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 --> 00:00:10,130 to offer high quality educational resources for free. 5 00:00:10,130 --> 00:00:12,690 To make a donation, or to view additional materials 6 00:00:12,690 --> 00:00:16,315 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,315 --> 00:00:16,940 at ocw.mit.edu. 8 00:00:21,880 --> 00:00:23,510 PROFESSOR: So let's get started. 9 00:00:23,510 --> 00:00:28,160 So I'm going to lecture today, Professor Zweback's away. 10 00:00:28,160 --> 00:00:30,180 And I just wanted to say a couple of things, 11 00:00:30,180 --> 00:00:32,140 just in case you haven't noticed. 12 00:00:32,140 --> 00:00:34,950 We posted the solutions for P-set 11. 13 00:00:34,950 --> 00:00:38,410 And then also later in the week, we'll 14 00:00:38,410 --> 00:00:40,920 post the solutions for the extra problems that came along 15 00:00:40,920 --> 00:00:44,330 with P-set 11, so you can look at those. 16 00:00:44,330 --> 00:00:47,220 And also, there's two past exams with solutions 17 00:00:47,220 --> 00:00:48,220 also on the website now. 18 00:00:48,220 --> 00:00:51,110 So you can start going through those. 19 00:00:51,110 --> 00:00:54,020 And also, there's a formula sheet there. 20 00:00:54,020 --> 00:00:55,900 And if you've got suggestions for things 21 00:00:55,900 --> 00:00:57,983 that you think should be on there that aren't, let 22 00:00:57,983 --> 00:01:01,810 us know and they probably can be put on there. 23 00:01:01,810 --> 00:01:03,477 So I want to turn back to what we 24 00:01:03,477 --> 00:01:05,810 were doing at the end of last lecture, which was talking 25 00:01:05,810 --> 00:01:07,850 about the spin-orbit coupling. 26 00:01:07,850 --> 00:01:10,240 And so this is a contribution to our Hamiltonian 27 00:01:10,240 --> 00:01:12,310 that looks like spin of the electron 28 00:01:12,310 --> 00:01:14,930 dotted into the angular momentum that the electron has 29 00:01:14,930 --> 00:01:19,310 around the proton in the hydrogen atom. 30 00:01:19,310 --> 00:01:22,320 And so because of this term we had 31 00:01:22,320 --> 00:01:27,600 to change the complete set of commuting observables 32 00:01:27,600 --> 00:01:30,220 that we wanted to talk about. 33 00:01:30,220 --> 00:01:32,400 So we have now this full Hamiltonian 34 00:01:32,400 --> 00:01:36,570 that includes this piece that has the Se dot L term in it. 35 00:01:36,570 --> 00:01:39,870 We have L squared, we have the spin squared. 36 00:01:39,870 --> 00:01:43,480 But because of this piece, Lz, which was previously 37 00:01:43,480 --> 00:01:46,519 one of the quantum numbers we used to classify things by, 38 00:01:46,519 --> 00:01:48,310 that doesn't commute with this term, right? 39 00:01:48,310 --> 00:01:53,600 So here we have to throw that one away. 40 00:01:53,600 --> 00:01:55,230 Similarly, we have this throw away 41 00:01:55,230 --> 00:01:58,210 the z component of the electron spin. 42 00:01:58,210 --> 00:02:00,800 That doesn't commute with this either. 43 00:02:00,800 --> 00:02:03,750 And what we replace those by is actually 44 00:02:03,750 --> 00:02:08,169 the J squared and the Z component of J. 45 00:02:08,169 --> 00:02:13,582 So J is the vector sum of the angular momentum 46 00:02:13,582 --> 00:02:14,790 and the spin of the electron. 47 00:02:17,367 --> 00:02:18,575 And this is very interesting. 48 00:02:21,290 --> 00:02:24,230 This term does something interesting. 49 00:02:24,230 --> 00:02:27,870 So if we look at-- let me go up here. 50 00:02:31,720 --> 00:02:35,469 If we remember the hydrogen states 51 00:02:35,469 --> 00:02:37,010 when we don't have this term, there's 52 00:02:37,010 --> 00:02:41,950 a state that has n equals 2 and l equals 1. 53 00:02:41,950 --> 00:02:45,595 And you can think of that as three states. 54 00:02:48,660 --> 00:02:52,360 And then we've got to tensor that with the spin, 55 00:02:52,360 --> 00:02:55,400 so the spin of the electron could be spin up or spin down. 56 00:02:55,400 --> 00:02:59,630 So there's a spin a half, so this is two states. 57 00:03:04,280 --> 00:03:06,210 And so you've got a total of six states 58 00:03:06,210 --> 00:03:07,790 you're going to talk about. 59 00:03:07,790 --> 00:03:10,237 And now what we have to do is classify these 60 00:03:10,237 --> 00:03:12,320 according to the quantum numbers that are actually 61 00:03:12,320 --> 00:03:13,360 preserved by the system. 62 00:03:13,360 --> 00:03:17,630 So we can't use Lz or Sz. 63 00:03:17,630 --> 00:03:21,330 We have to use J squared and Jz. 64 00:03:21,330 --> 00:03:29,150 So we've got a J equals 3/2 multiplet-- 65 00:03:29,150 --> 00:03:35,680 and that's four states-- plus a J equals 1/2. 66 00:03:42,442 --> 00:03:44,400 And you can see the number of states works out. 67 00:03:44,400 --> 00:03:48,180 We've got 3 times 2 is equal to 4 plus 2. 68 00:03:48,180 --> 00:03:56,150 And so this L dot S term takes these original six states, 69 00:03:56,150 --> 00:04:00,190 which without this interaction degenerate, 70 00:04:00,190 --> 00:04:03,770 and it splits them into the four states 71 00:04:03,770 --> 00:04:10,340 up here, and then two states down here. 72 00:04:10,340 --> 00:04:15,050 The J equals 1/2, J equals 3/2. 73 00:04:15,050 --> 00:04:17,470 And we also worked out the splittings. 74 00:04:20,540 --> 00:04:26,620 If I do this, this is plus h bar squared over 2. 75 00:04:26,620 --> 00:04:30,710 And this is minus h bar squared. 76 00:04:30,710 --> 00:04:33,340 So this gives you a splitting. 77 00:04:33,340 --> 00:04:37,630 Now this is not the only thing that happens in hydrogen, 78 00:04:37,630 --> 00:04:40,680 because you probably all know that the proton itself 79 00:04:40,680 --> 00:04:41,650 has spin. 80 00:04:41,650 --> 00:04:43,580 The proton has a spin 1/2 particle, just 81 00:04:43,580 --> 00:04:45,190 like the electron. 82 00:04:45,190 --> 00:04:46,890 It's even more complicated because it's 83 00:04:46,890 --> 00:04:49,760 a composite object. 84 00:04:49,760 --> 00:04:56,010 But that leads to additional splittings in hydrogen. 85 00:04:56,010 --> 00:04:58,740 And so these ones, this one here is 86 00:04:58,740 --> 00:05:00,609 called the defined structure. 87 00:05:00,609 --> 00:05:02,900 Or we can also talk about the type hyperfine structure. 88 00:05:11,800 --> 00:05:14,180 So this is going to be a small effect on top of this one. 89 00:05:23,590 --> 00:05:25,180 So we have the proton that's spin 1/2, 90 00:05:25,180 --> 00:05:26,555 we have the electron spin a half, 91 00:05:26,555 --> 00:05:30,350 and then we have the relative orbital angular momentum. 92 00:05:30,350 --> 00:05:39,070 And so the total angular momentum, 93 00:05:39,070 --> 00:05:46,640 which is J, which is going to be the sum of L plus the spin 94 00:05:46,640 --> 00:05:51,480 of the electron plus the spin of the proton, this is conserved. 95 00:05:54,630 --> 00:05:56,510 And the thing we were talking about here 96 00:05:56,510 --> 00:05:59,090 is actually not conserved. 97 00:05:59,090 --> 00:06:01,250 So once you worry about the spin of the proton 98 00:06:01,250 --> 00:06:03,370 you've got to look at the total angular momentum. 99 00:06:03,370 --> 00:06:06,930 And that's what will be conserved. 100 00:06:06,930 --> 00:06:11,200 And so our complete set of commuting observables 101 00:06:11,200 --> 00:06:15,810 is going to be a four Hamiltonian, which we'll get to 102 00:06:15,810 --> 00:06:23,000 in a moment, L squared the spin squareds 103 00:06:23,000 --> 00:06:27,800 of the proton and the electron, and then J squared, 104 00:06:27,800 --> 00:06:31,850 and finally Jz is the things that we're 105 00:06:31,850 --> 00:06:33,666 going to end up classifying states by. 106 00:06:38,250 --> 00:06:40,600 So we originally thought about these two 107 00:06:40,600 --> 00:06:43,790 here, and did a coupling between those. 108 00:06:43,790 --> 00:06:45,970 It's pretty natural to assume that there maybe 109 00:06:45,970 --> 00:06:48,460 couplings between the angular momentum and the spin 110 00:06:48,460 --> 00:06:50,895 of the proton, which there are. 111 00:06:50,895 --> 00:06:53,270 But also there's going to be a coupling between the spins 112 00:06:53,270 --> 00:06:54,692 of the electron and the proton. 113 00:06:54,692 --> 00:06:57,150 And that's the one we're going to talk about at the moment. 114 00:06:57,150 --> 00:07:03,200 The other one is there but we won't go over it in any detail. 115 00:07:03,200 --> 00:07:08,770 So the proton and the electron both spin 1/2 particles, 116 00:07:08,770 --> 00:07:19,970 and they both have magnetic dipole moments, 117 00:07:19,970 --> 00:07:22,610 which are proportional to their spin. 118 00:07:22,610 --> 00:07:25,090 And so it's really a coupling between these moments that 119 00:07:25,090 --> 00:07:29,400 tells us what the effect of this interaction is going to be. 120 00:07:29,400 --> 00:07:32,880 So we have the mu of the electron 121 00:07:32,880 --> 00:07:38,790 is equal to e over me-- minus me-- 122 00:07:38,790 --> 00:07:41,900 times the spin of the electron. 123 00:07:41,900 --> 00:07:48,940 And mu of the proton is, let me just write it as gp. 124 00:07:59,290 --> 00:08:07,550 And gp happens to have the value of about 5.6. 125 00:08:07,550 --> 00:08:09,410 And this is actually kind of interesting. 126 00:08:09,410 --> 00:08:11,320 So if you look at the formula up here, 127 00:08:11,320 --> 00:08:15,990 really I could have written this as a g over 2, with g being 2. 128 00:08:15,990 --> 00:08:21,259 So for the electron, the g factor is very close to 2. 129 00:08:21,259 --> 00:08:23,050 This is because the electron is essentially 130 00:08:23,050 --> 00:08:26,120 a fundamental particle, with no substructure. 