you compute the commutator. 48 00:03:20,230 --> 00:03:24,790 So r and p are vectors under rotation. 49 00:03:24,790 --> 00:03:29,950 Then comes that little theorem, that is awfully important, 50 00:03:29,950 --> 00:03:37,505 that shows that if u and v are vectors under rotations-- u 51 00:03:37,505 --> 00:03:52,065 and v vectors under rotations-- then u dot v is a scalar. 52 00:03:54,870 --> 00:04:01,960 And u cross v is a vector. 53 00:04:01,960 --> 00:04:04,746 And in both cases, under rotations. 54 00:04:08,840 --> 00:04:14,610 So this is something you must prove, 55 00:04:14,610 --> 00:04:18,350 because if you know how u and v commute with the angular 56 00:04:18,350 --> 00:04:22,412 momentum, you know how u times v, in either the dot 57 00:04:22,412 --> 00:04:27,110 combination or the cross combination, commute with j, 58 00:04:27,110 --> 00:04:31,840 with L. So to say that something is a scalar, 59 00:04:31,840 --> 00:04:38,380 the translation is that Li with u dot v will be 0. 60 00:04:38,380 --> 00:04:40,630 You don't have to calculate it again. 61 00:04:40,630 --> 00:04:42,815 If you've shown that u and v are vectors, 62 00:04:42,815 --> 00:04:46,850 that they transform like that, this commutes with this. 63 00:04:46,850 --> 00:04:50,200 So r-- so what do you conclude from this? 64 00:04:50,200 --> 00:04:59,840 That Li commutes with r squared, commutes-- it's equal to p 65 00:04:59,840 --> 00:05:02,070 squared. 66 00:05:02,070 --> 00:05:07,140 And it's equal to Li, r dot p. 67 00:05:07,140 --> 00:05:10,320 They all are 0. 68 00:05:10,320 --> 00:05:13,040 Because r and p are vectors under rotation, 69 00:05:13,040 --> 00:05:16,910 so you don't have to compute those ones anymore. 70 00:05:16,910 --> 00:05:21,290 Li will commute with r squared, with p squared r cross p. 71 00:05:21,290 --> 00:05:24,620 And also, the fact that u cross v 72 00:05:24,620 --> 00:05:31,940 is a vector means that Li commutated with u cross v, 73 00:05:31,940 --> 00:05:39,860 j-- the j component of u cross v is ih bar, u cross v. I'm 74 00:05:39,860 --> 00:05:45,900 sorry-- epsilon, ijk, u cross v, k. 75 00:05:49,160 --> 00:05:56,560 Which is to say that u cross v is a vector under rotations. 76 00:05:56,560 --> 00:06:00,920 This has a lot of important corollaries. 77 00:06:00,920 --> 00:06:04,080 The most important perhaps is the commutation 78 00:06:04,080 --> 00:06:06,900 of angular momentum with itself. 79 00:06:06,900 --> 00:06:12,540 That is since you've shown that r and p satisfy this, 80 00:06:12,540 --> 00:06:17,790 r cross p, which is angular momentum, 81 00:06:17,790 --> 00:06:20,100 is also a vector under rotation. 82 00:06:20,100 --> 00:06:27,640 So here choosing u equal r, and v equal p, 83 00:06:27,640 --> 00:06:37,485 you get that Li, Lj is equal to ih bar, epsilon, ijk, Lk. 84 00:06:37,485 --> 00:06:39,310 And it's the end of the story. 85 00:06:39,310 --> 00:06:42,850 You got this commutation. 86 00:06:42,850 --> 00:06:46,020 The commutation you wanted. 87 00:06:46,020 --> 00:06:50,470 In earlier courses, you probably found 88 00:06:50,470 --> 00:06:54,080 that this was a fairly complicated calculation. 89 00:06:54,080 --> 00:06:57,690 Which you had to put the x's and the p's, the x's and the p's, 90 00:06:57,690 --> 00:07:00,040 and start moving them. 91 00:07:00,040 --> 00:07:03,800 And it takes quite a while to do it. 92 00:07:03,800 --> 00:07:06,560 So, that's important. 93 00:07:06,560 --> 00:07:09,480 Another property that follows from all 94 00:07:09,480 --> 00:07:13,070 of this, which is sort of interesting, that since L 95 00:07:13,070 --> 00:07:19,116 is now also a vector under rotations, Li commutes with L 96 00:07:19,116 --> 00:07:19,615 squared. 97 00:07:27,720 --> 00:07:33,850 Because l squared is L dot L, therefore it's a scalar. 98 00:07:33,850 --> 00:07:38,320 So Li commutes with L squared. 99 00:07:38,320 --> 00:07:44,100 And that property is absolutely crucial. 100 00:07:44,100 --> 00:07:48,600 It's important that it's worth checking that in fact, it 101 00:07:48,600 --> 00:07:50,920 follows just from this algebra. 102 00:07:56,290 --> 00:07:58,490 You see, the only thing you need to know 103 00:07:58,490 --> 00:08:01,240 to compute the commutator of Li with L squared 104 00:08:01,240 --> 00:08:03,850 is how L's commute. 105 00:08:03,850 --> 00:08:07,070 Therefore it should be possible to calculate this 106 00:08:07,070 --> 00:08:09,750 based on this algebra. 107 00:08:09,750 --> 00:08:18,200 So this property is true just because of this algebra, not 108 00:08:18,200 --> 00:08:21,460 because of anything we've said before. 109 00:08:21,460 --> 00:08:24,120 And that's important to realize it. 110 00:08:24,120 --> 00:08:28,140 Because you have algebra like si, 111 00:08:28,140 --> 00:08:34,350 sj, ih bar, epsilon, ijk, sk, which 112 00:08:34,350 --> 00:08:37,440 was the algebra of spin angular momentum. 113 00:08:37,440 --> 00:08:41,039 And we claim that for that same reason 114 00:08:41,039 --> 00:08:44,960 that this algebra leads to this result, 115 00:08:44,960 --> 00:08:50,700 that si should commute with s squared. 116 00:08:50,700 --> 00:08:55,540 And you may remember that in the particular case 117 00:08:55,540 --> 00:08:59,910 we examined in this course, s squared-- 118 00:08:59,910 --> 00:09:06,520 that would be sx squared plus sy squared plus sz squared-- was 119 00:09:06,520 --> 00:09:12,260 in fact h bar over 2 squared. 120 00:09:12,260 --> 00:09:15,810 And each matrix was proportional to the identity. 121 00:09:15,810 --> 00:09:18,780 So there's a 3 in the identity matrix. 122 00:09:18,780 --> 00:09:23,750 And s squared is really in the way we represent that spin, 123 00:09:23,750 --> 00:09:29,050 by 2 by 2 matrices, commutes with si. 124 00:09:29,050 --> 00:09:30,600 Because it is the identity. 125 00:09:30,600 --> 00:09:35,870 So it's no accident that this thing is 0. 126 00:09:35,870 --> 00:09:41,220 Because this algebra, whatever l is, 127 00:09:41,220 --> 00:09:47,940 implies that this with the thing squared is equal to zero. 128 00:09:47,940 --> 00:09:51,550 So whenever we'll be talking about spin angular 129 00:09:51,550 --> 00:09:56,400 momentum, orbital angular momentum, total angular 130 00:09:56,400 --> 00:09:58,280 momentum, when we add them, there's 131 00:09:58,280 --> 00:09:59,810 all kinds of angular momentum. 132 00:09:59,810 --> 00:10:05,610 And our another generic name for angular momentum will be j. 133 00:10:05,610 --> 00:10:15,460 And we'll say that ji, jj, equal ih bar, epsilon, ijk, 134 00:10:15,460 --> 00:10:20,485 jk is the algebra of angular momentum. 135 00:10:25,590 --> 00:10:29,200 And by using j, you're sending the signal 136 00:10:29,200 --> 00:10:31,906 that you may be talking about l. 137 00:10:31,906 --> 00:10:35,150 Or may be talking about s, but it's not obvious 138 00:10:35,150 --> 00:10:36,650 which you're talking about. 139 00:10:36,650 --> 00:10:41,320 And you're focusing on those properties of angular momentum 140 00:10:41,320 --> 00:10:49,320 that hold just because this algebra is supposed to be true. 141 00:10:49,320 --> 00:10:54,460 So in this algebra, you will have that ji commutes with j 142 00:10:54,460 --> 00:10:55,100 squared. 143 00:10:55,100 --> 00:10:56,720 And what is j squared? 144 00:10:56,720 --> 00:11:00,850 Of course, j squared is j1 squared 145 00:11:00,850 --> 00:11:05,860 plus j2 squared plus j3 squared. 146 00:11:05,860 --> 00:11:08,560 Now this is so important, and this derivation 147 00:11:08,560 --> 00:11:13,300 is a little bit indirect, that I encourage you all 148 00:11:13,300 --> 00:11:16,230 to just do it. 149 00:11:16,230 --> 00:11:19,820 Without using any formula, put the jx here, 150 00:11:19,820 --> 00:11:21,930 and compute this commutator. 151 00:11:21,930 --> 00:11:26,430 And it takes a couple of lines, but just convince yourself 152 00:11:26,430 --> 00:11:29,460 that this is true. 153 00:11:29,460 --> 00:11:38,580 OK, now we did have a little more discussion. 154 00:11:38,580 --> 00:11:41,760 And these are all things that are basically related 155 00:11:41,760 --> 00:11:43,610 to what you've been doing in the homework. 156 00:11:46,380 --> 00:11:51,330 Another fact is that this algebra 157 00:11:51,330 --> 00:11:58,780 is translated into j cross j equal ih bar, j. 158 00:12:03,030 --> 00:12:10,850 Another result in transcription of equations 159 00:12:10,850 --> 00:12:24,370 is that the statement that u is a vector under rotations 160 00:12:24,370 --> 00:12:26,900 corresponds to a vector identity. 161 00:12:26,900 --> 00:12:29,060 Just the fact that the algebra here 162 00:12:29,060 --> 00:12:35,460 is this, the fact that l with u is this, 163 00:12:35,460 --> 00:12:37,640 implies the following algebra. 164 00:12:37,640 --> 00:12:51,276 j cross u plus u cross j equal 2i h bar. 165 00:12:56,440 --> 00:12:59,776 So this is for a vector under rotations. 166 00:13:04,450 --> 00:13:06,490 Under rotations. 167 00:13:06,490 --> 00:13:09,760 So this I think is in the notes. 168 00:13:09,760 --> 00:13:11,380 It's basically saying that if you 169 00:13:11,380 --> 00:13:16,210 want to translate this equation into vector form, which 170 00:13:16,210 --> 00:13:20,370 is a nice thing to have, it reads like this. 171 00:13:20,370 --> 00:13:24,280 And the way to do that is to just calculate the left hand 172 00:13:24,280 --> 00:13:24,780 side. 173 00:13:24,780 --> 00:13:27,500 Put and index, i. 174 00:13:27,500 --> 00:13:30,050 And just try to get the right hand side. 175 00:13:30,050 --> 00:13:31,360 It will work out. 176 00:13:34,130 --> 00:13:34,710 OK. 177 00:13:34,710 --> 00:13:37,355 Any questions so far with these identities? 178 00:13:48,710 --> 00:13:49,330 OK. 179 00:13:49,330 --> 00:13:52,370 So we move on to another identity 180 00:13:52,370 --> 00:13:57,470 that you've been working on, based 181 00:13:57,470 --> 00:14:04,540 on the calculation of what is a cross b dot a cross b. 182 00:14:07,550 --> 00:14:11,540 If these things are operators, there's 183 00:14:11,540 --> 00:14:14,020 corrections to the classical formula 184 00:14:14,020 --> 00:14:21,390 for the answer of of what this product is supposed to be. 185 00:14:21,390 --> 00:14:26,990 Actually, the classical formula, so it's not 186 00:14:26,990 --> 00:14:33,820 equal to a squared, b squared, minus a dot b squared. 