1 00:00:00,060 --> 00:00:01,770 The following content is provided 2 00:00:01,770 --> 00:00:04,010 under a Creative Commons license. 3 00:00:04,010 --> 00:00:06,860 Your support will help MIT OpenCourseWare continue 4 00:00:06,860 --> 00:00:10,720 to offer high-quality educational resources for free. 5 00:00:10,720 --> 00:00:13,340 To make a donation, or view additional materials 6 00:00:13,340 --> 00:00:17,195 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,195 --> 00:00:17,820 at ocw.mit.edu. 8 00:00:20,700 --> 00:00:25,600 PROFESSOR: OK, let's get started. 9 00:00:25,600 --> 00:00:28,780 Last class, which also means last week, 10 00:00:28,780 --> 00:00:31,465 we discussed what happens when atom 11 00:00:31,465 --> 00:00:34,560 are exposed to external fields. 12 00:00:34,560 --> 00:00:38,550 Well, you would say, isn't it enough 13 00:00:38,550 --> 00:00:41,190 if you understand atoms in isolation? 14 00:00:41,190 --> 00:00:42,470 Well, not quite. 15 00:00:42,470 --> 00:00:46,670 Because whenever we want to talk to the atoms, whenever 16 00:00:46,670 --> 00:00:50,150 we want to manipulate them or find out in what states 17 00:00:50,150 --> 00:00:53,680 they are, we have to apply external fields. 18 00:00:53,680 --> 00:00:55,555 The way how we communicate with atoms 19 00:00:55,555 --> 00:00:59,071 is through electric, magnetic, and electromagnetic fields. 20 00:00:59,071 --> 00:01:00,570 And therefore, we have to understand 21 00:01:00,570 --> 00:01:03,630 what happens to the structure of atoms 22 00:01:03,630 --> 00:01:06,330 when we expose them to such fields. 23 00:01:06,330 --> 00:01:10,690 We started out with structure in magnetic fields. 24 00:01:10,690 --> 00:01:13,060 And if I just show you this picture, 25 00:01:13,060 --> 00:01:16,010 this is what we discussed last week. 26 00:01:16,010 --> 00:01:19,160 However, I noticed that our discussion 27 00:01:19,160 --> 00:01:20,850 with the different coupling cases-- 28 00:01:20,850 --> 00:01:23,990 fine structure plus magnetic field, hyperfine structure, 29 00:01:23,990 --> 00:01:25,900 strong fields, weak fields. 30 00:01:25,900 --> 00:01:29,490 I noticed that when I was teaching, it's a lot of details 31 00:01:29,490 --> 00:01:32,040 and it looks a little bit messy. 32 00:01:32,040 --> 00:01:35,680 So what I want to do, therefore at the beginning of today, 33 00:01:35,680 --> 00:01:38,380 is I want to give you sort of a summary 34 00:01:38,380 --> 00:01:41,130 that you see the bigger picture. 35 00:01:41,130 --> 00:01:45,060 That you see beyond the details, that what I actually taught you 36 00:01:45,060 --> 00:01:51,360 about atoms in magnetic field is some paradigmatic example 37 00:01:51,360 --> 00:01:52,620 of quantum physics. 38 00:01:52,620 --> 00:01:59,540 What happens if you have two different terms in Hamiltonian 39 00:01:59,540 --> 00:02:02,570 and you have to interpolate between one and the other? 40 00:02:02,570 --> 00:02:04,790 But before I do that, do you have any questions 41 00:02:04,790 --> 00:02:07,754 about magnetic fields, magnetic structure? 42 00:02:11,400 --> 00:02:18,965 Well, then let's try to summarize as follows. 43 00:02:46,270 --> 00:02:49,300 What we have is we have a Hamiltonian. 44 00:02:51,830 --> 00:02:57,740 And it has one part, the hyperfine interaction, 45 00:02:57,740 --> 00:03:03,970 which depends on I dot J. And then 46 00:03:03,970 --> 00:03:07,860 it has an external magnetic field part. 47 00:03:07,860 --> 00:03:11,360 And what couples to the magnetic field, which we assume 48 00:03:11,360 --> 00:03:16,565 is in the J-direction-- in the z-direction 49 00:03:16,565 --> 00:03:20,310 are the z-components of the magnetic moment. 