131 00:08:26,120 --> 00:08:28,940 But the proton, which is made up of quarks and gluons 132 00:08:28,940 --> 00:08:32,770 flying around inside some region, has a lot of structure. 133 00:08:32,770 --> 00:08:42,190 And so this is really indicative of it 134 00:08:42,190 --> 00:08:43,450 being a composite particle. 135 00:08:49,640 --> 00:08:52,410 Because a fundamental spin 1/2 particle 136 00:08:52,410 --> 00:08:54,040 should have this g being 2. 137 00:08:57,260 --> 00:09:00,420 So we've got these two dipole moments. 138 00:09:00,420 --> 00:09:01,970 And one way to think about this is 139 00:09:01,970 --> 00:09:04,360 you've got this dipole of the proton. 140 00:09:04,360 --> 00:09:06,770 We're going to think about the proton having 141 00:09:06,770 --> 00:09:11,160 a little dipole charge-- sorry, dipole magnetic moment-- 142 00:09:11,160 --> 00:09:13,670 and this produces a magnetic field. 143 00:09:13,670 --> 00:09:16,560 And the electron is sitting in that magnetic field. 144 00:09:16,560 --> 00:09:20,290 And its spin can couple to the field. 145 00:09:20,290 --> 00:09:23,851 So we're going to have a Hamiltonian, 146 00:09:23,851 --> 00:09:26,040 a hyperfine Hamiltonian, that looks 147 00:09:26,040 --> 00:09:29,530 like minus mu of the electron dotted 148 00:09:29,530 --> 00:09:37,460 into a magnetic field produced by the proton, which is going 149 00:09:37,460 --> 00:09:40,840 to depend on r, on where the electron is. 150 00:09:44,470 --> 00:09:46,850 And you can simplify this as just 151 00:09:46,850 --> 00:09:50,930 e over m spin of the electron dotted 152 00:09:50,930 --> 00:09:52,600 into this B of the proton. 153 00:09:55,980 --> 00:09:58,890 So we need to know what this dipole field is. 154 00:09:58,890 --> 00:10:03,730 And for that you really have to go back to electromagnetism. 155 00:10:03,730 --> 00:10:06,780 And you've probably seen this before. 156 00:10:06,780 --> 00:10:10,640 But let me just write it down, and we won't derive it here. 157 00:10:13,320 --> 00:10:16,240 But let's go down here. 158 00:10:24,310 --> 00:10:26,030 This has a kind of complicated form. 159 00:10:48,956 --> 00:10:50,580 So there's this piece, and then there's 160 00:10:50,580 --> 00:11:03,330 another piece that looks like 8 pi over 3c squared mu p times 161 00:11:03,330 --> 00:11:05,600 the delta function at the origin. 162 00:11:05,600 --> 00:11:10,070 And so you think about the dipole field 163 00:11:10,070 --> 00:11:15,010 arising from a spinning charge distribution here. 164 00:11:15,010 --> 00:11:18,010 So we've got a magnetic dipole moment pointing up. 165 00:11:18,010 --> 00:11:22,710 This produces a field like this, a dipole type 166 00:11:22,710 --> 00:11:24,930 field going this way. 167 00:11:24,930 --> 00:11:26,630 So this is our B. 168 00:11:26,630 --> 00:11:30,430 And then inside here, you should really 169 00:11:30,430 --> 00:11:34,160 think of taking the limit as this thing goes to 0 size. 170 00:11:34,160 --> 00:11:37,710 And so in order to get the right field in the middle, 171 00:11:37,710 --> 00:11:39,153 you need to have this term here. 172 00:11:57,300 --> 00:12:00,007 And so if you want to see this being derived 173 00:12:00,007 --> 00:12:01,090 you can look in Griffiths. 174 00:12:01,090 --> 00:12:03,170 That does the derivation of this. 175 00:12:03,170 --> 00:12:05,320 But we will skip that. 176 00:12:10,222 --> 00:12:11,180 So we've got the field. 177 00:12:11,180 --> 00:12:13,160 And now we can put it into our Hamiltonian. 178 00:12:17,560 --> 00:12:19,010 So it's mu e. 179 00:12:26,430 --> 00:12:30,670 So I could replace my mu's with the spins. 180 00:12:30,670 --> 00:12:32,270 So I get some factor out the front 181 00:12:32,270 --> 00:12:42,870 that looks like ge squared over 2 Me Mp c squared. 182 00:12:42,870 --> 00:13:08,170 And then I get 1 over r cubed plus-- 183 00:13:23,090 --> 00:13:26,335 So just plugging those in we get this Hamiltonian here. 184 00:13:29,820 --> 00:13:31,650 And let me just simplify a little bit. 185 00:13:31,650 --> 00:13:35,520 Let's just call this thing q. 186 00:13:35,520 --> 00:13:40,150 And so this Hamiltonian is going to be given by q. 187 00:13:40,150 --> 00:13:45,810 And I can write it as the i-th component of the electron spin, 188 00:13:45,810 --> 00:13:49,490 the j-th component of the proton spin, 189 00:13:49,490 --> 00:14:17,850 dotted into r hat i hat j minus-- So just taking 190 00:14:17,850 --> 00:14:21,210 the common factors of the spins components out the front. 191 00:14:24,280 --> 00:14:26,319 So if we've got this, we want to ask 192 00:14:26,319 --> 00:14:28,360 what it's going to do to the energy of the ground 193 00:14:28,360 --> 00:14:30,320 state of hydrogen. 194 00:14:30,320 --> 00:14:34,280 So we're going to take matrix elements of this 195 00:14:34,280 --> 00:14:37,830 between the hydrogen wave functions. 196 00:14:37,830 --> 00:14:40,767 So does anyone have questions so far? 197 00:14:40,767 --> 00:14:41,267 Yes. 198 00:14:41,267 --> 00:14:43,022 AUDIENCE: Can you use r as a [INAUDIBLE]? 199 00:14:43,022 --> 00:14:43,980 PROFESSOR: Right right. 200 00:14:43,980 --> 00:14:47,310 So these are unit vectors in the r direction. 201 00:14:47,310 --> 00:14:52,220 And this r is the length of the vector, r vector. 202 00:14:52,220 --> 00:14:54,160 The usual thing. 203 00:14:54,160 --> 00:14:57,490 So what we're going to try and evaluate 204 00:14:57,490 --> 00:15:02,815 is the expectation value. 205 00:15:16,430 --> 00:15:18,220 So we're going to do this. 206 00:15:18,220 --> 00:15:21,880 Because going back to the start of last lecture, 207 00:15:21,880 --> 00:15:23,702 this is going to be a small correction. 208 00:15:23,702 --> 00:15:25,910 And so we can work out its contribution to the energy 209 00:15:25,910 --> 00:15:28,300 by using the original wave functions, 210 00:15:28,300 --> 00:15:30,560 but just calculating its matrix elements. 211 00:15:30,560 --> 00:15:43,320 So we're going to calculate-- and let me just 212 00:15:43,320 --> 00:15:44,650 give this a name. 213 00:15:44,650 --> 00:15:45,370 This can be-- 214 00:15:55,560 --> 00:16:26,220 So this is q, and the ground state 215 00:16:26,220 --> 00:16:27,580 has no angular dependents. 216 00:16:27,580 --> 00:16:29,320 So in fact, for the ground state, 217 00:16:29,320 --> 00:16:32,891 I can just write this is a function of r squared. 218 00:16:32,891 --> 00:16:35,250 For overtly excited states I can't do that. 219 00:16:35,250 --> 00:16:37,589 But for the ground state that works. 220 00:16:37,589 --> 00:16:39,630 And then we have, so we've got the wave function. 221 00:16:39,630 --> 00:16:44,170 And then in between them we have to put this stuff over here. 222 00:16:44,170 --> 00:16:45,440 So let's put the there. 223 00:17:05,790 --> 00:17:09,528 So one of these terms is very easy to evaluate. 224 00:17:09,528 --> 00:17:11,069 With this [INAUDIBLE] function I just 225 00:17:11,069 --> 00:17:14,339 get the wave function at the origin. 226 00:17:14,339 --> 00:17:16,960 And the second term is actually also 227 00:17:16,960 --> 00:17:18,855 relatively easy to evaluate. 228 00:17:29,348 --> 00:17:40,780 Who can tell me what this integral over all three 229 00:17:40,780 --> 00:17:44,030 directions of just one direction? 230 00:17:44,030 --> 00:17:45,200 What's that? 231 00:17:45,200 --> 00:17:46,100 AUDIENCE: 0. 232 00:17:46,100 --> 00:17:48,130 PROFESSOR: 0. 233 00:17:48,130 --> 00:17:51,380 And you can argue that by just asking, well what can it be? 234 00:17:51,380 --> 00:17:54,525 It's got to carry an index, because there's 235 00:17:54,525 --> 00:17:56,494 an index on this side of the equation. 236 00:17:56,494 --> 00:17:58,660 And there's no other vectors around in this problem. 237 00:17:58,660 --> 00:18:02,440 So the only thing it can be is 0. 238 00:18:02,440 --> 00:18:11,460 So if I do integral d3r of ri rj, what can that be? 239 00:18:15,220 --> 00:18:15,720 Sorry? 240 00:18:15,720 --> 00:18:17,130 AUDIENCE: 1. 241 00:18:17,130 --> 00:18:18,380 PROFESSOR: 1? 242 00:18:18,380 --> 00:18:19,110 No. 243 00:18:19,110 --> 00:18:21,530 So it's got two indices. 244 00:18:21,530 --> 00:18:23,280 So the thing on this side of the equation 245 00:18:23,280 --> 00:18:25,962 also has to have two indices. 246 00:18:25,962 --> 00:18:26,940 AUDIENCE: Delta ij? 247 00:18:26,940 --> 00:18:28,630 PROFESSOR: Delta ij, very good. 248 00:18:28,630 --> 00:18:32,300 So the only thing that can carry two indices is delta ij. 249 00:18:32,300 --> 00:18:34,090 And then there might be some number here. 250 00:18:41,980 --> 00:18:48,010 And it actually turns out that you 251 00:18:48,010 --> 00:18:49,720 can do an even more complicated integral. 252 00:18:49,720 --> 00:18:58,370 We can look at integral d3r of ri rj sum f of r squared. 253 00:18:58,370 --> 00:19:00,150 And that is also just some number, 254 00:19:00,150 --> 00:19:05,550 which depends on what f is, times delta ij. 255 00:19:05,550 --> 00:19:12,160 And if you go along these lines and actually look at this, 256 00:19:12,160 --> 00:19:14,370 the difference between the integral of this piece 257 00:19:14,370 --> 00:19:18,400 and the integral of this piece is actually a factor of 1/3. 258 00:19:18,400 --> 00:19:24,760 And so this actually integrates to 0. 259 00:19:27,410 --> 00:19:29,570 So when I integrate over this one, 260 00:19:29,570 --> 00:19:32,090 I get something times delta ij. 261 00:19:32,090 --> 00:19:33,610 And that something is actually 1/3. 262 00:19:37,240 --> 00:19:41,310 And so this term and this term cancel in the integral. 