187 00:14:33,820 --> 00:14:38,020 But it's actually equal to this, plus dot dot dot. 188 00:14:38,020 --> 00:14:40,000 A few more things. 189 00:14:40,000 --> 00:14:41,960 Classically it's just that. 190 00:14:41,960 --> 00:14:44,640 You put 2 epsilons. 191 00:14:44,640 --> 00:14:46,240 Calculate the left hand side. 192 00:14:46,240 --> 00:14:49,080 And it's just these 2 terms. 193 00:14:49,080 --> 00:14:52,280 Since there are more terms, let's 194 00:14:52,280 --> 00:14:55,590 look what they are for a particular case of interest. 195 00:14:55,590 --> 00:15:01,060 So our case of interest is L squared, that corresponds to r 196 00:15:01,060 --> 00:15:03,380 cross b, times r cross b. 197 00:15:08,100 --> 00:15:12,780 And indeed, it's not just r squared, 198 00:15:12,780 --> 00:15:18,810 p squared, minus r dot p squared. 199 00:15:18,810 --> 00:15:21,610 But there's a little extra. 200 00:15:21,610 --> 00:15:25,660 And perhaps you have computed that little extra by now. 201 00:15:25,660 --> 00:15:29,258 It's ih bar r dot p. 202 00:15:32,680 --> 00:15:40,310 So that's a pretty useful result. 203 00:15:40,310 --> 00:15:43,925 And from here, we typically look for what is p squared. 204 00:15:46,720 --> 00:15:50,280 So for p squared-- so what we do is pass these other terms 205 00:15:50,280 --> 00:15:52,180 to the other side. 206 00:15:52,180 --> 00:15:57,390 And therefore we have 1 over r squared, 207 00:15:57,390 --> 00:16:04,660 r dot p squared, minus ih bar, r dot p. 208 00:16:08,600 --> 00:16:11,280 Yes. 209 00:16:11,280 --> 00:16:14,185 Plus 1 over r squared, l squared. 210 00:16:17,890 --> 00:16:22,520 And, we've done this with some prudence. 211 00:16:22,520 --> 00:16:25,620 The r squared is here in front of the p squared. 212 00:16:25,620 --> 00:16:29,415 It may be fairly different from having it to the other side. 213 00:16:31,930 --> 00:16:36,135 And therefore, when I apply the inverse 1 over r squared, 214 00:16:36,135 --> 00:16:37,860 I apply it from the left. 215 00:16:37,860 --> 00:16:41,190 So I write it like that. 216 00:16:41,190 --> 00:16:42,970 And that's very different for having 217 00:16:42,970 --> 00:16:45,220 the r squared on the other side. 218 00:16:45,220 --> 00:16:48,550 Could be completely different. 219 00:16:48,550 --> 00:16:52,010 Now, what is this? 220 00:16:52,010 --> 00:17:00,590 Well this is a simple computation, 221 00:17:00,590 --> 00:17:08,899 when you remember that p vector is h bar over i gradient. 222 00:17:11,839 --> 00:17:21,956 And r dot p, therefore is h bar over i, r, dvr. 223 00:17:26,410 --> 00:17:29,980 Because r vector is r magnitude times 224 00:17:29,980 --> 00:17:32,620 the unit vector in the radial direction. 225 00:17:32,620 --> 00:17:36,550 And the radial direction of gradient is dvr. 226 00:17:36,550 --> 00:17:38,920 So this can be simplified. 227 00:17:38,920 --> 00:17:42,300 I will not do it because it's in the notes. 228 00:17:42,300 --> 00:17:49,280 And you get minus h squared, 1 over r, d second, d r squared, 229 00:17:49,280 --> 00:17:51,050 r. 230 00:17:51,050 --> 00:17:56,300 In a funny notation, the r is on the right. 231 00:17:56,300 --> 00:17:58,750 And the 1 over r is on the left. 232 00:17:58,750 --> 00:18:03,990 And you would say, this doesn't sound right. 233 00:18:03,990 --> 00:18:06,840 You have here all this derivatives 234 00:18:06,840 --> 00:18:10,260 and what is an r doing to the right of the derivatives. 235 00:18:10,260 --> 00:18:11,690 I see no r. 236 00:18:11,690 --> 00:18:14,775 But this is a kind of a trick to rewrite everything 237 00:18:14,775 --> 00:18:17,010 in a short way. 238 00:18:17,010 --> 00:18:19,800 So if you want, think of this being 239 00:18:19,800 --> 00:18:22,753 acting on some function of r. 240 00:18:22,753 --> 00:18:24,420 And see what it is. 241 00:18:24,420 --> 00:18:28,560 And then you put a function of r here, and calculate it. 242 00:18:28,560 --> 00:18:30,980 And you will see, you get the same. 243 00:18:30,980 --> 00:18:35,770 So it's a good thing to try that. 244 00:18:35,770 --> 00:18:38,930 So p squared is given by this. 245 00:18:38,930 --> 00:18:42,480 There's another formula for p squared. 246 00:18:42,480 --> 00:18:49,470 p squared is, of course, the Laplacian. 247 00:18:52,890 --> 00:19:01,530 So p squared is also equal to minus h squared 248 00:19:01,530 --> 00:19:04,730 times the Laplacian operator. 249 00:19:04,730 --> 00:19:10,600 And that's equal to minus h squared times-- in fact, 250 00:19:10,600 --> 00:19:17,270 the Laplacian operator is 1 over r, d second, dr squared, r, 251 00:19:17,270 --> 00:19:23,680 plus 1 over r squared, 1 over sine theta, dd 252 00:19:23,680 --> 00:19:28,400 theta, sine theta, dd theta. 253 00:19:28,400 --> 00:19:30,380 It's a little bit messy. 254 00:19:30,380 --> 00:19:40,090 Plus 1 over sine squared theta, d second, d phi squared, 255 00:19:40,090 --> 00:19:41,560 times closing this. 256 00:19:41,560 --> 00:19:43,955 So a few things are there to learn. 257 00:19:47,220 --> 00:19:52,710 And the first thing is if you compare these 2 expressions, 258 00:19:52,710 --> 00:19:55,130 you have a formula for l squared. 259 00:19:57,640 --> 00:20:02,310 You have l squared is 1 over r squared on the upper right. 260 00:20:02,310 --> 00:20:06,790 And here you have minus h squared times this thing. 261 00:20:06,790 --> 00:20:12,165 So l squared, that scalar operator 262 00:20:12,165 --> 00:20:18,830 is minus h squared, 1 over sine theta, 263 00:20:18,830 --> 00:20:26,380 dd theta, sine theta, dd theta, plus 1 over sine 264 00:20:26,380 --> 00:20:30,223 squared theta, d second, d phi squared. 265 00:20:35,720 --> 00:20:40,905 So in terms of functions of 3 variables, x, y, and z, 266 00:20:40,905 --> 00:20:45,700 L squared, which is a very complicated object, 267 00:20:45,700 --> 00:20:49,080 has become just a function of the angular variables. 268 00:20:49,080 --> 00:20:52,590 And that this a very important intuitive fact. 269 00:20:52,590 --> 00:20:54,170 L squared. 270 00:20:54,170 --> 00:20:55,280 L is operator. 271 00:20:55,280 --> 00:20:56,570 That's rotation. 272 00:20:56,570 --> 00:21:00,790 So it shouldn't really affect the r, shouldn't change r, 273 00:21:00,790 --> 00:21:02,710 modify r in any way. 274 00:21:02,710 --> 00:21:05,860 So it's a nice thing to confirm here 275 00:21:05,860 --> 00:21:11,900 that this operator can be thought as an operator acting 276 00:21:11,900 --> 00:21:14,050 on the angular variables. 277 00:21:14,050 --> 00:21:21,450 Or you could say, on functions, on the units here for example. 278 00:21:21,450 --> 00:21:22,780 It's a good thing. 279 00:21:22,780 --> 00:21:24,660 The other thing that you've learned here-- 280 00:21:24,660 --> 00:21:27,620 so this is a very nice result. 281 00:21:27,620 --> 00:21:32,170 It's not all that easy to get by direct computation. 282 00:21:32,170 --> 00:21:37,420 If you had to do Lx squared plus Ly squared plus Lz 283 00:21:37,420 --> 00:21:41,950 squared, first all this possible order-- well, 284 00:21:41,950 --> 00:21:43,770 there's no ordering problems here. 285 00:21:43,770 --> 00:21:48,430 But you would have to write this in terms of x, and py, and pz, 286 00:21:48,430 --> 00:21:52,260 and xy, and z, then pass to angular variables. 287 00:21:52,260 --> 00:21:53,380 Simplify all that. 288 00:21:53,380 --> 00:21:57,530 It's a very bad way to do it. 289 00:21:57,530 --> 00:21:59,240 And it's painful. 290 00:21:59,240 --> 00:22:02,910 So the fact that we got this like that is very nice. 291 00:22:02,910 --> 00:22:06,750 The other thing that we've got is some understanding 292 00:22:06,750 --> 00:22:10,100 of the Hamiltonian for a central potential, what 293 00:22:10,100 --> 00:22:13,360 we call a central potential problem. 294 00:22:13,360 --> 00:22:14,305 v of r. 295 00:22:17,530 --> 00:22:21,710 Now, I will write a v of r like this. 296 00:22:21,710 --> 00:22:24,690 But then we'll simplify it. 297 00:22:24,690 --> 00:22:29,060 In fact, let me just go to a central potential case, which 298 00:22:29,060 --> 00:22:32,960 means that the potential just depends on the magnitude of r. 299 00:22:32,960 --> 00:22:38,540 So r is the magnitude of the vector r. 300 00:22:38,540 --> 00:22:44,220 So at this moment, you have p squared over there. 301 00:22:44,220 --> 00:22:50,120 So this whole Hamiltonian is minus h 302 00:22:50,120 --> 00:23:02,850 squared over 2m, 1 over r, d second, dr squared, r, plus p 303 00:23:02,850 --> 00:23:04,080 squared over 2m. 304 00:23:04,080 --> 00:23:11,850 So 1 over 2m, r squared, l squared plus v of r. 305 00:23:14,930 --> 00:23:19,506 So our Hamiltonian has also been simplified. 306 00:23:23,960 --> 00:23:26,290 So this will be the starting point 307 00:23:26,290 --> 00:23:30,350 for writing the Schrodinger equation 308 00:23:30,350 --> 00:23:33,740 for central potentials. 309 00:23:33,740 --> 00:23:37,170 And you have the operator l squared. 310 00:23:37,170 --> 00:23:41,930 And as far as we can, we'll try to avoid computations 311 00:23:41,930 --> 00:23:45,060 in theta and phi very explicitly, 312 00:23:45,060 --> 00:23:49,180 but try to do things algebraically. 313 00:23:49,180 --> 00:23:53,500 So at this moment, the last comment 314 00:23:53,500 --> 00:23:56,980 I want to make on this subject is the issue 315 00:23:56,980 --> 00:24:01,790 of set of commuting observables. 316 00:24:01,790 --> 00:24:05,730 So if you have a Hamiltonian like that, 317 00:24:05,730 --> 00:24:09,390 you can try to form a set of commuting observables that 318 00:24:09,390 --> 00:24:13,180 are going to help you understand the physics 319 00:24:13,180 --> 00:24:14,940 of your particular problem. 320 00:24:14,940 --> 00:24:17,140 So the first thing that you would 321 00:24:17,140 --> 00:24:21,310 want to put in the list of complete set of observables 322 00:24:21,310 --> 00:24:22,690 is the Hamiltonian. 323 00:24:22,690 --> 00:24:25,960 We really want to know the energies of this thing. 324 00:24:25,960 --> 00:24:29,500 So what other operators do I have? 325 00:24:29,500 --> 00:24:32,960 Well I have x1, x2, and x3. 326 00:24:37,820 --> 00:24:42,950 And well, can I add them to the Hamiltonian 327 00:24:42,950 --> 00:24:45,880 to have a complete set of commuting observables? 328 00:24:45,880 --> 00:24:47,915 Well, the x's commute among themselves. 329 00:24:51,560 --> 00:24:52,720 So can I add them? 330 00:24:58,280 --> 00:24:59,930 Yes or no? 331 00:24:59,930 --> 00:25:00,590 No. 332 00:25:00,590 --> 00:25:03,450 No you can't add them, because the x's 333 00:25:03,450 --> 00:25:05,820 don't commute with the Hamiltonian. 