50 00:03:20,310 --> 00:03:23,210 And the z-component of the magnetic moment 51 00:03:23,210 --> 00:03:30,410 are proportional to the mJ or mI quantum number, 52 00:03:30,410 --> 00:03:35,590 to the magnetic quantum number, of the atom and the nucleus. 53 00:03:38,280 --> 00:03:55,130 So in a weak field, it is the hyperfine structure 54 00:03:55,130 --> 00:03:56,690 which dominates. 55 00:03:56,690 --> 00:03:59,970 So in a weak field, we first solve 56 00:03:59,970 --> 00:04:02,470 for the hyperfine structure. 57 00:04:02,470 --> 00:04:08,830 And then we use the eigenfunction 58 00:04:08,830 --> 00:04:11,570 of the hyperfine structure. 59 00:04:11,570 --> 00:04:14,530 And the eigenfunction of the hyperfine structure 60 00:04:14,530 --> 00:04:23,660 have the quantum number F where J and I have coupled to F. 61 00:04:23,660 --> 00:04:27,530 And then we treat the magnetic Zeeman Hamiltonian 62 00:04:27,530 --> 00:04:29,510 perturbatively. 63 00:04:29,510 --> 00:04:33,350 And that led us to the formulation 64 00:04:33,350 --> 00:04:35,890 of the Lande g-factor, gF. 65 00:04:40,970 --> 00:04:49,040 The other case is the strong field case 66 00:04:49,040 --> 00:04:51,775 where the magnetic field dominates. 67 00:04:58,260 --> 00:05:02,470 Then, we simply solve for the hyperfine structure 68 00:05:02,470 --> 00:05:04,430 in the magnetic field. 69 00:05:04,430 --> 00:05:07,070 It's one of those rules in quantum physics, or in physics, 70 00:05:07,070 --> 00:05:09,740 or maybe even in life, first things first. 71 00:05:09,740 --> 00:05:12,170 You should first take care of the big things. 72 00:05:12,170 --> 00:05:14,450 And this is now the magnetic field. 73 00:05:14,450 --> 00:05:18,450 And since the magnetic field Hamiltonian 74 00:05:18,450 --> 00:05:22,520 is diagonalized when we have eigenfunctions where 75 00:05:22,520 --> 00:05:28,900 mJ and mI are good quantum numbers, 76 00:05:28,900 --> 00:05:32,430 this is sort of-- if you ignore the hyperfine coupling, 77 00:05:32,430 --> 00:05:36,130 this is the exact [INAUDIBLE] of the Zeeman term. 78 00:05:36,130 --> 00:05:39,770 And then in perturbation theory, we 79 00:05:39,770 --> 00:05:42,250 look for the hyperfine coupling. 80 00:05:42,250 --> 00:05:44,100 And well, we do perturbation theory 81 00:05:44,100 --> 00:05:47,060 in eigenfunctions with mI and mJ. 82 00:05:47,060 --> 00:05:49,600 And that means if you have the I dot J term, 83 00:05:49,600 --> 00:05:56,340 it is only the component mI mJ which remains. 84 00:05:56,340 --> 00:05:58,650 So I've given you those two cases. 85 00:05:58,650 --> 00:06:01,540 Now, what you should also learn here in this example 86 00:06:01,540 --> 00:06:03,390 is the language which we use. 87 00:06:03,390 --> 00:06:05,240 And sometimes, I would say, the language 88 00:06:05,240 --> 00:06:12,270 can be more confusing than the equations. 89 00:06:12,270 --> 00:06:18,960 What we say here is we say that the angular 90 00:06:18,960 --> 00:06:21,980 momentum of the electron and of the nucleus 91 00:06:21,980 --> 00:06:24,580 are coupled to the magnetic field axis. 92 00:06:24,580 --> 00:06:26,350 They are quantized. 93 00:06:26,350 --> 00:06:28,520 The approximate eigenstates are those 94 00:06:28,520 --> 00:06:32,710 which have a specific quantum number in the z-direction 95 00:06:32,710 --> 00:06:35,620 because the magnetic field points to the z-direction. 96 00:06:35,620 --> 00:06:41,500 So we're saying I and J are strongly coupled to the z-axis 97 00:06:41,500 --> 00:06:43,010 by the magnetic field. 