263 00:19:41,310 --> 00:19:43,950 And so you just get the delta function contributions. 264 00:19:43,950 --> 00:19:51,540 So you get some number times Sei S delta ij. 265 00:19:51,540 --> 00:19:54,365 So it becomes Se dotted into Sp. 266 00:20:02,520 --> 00:20:04,590 8 pi over 3. 267 00:20:04,590 --> 00:20:09,090 And then it's psi 100 at the origin. 268 00:20:14,710 --> 00:20:18,900 So this we know, we've already computed these radial wave 269 00:20:18,900 --> 00:20:22,260 functions, and saw at the origin this one 270 00:20:22,260 --> 00:20:28,220 is actually 1 over pi times the Bohr constant. 271 00:20:30,910 --> 00:20:35,270 And if you plug-in what Q is, and what the Bohr constant is, 272 00:20:35,270 --> 00:20:38,540 you can just find out that this whole thing ends up 273 00:20:38,540 --> 00:21:01,570 looking like 4/3 this gp and this 274 00:21:01,570 --> 00:21:07,100 we can call delta e hyperfine. 275 00:21:07,100 --> 00:21:09,010 So you end up with a very simple thing. 276 00:21:09,010 --> 00:21:11,390 And it's just proportional to the dot product of the two 277 00:21:11,390 --> 00:21:11,890 spins. 278 00:21:20,440 --> 00:21:26,500 So you've seen, essentially, you saw this term in your homework. 279 00:21:26,500 --> 00:21:31,100 So we just assume that this thing here came out of nowhere 280 00:21:31,100 --> 00:21:33,850 and was just some number times Se dot Sp, 281 00:21:33,850 --> 00:21:36,740 and this was a contribution to your Hamiltonian. 282 00:21:36,740 --> 00:21:40,420 But now we actually know where that comes from. 283 00:21:40,420 --> 00:21:48,250 And interestingly, this thing here, this whole thing, it's 284 00:21:48,250 --> 00:21:52,760 still an operator because it's got these spins in it. 285 00:21:56,530 --> 00:22:01,580 And that's-- put a star next to that because it's important. 286 00:22:01,580 --> 00:22:07,940 So now we need to ask, well what are the real states of hydrogen 287 00:22:07,940 --> 00:22:09,784 so they're where we've got two spins? 288 00:22:09,784 --> 00:22:11,700 The spin of the proton, they could be aligned, 289 00:22:11,700 --> 00:22:13,460 or they could be anti-aligned. 290 00:22:16,300 --> 00:22:17,000 Oh, sorry. 291 00:22:17,000 --> 00:22:18,770 We have a question up there. 292 00:22:18,770 --> 00:22:22,040 AUDIENCE: Is that np over np, or mu u? 293 00:22:22,040 --> 00:22:23,490 PROFESSOR: No, me, mass. 294 00:22:23,490 --> 00:22:25,364 Mass of the electron over mass of the proton. 295 00:22:30,350 --> 00:22:34,660 So you have to remember that the spins of the proton 296 00:22:34,660 --> 00:22:37,335 and the electron to can parallel or they can be anti-parallel. 297 00:22:37,335 --> 00:22:40,090 Or they can be both down. 298 00:22:40,090 --> 00:22:43,780 And so we have to go back and work out-- we 299 00:22:43,780 --> 00:22:47,340 have to realize that because of these terms 300 00:22:47,340 --> 00:22:53,290 the z components of those spins are not good quantum numbers. 301 00:22:53,290 --> 00:22:55,400 The only z component that appears in our list 302 00:22:55,400 --> 00:22:59,375 is Jz, so the total z component of angular momentum. 303 00:23:01,910 --> 00:23:06,620 So we need to go back and do what you-- you probably 304 00:23:06,620 --> 00:23:08,440 have done this to the P-set. 305 00:23:08,440 --> 00:23:10,080 But let's just do it very quickly. 306 00:23:10,080 --> 00:23:17,310 We'll take those two spin 1/2 things 307 00:23:17,310 --> 00:23:23,020 and so let's make this J1 and this is J2. 308 00:23:23,020 --> 00:23:32,970 And we're going to have J. 309 00:23:32,970 --> 00:23:35,520 So if I've got these two spins I can make various things. 310 00:23:35,520 --> 00:23:45,770 I can write down-- And if I've done this than 311 00:23:45,770 --> 00:23:49,405 I should also write that the m, the m quantum number that 312 00:23:49,405 --> 00:23:50,780 goes with the J quantum number is 313 00:23:50,780 --> 00:23:54,650 going to be equal to m1 plus m2. 314 00:23:54,650 --> 00:23:57,780 So this state here, because both of the spins are pointing up, 315 00:23:57,780 --> 00:23:59,490 this is an m equals 1 state. 316 00:24:02,210 --> 00:24:06,650 And then we can also have something like 1/2, 1/2. 317 00:24:22,620 --> 00:24:24,090 You could have these two states. 318 00:24:24,090 --> 00:24:27,740 So they both have m equals 0. 319 00:24:27,740 --> 00:24:38,470 And then there's an m equals minus 1, which is 1/2 this guy. 320 00:24:41,110 --> 00:24:45,500 So since this has m equals 1, and 1/2 cross 321 00:24:45,500 --> 00:24:51,770 1/2 is going to give us a spin 0 multiplet and a spin 1 322 00:24:51,770 --> 00:24:56,180 multiplet, because it's got m equals 1, 323 00:24:56,180 --> 00:24:58,950 this has to be J equals 1 as well. 324 00:24:58,950 --> 00:25:01,190 And this one has to be J equals 1. 325 00:25:01,190 --> 00:25:03,970 But the two states in the middle, 326 00:25:03,970 --> 00:25:05,830 we don't know what those are. 327 00:25:08,750 --> 00:25:10,300 There's going to be a J equals 1, 328 00:25:10,300 --> 00:25:12,110 m equals 0 state, which is going to be 329 00:25:12,110 --> 00:25:15,350 some linear combination of these two. 330 00:25:15,350 --> 00:25:20,955 So let's just go over here. 331 00:25:20,955 --> 00:25:22,060 We don't need any of this. 332 00:25:27,710 --> 00:25:31,290 And we need to work out what the linear combination is. 333 00:25:31,290 --> 00:25:34,740 So something to remember is this. 334 00:25:34,740 --> 00:25:39,451 The J plus or minus acting on Jm is this funny square root 335 00:25:39,451 --> 00:25:39,950 thing. 336 00:25:52,111 --> 00:25:54,110 So these are the raising and lowering operators. 337 00:25:54,110 --> 00:25:57,346 They take us from one state to the one with a different m 338 00:25:57,346 --> 00:25:57,845 value. 339 00:26:00,410 --> 00:26:03,130 And so we can use that to start with. 340 00:26:03,130 --> 00:26:05,925 We could basically take J minus on our state. 341 00:26:10,390 --> 00:26:12,960 And according to this formula, this 342 00:26:12,960 --> 00:26:23,160 will give us the square root of 1 times 2 minus m is 1. 343 00:26:32,230 --> 00:26:36,030 And this should be 0, right? 344 00:26:36,030 --> 00:26:38,490 I think I've got this sign up the wrong way. 345 00:26:38,490 --> 00:26:40,460 I think this is minus plus. 346 00:26:44,470 --> 00:26:46,950 No, sorry, that's right. 347 00:26:46,950 --> 00:26:49,980 It should be-- I'm doing the J minus 348 00:26:49,980 --> 00:26:57,300 so I have 1 times 1 minus-- yeah, right, so it's this. 349 00:26:57,300 --> 00:27:03,370 So this is square root 2 times Jm minus 1. 350 00:27:03,370 --> 00:27:10,150 But we also know that J minus a is equal to J1 minus plus J2 351 00:27:10,150 --> 00:27:17,310 minus because J is just the vector sum of the two J's. 352 00:27:17,310 --> 00:27:26,230 So we can ask what J minus on the state is. 353 00:27:26,230 --> 00:27:30,440 But this state we can write in terms of the tensor product. 354 00:27:38,612 --> 00:27:45,220 So this is equal to J1 minus 1. 355 00:28:09,030 --> 00:28:11,140 If we use this formula for lowering something 356 00:28:11,140 --> 00:28:18,100 with spin 1/2 we get 1/2 times 3/2 minus 1/2 times minus 1/2, 357 00:28:18,100 --> 00:28:21,360 which is actually 1 under that square root. 358 00:28:21,360 --> 00:28:28,740 And so this actually equals 1/2 minus 1/2 tensor 1/2, 1/2. 359 00:28:41,300 --> 00:28:45,520 So these two things are equal. 360 00:28:45,520 --> 00:28:50,220 And so that tells us, in fact, that the 1, 0 state, 361 00:28:50,220 --> 00:28:54,331 which is what's over here-- oh, sorry. 362 00:28:54,331 --> 00:28:55,350 Oh, why did I do that? 363 00:28:55,350 --> 00:28:59,610 This should be J equals 1 and M equals 364 00:28:59,610 --> 00:29:03,600 what it was, minus 1, this. 365 00:29:03,600 --> 00:29:06,830 So the 1, 0 state, if we bring that 1 over root 2 366 00:29:06,830 --> 00:29:26,860 on the other side is this combination. 367 00:29:26,860 --> 00:29:28,995 So it's one linear combination of those two pieces. 368 00:29:32,600 --> 00:29:34,510 We also want the other one. 369 00:29:34,510 --> 00:29:36,990 So we've got three of our states. 370 00:29:36,990 --> 00:29:38,540 The fourth state is then, of course, 371 00:29:38,540 --> 00:29:44,480 the other linear combination of the two states over there. 372 00:29:44,480 --> 00:29:49,050 And so that's going to be our J equals 0, M equals 0 state. 373 00:29:53,660 --> 00:30:01,070 So this state is going to be orthogonal to the one we've 374 00:30:01,070 --> 00:30:02,450 just written here. 375 00:30:02,450 --> 00:30:04,480 And so this is pretty easy to work out. 376 00:30:04,480 --> 00:30:06,000 Since there's only two terms, all we 377 00:30:06,000 --> 00:30:07,840 do is change the sign of one of them 378 00:30:07,840 --> 00:30:11,705 and it becomes orthogonal, because these states here 379 00:30:11,705 --> 00:30:12,560 are normalized. 380 00:30:15,660 --> 00:30:20,602 So this becomes 1/2, 1/2 tensor-- 381 00:30:20,602 --> 00:30:24,070 let me just write it in the same way that-- 382 00:30:24,070 --> 00:30:28,125 1/2 minus 1/2 minus-- 383 00:30:38,150 --> 00:30:42,410 And so our four states, so we can condense our notation 384 00:30:42,410 --> 00:30:49,340 so we can say that this state we can just label as this. 385 00:30:49,340 --> 00:30:57,710 And we can just label as a down arrow. 386 00:30:57,710 --> 00:31:11,240 And then something like we can label as just up down, just 387 00:31:11,240 --> 00:31:14,140 to make everything compact. 388 00:31:14,140 --> 00:31:16,230 You just have to remember that this is referring 389 00:31:16,230 --> 00:31:17,605 to the first spin, this is always 390 00:31:17,605 --> 00:31:19,760 referring to the second spin. 391 00:31:19,760 --> 00:31:21,500 And so then we can write our multiplets. 