334 00:25:05,820 --> 00:25:07,660 There's a p here. 335 00:25:07,660 --> 00:25:09,180 p doesn't commute with x's. 336 00:25:09,180 --> 00:25:11,750 So that's out of the question. 337 00:25:11,750 --> 00:25:15,500 They cannot be added to our list. 338 00:25:15,500 --> 00:25:16,850 How about the p's? 339 00:25:16,850 --> 00:25:20,690 p1, p2, and p3. 340 00:25:20,690 --> 00:25:23,240 Not good either, because they don't 341 00:25:23,240 --> 00:25:25,370 commune with the potential term. 342 00:25:25,370 --> 00:25:30,310 The potential has x dependents, and will take a miracle for it 343 00:25:30,310 --> 00:25:30,970 to commute. 344 00:25:30,970 --> 00:25:32,370 In general, it won't commute. 345 00:25:32,370 --> 00:25:37,890 So no reason for it to commute, unless the potential is 0. 346 00:25:37,890 --> 00:25:39,140 So this is not good. 347 00:25:41,760 --> 00:25:51,235 Nor is good to have r squared, or p squared, or r dot p. 348 00:25:51,235 --> 00:25:56,870 r squared, p squared, r dot p. 349 00:25:56,870 --> 00:25:58,620 No good either. 350 00:25:58,620 --> 00:26:05,440 On the other hand, r cross p is interesting. 351 00:26:05,440 --> 00:26:10,135 You have the angular momentum, L1, L2, and L3. 352 00:26:14,660 --> 00:26:23,410 Well, the angular momentum will commute, I think, 353 00:26:23,410 --> 00:26:24,950 with the Hamiltonian. 354 00:26:24,950 --> 00:26:27,600 You can see it here. 355 00:26:27,600 --> 00:26:34,990 You have p squared, and Li's commute with p 356 00:26:34,990 --> 00:26:39,050 squared because p is a vector under rotations. 357 00:26:39,050 --> 00:26:42,900 p doesn't communicate with Li, but p squared does. 358 00:26:42,900 --> 00:26:44,580 Because that was a scalar. 359 00:26:44,580 --> 00:26:50,860 So this term commutes with any angular momentum operator. 360 00:26:50,860 --> 00:26:54,350 Moreover, v or r, r is this. 361 00:26:54,350 --> 00:26:58,440 So a v of r is a function of r squared. 362 00:26:58,440 --> 00:27:02,680 And r squared is the vector r squared. 363 00:27:02,680 --> 00:27:07,280 So ultimately, anything that is a function of r is a function 364 00:27:07,280 --> 00:27:10,650 of r squared that involves the operator r squared, 365 00:27:10,650 --> 00:27:13,890 that also commutes with all the Li's. 366 00:27:13,890 --> 00:27:17,170 So h commutes with all the Li's. 367 00:27:17,170 --> 00:27:19,490 And that's a great thing. 368 00:27:19,490 --> 00:27:23,010 So this is absolutely important. 369 00:27:23,010 --> 00:27:27,390 h commutes with all the Li's. 370 00:27:27,390 --> 00:27:32,780 That's angular momentum conservation. 371 00:27:32,780 --> 00:27:41,310 As we've seen, the rate of change of any operator 372 00:27:41,310 --> 00:27:44,450 is equal to expectation value of the commutator 373 00:27:44,450 --> 00:27:47,460 of the operator with the Hamiltonian. 374 00:27:47,460 --> 00:27:54,960 So if you put any Li, this commutator is 0. 375 00:27:54,960 --> 00:27:58,250 And the operator is conserved in the sense 376 00:27:58,250 --> 00:28:01,870 of expectation values. 377 00:28:01,870 --> 00:28:05,580 Now this conservation law is great. 378 00:28:05,580 --> 00:28:08,890 You could add this operators to the commuting set 379 00:28:08,890 --> 00:28:10,800 of observables. 380 00:28:10,800 --> 00:28:16,100 But this time, you have a different problem. 381 00:28:16,100 --> 00:28:17,800 Yes, this commutes with h. 382 00:28:17,800 --> 00:28:18,945 This commutes with h. 383 00:28:18,945 --> 00:28:20,810 And this commutes with h. 384 00:28:20,810 --> 00:28:23,270 But these one's don't commute with each other. 385 00:28:23,270 --> 00:28:26,090 So not quite good enough. 386 00:28:26,090 --> 00:28:27,515 You cannot add them all. 387 00:28:30,070 --> 00:28:33,750 So let's see how many can we add. 388 00:28:33,750 --> 00:28:35,640 We can only add 1. 389 00:28:35,640 --> 00:28:39,360 Because once you have 2 of them, they don't commute. 390 00:28:39,360 --> 00:28:46,000 So you're going to add 1, and everybody has agreed to add L3. 391 00:28:46,000 --> 00:28:48,590 So we have H, L3. 392 00:28:51,420 --> 00:28:56,630 And happily we have 1 more is L squared. 393 00:28:56,630 --> 00:29:01,430 Remember, L squared commutes with all the Li's, so that's 394 00:29:01,430 --> 00:29:02,240 another operator. 395 00:29:06,720 --> 00:29:10,010 And for a central potential problem, 396 00:29:10,010 --> 00:29:18,094 this will be sufficient to label all of our states some. 397 00:29:18,094 --> 00:29:21,976 AUDIENCE: So how do we know that we need the L squared? 398 00:29:21,976 --> 00:29:23,805 How do we know that we can't get-- 399 00:29:23,805 --> 00:29:26,986 how do we know that just H and L3 isn't already 400 00:29:26,986 --> 00:29:30,142 a complete set? 401 00:29:30,142 --> 00:29:34,010 PROFESSOR: I probably wouldn't know now, but in a little bit, 402 00:29:34,010 --> 00:29:36,960 as we calculate the kind of states 403 00:29:36,960 --> 00:29:40,040 that we get with angular momentum, 404 00:29:40,040 --> 00:29:44,260 I will see that there are many states with the same value 405 00:29:44,260 --> 00:29:51,260 of L3 that don't correspond to the same value of the total 406 00:29:51,260 --> 00:29:53,920 or length of the angular momentum. 407 00:29:53,920 --> 00:30:00,810 So it's almost like saying that there are angular 408 00:30:00,810 --> 00:30:07,360 momenta-- here is-- let me draw a plane. 409 00:30:07,360 --> 00:30:12,100 Here is z component of angular momentum, Lz. 410 00:30:12,100 --> 00:30:13,360 And here you got it. 411 00:30:13,360 --> 00:30:18,210 You can have an angular momentum that is like that, 412 00:30:18,210 --> 00:30:20,400 and has this Lz. 413 00:30:20,400 --> 00:30:22,430 Or you can have an angular momentum 414 00:30:22,430 --> 00:30:28,750 that is like this, L prime, that has the same Lz. 415 00:30:28,750 --> 00:30:32,435 And then it will be difficult to tell these 2 states apart. 416 00:30:32,435 --> 00:30:35,250 And they will correspond to states of this angular 417 00:30:35,250 --> 00:30:38,390 momentum, or this angular momentum, have the same Lz. 418 00:30:38,390 --> 00:30:45,480 Now drawing these arrows is extraordinarily misleading. 419 00:30:45,480 --> 00:30:48,640 Hope you don't get upset that I did it. 420 00:30:48,640 --> 00:30:53,120 It's misleading because this vector you cannot measure 421 00:30:53,120 --> 00:30:55,080 simultaneously the 3 components. 422 00:30:55,080 --> 00:30:56,280 Because they don't commute. 423 00:30:56,280 --> 00:30:59,700 So what do I mean by drawing an arrow? 424 00:30:59,700 --> 00:31:02,780 Nevertheless, the intuition is sort of there. 425 00:31:02,780 --> 00:31:05,450 And it's not wrong, the intuition. 426 00:31:05,450 --> 00:31:08,950 It will happen to be the case that states 427 00:31:08,950 --> 00:31:16,320 that have same amount of Lz will not be distinguished. 428 00:31:16,320 --> 00:31:21,530 But by the time we have this, we will distinguish them. 429 00:31:21,530 --> 00:31:25,860 And that's also a peculiarity of a result but we'll use. 430 00:31:25,860 --> 00:31:28,390 Even though we're talking about 3 dimensions, 431 00:31:28,390 --> 00:31:33,040 the fact that the 1 dimensional Schrodinger equation 432 00:31:33,040 --> 00:31:36,270 has non degenerate bound states. 433 00:31:36,270 --> 00:31:40,410 You say, what does that have to do with 3 dimensions? 434 00:31:40,410 --> 00:31:44,600 What will happen is that the 3 dimensional Schrodinger 435 00:31:44,600 --> 00:31:48,180 equation will reduce to a 1 dimensional radial equation. 436 00:31:48,180 --> 00:31:53,100 And the fact that that doesn't have degeneracies 437 00:31:53,100 --> 00:31:55,630 tells you that for bound state problems, 438 00:31:55,630 --> 00:31:57,530 this will be enough to do it. 439 00:31:57,530 --> 00:31:59,950 So you will have to wait a little 440 00:31:59,950 --> 00:32:01,970 to be sure that this will do it. 441 00:32:01,970 --> 00:32:05,120 But this is pretty much the best we can do now. 442 00:32:05,120 --> 00:32:08,400 And I don't think you will be able to add anything 443 00:32:08,400 --> 00:32:12,370 else to this at this stage. 444 00:32:12,370 --> 00:32:14,110 Now there's of course funny things 445 00:32:14,110 --> 00:32:16,660 that you could add like-- if there's spin, 446 00:32:16,660 --> 00:32:19,890 the particles have spin, well we can add spin and things 447 00:32:19,890 --> 00:32:21,390 like that. 448 00:32:21,390 --> 00:32:26,930 But let's leave it at that and now 449 00:32:26,930 --> 00:32:32,090 begin really our calculation, algebraic calculation, 450 00:32:32,090 --> 00:32:36,260 of the angular momentum representations. 451 00:32:36,260 --> 00:32:40,370 So at this moment, we really want 452 00:32:40,370 --> 00:32:44,730 to make sure we work with this. 453 00:32:44,730 --> 00:32:49,075 Only this formula over here. 454 00:32:58,680 --> 00:33:04,470 And learn things about the kind of states 455 00:33:04,470 --> 00:33:08,160 that can exist in a system in which there 456 00:33:08,160 --> 00:33:10,570 are operators like that. 457 00:33:10,570 --> 00:33:12,360 So it's a funny thing. 458 00:33:12,360 --> 00:33:14,810 You're talking about a vector space. 459 00:33:14,810 --> 00:33:17,570 And in fact, you don't know almost anything 460 00:33:17,570 --> 00:33:20,200 about this vector space so far. 461 00:33:20,200 --> 00:33:23,780 But there is an action of those operators. 462 00:33:23,780 --> 00:33:29,640 From that fact alone, and one more important fact-- 463 00:33:29,640 --> 00:33:33,435 the j's are Hermitian. 464 00:33:38,950 --> 00:33:42,690 From these 2 facts, we're going to derive 465 00:33:42,690 --> 00:33:49,420 incredibly powerful results, extremely powerful things. 466 00:33:49,420 --> 00:33:55,970 And as we'll see, they have applications even 467 00:33:55,970 --> 00:33:59,120 in cases that you would imagine they have nothing 468 00:33:59,120 --> 00:34:04,060 to do with angular momentum, which is really surprising. 469 00:34:04,060 --> 00:34:08,949 So how do we proceed with this stuff? 470 00:34:08,949 --> 00:34:10,639 Well, there's a hermeticity. 471 00:34:10,639 --> 00:34:13,750 And you immediately introduce things 472 00:34:13,750 --> 00:34:22,290 called J plus minus, which are J1 plus minus i J 2. 