98 00:06:43,010 --> 00:06:45,340 And then we treat the coupling of I 99 00:06:45,340 --> 00:06:48,085 and J with each other in perturbation theory. 100 00:06:48,085 --> 00:06:52,660 Whereas in the previous case, we say I and J strongly couple. 101 00:06:52,660 --> 00:06:54,800 And when I and J strongly couple, 102 00:06:54,800 --> 00:06:56,760 F becomes a good quantum number. 103 00:06:56,760 --> 00:06:59,970 And that means I and J both precess 104 00:06:59,970 --> 00:07:04,070 around the axis of the total angular momentum f. 105 00:07:04,070 --> 00:07:07,920 And therefore, we say I and J couple to F. 106 00:07:07,920 --> 00:07:11,790 And then we solve for this coupling of F 107 00:07:11,790 --> 00:07:15,360 to the magnetic field in a second step. 108 00:07:15,360 --> 00:07:20,100 But I hope you see that there's two limiting cases. 109 00:07:20,100 --> 00:07:23,840 We can exactly diagonalize one term, 110 00:07:23,840 --> 00:07:28,520 and then we perturbatively add on the result 111 00:07:28,520 --> 00:07:31,120 for the second term. 112 00:07:31,120 --> 00:07:33,570 Of course, in the age of computers 113 00:07:33,570 --> 00:07:38,230 I could have simply written down for you a Hamiltonian and said, 114 00:07:38,230 --> 00:07:41,400 well, it has to be numerically diagonalized. 115 00:07:41,400 --> 00:07:43,880 What I discussed instead were the two limiting cases. 116 00:07:47,430 --> 00:07:55,650 Now, this discussion now allows me 117 00:07:55,650 --> 00:08:02,470 to discuss what happens when we go to even stronger fields. 118 00:08:02,470 --> 00:08:06,940 Well, when we go to even stronger fields, 119 00:08:06,940 --> 00:08:09,440 then we may have fields which are 120 00:08:09,440 --> 00:08:12,230 even stronger than the fine structure coupling, 121 00:08:12,230 --> 00:08:15,770 the coupling of the orbital angular momentum 122 00:08:15,770 --> 00:08:18,970 of the electron and the spin angular momentum to J. 123 00:08:18,970 --> 00:08:23,550 And well, without even any deviation, which is obvious, 124 00:08:23,550 --> 00:08:28,500 you know what happens now is that each component which 125 00:08:28,500 --> 00:08:32,580 provides magnetic moment-- the spin, the orbital angular 126 00:08:32,580 --> 00:08:37,130 momentum, and the nucleus-- the dominant term for each of them 127 00:08:37,130 --> 00:08:43,490 is the coupling to the magnetic field. 128 00:08:43,490 --> 00:08:51,726 So in strong magnetic fields, these are the eigenstates. 129 00:08:51,726 --> 00:08:57,490 The eigenstates are labeled by mI, mL, mS. 130 00:08:57,490 --> 00:09:03,100 So we have taken care of the strong coupling term. 131 00:09:03,100 --> 00:09:09,900 And now in addition, we are now treating in perturbation theory 132 00:09:09,900 --> 00:09:13,860 some fine structure coupling, but the quantum numbers 133 00:09:13,860 --> 00:09:22,770 are already distributed, mL, mS. There 134 00:09:22,770 --> 00:09:25,870 is a coupling become between mI mS 135 00:09:25,870 --> 00:09:29,310 and a coupling between mI and mL. 136 00:09:36,530 --> 00:09:40,690 So this is sort of the limiting cases. 137 00:09:40,690 --> 00:09:45,140 But as a general illustration of quantum mechanics, 138 00:09:45,140 --> 00:09:59,100 I thought this was a nice example for a Hamiltonian 139 00:09:59,100 --> 00:10:08,990 where we have different scalar products, like B times S, 140 00:10:08,990 --> 00:10:19,360 B dot L, S dot L, I dot J. And the question is, how do we 141 00:10:19,360 --> 00:10:21,460 take care of those different parts 142 00:10:21,460 --> 00:10:24,050 because they do not commute? 