392 00:31:24,105 --> 00:31:25,840 J equals 1 has three states. 393 00:31:25,840 --> 00:31:28,720 It has up, up. 394 00:31:28,720 --> 00:31:37,540 It has up, down plus down, up. 395 00:31:37,540 --> 00:31:41,200 And it has down, down. 396 00:31:41,200 --> 00:31:44,830 So those are our three states that have J equals 1. 397 00:31:44,830 --> 00:31:48,320 And then we have J equals 0, which 398 00:31:48,320 --> 00:31:57,861 just has 1 over square root 2 up, down minus down, up. 399 00:32:00,950 --> 00:32:03,050 And so the two spins in our hydrogen 400 00:32:03,050 --> 00:32:05,270 atom, the spin of the proton and the electron, 401 00:32:05,270 --> 00:32:10,020 can combine to be a J equals 1 or a J equals 0 system. 402 00:32:10,020 --> 00:32:12,600 And since we're talking about the ground state of hydrogen, 403 00:32:12,600 --> 00:32:15,680 it has 0 angular momentum. 404 00:32:15,680 --> 00:32:20,610 And so I'm really just talking about J total, here. 405 00:32:26,360 --> 00:32:31,490 So if we now have this Hamiltonian, which is still 406 00:32:31,490 --> 00:32:33,540 an operator in spin-- we've dealt 407 00:32:33,540 --> 00:32:36,860 with the spacial dependence of the wave functions, 408 00:32:36,860 --> 00:32:39,210 but it's still an operator in spin-- 409 00:32:39,210 --> 00:32:42,090 we can now evaluate this. 410 00:32:42,090 --> 00:32:51,840 So we can take its expectation value in either the J equals 1 411 00:32:51,840 --> 00:32:53,940 multiplet or the J equals 0 multiplet. 412 00:32:56,880 --> 00:32:59,380 So let's just write it out again. 413 00:32:59,380 --> 00:33:04,610 So we have h hyperfine 1, 0, 0. 414 00:33:04,610 --> 00:33:12,039 This is equal to some delta E HF spin of the electron dotted 415 00:33:12,039 --> 00:33:13,205 into the spin of the proton. 416 00:33:16,070 --> 00:33:20,260 We can rewrite this using something we did last time. 417 00:33:24,540 --> 00:33:31,520 We can write this as J squared minus Se squared 418 00:33:31,520 --> 00:33:36,730 minus Sp squared, with the 1/2 out the front. 419 00:33:36,730 --> 00:33:42,790 So here, because l equals 0 because we're in the ground 420 00:33:42,790 --> 00:33:48,634 state, then J equals Se plus Sp. 421 00:33:48,634 --> 00:33:51,030 And so J squared is going to give us 422 00:33:51,030 --> 00:33:55,860 Se squared, Sp squared, and then the dot product. 423 00:33:55,860 --> 00:33:58,910 So great. 424 00:33:58,910 --> 00:34:01,650 So what is this, the spin squared of the electron? 425 00:34:07,530 --> 00:34:11,880 What's the eigenvalue of J squared, always? 426 00:34:11,880 --> 00:34:16,170 J, J plus 1 times h bar squared. 427 00:34:16,170 --> 00:34:18,376 And what is J for the electron? 428 00:34:18,376 --> 00:34:19,079 1/2. 429 00:34:19,079 --> 00:34:21,302 And what about the proton? 430 00:34:21,302 --> 00:34:22,218 AUDIENCE: 1/2 as well. 431 00:34:22,218 --> 00:34:22,843 PROFESSOR: 1/2. 432 00:34:22,843 --> 00:34:25,949 So we've got 1/2 times 1/2 plus 1, so 3/2. 433 00:34:25,949 --> 00:34:28,330 So this gives us minus 3/4. 434 00:34:28,330 --> 00:34:30,790 This gives us minus 3/4. 435 00:34:30,790 --> 00:34:35,399 So this just looks like delta e HF 436 00:34:35,399 --> 00:34:44,989 over 2 J squared minus 3/2 h bar squared. 437 00:34:44,989 --> 00:34:45,489 OK? 438 00:34:48,179 --> 00:34:49,420 Anyone lost doing that? 439 00:34:49,420 --> 00:34:50,587 Or is that OK? 440 00:34:50,587 --> 00:34:51,900 AUDIENCE: [INAUDIBLE] 441 00:34:51,900 --> 00:34:52,980 PROFESSOR: Yep. 442 00:34:52,980 --> 00:34:58,150 AUDIENCE: So, when you define delta e [INAUDIBLE] 443 00:34:58,150 --> 00:35:00,564 over there, that exudes energy? 444 00:35:00,564 --> 00:35:01,730 PROFESSOR: Oh, you're right. 445 00:35:01,730 --> 00:35:02,840 You're very right. 446 00:35:02,840 --> 00:35:03,471 I've messed up. 447 00:35:03,471 --> 00:35:03,970 I've-- 448 00:35:03,970 --> 00:35:06,470 AUDIENCE: [INAUDIBLE] 449 00:35:06,470 --> 00:35:08,750 PROFESSOR: Let me see. 450 00:35:08,750 --> 00:35:15,400 Yeah, really I have an h bar squared here. 451 00:35:18,640 --> 00:35:21,460 I think I should have had an h bar squared over there as well. 452 00:35:25,790 --> 00:35:27,545 Yeah. 453 00:35:27,545 --> 00:35:32,180 That should be over h bar squared here. 454 00:35:32,180 --> 00:35:35,110 Thank you. 455 00:35:35,110 --> 00:35:36,396 OK so-- 456 00:35:36,396 --> 00:35:38,622 AUDIENCE: [INAUDIBLE] 457 00:35:38,622 --> 00:35:39,330 PROFESSOR: Sorry? 458 00:35:39,330 --> 00:35:43,016 AUDIENCE: When does it get [INAUDIBLE] 459 00:35:43,016 --> 00:35:44,640 PROFESSOR: That was just in the algebra 460 00:35:44,640 --> 00:35:48,820 going from this expression, writing it in terms of alpha, 461 00:35:48,820 --> 00:35:49,560 things like that. 462 00:35:49,560 --> 00:35:53,850 So it's just some algebra. 463 00:35:53,850 --> 00:35:56,585 OK, anything else? 464 00:35:56,585 --> 00:35:57,085 No? 465 00:35:57,085 --> 00:35:57,585 Good. 466 00:35:57,585 --> 00:36:02,330 OK so now we can easily evaluate these things. 467 00:36:02,330 --> 00:36:17,040 We can now take J equals 1 and some M-- 468 00:36:17,040 --> 00:36:22,650 and this is for M equals all three states here-- and just 469 00:36:22,650 --> 00:36:24,310 evaluate this. 470 00:36:24,310 --> 00:36:26,250 And all that means is we have this J squared 471 00:36:26,250 --> 00:36:29,170 operator acting on this state here. 472 00:36:29,170 --> 00:36:32,680 And this gives us h bar squared 1 times 1 plus 1, 473 00:36:32,680 --> 00:36:34,730 or 2h bar squared. 474 00:36:34,730 --> 00:36:42,390 So this will give us delta e hyperfine over 2. 475 00:36:42,390 --> 00:36:45,010 And then we've got, let's pull the-- sorry there's still 476 00:36:45,010 --> 00:36:51,290 and h bar squared here, and an h bar squared there. 477 00:36:51,290 --> 00:36:54,130 But now we can evaluate. 478 00:36:54,130 --> 00:36:56,060 The h bar squared here cancels that one, 479 00:36:56,060 --> 00:37:02,750 and we get a 1 times a 1 plus 1 minus 3/2. 480 00:37:02,750 --> 00:37:05,730 And that's just one quarter, which is-- 481 00:37:12,430 --> 00:37:27,050 And similarly we can take the J equals 0 state, 482 00:37:27,050 --> 00:37:33,880 and this one gives us delta e hyperfine over 2. 483 00:37:33,880 --> 00:37:38,515 And then it's 0 time 1 minus 3/2. 484 00:37:38,515 --> 00:37:43,180 And so that equals minus 3/4 EHF. 485 00:37:48,206 --> 00:37:49,580 So what we're doing is evaluating 486 00:37:49,580 --> 00:37:51,151 these in these particular J states. 487 00:37:51,151 --> 00:37:53,400 And now we end up with something that's just a number. 488 00:37:53,400 --> 00:37:54,690 It's no longer an operator. 489 00:37:54,690 --> 00:37:56,650 It's an energy that we can measure. 490 00:37:56,650 --> 00:37:58,400 Yeah? 491 00:37:58,400 --> 00:38:01,400 AUDIENCE: So, this expectation value 492 00:38:01,400 --> 00:38:04,817 h hyperfine 1, 0, 0, is still an operator. 493 00:38:04,817 --> 00:38:06,900 Is that because we only took the expectation value 494 00:38:06,900 --> 00:38:08,900 over the angular [INAUDIBLE] 495 00:38:08,900 --> 00:38:12,060 PROFESSOR: We took over the spacial wave function. 496 00:38:12,060 --> 00:38:13,780 We did the r integral, right? 497 00:38:13,780 --> 00:38:14,480 But we didn't-- 498 00:38:14,480 --> 00:38:15,355 AUDIENCE: [INAUDIBLE] 499 00:38:15,355 --> 00:38:16,831 PROFESSOR: Right, right. 500 00:38:16,831 --> 00:38:17,330 Yeah. 501 00:38:20,100 --> 00:38:22,935 So this is actually a really important system. 502 00:38:28,400 --> 00:38:35,880 So let's just draw the energy level diagram here. 503 00:38:35,880 --> 00:38:37,800 And here we have four states. 504 00:38:37,800 --> 00:38:41,860 We have the spin 1/2 times spin 1/2. 505 00:38:41,860 --> 00:38:43,230 So 2 times 2 states. 506 00:38:43,230 --> 00:38:45,210 So we get a triplet and a singlet. 507 00:38:45,210 --> 00:38:47,040 And what this hyperfine splitting does 508 00:38:47,040 --> 00:38:53,070 is take those four states and split the triplet up here, 509 00:38:53,070 --> 00:38:56,270 and split the singlet down here. 510 00:38:56,270 --> 00:38:59,792 And this gap we can see is-- oops, 511 00:38:59,792 --> 00:39:03,380 so this should be a delta HF. 512 00:39:03,380 --> 00:39:08,150 So this gap is delta E HF. 513 00:39:08,150 --> 00:39:10,245 So it's 1/4 and minus 3/4. 514 00:39:13,150 --> 00:39:19,070 And if you plug numbers in, delta E HF, 515 00:39:19,070 --> 00:39:26,330 this actually ends up being 5.9 times 10 to the minus 6 516 00:39:26,330 --> 00:39:32,620 electron volts, which is a pretty small scale. 517 00:39:32,620 --> 00:39:35,590 So you should be comparing that to the binding energy 518 00:39:35,590 --> 00:39:39,550 of the ground state of hydrogen of 13.6 electron volts. 519 00:39:39,550 --> 00:39:41,750 So this is a very small effect. 520 00:39:41,750 --> 00:39:44,890 And you can really think about the relative size. 521 00:39:44,890 --> 00:39:52,550 So the Bohr energy, so that 13.6, formally this 522 00:39:52,550 --> 00:39:58,060 goes like, alpha squared times Me c squared. 523 00:40:01,060 --> 00:40:06,200 Then last time we talked about the S coupling, so the spin 524 00:40:06,200 --> 00:40:10,625 orbit, or fine structure. 525 00:40:14,120 --> 00:40:19,162 And so this one we found went like alpha to the fourth Me C 526 00:40:19,162 --> 00:40:20,660 squared. 527 00:40:20,660 --> 00:40:23,180 So smaller than the binding energy 528 00:40:23,180 --> 00:40:28,070 by a factor of 1 over 137 squared, or about 20,000. 