473 00:34:22,290 --> 00:34:26,278 Or Jx plus minus y Jc. 474 00:34:26,278 --> 00:34:33,409 Then you calculate what is J plus J minus. 475 00:34:33,409 --> 00:34:40,960 Well J plus J minus will be a J1 squared plus J2 squared. 476 00:34:40,960 --> 00:34:44,120 And then you have the cross product 477 00:34:44,120 --> 00:34:45,502 that this doesn't cancel. 478 00:34:51,530 --> 00:35:01,561 So J plus times J minus would be J1 plus i J2, J1 minus i J2. 479 00:35:01,561 --> 00:35:11,140 So the next term would be minus i, J1, J2. 480 00:35:11,140 --> 00:35:15,400 And that's i h bar, J3. 481 00:35:15,400 --> 00:35:24,185 So this is J1 squared plus J2 squared plus h bar J3. 482 00:35:27,050 --> 00:35:32,320 So that's a nice formula for J plus, J minus. 483 00:35:32,320 --> 00:35:37,510 J minus, J plus would be J1 squared plus J2 484 00:35:37,510 --> 00:35:42,700 squared minus h bar J3. 485 00:35:42,700 --> 00:35:52,740 These 2 formulas are summarized by J plus, J minus-- minus, 486 00:35:52,740 --> 00:35:57,210 plus-- is equal to J1 squared plus J2 487 00:35:57,210 --> 00:36:01,120 squared plus minus h bar J3. 488 00:36:08,260 --> 00:36:10,520 OK. 489 00:36:10,520 --> 00:36:12,215 Things to learn from this. 490 00:36:15,100 --> 00:36:18,440 Maybe I'll continue here for a little while 491 00:36:18,440 --> 00:36:22,960 to use the blackboards, up to here only. 492 00:36:22,960 --> 00:36:29,040 The commutator of J plus and J minus 493 00:36:29,040 --> 00:36:31,540 can be obtained from this equation. 494 00:36:31,540 --> 00:36:33,740 You just subtract them. 495 00:36:33,740 --> 00:36:37,820 And that's 2h bar, J3. 496 00:36:41,450 --> 00:36:47,190 And finally, one last thing that we like to know 497 00:36:47,190 --> 00:36:51,600 is how to write J squared. 498 00:36:51,600 --> 00:37:01,930 So J squared is J1 squared plus J2 squared plus J3 499 00:37:01,930 --> 00:37:06,500 squared, which then show up here. 500 00:37:06,500 --> 00:37:10,970 So we might as well add it and subtract it. 501 00:37:10,970 --> 00:37:16,440 So I add a J3 squared, and I add it on the left hand side. 502 00:37:16,440 --> 00:37:19,400 And pass this term to the other side. 503 00:37:19,400 --> 00:37:29,010 So J squared would be J plus, J minus, plus J3 squared, 504 00:37:29,010 --> 00:37:32,890 minus h bar, J3. 505 00:37:32,890 --> 00:37:40,741 Or J minus, J plus, plus J3 squared, plus h bar, J3. 506 00:37:44,800 --> 00:37:45,730 OK. 507 00:37:45,730 --> 00:37:47,110 So that's J squared. 508 00:37:54,770 --> 00:37:56,240 OK. 509 00:37:56,240 --> 00:37:59,680 So we're doing sort of simple things. 510 00:37:59,680 --> 00:38:02,130 Basically at this moment, we decided 511 00:38:02,130 --> 00:38:06,590 that we like better J plus and J minus. 512 00:38:06,590 --> 00:38:09,390 And we tried to figure out everything 513 00:38:09,390 --> 00:38:11,870 that we should know about J plus, J minus. 514 00:38:11,870 --> 00:38:18,270 If we substitute Lx, and Jx, and Jy for J plus and J minus, 515 00:38:18,270 --> 00:38:21,080 you better know what is the commutator 516 00:38:21,080 --> 00:38:23,650 of J plus and J minus. 517 00:38:23,650 --> 00:38:28,000 And how to write J squared in terms of J plus and J minus. 518 00:38:28,000 --> 00:38:30,640 And this is what we've done here. 519 00:38:30,640 --> 00:38:35,910 And in particular, we have a whole lot of nice formulas. 520 00:38:35,910 --> 00:38:39,515 So one more formula is probably useful. 521 00:38:42,680 --> 00:38:48,790 And it's the formula for the commutator of J plus and J 522 00:38:48,790 --> 00:38:51,050 minus with Jz. 523 00:38:51,050 --> 00:38:54,570 Because after all, the J plus, J minus commutator, 524 00:38:54,570 --> 00:38:55,450 you've got it. 525 00:38:55,450 --> 00:38:58,700 So if you're systematic about these things 526 00:38:58,700 --> 00:39:01,960 you should figure out that at this I 527 00:39:01,960 --> 00:39:05,165 would like to know what is the commutator of J plus and J 528 00:39:05,165 --> 00:39:06,250 minus with Jz. 529 00:39:06,250 --> 00:39:12,720 So I can do Jz, J plus. 530 00:39:12,720 --> 00:39:13,840 It's not hard. 531 00:39:13,840 --> 00:39:16,020 It's Jz. 532 00:39:16,020 --> 00:39:16,700 I'm sorry. 533 00:39:16,700 --> 00:39:17,990 I'm calling it 3. 534 00:39:17,990 --> 00:39:21,370 So, I think in the notes I call them x, y, and z. 535 00:39:21,370 --> 00:39:24,550 But never mind. 536 00:39:24,550 --> 00:39:28,910 J1 plus i, J2. 537 00:39:28,910 --> 00:39:32,330 The plus is really with a plus i. 538 00:39:32,330 --> 00:39:40,310 So J3 with J1 by the cyclic ordering is ih bar, J2. 539 00:39:40,310 --> 00:39:52,670 And here you have plus i, and J3 with J2 is minus ih bar, J1. 540 00:39:52,670 --> 00:40:02,770 So this is h bar, J1, plus i, J2, which is h bar, J plus. 541 00:40:02,770 --> 00:40:09,290 So what you've learned is that J3 with J plus 542 00:40:09,290 --> 00:40:12,340 is equal to h bar, J plus. 543 00:40:12,340 --> 00:40:16,190 And if you did it with J minus, you'll 544 00:40:16,190 --> 00:40:20,580 find a minus, and a plus minus here. 545 00:40:20,580 --> 00:40:25,120 So that is the complete result. 546 00:40:25,120 --> 00:40:30,970 And that should remind you of the analogous relation 547 00:40:30,970 --> 00:40:35,130 in which you have in the harmonic oscillator, N 548 00:40:35,130 --> 00:40:38,160 commutator, with a dagger. 549 00:40:38,160 --> 00:40:39,895 With a dagger. 550 00:40:39,895 --> 00:40:48,280 And N commutator with a was minus a. 551 00:40:48,280 --> 00:40:52,120 Because of the fact that I maybe didn't say it here, 552 00:40:52,120 --> 00:41:01,090 and I should have, that the dagger of J plus is J minus. 553 00:41:01,090 --> 00:41:03,190 Because the operators are Hermitians. 554 00:41:03,190 --> 00:41:06,890 So J plus and J minus are daggers of each other, 555 00:41:06,890 --> 00:41:09,180 are adjoins of each other. 556 00:41:09,180 --> 00:41:11,770 And here you see a very analogous situation. 557 00:41:11,770 --> 00:41:15,460 a and a dagger were adjoins of each other. 558 00:41:15,460 --> 00:41:19,940 And with respect to N, a counting number operator. 559 00:41:19,940 --> 00:41:21,160 One increased it. 560 00:41:21,160 --> 00:41:23,160 One decreased it. 561 00:41:23,160 --> 00:41:28,690 a dagger increased the number eigenvalue of N. a 562 00:41:28,690 --> 00:41:31,580 decreased it, the same way it's going to happen here. 563 00:41:31,580 --> 00:41:34,260 J plus is going to increase the C 564 00:41:34,260 --> 00:41:36,290 component of angular momentum. 565 00:41:36,290 --> 00:41:38,320 And J minus is going to decrease it. 566 00:41:40,930 --> 00:41:41,920 OK. 567 00:41:41,920 --> 00:41:47,380 So we've done most of the calculations that we need. 568 00:41:47,380 --> 00:41:50,680 The rest is pretty easy work. 569 00:41:50,680 --> 00:41:52,920 Not that it was difficult so far. 570 00:41:52,920 --> 00:41:57,970 But it took a little time. 571 00:41:57,970 --> 00:42:04,960 So what happens next is the following. 572 00:42:04,960 --> 00:42:08,820 You must make a declaration. 573 00:42:08,820 --> 00:42:13,590 There should exist states, basically. 574 00:42:13,590 --> 00:42:15,430 We have a vector space. 575 00:42:15,430 --> 00:42:17,370 It's very large. 576 00:42:17,370 --> 00:42:21,180 It's actually infinite dimensional. 577 00:42:21,180 --> 00:42:24,360 Because they will be related to all kinds of functions 578 00:42:24,360 --> 00:42:25,620 on the unit sphere. 579 00:42:25,620 --> 00:42:28,460 All these angular variables. 580 00:42:28,460 --> 00:42:30,400 So it's infinite dimensional. 581 00:42:30,400 --> 00:42:32,810 So it's a little scary. 582 00:42:32,810 --> 00:42:37,130 But let's not worry about that. 583 00:42:37,130 --> 00:42:40,510 Something very nice happens with angular momentum. 584 00:42:40,510 --> 00:42:44,330 Something so nice that it didn't happen actually 585 00:42:44,330 --> 00:42:47,370 with a and a dagger. 586 00:42:47,370 --> 00:42:49,650 With a and a dagger, you build states 587 00:42:49,650 --> 00:42:51,090 in the harmonic oscillator. 588 00:42:51,090 --> 00:42:54,510 And you build infinitely many ones. 589 00:42:54,510 --> 00:42:59,640 The operators x and p, you've learned you cannot represent 590 00:42:59,640 --> 00:43:02,830 them by finite dimensional matrices. 591 00:43:02,830 --> 00:43:07,110 So this is a lot more complicated, you would say. 592 00:43:07,110 --> 00:43:11,080 And you would say, well, this is just much harder. 593 00:43:11,080 --> 00:43:15,480 This algebra is so much harder than this algebra. 594 00:43:18,650 --> 00:43:21,850 Nevertheless, this algebra is the difficult one. 595 00:43:21,850 --> 00:43:25,182 Gives you infinite dimensional representations. 596 00:43:25,182 --> 00:43:28,830 You can keep piling the a daggers. 597 00:43:28,830 --> 00:43:31,670 Here, this is a very dense algebra. 598 00:43:31,670 --> 00:43:36,190 Mathematicians would say this is much simpler than this one. 599 00:43:36,190 --> 00:43:39,940 And we'll see the simplicity of this one, in that you 600 00:43:39,940 --> 00:43:42,820 will manage to get representations 601 00:43:42,820 --> 00:43:45,400 and matrices that are finite dimensional 602 00:43:45,400 --> 00:43:47,550 to work these things out. 603 00:43:47,550 --> 00:43:51,170 So it's going to be nicer in that sense. 604 00:43:51,170 --> 00:43:54,315 So what do we have? 605 00:43:54,315 --> 00:43:58,400 We have to think of our commuting observables 606 00:43:58,400 --> 00:44:02,650 and the set of Hermitian operators that commute. 607 00:44:02,650 --> 00:44:11,450 So we have J squared, and J3-- I call it Jz now, apologies. 608 00:44:11,450 --> 00:44:15,040 And we'll declare that there are states. 609 00:44:15,040 --> 00:44:17,850 These are Hermitian, and they commute. 610 00:44:17,850 --> 00:44:22,040 So they must be diagonalized simultaneously. 611 00:44:22,040 --> 00:44:25,150 And there should exist states that 612 00:44:25,150 --> 00:44:27,320 represent the diagonalization. 613 00:44:27,320 --> 00:44:31,720 In fact, since they commute, and can be diagonalized 614 00:44:31,720 --> 00:44:35,500 simultaneously, the vector space must 615 00:44:35,500 --> 00:44:37,930 break into a list of vectors. 616 00:44:37,930 --> 00:44:41,900 All of them eigenstates of these 2 operators. 617 00:44:41,900 --> 00:44:44,930 And all of them orthogonal to each other. 618 00:44:44,930 --> 00:44:46,775 Matthew, you had a question? 619 00:44:46,775 --> 00:44:48,472 AUDIENCE: I was just wondering when 620 00:44:48,472 --> 00:44:53,080 we showed that Jz is Hermitian? 