143 00:10:24,050 --> 00:10:27,820 Of course, the theorist would just say, 144 00:10:27,820 --> 00:10:29,860 I simply diagonalize it and that's it. 145 00:10:29,860 --> 00:10:32,210 But if you want to develop intuition, 146 00:10:32,210 --> 00:10:35,410 then you have to discuss the limiting cases. 147 00:10:35,410 --> 00:10:39,860 And in particular, the approach which allows an intuitive 148 00:10:39,860 --> 00:10:44,210 understanding is first things first. 149 00:10:44,210 --> 00:10:52,020 And we first treat the stronger terms and then 150 00:10:52,020 --> 00:10:53,480 the weaker terms. 151 00:10:53,480 --> 00:10:58,040 And we can quantitatively derive, analytically derive 152 00:10:58,040 --> 00:11:00,800 expressions, for instance, for the Lande g-factor 153 00:11:00,800 --> 00:11:04,000 in this vector model. 154 00:11:04,000 --> 00:11:08,720 This vector model assumes, so to speak, 155 00:11:08,720 --> 00:11:13,760 that a state which has an eigenfunction of mJ 156 00:11:13,760 --> 00:11:16,080 rapidly precesses around the z-axis. 157 00:11:16,080 --> 00:11:18,550 And this vector model actually allows 158 00:11:18,550 --> 00:11:21,530 you to do easy calculation without 159 00:11:21,530 --> 00:11:22,655 Clebsch-Gordan coefficient. 160 00:11:25,710 --> 00:11:28,060 So the concept of the vector model 161 00:11:28,060 --> 00:11:39,490 is rapid precession for transverse components 162 00:11:39,490 --> 00:11:47,590 and projecting of vectors onto the axis around which you 163 00:11:47,590 --> 00:11:48,610 have rapid precession. 164 00:11:52,450 --> 00:12:00,190 But this is simply a tool to do calculations 165 00:12:00,190 --> 00:12:06,615 without the explicit use of Clebsch-Gordan coefficients. 166 00:12:12,670 --> 00:12:14,870 OK, so this is what I wanted to tell you 167 00:12:14,870 --> 00:12:19,660 about atoms in magnetic field. 168 00:12:19,660 --> 00:12:20,315 Any questions? 169 00:12:27,340 --> 00:12:36,290 OK then, we can actually move onto atoms in electric field. 170 00:12:47,260 --> 00:12:52,750 But before we do that, we should have some clicker 171 00:12:52,750 --> 00:12:55,670 questions about atomic structure and atoms 172 00:12:55,670 --> 00:12:58,110 comes in external magnetic fields. 173 00:12:58,110 --> 00:13:01,960 So get out the clickers. 174 00:13:13,900 --> 00:13:17,900 So the first questions take us back to electronic structure. 175 00:13:21,250 --> 00:13:27,440 It's a question about, how do wave functions, 176 00:13:27,440 --> 00:13:30,880 how does density, and how does inverse size 177 00:13:30,880 --> 00:13:33,440 scale with principal quantum number? 178 00:13:33,440 --> 00:13:35,340 So the first question is, how does 179 00:13:35,340 --> 00:13:44,180 1/r, 1 over the size of the electronic wave function, 180 00:13:44,180 --> 00:13:46,490 how does it scale with n? 181 00:13:46,490 --> 00:13:49,852 1/n, 1 over n squared, 1 over n cubed. 182 00:14:01,851 --> 00:14:02,350 OK. 183 00:14:10,420 --> 00:14:11,980 OK, yes. 184 00:14:11,980 --> 00:14:13,720 It's 1 over n squared. 185 00:14:13,720 --> 00:14:15,780 But I would have hope that 100% of you 186 00:14:15,780 --> 00:14:20,210 would know it because 1 over is the Coulomb energy. 187 00:14:20,210 --> 00:14:23,460 The Coulomb energy is 1/2 of the total binding 188 00:14:23,460 --> 00:14:25,480 energy because of the Virial theorem. 189 00:14:25,480 --> 00:14:29,610 So when you see 1/r, there should be a flash in your head 190 00:14:29,610 --> 00:14:31,320 which says energy. 191 00:14:31,320 --> 00:14:34,980 And Rydberg energy is 1 over n squared. 192 00:14:34,980 --> 00:14:40,721 The energy levels of hydrogen are 1 over n squared. 