529 00:40:28,070 --> 00:40:29,800 And then this one that we're talking 530 00:40:29,800 --> 00:40:39,210 about here, the hyperfine, this, if you look over here, 531 00:40:39,210 --> 00:40:42,660 this is going like alpha to the fourth Me C 532 00:40:42,660 --> 00:40:46,780 squared times an additional factor of Me over Mp. 533 00:40:49,430 --> 00:40:51,820 And the mass of the proton is about 2,000 times 534 00:40:51,820 --> 00:40:53,240 the mass of the electron. 535 00:40:53,240 --> 00:40:55,710 And so this again is-- oh, sorry. 536 00:40:55,710 --> 00:40:58,320 This is alpha to the fourth. 537 00:40:58,320 --> 00:41:03,210 So this is suppressed by about another factor of 2,000. 538 00:41:03,210 --> 00:41:05,270 You can go further. 539 00:41:05,270 --> 00:41:07,440 There are further corrections to this 540 00:41:07,440 --> 00:41:10,620 in something called the Lamb shift, which 541 00:41:10,620 --> 00:41:12,740 we won't say anything else about. 542 00:41:12,740 --> 00:41:15,700 This goes like alpha to the fifth Me C squared. 543 00:41:15,700 --> 00:41:18,260 And there's a whole host of higher order corrections. 544 00:41:18,260 --> 00:41:20,520 People actually calculate these energy levels 545 00:41:20,520 --> 00:41:23,530 to very high precision. 546 00:41:23,530 --> 00:41:25,650 But we won't do any more. 547 00:41:25,650 --> 00:41:33,850 So this transition here is actually 548 00:41:33,850 --> 00:41:36,200 astrophysically extremely important. 549 00:41:36,200 --> 00:41:39,170 So if we think about something sitting in the state here, 550 00:41:39,170 --> 00:41:43,890 it can decay down to the ground state by emitting a photon. 551 00:41:43,890 --> 00:41:51,765 So we can decay from J equals 1 to J equals 0 by a photon. 552 00:41:54,500 --> 00:42:01,030 And that photon will have a wavelength 553 00:42:01,030 --> 00:42:04,830 that corresponds exactly to this energy difference. 554 00:42:04,830 --> 00:42:07,230 And so that wavelength is going to be, 555 00:42:07,230 --> 00:42:10,970 we can write it as c over the frequency, 556 00:42:10,970 --> 00:42:18,040 or hc-- oh, hc not h bar c-- hc over this delta e hyperfine. 557 00:42:18,040 --> 00:42:21,510 If you plug numbers into this you 558 00:42:21,510 --> 00:42:29,660 find out that this is approximately 21.1 centimeters. 559 00:42:29,660 --> 00:42:38,540 And the frequency is 1,420 megahertz. 560 00:42:38,540 --> 00:42:40,880 And so right in the middle-- well, 561 00:42:40,880 --> 00:42:43,740 at the end-- of the FM band in radio. 562 00:42:43,740 --> 00:42:46,400 So theses are radio waves. 563 00:42:46,400 --> 00:42:52,750 So the size of this wavelength is firstly important 564 00:42:52,750 --> 00:42:55,690 because it's large compared to the size of dust 565 00:42:55,690 --> 00:42:56,910 in the universe. 566 00:42:56,910 --> 00:42:58,640 So dust is little stuff. 567 00:42:58,640 --> 00:43:02,230 So this is essentially goes straight through dust. 568 00:43:02,230 --> 00:43:04,860 So these photons will go straight through dust. 569 00:43:04,860 --> 00:43:10,910 The other important thing is that you probably 570 00:43:10,910 --> 00:43:14,170 know that there's a cosmic microwave background 571 00:43:14,170 --> 00:43:17,490 radiation in the universe, that's essentially very 572 00:43:17,490 --> 00:43:18,710 close to constant everywhere. 573 00:43:28,181 --> 00:43:29,680 And so we have, essentially, we have 574 00:43:29,680 --> 00:43:36,070 a temperature of 2.7 Kelvin. 575 00:43:36,070 --> 00:43:43,760 That corresponds to photons with an energy kT, which 576 00:43:43,760 --> 00:43:48,790 is about 0.2 times 10 to the minus 3 electron volts. 577 00:43:48,790 --> 00:43:51,830 So milli electron volts. 578 00:43:51,830 --> 00:43:56,550 But if you compare this number to what we have here, 579 00:43:56,550 --> 00:44:00,250 this cosmic background microwave radiation 580 00:44:00,250 --> 00:44:03,570 can excite hydrogen from here up to here. 581 00:44:03,570 --> 00:44:05,555 There's enough energy for one of those photons 582 00:44:05,555 --> 00:44:07,380 to come along, knock the hydrogen atom, 583 00:44:07,380 --> 00:44:09,270 and excite it up to here. 584 00:44:09,270 --> 00:44:11,940 And then it will decay and will emit 585 00:44:11,940 --> 00:44:18,110 this beautiful 21 centimeter line, 586 00:44:18,110 --> 00:44:19,680 which will go through all the dust. 587 00:44:19,680 --> 00:44:22,721 And so we can actually see the universe in this 21 centimeter 588 00:44:22,721 --> 00:44:23,220 line. 589 00:44:27,270 --> 00:44:31,490 Even more remarkable is, we can't calculate this 590 00:44:31,490 --> 00:44:34,360 at the moment, but you can show that the lifetime 591 00:44:34,360 --> 00:44:42,470 for this transition to happen is about 10 to the 7 years. 592 00:44:46,380 --> 00:44:49,740 So we can never measure that in a lab. 593 00:44:49,740 --> 00:44:53,985 But because these hydrogen atoms can 594 00:44:53,985 --> 00:44:55,360 be wandering around the universe, 595 00:44:55,360 --> 00:44:58,870 not interacting for that long, then they can emit. 596 00:44:58,870 --> 00:45:01,620 And so we can see that. 597 00:45:01,620 --> 00:45:05,120 This was first observed in about 1951, 598 00:45:05,120 --> 00:45:07,340 and is the first way that we actually 599 00:45:07,340 --> 00:45:12,130 saw that the galaxy had spiral shaped arms. 600 00:45:12,130 --> 00:45:13,700 So it's pretty important. 601 00:45:13,700 --> 00:45:16,120 And another nice thing about this 602 00:45:16,120 --> 00:45:19,020 is if you think about another galaxy-- 603 00:45:19,020 --> 00:45:21,570 so let me just draw another galaxy, a spiral 604 00:45:21,570 --> 00:45:25,580 galaxy somewhere else, like this. 605 00:45:25,580 --> 00:45:29,750 Let's have us over here looking at this galaxy from side on. 606 00:45:29,750 --> 00:45:31,660 This galaxy is rotating. 607 00:45:31,660 --> 00:45:34,560 So this one's moving this way, this one to moving this way. 608 00:45:34,560 --> 00:45:36,870 There's hydrogen over here and over here. 609 00:45:36,870 --> 00:45:39,170 And so we get these photons coming over here, 610 00:45:39,170 --> 00:45:41,840 and photons coming to us from there. 611 00:45:41,840 --> 00:45:43,933 But what's going to be different about these? 612 00:45:43,933 --> 00:45:44,920 AUDIENCE: [INAUDIBLE] 613 00:45:44,920 --> 00:45:45,810 PROFESSOR: Their rate shifted, right? 614 00:45:45,810 --> 00:45:46,770 The Doppler shifted. 615 00:45:46,770 --> 00:45:51,502 So this is my 21 centimeter photon. 616 00:45:51,502 --> 00:45:52,710 But they get Doppler shifted. 617 00:45:52,710 --> 00:45:54,500 And so we can measure the difference 618 00:45:54,500 --> 00:45:57,190 in the frequencies of those. 619 00:45:57,190 --> 00:45:58,725 What does that tell us? 620 00:45:58,725 --> 00:46:00,210 AUDIENCE: [INAUDIBLE] 621 00:46:00,210 --> 00:46:03,710 PROFESSOR: How fast this galaxy is spinning, right? 622 00:46:03,710 --> 00:46:05,090 And so one very interesting thing 623 00:46:05,090 --> 00:46:08,050 you find from that is if you look at the galaxy 624 00:46:08,050 --> 00:46:09,820 and count how many stars are in it, 625 00:46:09,820 --> 00:46:13,130 and essentially work out how massive 626 00:46:13,130 --> 00:46:15,910 that galaxy is, the speed of rotation 627 00:46:15,910 --> 00:46:19,750 here is actually-- that you measure from these hydrogen 628 00:46:19,750 --> 00:46:22,480 lines-- is that it's actually faster 629 00:46:22,480 --> 00:46:26,740 than the escape velocity of the matter. 630 00:46:26,740 --> 00:46:30,930 And so if all that was there was the visible matter, then 631 00:46:30,930 --> 00:46:32,630 the thing would just fall apart. 632 00:46:32,630 --> 00:46:34,421 And so this actually tells you that there's 633 00:46:34,421 --> 00:46:37,650 dark matter that doesn't interact 634 00:46:37,650 --> 00:46:43,860 with visible light, that's kind of all over here. 635 00:46:43,860 --> 00:46:45,740 So that's kind of a pretty interesting thing. 636 00:46:48,580 --> 00:46:53,450 So that, I think, yeah. 637 00:46:53,450 --> 00:46:55,074 So any questions about that? 638 00:46:55,074 --> 00:46:56,740 We're going to move on to another topic. 639 00:46:59,770 --> 00:47:00,270 Yep? 640 00:47:00,270 --> 00:47:01,853 AUDIENCE: You said earlier [INAUDIBLE] 641 00:47:01,853 --> 00:47:02,820 it's 10 to the 7 years. 642 00:47:02,820 --> 00:47:05,907 Does that mean it takes an average of 10 643 00:47:05,907 --> 00:47:09,792 to the 7 years for the cosmic microwave background energy 644 00:47:09,792 --> 00:47:13,450 to shift back [INAUDIBLE] 645 00:47:13,450 --> 00:47:15,600 PROFESSOR: No, it's really, if I just 646 00:47:15,600 --> 00:47:18,487 took hydrogen in this state, and took a sample of it, 647 00:47:18,487 --> 00:47:20,320 that's how long it would take for half of it 648 00:47:20,320 --> 00:47:23,240 to have gone and made the decay. 649 00:47:23,240 --> 00:47:25,125 So it can happen much faster. 650 00:47:28,250 --> 00:47:30,090 And there's a lot of it in the universe. 651 00:47:30,090 --> 00:47:33,730 So there's many more than 10 to the 7 atoms 652 00:47:33,730 --> 00:47:34,900 of hydrogen in the universe. 653 00:47:34,900 --> 00:47:37,800 So we see more than one of these things every year. 654 00:47:37,800 --> 00:47:40,870 So if you were just looking at one of them 655 00:47:40,870 --> 00:47:42,794 you would have to wait a long time. 656 00:47:42,794 --> 00:47:45,335 AUDIENCE: But the thing about the cosmic microwave background 657 00:47:45,335 --> 00:47:49,220 is to go from [INAUDIBLE] from 0 to 0 up to get [INAUDIBLE] 658 00:47:49,220 --> 00:47:51,282 PROFESSOR: Right so I mean the energy is large 659 00:47:51,282 --> 00:47:51,990 compared to that. 660 00:47:51,990 --> 00:47:54,709 So it will typically knock you up into an even higher state. 