621 00:44:53,080 --> 00:44:54,660 PROFESSOR: We didn't show it. 622 00:44:54,660 --> 00:44:59,400 We postulated that J's are Hermitian operators. 623 00:44:59,400 --> 00:45:03,550 So you know that when J is L, yes it's Hermitian. 624 00:45:03,550 --> 00:45:08,760 You know when J is spin, yes it's Hermitian. 625 00:45:08,760 --> 00:45:11,625 Whatever you're doing we'll use Hermitian operators. 626 00:45:15,010 --> 00:45:21,040 So not only they can diagonalize simultaneously, 627 00:45:21,040 --> 00:45:25,830 by our main theorem about Hermitian operators, 628 00:45:25,830 --> 00:45:29,600 this should provide an orthonormal basis 629 00:45:29,600 --> 00:45:33,770 for the full vector space. 630 00:45:33,770 --> 00:45:36,390 So the whole answer is supposed to be here. 631 00:45:36,390 --> 00:45:37,380 Let's see. 632 00:45:37,380 --> 00:45:41,970 So I'll define states, Jm, that are 633 00:45:41,970 --> 00:45:44,540 eigenstates of both of these things. 634 00:45:44,540 --> 00:45:48,410 And I have 2 numbers to declare those eigenvalues. 635 00:45:48,410 --> 00:45:52,300 You would say J squared. 636 00:45:52,300 --> 00:45:56,970 Now, any normal person would put here maybe h 637 00:45:56,970 --> 00:45:59,830 squared, for units, time J squared. 638 00:46:02,900 --> 00:46:03,890 And then Jm. 639 00:46:06,680 --> 00:46:09,360 Don't copy it yet. 640 00:46:09,360 --> 00:46:14,070 And Jz for Jm. 641 00:46:14,070 --> 00:46:15,800 It has units of angular momentum. 642 00:46:15,800 --> 00:46:19,650 So an h, times m, times Jm. 643 00:46:22,830 --> 00:46:25,900 But that turns out not to be very 644 00:46:25,900 --> 00:46:29,590 convenient to put the J squared there. 645 00:46:29,590 --> 00:46:31,660 It ruins the algebra later. 646 00:46:31,660 --> 00:46:33,780 So we'll put something different that we 647 00:46:33,780 --> 00:46:35,880 hope has the same effect. 648 00:46:35,880 --> 00:46:38,200 And I will discuss that. 649 00:46:38,200 --> 00:46:44,480 I'll put h squared, J times J plus 1. 650 00:46:44,480 --> 00:46:47,200 It's a funny way of declaring how 651 00:46:47,200 --> 00:46:48,760 you're going to build the states. 652 00:46:48,760 --> 00:46:53,850 But it's a possible thing to do. 653 00:46:53,850 --> 00:46:56,910 So here are the states, J and m. 654 00:46:56,910 --> 00:46:59,660 And the only thing I know at this moment 655 00:46:59,660 --> 00:47:02,430 is that since these are Hermitian operators, 656 00:47:02,430 --> 00:47:04,820 their eigenvalues must be real. 657 00:47:04,820 --> 00:47:07,480 So J times J plus 1 is real. 658 00:47:07,480 --> 00:47:08,890 And m is real. 659 00:47:08,890 --> 00:47:11,470 So J and m belong to the reals. 660 00:47:18,670 --> 00:47:22,030 And they are orthogonal to-- we can 661 00:47:22,030 --> 00:47:25,290 say they're orthonormal states. 662 00:47:25,290 --> 00:47:29,430 We will see very soon that these things get quantized. 663 00:47:29,430 --> 00:47:34,590 But basically, the overlap of a Jm with a J prime, m prime 664 00:47:34,590 --> 00:47:38,710 would be 0 whenever the J's and the m's are different. 665 00:47:38,710 --> 00:47:45,460 As you know from our theory, any 2 eigenstates 666 00:47:45,460 --> 00:47:48,070 with different eigenvalues are orthonormal. 667 00:47:48,070 --> 00:47:51,020 And in fact, you can choose a basis 668 00:47:51,020 --> 00:47:53,670 so that in fact, everything is orthonormal. 669 00:47:53,670 --> 00:47:57,050 So there's no question like that. 670 00:47:57,050 --> 00:48:00,250 So let's explain a little what's happening with this thing. 671 00:48:00,250 --> 00:48:02,880 Why do we put this like that? 672 00:48:02,880 --> 00:48:05,640 Or why can we get away with this? 673 00:48:05,640 --> 00:48:09,590 And the reason is the following. 674 00:48:09,590 --> 00:48:19,470 Let's consider Jm, J squared, Jm. 675 00:48:19,470 --> 00:48:27,080 If I use this, J squared on this is this number. 676 00:48:27,080 --> 00:48:31,980 And Jm with itself will be 1. 677 00:48:31,980 --> 00:48:34,400 And therefore I'll put here h-- I'm sorry. 678 00:48:34,400 --> 00:48:35,950 This should be an h squared. 679 00:48:35,950 --> 00:48:38,610 J has units of angular momentum. 680 00:48:38,610 --> 00:48:44,020 h squared, J times J plus 1. 681 00:48:44,020 --> 00:48:47,930 And I'm assuming that this will be discretized 682 00:48:47,930 --> 00:48:51,680 so I don't have to put the delta function normalization. 683 00:48:51,680 --> 00:48:56,030 At any rate, this thing is equal to this. 684 00:48:56,030 --> 00:48:59,300 And moreover, it's equal to the following. 685 00:48:59,300 --> 00:49:12,990 Jm sum over i, Jm, Ji, Ji, Jm. 686 00:49:12,990 --> 00:49:18,950 But since J is Hermitian, this is nothing but the sum over i 687 00:49:18,950 --> 00:49:25,300 of the norm squared of Ji with J acting on Jm. 688 00:49:28,930 --> 00:49:31,250 The norm squared of this state. 689 00:49:31,250 --> 00:49:34,440 Because this times the bra with Ji Hermitian 690 00:49:34,440 --> 00:49:35,760 is the norm squared. 691 00:49:35,760 --> 00:49:39,030 So this is greater or equal than 0. 692 00:49:39,030 --> 00:49:43,370 Perhaps no surprise, this is a vector operator, 693 00:49:43,370 --> 00:49:47,010 which is the sum of squares of Hermitian operators. 694 00:49:47,010 --> 00:49:49,600 And therefore it should be like that. 695 00:49:49,600 --> 00:50:00,250 Now, given that, we have the following-- oops-- 696 00:50:00,250 --> 00:50:07,200 the following fact that L times L plus-- no. 697 00:50:07,200 --> 00:50:11,270 J times J plus 1 must be greater or equal than 0. 698 00:50:11,270 --> 00:50:16,050 J times J plus 1 must be greater or equal than 0. 699 00:50:16,050 --> 00:50:21,580 Well, plot it as a function of J. It vanishes at 0. 700 00:50:21,580 --> 00:50:26,770 J times J plus 1 vanishes at 0, and vanishes at minus 1. 701 00:50:26,770 --> 00:50:29,440 It's a function like this. 702 00:50:29,440 --> 00:50:32,270 The function J times J plus 1. 703 00:50:32,270 --> 00:50:42,450 And this shows that all you need is this thing to be positive. 704 00:50:42,450 --> 00:50:47,650 So to represent all the states that have J times J plus 1 705 00:50:47,650 --> 00:50:52,680 positive, I could label them with J's that are positive. 706 00:50:52,680 --> 00:50:56,380 Or J's that are smaller than minus 1. 707 00:50:56,380 --> 00:51:00,510 So each way, I can label uniquely those states. 708 00:51:00,510 --> 00:51:03,410 So if I get J times J plus 1 equals 3, 709 00:51:03,410 --> 00:51:05,930 it may correspond to a J of something 710 00:51:05,930 --> 00:51:08,720 and a J of some other thing. 711 00:51:08,720 --> 00:51:13,250 I will have just 1 state, so I will choose J positive. 712 00:51:13,250 --> 00:51:17,080 So given that J times J plus 1 is positive, 713 00:51:17,080 --> 00:51:28,320 I can label states with J positive, or 0. 714 00:51:28,320 --> 00:51:32,720 So it allows you to do this. 715 00:51:32,720 --> 00:51:37,900 Whatever value of this quantity that is positive 716 00:51:37,900 --> 00:51:41,540 corresponds to some J positive that you can put in here. 717 00:51:41,540 --> 00:51:42,990 A unique J positive. 718 00:51:42,990 --> 00:51:48,450 So this is a fine parametrization of the problem. 719 00:51:48,450 --> 00:51:49,460 OK. 720 00:51:49,460 --> 00:51:54,255 Now what's next? 721 00:51:56,925 --> 00:52:00,840 Next, we have to understand what the J plus operators and J 722 00:52:00,840 --> 00:52:04,360 minus operators do to the states. 723 00:52:04,360 --> 00:52:15,790 So, first thing is that J plus and J minus commute with J 724 00:52:15,790 --> 00:52:16,290 squared. 725 00:52:19,400 --> 00:52:21,970 That should not be a surprise. 726 00:52:21,970 --> 00:52:24,160 J1 and J2 commute. 727 00:52:24,160 --> 00:52:26,660 Every J commutes with J squared. 728 00:52:26,660 --> 00:52:29,970 So J plus and J minus commute with J squared. 729 00:52:29,970 --> 00:52:36,540 What this means in words is that J plus and J minus 730 00:52:36,540 --> 00:52:42,220 do not change the eigenvalue of J squared on a state. 731 00:52:42,220 --> 00:52:49,330 That is, if I would have J squared on J plus or minus 732 00:52:49,330 --> 00:53:00,270 on Jm-- since I can move the J squared up across the J plus, 733 00:53:00,270 --> 00:53:02,220 minus-- it hits here. 734 00:53:02,220 --> 00:53:07,690 Then I have J plus minus, J squared, Jm. 735 00:53:07,690 --> 00:53:12,330 And that's there for h squared, J times J plus 1, 736 00:53:12,330 --> 00:53:15,080 times J plus minus on Jm. 737 00:53:18,020 --> 00:53:22,590 So this state is also a state with the same value 738 00:53:22,590 --> 00:53:25,040 of J squared. 739 00:53:25,040 --> 00:53:39,410 Therefore, it must have the same value of J. In other words, 740 00:53:39,410 --> 00:53:46,670 this state J plus minus of Jm must be proportional 741 00:53:46,670 --> 00:53:50,350 to a state with J and maybe some different value of m, 742 00:53:50,350 --> 00:53:52,390 but the same value of J. 743 00:53:52,390 --> 00:53:53,860 J cannot have changed. 744 00:53:56,380 --> 00:53:57,570 J must be the same. 745 00:54:01,950 --> 00:54:10,280 Then we have to see who changes m, 746 00:54:10,280 --> 00:54:14,220 or how does J plus minus changes m. 747 00:54:14,220 --> 00:54:17,770 So here comes a little bit of a same calculation. 748 00:54:17,770 --> 00:54:21,900 You want to see what is the m value of this thing. 749 00:54:21,900 --> 00:54:26,850 So you have J plus minus on Jm. 750 00:54:26,850 --> 00:54:32,630 And you act with it with a Jz, to see what it is. 751 00:54:32,630 --> 00:54:38,030 And then, you put, well, the commutator first. 752 00:54:38,030 --> 00:54:46,830 Jz, J plus minus, plus J plus minus, Jz on the state. 753 00:54:46,830 --> 00:54:49,370 The commutator, you've calculated it 754 00:54:49,370 --> 00:54:53,770 before, was Jz with J plus minus is there, 755 00:54:53,770 --> 00:55:01,490 is plus minus h bar, J plus minus. 756 00:55:01,490 --> 00:55:04,360 And this Jz already act. 757 00:55:04,360 --> 00:55:16,900 So this is plus h bar m, J plus minus on Jm. 758 00:55:16,900 --> 00:55:20,350 So we can get the J plus minus out. 759 00:55:20,350 --> 00:55:32,180 And this h bar m plus minus 1, j plus minus, Jm. 760 00:55:32,180 --> 00:55:34,220 So look what you got. 761 00:55:34,220 --> 00:55:43,940 Jz acting on this state is h bar, m plus minus 1, Jm. 762 00:55:43,940 --> 00:55:51,520 So this state has m equal to either m plus 1, or m minus 1. 