193 00:14:40,721 --> 00:14:41,220 OK. 194 00:14:48,220 --> 00:14:50,030 Yeah, next question. 195 00:14:50,030 --> 00:14:54,130 How does psi 0 of 0 square, how does 196 00:14:54,130 --> 00:14:56,580 the density of the electron at the origin 197 00:14:56,580 --> 00:15:00,570 scale with principal quantum number? 198 00:15:00,570 --> 00:15:03,890 1/n, 1 over n squared, 1 over n cubed, or 1 over n 199 00:15:03,890 --> 00:15:04,540 to the sixth? 200 00:15:19,380 --> 00:15:20,940 OK, yes, very good. 201 00:15:20,940 --> 00:15:23,040 So what I try to remind you here is 202 00:15:23,040 --> 00:15:26,790 that there are two different radii. 203 00:15:26,790 --> 00:15:31,270 There is one radius which scales with n squared. 204 00:15:31,270 --> 00:15:34,240 But if you calculate the density at the origin, 205 00:15:34,240 --> 00:15:37,790 if you would say, well, r scales with n squared. 206 00:15:37,790 --> 00:15:39,741 The volume scales with n to the sixth. 207 00:15:39,741 --> 00:15:41,740 And then you would say the density scales with n 208 00:15:41,740 --> 00:15:43,660 to the minus 6, you're wrong. 209 00:15:43,660 --> 00:15:46,460 And we had a discussion that there are two different lengths 210 00:15:46,460 --> 00:15:49,325 case in the hydrogen atom and in hydrogenic wave function. 211 00:15:51,880 --> 00:15:56,010 One scales with n squared and the other ones scales with n. 212 00:15:56,010 --> 00:15:59,310 And therefore, the density at the origin is n to the minus 3. 213 00:16:02,961 --> 00:16:03,460 Good. 214 00:16:06,510 --> 00:16:10,210 Next question, from hydrogen to helium. 215 00:16:10,210 --> 00:16:13,892 In helium, for the same electronic configuration, , 216 00:16:13,892 --> 00:16:17,310 are singlet or triplet states more tightly bound? 217 00:16:21,830 --> 00:16:25,910 So we talked about a shift or splitting between singlet 218 00:16:25,910 --> 00:16:29,430 and triplet states. 219 00:16:29,430 --> 00:16:30,970 Which way does it go? 220 00:16:47,600 --> 00:16:48,720 You want to try again? 221 00:16:53,470 --> 00:16:57,610 For the same electronic configuration-- 222 00:16:57,610 --> 00:16:59,531 the ground state has only one configuration. 223 00:16:59,531 --> 00:17:01,280 It has only one state in the ground state. 224 00:17:01,280 --> 00:17:05,660 But now we go to the excited state, to the excited states, 225 00:17:05,660 --> 00:17:09,480 and there are a number of excited states because they 226 00:17:09,480 --> 00:17:12,109 have the same configuration, but they 227 00:17:12,109 --> 00:17:15,100 can be classified by singlet and triplet states. 228 00:17:24,859 --> 00:17:26,566 OK, we are converging. 229 00:17:26,566 --> 00:17:29,270 It is the triplet state. 230 00:17:29,270 --> 00:17:32,400 Some people are confused when they think about molecules. 231 00:17:32,400 --> 00:17:34,530 usually, in molecules the singlet state 232 00:17:34,530 --> 00:17:37,090 is more tightly bound than the triplet state. 233 00:17:37,090 --> 00:17:40,790 But the magic word is for the same electronic configuration. 234 00:17:40,790 --> 00:17:45,780 You can have one orbital filled with two electrons only 235 00:17:45,780 --> 00:17:49,390 in a singlet state because of the Pauli exclusion principle. 236 00:17:49,390 --> 00:17:51,890 It is only in the first excited state 237 00:17:51,890 --> 00:17:54,000 or in an excited state of a molecule 238 00:17:54,000 --> 00:17:59,590 or of the helium atom that you have two orbitals, 1s and 2s. 239 00:17:59,590 --> 00:18:02,670 And you can now put the electrons in with the same spin 240 00:18:02,670 --> 00:18:04,660 or with opposite spin. 241 00:18:04,660 --> 00:18:07,590 So usually, it's only in an excited state 242 00:18:07,590 --> 00:18:10,840 that the question singlet versus triplet arises. 