661 00:47:54,709 --> 00:47:56,250 And then you will kind of decay down. 662 00:47:56,250 --> 00:47:59,000 But then this last decay is-- because this lifetime 663 00:47:59,000 --> 00:48:02,500 is very long, the width of this line is also very, very narrow. 664 00:48:06,240 --> 00:48:10,510 So now let's talk more about adding angular momenta. 665 00:48:13,235 --> 00:48:14,855 Oh, maybe I should have left that up. 666 00:48:14,855 --> 00:48:15,355 Too late. 667 00:48:30,860 --> 00:48:35,060 So we're going to do this in a more general sense. 668 00:48:35,060 --> 00:48:38,050 So we're going to take J1, some spin 669 00:48:38,050 --> 00:48:51,520 J1 that has states J1, M1, with M1 equals minus J up to J. 670 00:48:51,520 --> 00:48:53,680 And so we're sort of talking about something 671 00:48:53,680 --> 00:48:56,600 like the electron in the hydrogen atom. 672 00:48:56,600 --> 00:48:59,967 And so that's not in any particular orbital angular 673 00:48:59,967 --> 00:49:00,466 momentum. 674 00:49:03,090 --> 00:49:05,450 So we can talk about that the Hilbert space 675 00:49:05,450 --> 00:49:06,600 that this thing lives in. 676 00:49:10,150 --> 00:49:15,710 So we can think about particle 1 with angular momentum J1. 677 00:49:15,710 --> 00:49:20,430 And this is basically spanned by the states 678 00:49:20,430 --> 00:49:25,940 J1, M1 of these [INAUDIBLE]. 679 00:49:25,940 --> 00:49:32,640 We can take another system with another J2, 680 00:49:32,640 --> 00:49:41,198 and this is going to have states J2, M2, with M2-- that 681 00:49:41,198 --> 00:49:44,800 should be J1's there. 682 00:49:44,800 --> 00:49:47,600 And that would similarly talk about some Hilbert space 683 00:49:47,600 --> 00:49:50,110 of some fixed angular momentum. 684 00:49:50,110 --> 00:49:54,110 If we want to talk about the electron in a hydrogen 685 00:49:54,110 --> 00:49:56,690 atom, where it doesn't have a fixed angular momentum, what we 686 00:49:56,690 --> 00:50:00,140 really want to talk about is the Hilbert space H1, 687 00:50:00,140 --> 00:50:03,470 which is the sum over J1 of these Hilbert spaces. 688 00:50:06,680 --> 00:50:09,870 And so this is talking about-- this Hilbert space 689 00:50:09,870 --> 00:50:12,610 contains every state the electron 690 00:50:12,610 --> 00:50:14,920 can have in a hydrogen atom. 691 00:50:14,920 --> 00:50:18,350 It can have all the different angular momenta. 692 00:50:18,350 --> 00:50:21,430 And similarly we could do that for J2. 693 00:50:21,430 --> 00:50:29,850 We can define J, which is J1 plus J2, as you might guess. 694 00:50:29,850 --> 00:50:35,440 And really this you should think of as J1 tensor 695 00:50:35,440 --> 00:50:45,290 the identity plus the identity tensor J2, where this one is 696 00:50:45,290 --> 00:50:47,370 acting on things in this Hilbert space, 697 00:50:47,370 --> 00:50:50,170 and the 1 here is acting on things in this Hilbert space. 698 00:50:50,170 --> 00:50:55,830 And similarly there's an H2, J2 that 699 00:50:55,830 --> 00:50:59,090 goes along with these guys. 700 00:50:59,090 --> 00:51:01,690 And so this operator, this big J, 701 00:51:01,690 --> 00:51:10,660 is something that acts on vectors 702 00:51:10,660 --> 00:51:21,835 in things in this tensor product space. 703 00:51:24,730 --> 00:51:28,654 Actually I should label this with a J1. 704 00:51:28,654 --> 00:51:30,880 It also acts on things in the full space, 705 00:51:30,880 --> 00:51:33,070 but we can talk just about that one. 706 00:51:45,610 --> 00:51:48,140 So now we might want to construct 707 00:51:48,140 --> 00:51:49,330 a basis for this space. 708 00:51:52,320 --> 00:51:57,710 And we conversely construct an uncoupled basis 709 00:51:57,710 --> 00:52:01,230 which is just take the basis elements of each of the spaces 710 00:52:01,230 --> 00:52:03,700 and multiply them. 711 00:52:03,700 --> 00:52:10,000 So we would have J1, J2, M1, M2. 712 00:52:17,670 --> 00:52:20,240 We'd have the states here. 713 00:52:20,240 --> 00:52:24,285 And if we just ask what our various-- 714 00:52:39,430 --> 00:52:43,230 J1 just gives us h bar squared J, J plus 1. 715 00:52:48,020 --> 00:53:01,560 And this one gives us h bar M1 h bar squared times our state. 716 00:53:08,690 --> 00:53:11,420 And so we can think about all of these. 717 00:53:11,420 --> 00:53:13,347 And this is what we label our state with. 718 00:53:13,347 --> 00:53:15,180 And that's because these form a complete set 719 00:53:15,180 --> 00:53:16,221 of commuting observables. 720 00:53:18,720 --> 00:53:20,820 And we'll just call this the A set. 721 00:53:33,500 --> 00:53:38,090 We can also talk about our operator J 722 00:53:38,090 --> 00:53:39,620 and use that to define our basis. 723 00:53:47,420 --> 00:53:49,690 And let's just be a little explicit 724 00:53:49,690 --> 00:53:51,760 about what J squared is going to be. 725 00:53:51,760 --> 00:54:01,620 So this is J1 tensor identity plus 1 tensor J2. 726 00:54:06,004 --> 00:54:07,590 And the same thing here. 727 00:54:14,500 --> 00:54:19,530 If you expand this out you get J1 squared tensor identity 728 00:54:19,530 --> 00:54:26,380 plus 1 tensor J2 squared plus the dot product, which 729 00:54:26,380 --> 00:54:34,020 we can write as the sum of J1k tensor J2k. 730 00:54:37,880 --> 00:54:42,030 And because of this piece here, J squared 731 00:54:42,030 --> 00:54:45,857 doesn't commute with J1z, for example. 732 00:54:52,330 --> 00:54:55,410 So we can't add this operator to our list 733 00:54:55,410 --> 00:54:56,600 of operators over there. 734 00:55:00,420 --> 00:55:07,262 And similarly J2z J squared is not equal to 0. 735 00:55:07,262 --> 00:55:08,970 So if we want to talk about this operator 736 00:55:08,970 --> 00:55:11,740 we have to throw both of those away. 737 00:55:11,740 --> 00:55:22,290 But there is an operator total Jz that commutes with J 738 00:55:22,290 --> 00:55:23,060 squared. 739 00:55:23,060 --> 00:55:27,320 And it also commutes with J1 squared and J2 squared. 740 00:55:27,320 --> 00:55:32,470 And so we can have another complete set 741 00:55:32,470 --> 00:55:35,110 of commuting observables B that's 742 00:55:35,110 --> 00:55:43,490 equal to J1 squared, J2 squared, J squared, and Jz. 743 00:55:50,960 --> 00:55:58,630 And so if there are observables then the natural basis 744 00:55:58,630 --> 00:56:01,540 is to label them by the eigenvalues here. 745 00:56:01,540 --> 00:56:20,230 So we're going to have a J1, a J2, then a J and an M. 746 00:56:20,230 --> 00:56:23,050 And so this is the coupled basis. 747 00:56:23,050 --> 00:56:27,550 Now both of these bases are equally good. 748 00:56:27,550 --> 00:56:30,940 They both span the full space. 749 00:56:30,940 --> 00:56:33,180 They're both orthogonal, orthonormal. 750 00:56:46,040 --> 00:56:47,810 And so we can actually write one basis 751 00:56:47,810 --> 00:56:50,110 into in terms of the other one. 752 00:56:50,110 --> 00:56:52,284 And that's the generic problem that we 753 00:56:52,284 --> 00:56:53,700 are trying to do when we're trying 754 00:56:53,700 --> 00:56:55,500 to write what we did over here before, 755 00:56:55,500 --> 00:56:58,070 when we did spin 1/2 cross spin 1/2. 756 00:56:58,070 --> 00:57:01,530 We're trying to write those products states 757 00:57:01,530 --> 00:57:03,530 in terms of the coupled basis. 758 00:57:21,430 --> 00:57:23,350 Well, they're both orthonormal basis. 759 00:57:27,590 --> 00:57:31,830 So I can expand J1, J2. 760 00:57:39,860 --> 00:57:44,470 Well actually, maybe I'll say one more thing first. 761 00:57:44,470 --> 00:57:49,850 So being orthonormal means that, for example, sum over J1, J2-- 762 00:58:03,375 --> 00:58:04,700 This is 1, right? 763 00:58:04,700 --> 00:58:08,920 You can resolve the identity in terms of these states. 764 00:58:08,920 --> 00:58:17,000 And this is the identity on this Hilbert space. 765 00:58:17,000 --> 00:58:19,100 I can also think about the identity just 766 00:58:19,100 --> 00:58:25,190 on this smaller Hilbert space, where the J1 and J2 are fixed. 767 00:58:25,190 --> 00:58:39,905 And so I can actually write it's the identity operator. 768 00:58:53,790 --> 00:58:56,750 So because every state in this space 769 00:58:56,750 --> 00:59:00,130 has J1 equal to some fixed value and J2 equal to some fixed 770 00:59:00,130 --> 00:59:03,090 value, then an identity in that thing 771 00:59:03,090 --> 00:59:07,240 is just somewhere over the M's, because they're the only things 772 00:59:07,240 --> 00:59:09,570 there. 773 00:59:09,570 --> 00:59:17,420 So using this, because I know that the state J1, J2, 774 00:59:17,420 --> 00:59:21,280 Jm has some fixed value of J1 and J2, 775 00:59:21,280 --> 00:59:23,160 I can write this as a sum. 776 00:59:25,860 --> 00:59:30,275 I can use this form of the identity. 777 00:59:45,940 --> 00:59:50,010 So I've written my coupled basis in terms of the uncoupled basis 778 00:59:50,010 --> 00:59:50,980 here. 779 00:59:50,980 --> 00:59:53,110 And these are just coefficients. 780 00:59:53,110 --> 01:00:02,005 These are called Clebsch-Gordan coefficients. 781 01:00:05,169 --> 01:00:07,710 They're just numbers like square root 2 and things like this. 782 01:00:07,710 --> 01:00:11,095 And I tell you how to do this decomposition. 783 01:00:15,690 --> 01:00:21,415 So they have various properties. 784 01:00:24,930 --> 01:00:26,810 Firstly, sometimes you also see them 785 01:00:26,810 --> 01:00:41,120 written as C of J1J2J colon M1M2M 786 01:00:41,120 --> 01:00:43,230 and various other notations. 