763 00:55:51,520 --> 00:55:52,875 Something that we can write. 764 00:55:56,170 --> 00:56:02,510 Clearly-- oops-- in this way, we'll 765 00:56:02,510 --> 00:56:08,370 say that J plus minus, Jm-- we know already 766 00:56:08,370 --> 00:56:13,150 it's a state with J and m plus minus 1. 767 00:56:13,150 --> 00:56:14,970 So it raises m. 768 00:56:14,970 --> 00:56:19,630 Just like what we said that the a's and a daggers 769 00:56:19,630 --> 00:56:21,860 raise or lower the number. 770 00:56:21,860 --> 00:56:27,420 J plus and J minus raise and lower Jz. 771 00:56:27,420 --> 00:56:30,940 Therefore, it's this is proportional to this state. 772 00:56:30,940 --> 00:56:33,430 But there's a constant of proportionality 773 00:56:33,430 --> 00:56:36,080 that we have to figure out. 774 00:56:36,080 --> 00:56:41,140 And we'll call it the constant C, Jm. 775 00:56:41,140 --> 00:56:41,890 To be calculated. 776 00:56:51,000 --> 00:56:55,350 So the way to calculate this constant-- and that 777 00:56:55,350 --> 00:56:59,870 will bring us almost pretty close to what we need-- 778 00:56:59,870 --> 00:57:03,130 is to take inner products. 779 00:57:03,130 --> 00:57:07,980 So we must take the dagger of this equation. 780 00:57:07,980 --> 00:57:16,544 So take the dagger, and you get Jm, the adjoin, J minus plus. 781 00:57:16,544 --> 00:57:19,280 And hit it with this equation. 782 00:57:22,690 --> 00:57:26,600 So you'll have here-- well maybe I'll write it. 783 00:57:26,600 --> 00:57:28,380 The dagger of this equation would 784 00:57:28,380 --> 00:57:34,239 be C plus minus star of Jm. 785 00:57:34,239 --> 00:57:37,545 Jm plus minus 1. 786 00:57:41,490 --> 00:57:46,390 And now, sandwich this with that. 787 00:57:46,390 --> 00:57:54,400 So you have Jm, J minus plus, J plus minus, 788 00:57:54,400 --> 00:58:05,310 Jm equals to norm of C plus minus Jm. 789 00:58:05,310 --> 00:58:08,900 And then you have this state times this state, but that's 1. 790 00:58:08,900 --> 00:58:11,910 Because it's J, J, m plus 1, m plus 1. 791 00:58:11,910 --> 00:58:13,910 So this is an orthonormal basis. 792 00:58:13,910 --> 00:58:15,430 So we have just 1. 793 00:58:15,430 --> 00:58:16,795 And I don't have to write more. 794 00:58:19,470 --> 00:58:24,770 Well the left hand side can be calculated. 795 00:58:24,770 --> 00:58:26,925 We have still that formula here. 796 00:58:30,110 --> 00:58:32,480 So let's calculate it. 797 00:58:37,210 --> 00:58:43,070 The left hand side, I'll write it like this. 798 00:58:43,070 --> 00:58:50,140 I will have C plus minus, Jm squared, 799 00:58:50,140 --> 00:58:56,500 which is equal to the norm squared of J plus minus, Jm. 800 00:58:59,960 --> 00:59:01,196 It's equal to what? 801 00:59:01,196 --> 00:59:04,300 Whatever this is, where you substitute 802 00:59:04,300 --> 00:59:07,300 that for this formula. 803 00:59:07,300 --> 00:59:10,270 So you'll put here Jm. 804 00:59:10,270 --> 00:59:17,470 And you'll have-- well, I want actually 805 00:59:17,470 --> 00:59:20,040 the formula I just erased. 806 00:59:20,040 --> 00:59:25,300 Because I actually would prefer to have J squared. 807 00:59:25,300 --> 00:59:34,020 So I would have this is equal to J squared, minus J3 squared, 808 00:59:34,020 --> 00:59:39,030 plus minus h, J3. 809 00:59:39,030 --> 00:59:40,350 So let's see. 810 00:59:40,350 --> 00:59:42,820 I have the sign minus plus, plus minus. 811 00:59:42,820 --> 00:59:45,560 So I should change the signs there. 812 00:59:45,560 --> 00:59:58,490 So it should be J squared, minus J3 squared, minus plus J3, 813 00:59:58,490 --> 01:00:05,010 and Jm, minus plus h bar, J3, Jm. 814 01:00:05,010 --> 01:00:11,540 So this is equal to h bar squared, J times J plus 1, 815 01:00:11,540 --> 01:00:18,000 minus an m squared, and a minus plus. 816 01:00:18,000 --> 01:00:21,635 So minus, plus, minus here. 817 01:00:26,380 --> 01:00:29,000 I think I have it here correct. 818 01:00:29,000 --> 01:00:31,680 Plus minus 1. 819 01:00:31,680 --> 01:00:34,060 And that's it. 820 01:00:34,060 --> 01:00:37,280 J squared is h squared this. 821 01:00:37,280 --> 01:00:39,430 J3 squared would give that. 822 01:00:39,430 --> 01:00:43,830 And the minus plus here is correctly with this one. 823 01:00:43,830 --> 01:00:46,080 So m should be here. 824 01:00:46,080 --> 01:00:49,420 Plus minus m. 825 01:00:49,420 --> 01:00:55,410 So this is h squared, J times J plus 1, 826 01:00:55,410 --> 01:00:59,445 minus m, times m plus minus 1. 827 01:01:05,611 --> 01:01:06,110 OK. 828 01:01:10,980 --> 01:01:15,980 So the C's have been already found. 829 01:01:15,980 --> 01:01:21,730 And you can take their square roots. 830 01:01:21,730 --> 01:01:26,400 In fact, we can ideally just take the square roots, 831 01:01:26,400 --> 01:01:31,150 because these things better be positive numbers 832 01:01:31,150 --> 01:01:32,990 because they're norms squared. 833 01:01:32,990 --> 01:01:36,270 So whenever we'll be able to do this, 834 01:01:36,270 --> 01:01:39,360 these things better be positive, being 835 01:01:39,360 --> 01:01:41,270 the square of some states. 836 01:01:41,270 --> 01:01:47,940 And therefore the C plus minus is-- C plus minus of Jm 837 01:01:47,940 --> 01:01:53,980 can be simply taken to be h bar, square root of J 838 01:01:53,980 --> 01:02:01,120 times J plus 1, minus m, times m plus 1. 839 01:02:01,120 --> 01:02:05,290 And it's because of this thing, this m times 840 01:02:05,290 --> 01:02:11,980 m plus 1, that it was convenient to have J times J plus 1. 841 01:02:11,980 --> 01:02:16,040 So that we can compare J's and m's better. 842 01:02:16,040 --> 01:02:20,310 Otherwise it would have been pretty disastrous. 843 01:02:20,310 --> 01:02:26,390 So, OK, we're almost done now with the calculation 844 01:02:26,390 --> 01:02:27,330 of the spectrum. 845 01:02:27,330 --> 01:02:32,110 You will say, well, we seem to be getting no where. 846 01:02:32,110 --> 01:02:34,600 Learned all these properties, these states, 847 01:02:34,600 --> 01:02:37,150 and now you're just manipulating the states. 848 01:02:37,150 --> 01:02:39,180 But the main thing is that we need 849 01:02:39,180 --> 01:02:42,830 these things to be positive. 850 01:02:42,830 --> 01:02:45,070 And that will give us the whole condition. 851 01:02:45,070 --> 01:02:50,220 So, for example, we need 1, that the states J 852 01:02:50,220 --> 01:02:56,750 plus, Jm, their norm squareds be positive. 853 01:02:56,750 --> 01:03:02,360 So for the plus sign-- so you should have J times J plus 1, 854 01:03:02,360 --> 01:03:07,480 minus m, times m plus 1 be positive. 855 01:03:07,480 --> 01:03:18,380 Or m times m plus 1 be smaller then J times J plus 1. 856 01:03:21,150 --> 01:03:26,960 The best way for my mind to solve these kind of things 857 01:03:26,960 --> 01:03:29,880 is to just plot them. 858 01:03:29,880 --> 01:03:34,820 So here is m. 859 01:03:37,520 --> 01:03:40,025 And here is m times m plus 1. 860 01:03:42,730 --> 01:03:45,010 So you plot this function. 861 01:03:45,010 --> 01:03:53,810 And you want it to be less than some value of J times J plus 1. 862 01:03:53,810 --> 01:03:57,050 So here's J times J plus 1, some value. 863 01:03:57,050 --> 01:04:00,150 So this is 0 here. 864 01:04:00,150 --> 01:04:02,400 This function is 0 at minus 1. 865 01:04:02,400 --> 01:04:07,260 So it will be something like this. 866 01:04:07,260 --> 01:04:15,400 And there's 2 values at which m becomes equal to this thing. 867 01:04:15,400 --> 01:04:22,670 And one is clearly J. When m is equal to J, 868 01:04:22,670 --> 01:04:25,390 it's saturates an inequality. 869 01:04:25,390 --> 01:04:28,755 And the other one is minus J, minus 1. 870 01:04:31,950 --> 01:04:34,910 If m is minus J, minus 1, you will 871 01:04:34,910 --> 01:04:38,280 have minus J, minus 1 here, and minus J 872 01:04:38,280 --> 01:04:41,920 here, which would be equal to this. 873 01:04:41,920 --> 01:04:49,350 So, in order for these states to be good, the value of m 874 01:04:49,350 --> 01:04:56,130 must be in between J and minus J, minus 1. 875 01:04:59,270 --> 01:05:11,280 Then the other case is that J plus on-- J minus on Jm. 876 01:05:11,280 --> 01:05:13,760 If you produce those states, they also 877 01:05:13,760 --> 01:05:15,380 must have positive norms. 878 01:05:15,380 --> 01:05:22,260 So J times J plus 1, minus m, times m minus 1 this time, 879 01:05:22,260 --> 01:05:23,940 must be greater than 0. 880 01:05:23,940 --> 01:05:28,900 So m times m minus 1 must be less than 881 01:05:28,900 --> 01:05:33,990 or equal then J times J plus 1. 882 01:05:33,990 --> 01:05:39,550 And again, we try to do it geometrically. 883 01:05:39,550 --> 01:05:43,420 So here it is. 884 01:05:43,420 --> 01:05:45,880 Here is m. 885 01:05:45,880 --> 01:05:48,280 And what values do you have? 886 01:05:48,280 --> 01:05:52,610 Well, if you plot here m times m minus 1. 887 01:05:56,260 --> 01:05:58,410 And that should be equal to some value 888 01:05:58,410 --> 01:06:02,845 that you get fixed, which is the value J times J plus 1. 889 01:06:06,370 --> 01:06:10,630 So you think in terms of m's, how far can they go? 890 01:06:10,630 --> 01:06:15,900 So if you take m equals J plus 1 that hits it. 891 01:06:15,900 --> 01:06:18,890 So this is 0 here, at 1, at 0. 892 01:06:18,890 --> 01:06:21,930 So it's some function like this. 893 01:06:21,930 --> 01:06:26,820 And here you have J plus 1. 894 01:06:26,820 --> 01:06:30,800 And here you have minus J. Both are 895 01:06:30,800 --> 01:06:36,570 the places for m equal J plus 1, and minus J 896 01:06:36,570 --> 01:06:41,230 that you get the states. 897 01:06:41,230 --> 01:06:42,630 You get the saturation. 898 01:06:42,630 --> 01:06:45,670 So you can run m from this range. 899 01:06:45,670 --> 01:06:51,210 Now, m can go less than or equal to J plus 1, 900 01:06:51,210 --> 01:06:57,000 and greater than or equal to minus J. 901 01:06:57,000 --> 01:07:00,520 But these 2 inequalities must hold at the same time. 902 01:07:00,520 --> 01:07:05,780 You cannot allow either one to go wrong for any set of states. 903 01:07:05,780 --> 01:07:13,120 So if both must hold at the same time for any state, 904 01:07:13,120 --> 01:07:18,620 because both things have to happen, you get constrained. 905 01:07:18,620 --> 01:07:24,900 This time for the upper range, this is the stronger value. 906 01:07:24,900 --> 01:07:28,220 For the lower range, this is the stronger value. 