787 01:00:43,230 --> 01:00:45,631 So basically things with six indices 788 01:00:45,631 --> 01:00:47,130 are probably going to be these guys. 789 01:00:51,990 --> 01:00:54,280 So they have various properties. 790 01:00:58,650 --> 01:01:11,830 The first property is they vanish 791 01:01:11,830 --> 01:01:19,210 if M is not equal to M1 plus M2. 792 01:01:19,210 --> 01:01:21,185 And this is actually very easy to prove. 793 01:01:24,560 --> 01:01:32,570 So remember that Jz is just going to be J1z plus J2z. 794 01:01:35,560 --> 01:01:42,895 So as an operator I can write Jz minus J1z minus J2z. 795 01:01:46,100 --> 01:01:47,205 And what is that operator? 796 01:01:53,054 --> 01:01:54,355 It's just 0, right? 797 01:01:57,119 --> 01:01:58,035 This is equal to that. 798 01:01:58,035 --> 01:01:59,300 So this is 0. 799 01:01:59,300 --> 01:02:03,010 So I can put this 0 anywhere I want and I'll still get 0. 800 01:02:03,010 --> 01:02:09,100 So let's put this between-- so this is 0-- put it 801 01:02:09,100 --> 01:02:15,610 between J1, J2, Jm on this side. 802 01:02:15,610 --> 01:02:17,510 So a coupled state here. 803 01:02:17,510 --> 01:02:20,140 And on this side I'll put it between the uncoupled state, 804 01:02:20,140 --> 01:02:22,920 J1, J2, M1, M2. 805 01:02:26,870 --> 01:02:32,520 So this state is an eigenstate of Jz. 806 01:02:32,520 --> 01:02:36,620 And this state is an eigenstate of J1z and J2z. 807 01:02:36,620 --> 01:02:40,600 So I can act to the right with Jz, 808 01:02:40,600 --> 01:02:44,100 and act to the left with these J1z and J2z, 809 01:02:44,100 --> 01:02:46,595 and they have mission operators. 810 01:02:46,595 --> 01:02:50,830 And I know because this is 0, this whole thing is 0. 811 01:02:50,830 --> 01:02:53,030 So then act this one on these guys 812 01:02:53,030 --> 01:02:56,040 and these two back this way. 813 01:02:56,040 --> 01:03:00,737 And so you see that gives me h bar. 814 01:03:00,737 --> 01:03:02,320 And then I get this one acting on here 815 01:03:02,320 --> 01:03:05,925 gives me M. And J1 acting on here gives me M1. 816 01:03:23,285 --> 01:03:29,820 And if M is not equal to M1 plus M2, then this term isn't 0. 817 01:03:29,820 --> 01:03:33,997 But the whole thing is so that has to be 0. 818 01:03:33,997 --> 01:03:36,310 So that's QED. 819 01:04:01,890 --> 01:04:29,760 The second property is that-- So they only allow values of J 820 01:04:29,760 --> 01:04:31,660 fall in this range here. 821 01:04:31,660 --> 01:04:36,590 And each J occurs once. 822 01:04:41,730 --> 01:04:43,640 Now one way to think about this is 823 01:04:43,640 --> 01:04:45,800 to think of these things as vectors. 824 01:04:45,800 --> 01:04:51,550 So you have vector J1, and then from the point of this 825 01:04:51,550 --> 01:04:52,785 you can have vector J2. 826 01:04:52,785 --> 01:04:55,690 But it can go into an arbitrary direction. 827 01:04:55,690 --> 01:05:00,400 So it can go up here, or it can go like this. 828 01:05:00,400 --> 01:05:02,570 These are meant to be the same length. 829 01:05:02,570 --> 01:05:04,430 And I can come all the way down here. 830 01:05:04,430 --> 01:05:10,390 But I can could only sit on integer points. 831 01:05:10,390 --> 01:05:12,090 And so this is kind of J2. 832 01:05:12,090 --> 01:05:15,660 And so the length of this thing here would be the length of J1 833 01:05:15,660 --> 01:05:16,700 plus the length of J2. 834 01:05:16,700 --> 01:05:17,870 So it would be this. 835 01:05:17,870 --> 01:05:21,460 And then the length of up to here would be this one. 836 01:05:21,460 --> 01:05:25,810 And then all of the other ones are in between. 837 01:05:25,810 --> 01:05:29,690 But you can also just look at the multiplicities 838 01:05:29,690 --> 01:05:30,840 of the different states. 839 01:05:30,840 --> 01:05:37,510 So if we look at the uncoupled basis-- 840 01:05:37,510 --> 01:05:42,242 so the first state, which was J equals J1, 841 01:05:42,242 --> 01:05:47,820 there are two J1 plus 1 states, because it 842 01:05:47,820 --> 01:05:52,480 can have all of the M values from J1 down to minus J1. 843 01:05:52,480 --> 01:05:57,600 And the other one can have two J2 plus 1 states. 844 01:05:57,600 --> 01:06:00,780 So that's the total number of states that I expect to have. 845 01:06:00,780 --> 01:06:05,070 So now let's assume this is correct 846 01:06:05,070 --> 01:06:09,120 and ask what the N coupled is. 847 01:06:09,120 --> 01:06:18,591 So this would be the sum over J equals mod J1 minus J2 up 848 01:06:18,591 --> 01:06:23,900 to J1 plus J2 of 2J plus 1. 849 01:06:23,900 --> 01:06:29,405 And let's assume that J1 is greater than or equal to J2, 850 01:06:29,405 --> 01:06:34,880 just to stop writing absolute values all the time. 851 01:06:34,880 --> 01:06:39,860 So we can write this as the difference of two sums, 852 01:06:39,860 --> 01:06:47,700 J equals 0 to J1 of-- J1 plus J2-- of 2J 853 01:06:47,700 --> 01:06:52,130 plus 1 minus the sum of J equals 0 854 01:06:52,130 --> 01:07:00,650 to J equals J1 minus J2 minus 1 of 2J plus 1. 855 01:07:00,650 --> 01:07:07,680 And if you go through-- so this is just N,N plus 1 over 2 856 01:07:07,680 --> 01:07:10,070 for each of these things. 857 01:07:10,070 --> 01:07:30,639 You end up with, well, you end up with this. 858 01:07:30,639 --> 01:07:31,930 You end up with the same thing. 859 01:07:31,930 --> 01:07:33,705 And so this is at least consistent 860 01:07:33,705 --> 01:07:36,730 that the number of states that we have 861 01:07:36,730 --> 01:07:38,370 is consistent with choosing this. 862 01:07:46,060 --> 01:07:49,010 One other thing we can do is look at the top state, 863 01:07:49,010 --> 01:07:50,135 and just see if that works. 864 01:07:53,040 --> 01:07:54,855 See if that has the right properties. 865 01:07:59,070 --> 01:08:09,008 So because we know that the J1, J2, J equals J1 plus J2, 866 01:08:09,008 --> 01:08:11,240 M equals J1 plus J2. 867 01:08:11,240 --> 01:08:16,140 So the maximal state, the only way we can make this 868 01:08:16,140 --> 01:08:25,600 is to take J1, J2, M1 equals J1, M2 equals J2. 869 01:08:25,600 --> 01:08:29,939 Our spins are completely aligned in the total up direction. 870 01:08:29,939 --> 01:08:30,594 Yeah? 871 01:08:30,594 --> 01:08:33,010 AUDIENCE: Sir, would you be able to write a little larger? 872 01:08:33,010 --> 01:08:34,560 PROFESSOR: Yes, sorry. 873 01:08:34,560 --> 01:08:35,420 OK, yeah. 874 01:08:35,420 --> 01:08:36,670 That's why I like a big chalk. 875 01:08:36,670 --> 01:08:39,844 But we've run out of big chalk, so I'll try. 876 01:08:44,450 --> 01:08:50,439 So we know, also, that J squared is 877 01:08:50,439 --> 01:08:55,125 equal to J1 squared plus J2 squared plus the dot product. 878 01:09:00,779 --> 01:09:06,300 We can write that out as J1 squared plus J2 squared 879 01:09:06,300 --> 01:09:22,531 plus 2J1z J2z plus J1 plus J2 minus plus J1 minus J2 plus. 880 01:09:25,279 --> 01:09:30,569 And then we can ask what does J squared on this state give? 881 01:09:33,740 --> 01:09:46,039 And this is J1, J2, J1 plus J2, J1 plus J2. 882 01:09:48,913 --> 01:09:53,490 So J squared, so we know what that should be. 883 01:09:53,490 --> 01:09:58,940 That should return J1 plus J2 times J1 plus J2 plus 1 times 884 01:09:58,940 --> 01:10:02,140 h bar squared, because J is the good quantum number. 885 01:10:02,140 --> 01:10:04,260 But let's let it act on this piece. 886 01:10:04,260 --> 01:10:10,300 So this equals J1 squared plus J2 887 01:10:10,300 --> 01:10:25,140 squared plus 2J1z J2z plus J1 plus J2 minus plus J1 minus J2 888 01:10:25,140 --> 01:10:32,406 plus acting on J1, J2, J1, J2. 889 01:10:32,406 --> 01:10:35,430 So this state here. 890 01:10:35,430 --> 01:10:37,410 So we know how that acts. 891 01:10:37,410 --> 01:10:39,520 So this one gives us-- everything 892 01:10:39,520 --> 01:10:41,290 gives us an h bar squared. 893 01:10:41,290 --> 01:10:46,390 This gives us J1 J1 plus 1, for this term, 894 01:10:46,390 --> 01:10:54,510 plus J2 J2 plus 1 for the second term. 895 01:10:54,510 --> 01:10:57,970 Each of these gives us the M quantum numbers. 896 01:10:57,970 --> 01:10:59,450 But that's J1 J2. 897 01:10:59,450 --> 01:11:05,930 So this is plus 2J1, J2. 898 01:11:05,930 --> 01:11:08,700 And now what does this one do? 899 01:11:08,700 --> 01:11:10,661 J1 plus on this state. 900 01:11:10,661 --> 01:11:11,452 AUDIENCE: Kills it. 901 01:11:11,452 --> 01:11:12,577 PROFESSOR: Kills it, right. 902 01:11:12,577 --> 01:11:16,200 Because it's trying to raise the M component of 1, 903 01:11:16,200 --> 01:11:17,800 and it's already maximal. 904 01:11:17,800 --> 01:11:20,800 And this one, J2 plus, also kills it. 905 01:11:20,800 --> 01:11:25,250 So you get plus 0 plus 0 times the state. 906 01:11:29,460 --> 01:11:33,590 So if you rearrange all of this you actually 907 01:11:33,590 --> 01:11:44,300 find you can write this as J1 plus J2, J1 908 01:11:44,300 --> 01:11:52,030 plus J2 plus 1 times the state, which is what you want. 909 01:11:55,400 --> 01:12:00,980 So the J squared operator acting in the coupled basis gives-- 910 01:12:00,980 --> 01:12:02,700 well, acting in the uncoupled basis 911 01:12:02,700 --> 01:12:04,658 gives you what you expect in the coupled basis. 912 01:12:10,180 --> 01:12:15,505 So now I need a big blackboard. 913 01:12:19,310 --> 01:12:23,400 So let's do an example of multiplying two things. 914 01:12:23,400 --> 01:12:25,280 So let's write out a multiplet. 915 01:12:25,280 --> 01:12:28,440 So we're going to take J1, and we're 916 01:12:28,440 --> 01:12:31,950 going to have J1 bigger than J2 here. 917 01:12:31,950 --> 01:12:38,130 So we've got J1 J1, J1 J1 minus 1. 918 01:12:44,610 --> 01:12:53,420 And then somewhere down here I've got J1 and 2J2 minus J1. 