907 01:07:28,220 --> 01:07:37,085 So m must go between J and minus J for both to hold. 908 01:07:40,470 --> 01:07:41,840 Oops. 909 01:07:41,840 --> 01:07:42,340 To hold. 910 01:07:46,950 --> 01:07:49,450 Now look what happens. 911 01:07:49,450 --> 01:07:54,890 Funny things happen if-- this is reasonable 912 01:07:54,890 --> 01:07:58,410 that the strongest value comes from this equation. 913 01:07:58,410 --> 01:08:01,800 Because J plus increases m. 914 01:08:01,800 --> 01:08:04,530 So at some point you run into trouble 915 01:08:04,530 --> 01:08:06,680 if you increase m too much. 916 01:08:06,680 --> 01:08:08,370 How much can you increase it? 917 01:08:08,370 --> 01:08:12,610 You cannot go beyond J, and that makes sense. 918 01:08:12,610 --> 01:08:14,250 In some sense, your intuition should 919 01:08:14,250 --> 01:08:18,330 be that J is the length of J squared. 920 01:08:18,330 --> 01:08:20,899 And m is mz. 921 01:08:20,899 --> 01:08:26,810 So m should not go beyond J. And that's reasonable here. 922 01:08:26,810 --> 01:08:32,930 And in fact, when m is equal to J, this whole thing vanishes. 923 01:08:32,930 --> 01:08:37,350 So if you reach that state when m is equal to J, 924 01:08:37,350 --> 01:08:42,920 only then for m equal to J, or for this state, you get 0. 925 01:08:42,920 --> 01:08:45,790 So you cannot raise the state anymore. 926 01:08:45,790 --> 01:08:55,700 So actually, you see if you choose some J over here, 927 01:08:55,700 --> 01:08:58,390 we need a few things to happen. 928 01:08:58,390 --> 01:09:03,620 You choose some J, and some m. 929 01:09:03,620 --> 01:09:07,460 Well you're going to be shifting the m's. 930 01:09:07,460 --> 01:09:12,830 And if you keep adding J pluses, eventually you 931 01:09:12,830 --> 01:09:15,500 will go beyond this point. 932 01:09:15,500 --> 01:09:18,899 The only way not to go beyond this point 933 01:09:18,899 --> 01:09:23,840 is if m reaches the value J. Because if m reaches the value 934 01:09:23,840 --> 01:09:27,029 J, the state is killed. 935 01:09:27,029 --> 01:09:32,229 So m should reach the value J over here at some stage. 936 01:09:32,229 --> 01:09:36,819 So you fix J, and you try to think what m can be. 937 01:09:36,819 --> 01:09:40,069 And m has to reach the value J. So 938 01:09:40,069 --> 01:09:43,939 m at some point, whatever m is, you add 1. 939 01:09:43,939 --> 01:09:44,689 You add 1. 940 01:09:44,689 --> 01:09:45,240 You add 1. 941 01:09:45,240 --> 01:09:50,260 And eventually you must reach the value J. Reach 942 01:09:50,260 --> 01:09:52,370 with some m prime. 943 01:09:52,370 --> 01:09:55,150 m here. 944 01:09:55,150 --> 01:09:57,530 You should reach the value J, so that you 945 01:09:57,530 --> 01:10:00,110 don't produce another state that is higher. 946 01:10:00,110 --> 01:10:05,480 If you reach something before that, that state is not killed. 947 01:10:05,480 --> 01:10:07,720 This number is not equal to 0. 948 01:10:07,720 --> 01:10:12,130 You produce a state and it's a bad state of bad norm. 949 01:10:12,130 --> 01:10:14,080 So you must reach this one. 950 01:10:14,080 --> 01:10:16,826 On the other hand, you can lower things. 951 01:10:16,826 --> 01:10:22,320 And if you go below minus J, you produce bad states. 952 01:10:22,320 --> 01:10:25,700 So you must also, when you decrease m, 953 01:10:25,700 --> 01:10:27,370 you must reach this point. 954 01:10:27,370 --> 01:10:31,700 Because if you didn't, and you stop half a unit away from it, 955 01:10:31,700 --> 01:10:34,570 the next state that you produce is bad. 956 01:10:34,570 --> 01:10:36,190 And that can't be. 957 01:10:36,190 --> 01:10:40,900 So you must reach this one too. 958 01:10:40,900 --> 01:10:44,120 And that's the key logical part of the argument 959 01:10:44,120 --> 01:10:57,881 in which this distance 2J plus 1-- no. 960 01:10:57,881 --> 01:10:58,380 I'm sorry. 961 01:10:58,380 --> 01:11:03,490 This 2J must be equal to some integer. 962 01:11:09,510 --> 01:11:11,880 And that's the key thing that must happen, 963 01:11:11,880 --> 01:11:15,200 because you must reach this and you must reach here. 964 01:11:15,200 --> 01:11:18,190 And m just varies by integers. 965 01:11:18,190 --> 01:11:21,630 So the distance between this J and minus J 966 01:11:21,630 --> 01:11:23,750 must be twice an integer. 967 01:11:23,750 --> 01:11:28,110 And you've discovered something remarkable by getting 968 01:11:28,110 --> 01:11:30,980 to that point, because now you see 969 01:11:30,980 --> 01:11:33,230 that if this has to be an integer, well 970 01:11:33,230 --> 01:11:37,230 it may be 0, 1, 2, 3. 971 01:11:37,230 --> 01:11:41,440 And when J-- then J-- this integer is equal to 0, 972 01:11:41,440 --> 01:11:43,910 then J is equal to 0. 973 01:11:43,910 --> 01:11:46,760 1/2, 1, 3/2. 974 01:11:46,760 --> 01:11:51,620 And you get all these spins with-- consider particles 975 01:11:51,620 --> 01:11:53,790 without spin having spin 0. 976 01:11:53,790 --> 01:11:55,720 Particles with spin 1/2. 977 01:11:55,720 --> 01:12:00,110 Particles of spin 1, or angular momentum 1, 978 01:12:00,110 --> 01:12:03,690 orbital angular momentum 1. 979 01:12:03,690 --> 01:12:07,630 And both these things have a reason for you. 980 01:12:07,630 --> 01:12:13,550 Now if you have 2J being an integer, the values of m 981 01:12:13,550 --> 01:12:19,310 go from J to J minus 1, up to minus J. 982 01:12:19,310 --> 01:12:21,730 And there are two J plus 1 values. 983 01:12:27,110 --> 01:12:29,630 And in fact, that is the main result 984 01:12:29,630 --> 01:12:33,080 of the theory of angular momentum. 985 01:12:33,080 --> 01:12:37,500 The values of the angular momentum are 0, 1, 1/2, 3/2. 986 01:12:37,500 --> 01:12:42,500 So for J equals 0, there's just one state. 987 01:12:42,500 --> 01:12:46,090 m is equal to 0. 988 01:12:46,090 --> 01:12:49,750 For J equals to 1, there's two states. 989 01:12:49,750 --> 01:12:52,710 I'm sorry for 1/2, two states. 990 01:12:52,710 --> 01:12:55,490 One with m equals 1/2. 991 01:12:55,490 --> 01:12:58,990 And m equals minus 1/2. 992 01:12:58,990 --> 01:13:01,890 J equals 1, there's three states. 993 01:13:01,890 --> 01:13:05,570 M equals 1, 0, and minus 1. 994 01:13:05,570 --> 01:13:06,995 And so on. 995 01:13:10,320 --> 01:13:12,440 OK. 996 01:13:12,440 --> 01:13:13,880 This is a great result. 997 01:13:13,880 --> 01:13:19,360 Let me give you an application in the last 10 minutes. 998 01:13:19,360 --> 01:13:22,690 It's a remarkable application. 999 01:13:22,690 --> 01:13:27,260 Now actually, you would say, so what 1000 01:13:27,260 --> 01:13:31,740 do you get-- what vector space were we talking about? 1001 01:13:31,740 --> 01:13:34,770 And what's sort of the punchline here 1002 01:13:34,770 --> 01:13:38,802 is that the vector space was infinite dimensional 1003 01:13:38,802 --> 01:13:45,050 and it breaks down into states with J equals 0. 1004 01:13:45,050 --> 01:13:47,120 States was J equal 1/2. 1005 01:13:47,120 --> 01:13:48,660 States with J equal 1. 1006 01:13:48,660 --> 01:13:51,580 States with J equal 3/2. 1007 01:13:51,580 --> 01:13:53,650 All these things are possibilities. 1008 01:13:53,650 --> 01:13:56,750 They can all be present in your vector space. 1009 01:13:56,750 --> 01:13:58,420 Maybe some are present. 1010 01:13:58,420 --> 01:13:59,560 Some are not. 1011 01:13:59,560 --> 01:14:03,560 That is part of figuring out what's going on. 1012 01:14:03,560 --> 01:14:08,980 When we do central potentials, 0, 1, 2, 4 1013 01:14:08,980 --> 01:14:13,270 will be present for the angular momentum theory. 1014 01:14:13,270 --> 01:14:16,430 When we do spins, we have 1/2. 1015 01:14:16,430 --> 01:14:20,260 And when we do other things, we can get some funny things 1016 01:14:20,260 --> 01:14:21,550 as well. 1017 01:14:21,550 --> 01:14:25,170 So let's do a case where you get something funny. 1018 01:14:25,170 --> 01:14:27,390 So the 2D, SHO. 1019 01:14:31,240 --> 01:14:41,790 You have ax's, and ay's, and a daggers, and ay daggers. 1020 01:14:41,790 --> 01:14:45,100 And this should seem very strange. 1021 01:14:45,100 --> 01:14:48,710 What are we talking about 2 dimensional oscillators 1022 01:14:48,710 --> 01:14:51,410 after talking about 3 dimensional angular 1023 01:14:51,410 --> 01:14:53,730 momentum and all that? 1024 01:14:53,730 --> 01:14:55,390 Doesn't make any sense. 1025 01:14:55,390 --> 01:15:00,160 Well, what's going to happen now is something more magical 1026 01:15:00,160 --> 01:15:04,090 than when a magician takes a bunny out of a hat. 1027 01:15:04,090 --> 01:15:08,780 Out of this problem, an angular momentum, a 3 dimensional 1028 01:15:08,780 --> 01:15:12,060 angular momentum, is going to pop out. 1029 01:15:12,060 --> 01:15:17,210 No reason whatsoever there should be there at first sight. 1030 01:15:17,210 --> 01:15:18,210 But it's there. 1031 01:15:18,210 --> 01:15:21,140 And it's an abstract angular momentum, 1032 01:15:21,140 --> 01:15:23,760 but it's a full angular momentum. 1033 01:15:23,760 --> 01:15:24,570 Let's see. 1034 01:15:29,450 --> 01:15:31,670 Let's look at the spectrum. 1035 01:15:31,670 --> 01:15:32,870 Ground state. 1036 01:15:32,870 --> 01:15:36,420 First excited state is isotropic. 1037 01:15:36,420 --> 01:15:43,010 So 2 states degenerate in energy. 1038 01:15:43,010 --> 01:15:44,080 Next state. 1039 01:15:44,080 --> 01:15:48,110 ax dagger, ax dagger. 1040 01:15:48,110 --> 01:15:51,210 ax, ay. 1041 01:15:51,210 --> 01:15:53,924 ay, ay. 1042 01:15:53,924 --> 01:15:57,620 3 states, degenerate. 1043 01:15:57,620 --> 01:16:04,780 Go up to ax dagger to the n, up to ax-- 1044 01:16:04,780 --> 01:16:07,570 no ax, or ax dagger to the 0. 1045 01:16:07,570 --> 01:16:10,075 And ay dagger to the n. 1046 01:16:12,880 --> 01:16:18,880 And that's n a daggers up to 0 a daggers, so n plus 1 states. 1047 01:16:22,150 --> 01:16:27,440 3 states, 2 states, 1 state. 1048 01:16:27,440 --> 01:16:32,650 And you'll come here and say, that's strange. 1049 01:16:32,650 --> 01:16:37,042 1 state, 2 states, 3 states, 4 states. 1050 01:16:37,042 --> 01:16:41,790 Does that have anything to do with it? 1051 01:16:41,790 --> 01:16:44,370 Well, the surprise is it has something to do with it. 1052 01:16:44,370 --> 01:16:45,900 Let's think about it. 1053 01:16:49,290 --> 01:16:56,470 Well, first thing is to put these aR's and aL oscillators-- 1054 01:16:56,470 --> 01:17:02,650 these were 1/2, 1 over square root of 2, ax plus iay. 1055 01:17:05,330 --> 01:17:12,960 And a left was 1 over square root of 2, ax minus iay. 1056 01:17:12,960 --> 01:17:16,940 I may have-- no, the signs are wrong. 1057 01:17:16,940 --> 01:17:20,500 Plus and minus. 1058 01:17:20,500 --> 01:17:22,730 And we had number operators. 1059 01:17:22,730 --> 01:17:26,960 n right, which were a right dagger, a right. 1060 01:17:26,960 --> 01:17:33,798 And n left, which was a left dagger, a left. 1061 01:17:33,798 --> 01:17:37,350 And they don't mix a lefts and a rights. 1062 01:17:37,350 --> 01:17:42,890 And now, we could build a state the following way. 1063 01:17:42,890 --> 01:17:44,720 0. 1064 01:17:44,720 --> 01:17:50,900 a right dagger on 0. a left dagger on 0. 1065 01:17:50,900 --> 01:17:55,580 A right dagger squared on 0. 1066 01:17:55,580 --> 01:17:59,730 a right, a left on 0. 1067 01:17:59,730 --> 01:18:04,764 and a left dagger, a left dagger on 0. 1068 01:18:04,764 --> 01:18:10,060 Up to a right dagger to the n on 0. 1069 01:18:10,060 --> 01:18:18,840 Up to a left dagger to the n on 0. 1070 01:18:18,840 --> 01:18:21,820 And this is completely analogous to what we had. 1071 01:18:24,990 --> 01:18:30,540 Now here comes the real thing. 1072 01:18:30,540 --> 01:18:35,440 You did compute the angular momentum in the z direction. 1073 01:18:41,140 --> 01:18:45,360 And the angular momentum in the z direction was Lz. 1074 01:18:45,360 --> 01:18:50,330 And you could compute this. xpy minus ypx. 1075 01:18:50,330 --> 01:18:52,670 And this was all legal. 1076 01:18:52,670 --> 01:19:00,763 And the answer was h bar, N right, minus NL. 1077 01:19:04,630 --> 01:19:09,320 That was the Lz component of angular momentum. 1078 01:19:09,320 --> 01:19:15,730 So, let's see what Lz's those states have. 1079 01:19:15,730 --> 01:19:22,660 This one has no n rights, or n lefts, so has Lz equals 0. 1080 01:19:22,660 --> 01:19:28,270 This state has Nz equal h bar. 1081 01:19:28,270 --> 01:19:32,580 And this has minus h bar. 1082 01:19:32,580 --> 01:19:33,280 OK. 1083 01:19:33,280 --> 01:19:35,450 h bar and minus h bar. 1084 01:19:35,450 --> 01:19:39,330 That doesn't quite seem to fit here, 1085 01:19:39,330 --> 01:19:42,750 because the z component of angular momentum 1086 01:19:42,750 --> 01:19:46,060 is 1/2 of h bar, and minus 1/2 of h bar. 1087 01:19:46,060 --> 01:19:48,950 That's-- something went wrong. 1088 01:19:48,950 --> 01:19:50,770 OK. 1089 01:19:50,770 --> 01:19:51,810 You go here. 1090 01:19:51,810 --> 01:19:54,796 You say, well, what is Lz? 1091 01:19:54,796 --> 01:19:59,990 Lz here was h bar, minus h bar. 1092 01:19:59,990 --> 01:20:06,140 Here is 2h bar, 0, and minus 2h bar. 1093 01:20:06,140 --> 01:20:11,210 And you look there, and say, no, that's not quite right either. 1094 01:20:11,210 --> 01:20:14,990 This-- if you would say these 3 states 1095 01:20:14,990 --> 01:20:18,350 should correspond to angular momentum, 1096 01:20:18,350 --> 01:20:24,160 they should have m equal plus 1, plus h bar, 0, and minus h bar. 1097 01:20:24,160 --> 01:20:25,490 So it's not right. 1098 01:20:28,680 --> 01:20:29,400 OK. 1099 01:20:29,400 --> 01:20:35,180 Well one other thing maybe we can make sense of this. 1100 01:20:35,180 --> 01:20:41,640 If we had L plus, should be the kind of thing 1101 01:20:41,640 --> 01:20:44,640 that you can't annihilate. 1102 01:20:44,640 --> 01:20:46,670 That you annihilate the top state. 1103 01:20:46,670 --> 01:20:51,840 Remember L plus, or J plus, kept increasing 1104 01:20:51,840 --> 01:20:54,885 so it should annihilate the top state. 1105 01:20:54,885 --> 01:20:57,510 And I could try to devise something 1106 01:20:57,510 --> 01:20:59,780 that annihilates the top state. 1107 01:20:59,780 --> 01:21:04,860 And it would be something like aR dagger, a left. 1108 01:21:04,860 --> 01:21:05,980 Why? 1109 01:21:05,980 --> 01:21:12,460 Because if aR dagger, a left, goes to the top state, 1110 01:21:12,460 --> 01:21:15,730 the top state has no a left daggers, 1111 01:21:15,730 --> 01:21:20,330 so the a left just zooms in, and hits the 0 and kills it. 1112 01:21:20,330 --> 01:21:21,450 Kills it here. 1113 01:21:21,450 --> 01:21:27,020 So actually I do have something like an L plus. 1114 01:21:27,020 --> 01:21:30,080 And I would have the dagger-- would be something 1115 01:21:30,080 --> 01:21:34,730 like an L minus-- would be aL dagger, a right. 1116 01:21:34,730 --> 01:21:37,630 And this one should annihilate the bottom one. 1117 01:21:37,630 --> 01:21:38,820 And it does. 1118 01:21:38,820 --> 01:21:41,692 Because the bottom state has no aR's, 1119 01:21:41,692 --> 01:21:45,160 and therefore has no aR daggers. 1120 01:21:45,160 --> 01:21:48,350 And therefore, the aR comes there, and hits the state, 1121 01:21:48,350 --> 01:21:49,820 and kills it. 1122 01:21:49,820 --> 01:21:52,930 So we seem to have more or less everything, 1123 01:21:52,930 --> 01:21:56,370 but nothing is working. 1124 01:21:56,370 --> 01:21:59,500 So we have to do a last conceptual step. 1125 01:21:59,500 --> 01:22:05,150 And say-- you see, this is moving in a plane. 1126 01:22:05,150 --> 01:22:08,600 There's no 3 dimensional angular momentum. 1127 01:22:08,600 --> 01:22:12,470 You are fooling yourself with this. 1128 01:22:12,470 --> 01:22:17,730 But what could exist is an abstract angular momentum. 1129 01:22:17,730 --> 01:22:22,350 And for that, in order to-- it's time 1130 01:22:22,350 --> 01:22:26,920 to change the letter from L to J. 1131 01:22:26,920 --> 01:22:30,770 That means some kind of abstract angular momentum. 1132 01:22:30,770 --> 01:22:36,020 And I'll put a 1/2 here, now a definition. 1133 01:22:36,020 --> 01:22:41,670 If this is what I called Jz, oh well, then thing's may 1134 01:22:41,670 --> 01:22:42,800 look good. 1135 01:22:42,800 --> 01:22:52,570 Because this one for Jz has now angular momentum 1/2 of h bar, 1136 01:22:52,570 --> 01:22:55,480 and minus a half of h bar. 1137 01:22:55,480 --> 01:23:01,940 And that fits with this, these 2 states. 1138 01:23:01,940 --> 01:23:07,180 And with the 1/2, the other ones, the Jz's, also 1139 01:23:07,180 --> 01:23:09,310 have something here. 1140 01:23:09,310 --> 01:23:15,220 So Jz here now becomes h bar, minus h bar, 1141 01:23:15,220 --> 01:23:16,225 and it looks right. 1142 01:23:19,030 --> 01:23:22,880 And now you put the 1/2 here, and in fact, 1143 01:23:22,880 --> 01:23:26,360 if you tried to make these things 1144 01:23:26,360 --> 01:23:29,230 J-- call it J plus and J minus. 1145 01:23:29,230 --> 01:23:32,920 Now you put a number here, and a number here. 1146 01:23:32,920 --> 01:23:36,160 If you would have put a number here, 1147 01:23:36,160 --> 01:23:39,470 if you try to enforce that the algebra be 1148 01:23:39,470 --> 01:23:42,850 the algebra of angular momentum, the number 1149 01:23:42,850 --> 01:23:45,380 would have come out to be 1/2. 1150 01:23:45,380 --> 01:23:50,040 But now we claim that in this 2 dimensional oscillator, 1151 01:23:50,040 --> 01:23:52,660 there is-- because there's a number here 1152 01:23:52,660 --> 01:23:54,370 that works with this 1/2. 1153 01:23:54,370 --> 01:23:56,900 Something you have to calculate. 1154 01:23:56,900 --> 01:24:03,570 And with this number, you have some sort of Jx, Jy, Jz, 1155 01:24:03,570 --> 01:24:07,260 where this is like 1/2 of Lz. 1156 01:24:07,260 --> 01:24:11,310 And those have come out of thin air. 1157 01:24:11,310 --> 01:24:14,590 But they form an algebra of angular momentum. 1158 01:24:14,590 --> 01:24:16,640 And what have we learned today, if you 1159 01:24:16,640 --> 01:24:18,960 have an algebra of angular momentum, 1160 01:24:18,960 --> 01:24:23,600 the states must organize themselves 1161 01:24:23,600 --> 01:24:28,040 into representations of angular momentum. 1162 01:24:28,040 --> 01:24:34,910 So the whole spectrum of the 2 dimensional harmonic oscillator 1163 01:24:34,910 --> 01:24:38,680 has in fact all spin representations. 1164 01:24:38,680 --> 01:24:40,530 J equals 0. 1165 01:24:40,530 --> 01:24:42,640 J equals 1/2. 1166 01:24:42,640 --> 01:24:44,400 J equals 1. 1167 01:24:44,400 --> 01:24:46,690 J equals 2. 1168 01:24:46,690 --> 01:24:49,510 J equals n, and all of them. 1169 01:24:49,510 --> 01:24:54,500 So the best example of all the representations 1170 01:24:54,500 --> 01:24:57,370 of angular momentum are in the states 1171 01:24:57,370 --> 01:25:00,960 of the 2 dimensional simple harmonic oscillator. 1172 01:25:00,960 --> 01:25:05,270 It's an abstract angular momentum, but it's very useful. 1173 01:25:05,270 --> 01:25:09,360 The one step I didn't do here for you is to check. 1174 01:25:09,360 --> 01:25:13,560 Although you check that all of these Ji commute 1175 01:25:13,560 --> 01:25:15,920 with the Hamiltonian. 1176 01:25:15,920 --> 01:25:18,360 Simple calculation to do it. 1177 01:25:18,360 --> 01:25:21,620 In fact, the Hamiltonian is NL plus N right, 1178 01:25:21,620 --> 01:25:23,130 and you can check it. 1179 01:25:23,130 --> 01:25:27,470 Since they commute with them, these operators act in states 1180 01:25:27,470 --> 01:25:29,600 and don't change the energy. 1181 01:25:29,600 --> 01:25:32,180 And they're a symmetry of the problem. 1182 01:25:32,180 --> 01:25:35,270 So that's why they fell into representations. 1183 01:25:35,270 --> 01:25:39,540 So this is our first example of a hidden symmetry. 1184 01:25:39,540 --> 01:25:42,890 A problem that there was no reason a priori 1185 01:25:42,890 --> 01:25:47,090 to expect an angular momentum to exist, 1186 01:25:47,090 --> 01:25:51,080 but it's there, and helps explain the degeneracies. 1187 01:25:51,080 --> 01:25:53,880 These degeneracies you could have said they're accidental. 1188 01:25:53,880 --> 01:25:56,180 But by the time you know they have 1189 01:25:56,180 --> 01:25:59,760 to fall into angular momentum representations, 1190 01:25:59,760 --> 01:26:02,180 you have great control over them. 1191 01:26:02,180 --> 01:26:05,100 You couldn't have found different number 1192 01:26:05,100 --> 01:26:08,090 of degenerate states at any level here. 1193 01:26:08,090 --> 01:26:11,270 This was in fact discovered by Julian Schwinger 1194 01:26:11,270 --> 01:26:13,535 in a very famous paper. 1195 01:26:13,535 --> 01:26:16,910 And is a classic example of angular momentum. 1196 01:26:16,910 --> 01:26:17,500 All right. 1197 01:26:17,500 --> 01:26:18,650 That's it for today. 1198 01:26:18,650 --> 01:26:25,580 See you on Wednesday if you come I'll be here.