919 01:12:53,420 --> 01:12:58,110 And then all the way down to J1 minus J1. 920 01:13:02,220 --> 01:13:05,925 So this has two J1 plus 1 states. 921 01:13:09,960 --> 01:13:12,660 And I'm going to tensor that was another multiplet, 922 01:13:12,660 --> 01:13:15,350 with my J2 multiplet, which is going to be smaller. 923 01:13:15,350 --> 01:13:17,910 So I'm going to have J2 J2. 924 01:13:23,929 --> 01:13:25,220 Oh, maybe I'll put one more in. 925 01:13:28,570 --> 01:13:32,220 And down here we've got J2 comma minus J2. 926 01:13:35,270 --> 01:13:40,175 And so here we have two J2 plus 1 states. 927 01:13:47,160 --> 01:13:54,400 And importantly, this left hand side 928 01:13:54,400 --> 01:14:03,775 has two J1 minus J2 more states than the right hand side. 929 01:14:06,820 --> 01:14:11,830 Just counting those states that's pretty obvious. 930 01:14:11,830 --> 01:14:14,840 So now let's start multiplying these things, 931 01:14:14,840 --> 01:14:22,680 and forming states of particular values of M, the total M. 932 01:14:22,680 --> 01:14:29,730 So if we say we want M equals J1 plus J2 what can we do? 933 01:14:29,730 --> 01:14:30,750 How can we make that? 934 01:14:35,590 --> 01:14:38,190 So we have to take the top state in each case. 935 01:14:38,190 --> 01:14:41,740 Because if I take this one and I take this M value, 936 01:14:41,740 --> 01:14:43,209 I can't get up to this, right? 937 01:14:43,209 --> 01:14:44,750 So there's only one way to make this. 938 01:14:48,840 --> 01:14:51,370 So I'm going to draw a diagram of this. 939 01:14:51,370 --> 01:14:54,120 We're going to have a one state there. 940 01:14:54,120 --> 01:15:00,425 The next M value, J1 plus J2 minus 1, how can I make that? 941 01:15:03,770 --> 01:15:05,640 So I can start with this state, and I 942 01:15:05,640 --> 01:15:09,696 will multiply it by this one, right? 943 01:15:09,696 --> 01:15:11,590 Or, what else can I do? 944 01:15:14,178 --> 01:15:15,678 AUDIENCE: Start with the second down 945 01:15:15,678 --> 01:15:17,810 on the left and tensor with the top? 946 01:15:17,810 --> 01:15:18,810 PROFESSOR: That's right. 947 01:15:18,810 --> 01:15:19,986 So I take those two. 948 01:15:19,986 --> 01:15:20,985 So there are two states. 949 01:15:24,820 --> 01:15:29,320 And those two states are just two linear combinations. 950 01:15:29,320 --> 01:15:33,165 So let me draw two dots here, I can form two states. 951 01:15:36,130 --> 01:15:40,565 Keep going-- minus 2, I get three states. 952 01:15:44,760 --> 01:15:51,470 And let me try and draw lines here to guide this stuff. 953 01:15:56,500 --> 01:15:57,910 OK, I'm not going to keep going. 954 01:15:57,910 --> 01:16:02,660 But at some point we get-- what's 955 01:16:02,660 --> 01:16:04,650 the largest number of states of a given M 956 01:16:04,650 --> 01:16:07,536 I can make going to be? 957 01:16:07,536 --> 01:16:08,630 Can anyone see that? 958 01:16:11,840 --> 01:16:13,458 AUDIENCE: 2J2 plus 1? 959 01:16:13,458 --> 01:16:15,840 PROFESSOR: 2J2 plus 2, right, because I've 960 01:16:15,840 --> 01:16:18,820 got 2J2 plus 1 states here. 961 01:16:18,820 --> 01:16:21,210 And I'm taking one of these plus one of these 962 01:16:21,210 --> 01:16:30,510 will give me-- so down here I'll have an M equals-- what is it-- 963 01:16:30,510 --> 01:16:32,342 J1 minus J2. 964 01:16:32,342 --> 01:16:37,530 And here I have 2J2 plus 1 states. 965 01:16:37,530 --> 01:16:41,320 And so let me kind of draw some of these states in. 966 01:16:41,320 --> 01:16:42,735 And then dot, dot, dot. 967 01:16:42,735 --> 01:16:49,170 And then over here we end up with this guy. 968 01:16:49,170 --> 01:16:51,955 So if I go down to the next one, how many states? 969 01:16:56,630 --> 01:17:02,910 So to form those states I was taking this top state 970 01:17:02,910 --> 01:17:04,280 with the bottom state here. 971 01:17:04,280 --> 01:17:06,640 That gives me J1 minus J2, right? 972 01:17:06,640 --> 01:17:09,590 Or I was taking the second state here with the second to bottom 973 01:17:09,590 --> 01:17:11,200 state here, and so forth. 974 01:17:11,200 --> 01:17:14,910 And then all the way up to here. 975 01:17:14,910 --> 01:17:20,020 Now if I then start shifting things down in this side, 976 01:17:20,020 --> 01:17:22,950 but leave exactly the same things over there, 977 01:17:22,950 --> 01:17:25,250 then I'll lower J by 1. 978 01:17:25,250 --> 01:17:29,030 And I'll keep doing it until I hit the bottom. 979 01:17:29,030 --> 01:17:32,860 And because there's this number of states 980 01:17:32,860 --> 01:17:35,906 more in the right hand side than the left hand side-- hang on, 981 01:17:35,906 --> 01:17:37,030 let me just write this one. 982 01:17:43,370 --> 01:17:45,980 OK, we might need to go onto the next board. 983 01:17:45,980 --> 01:17:47,740 So this keeps on going until I get 984 01:17:47,740 --> 01:17:53,750 to M equals-- I don't remember the number-- M 985 01:17:53,750 --> 01:17:58,780 equals J2 minus J1. 986 01:17:58,780 --> 01:18:01,715 And there are 2J plus 2 plus 1 states of this. 987 01:18:09,880 --> 01:18:14,560 And then once I do that, then I start 988 01:18:14,560 --> 01:18:15,810 having fewer and fewer states. 989 01:18:15,810 --> 01:18:17,190 Because I've gone basically moving out 990 01:18:17,190 --> 01:18:18,398 the bottom of this multiplet. 991 01:18:23,340 --> 01:18:30,950 And so here we have, this is 2 J1 minus J2 plus 1 rows. 992 01:18:33,522 --> 01:18:34,730 And then I start contracting. 993 01:18:39,590 --> 01:18:44,700 So the next one, M equals J2 minus J1 minus 1 994 01:18:44,700 --> 01:18:46,035 has two J2 states. 995 01:18:53,730 --> 01:18:54,820 So then we can keep going. 996 01:18:58,090 --> 01:19:01,410 And this is meant to continue this diagram up here. 997 01:19:01,410 --> 01:19:03,660 So then we keep going down, down, down. 998 01:19:03,660 --> 01:19:11,600 And then we'd have M equals minus J1 plus 1. 999 01:19:11,600 --> 01:19:13,545 And how many states can I make that have that? 1000 01:19:18,530 --> 01:19:24,460 So I need to take this one, and I could take the-- oh, 1001 01:19:24,460 --> 01:19:25,930 where is it? 1002 01:19:25,930 --> 01:19:27,220 Sorry, not this. 1003 01:19:27,220 --> 01:19:28,380 This is not what I mean. 1004 01:19:28,380 --> 01:19:32,620 Minus J1 minus J2 plus 1. 1005 01:19:32,620 --> 01:19:34,220 That's more obvious, right? 1006 01:19:34,220 --> 01:19:35,170 So there's two states. 1007 01:19:39,650 --> 01:19:42,120 And so this picture has kind of-- this line 1008 01:19:42,120 --> 01:19:43,340 starts coming in. 1009 01:19:43,340 --> 01:19:45,140 And now I've got my two states here. 1010 01:19:45,140 --> 01:19:47,430 I have the next one, I've got three states. 1011 01:19:47,430 --> 01:19:53,950 And then finally, M equals minus J1 minus J2, I have one state. 1012 01:19:53,950 --> 01:19:54,900 And so I get this. 1013 01:19:54,900 --> 01:19:58,620 And so you actually-- oh, that was not very well drawn. 1014 01:20:01,760 --> 01:20:07,160 So if you look how many states there are in this first column, 1015 01:20:07,160 --> 01:20:10,460 how many is going to be there? 1016 01:20:10,460 --> 01:20:16,050 So it goes from plus J1 plus J2 to minus J1 minus J2. 1017 01:20:16,050 --> 01:20:23,860 So there's two J1 plus J2 plus 1 states there. 1018 01:20:23,860 --> 01:20:30,660 And here there is two J1 plus J2 minus 1 1019 01:20:30,660 --> 01:20:32,850 plus 1 states in this guy. 1020 01:20:32,850 --> 01:20:36,730 And so this is a J equals J1 plus J2. 1021 01:20:36,730 --> 01:20:42,650 This one is J equals J1 plus J2 minus 1. 1022 01:20:42,650 --> 01:20:46,630 And if you are careful you'd find that this one here, 1023 01:20:46,630 --> 01:20:50,170 the last one here, this has this many states in it, 1024 01:20:50,170 --> 01:20:53,030 two J minus 1 plus 1 states. 1025 01:20:53,030 --> 01:20:59,224 So this is a J equals J1 minus J2. 1026 01:20:59,224 --> 01:21:02,130 And to be completely correct, we put 1027 01:21:02,130 --> 01:21:04,530 an absolute value in case J2 is bigger than J1. 1028 01:21:07,710 --> 01:21:11,955 So this is our full multiplet structure of this system. 1029 01:21:11,955 --> 01:21:15,480 So all of the states in this column 1030 01:21:15,480 --> 01:21:17,945 will transform into each other under rotations, 1031 01:21:17,945 --> 01:21:18,820 and things like this. 1032 01:21:18,820 --> 01:21:21,910 And same for each column, they all form separate multiplets. 1033 01:21:26,930 --> 01:21:36,000 So just some last things before we finish. 1034 01:21:36,000 --> 01:21:38,780 So another property of Clebsch-Gordan 1035 01:21:38,780 --> 01:21:49,330 coefficients we can choose them to be real. 1036 01:21:56,962 --> 01:22:05,580 They satisfy a recursion relation 1037 01:22:05,580 --> 01:22:07,430 but don't have a nice, closed form. 1038 01:22:20,780 --> 01:22:24,250 I think this is in Griffiths. 1039 01:22:24,250 --> 01:22:26,390 It gives you what this recursion relation is. 1040 01:22:26,390 --> 01:22:30,230 I think it does, at least many books do. 1041 01:22:30,230 --> 01:22:36,515 And also, they're tabulated in lots of places. 1042 01:22:40,930 --> 01:22:42,400 So if you need to know the values, 1043 01:22:42,400 --> 01:22:43,680 you can just go and look them up, 1044 01:22:43,680 --> 01:22:45,888 rather than trying to calculate them all necessarily. 1045 01:22:49,620 --> 01:22:52,720 And I think that's all we've got time for. 1046 01:22:52,720 --> 01:22:54,380 So are there any questions about that? 1047 01:22:56,994 --> 01:22:58,410 Any questions about anything else? 1048 01:23:00,980 --> 01:23:01,630 OK, great. 1049 01:23:01,630 --> 01:23:05,190 So we will see you on Wednesday for the last lecture.