243 00:18:10,840 --> 00:18:13,430 And then in the excited state manifold, 244 00:18:13,430 --> 00:18:18,640 the triplet state is lower because it has a symmetric spin 245 00:18:18,640 --> 00:18:23,450 wave function and anti-symmetric spatial wave function. 246 00:18:23,450 --> 00:18:28,716 OK, the next question. 247 00:18:28,716 --> 00:18:30,590 OK, so we understand now there's a difference 248 00:18:30,590 --> 00:18:32,110 between triplet and singlet state 249 00:18:32,110 --> 00:18:36,030 in excited states for the same electronic configuration. 250 00:18:36,030 --> 00:18:42,170 And the question is, what is the origin of the energy which 251 00:18:42,170 --> 00:18:45,600 is splitting the singlet from the triplet state? 252 00:18:45,600 --> 00:18:49,010 Magnetic energy, spin-spin interactions, 253 00:18:49,010 --> 00:18:50,315 or electrostatic interactions? 254 00:19:07,080 --> 00:19:11,280 Yes, the Coulomb interaction is electrostatic interactions. 255 00:19:11,280 --> 00:19:14,560 We discussed the singlet-triplet splitting and the structure 256 00:19:14,560 --> 00:19:18,640 of helium without any magnetic or spin-dependent interaction. 257 00:19:18,640 --> 00:19:21,570 All we had is the Coulomb interaction. 258 00:19:21,570 --> 00:19:25,320 And in the triplet state, which is the symmetric spin state, 259 00:19:25,320 --> 00:19:28,500 the spatial wave function has to be anti-symmetric. 260 00:19:28,500 --> 00:19:31,300 In the singlet state, the spatial wave function 261 00:19:31,300 --> 00:19:32,640 has to be symmetric. 262 00:19:32,640 --> 00:19:35,700 And the symmetric and the anti-symmetric spatial wave 263 00:19:35,700 --> 00:19:39,300 function have a different Coulomb energy. 264 00:19:39,300 --> 00:19:42,820 So the spin through the anti-symmetry through the Pauli 265 00:19:42,820 --> 00:19:46,000 exclusion principle determines the symmetry 266 00:19:46,000 --> 00:19:48,730 of the electronic wave function. 267 00:19:48,730 --> 00:19:52,320 And it is then purely the Coulomb energy. 268 00:19:52,320 --> 00:19:55,090 That's why the singlet-triplet splitting is so big. 269 00:19:55,090 --> 00:19:58,140 Because it's not magnetic, it's Coulomb in origin. 270 00:20:01,010 --> 00:20:04,930 OK, next question. 271 00:20:04,930 --> 00:20:10,000 Which interaction reflects-- oops, 272 00:20:10,000 --> 00:20:11,510 maybe you want to still read it. 273 00:20:19,310 --> 00:20:23,890 Which interaction reflects that the potential between nucleus 274 00:20:23,890 --> 00:20:28,260 and electron is not exactly a Coulomb potential 1/r? 275 00:20:28,260 --> 00:20:31,440 We've usually discussed Schrodinger equation, hydrogen, 276 00:20:31,440 --> 00:20:35,060 Bohr model for an exact 1/r potential. 277 00:20:35,060 --> 00:20:37,550 But then we discussed a lot of phenomena. 278 00:20:37,550 --> 00:20:43,835 And I want you to figure out now, which of those choices 279 00:20:43,835 --> 00:20:47,430 mean, in essence that you do not have a 1/r potential? 280 00:21:13,140 --> 00:21:15,330 OK, we have three choices. 281 00:21:15,330 --> 00:21:18,610 So the volume isotrope effect, I think, 282 00:21:18,610 --> 00:21:20,200 is a no-brainer, is trivial. 283 00:21:20,200 --> 00:21:23,130 It means explicitly that the nucleus is not 284 00:21:23,130 --> 00:21:25,910 a point, has an extended volume, and that 285 00:21:25,910 --> 00:21:29,940 means inside the nucleus the electron is not 286 00:21:29,940 --> 00:21:32,500 experiencing a 1/r potential. 287 00:21:32,500 --> 00:21:35,010 So it's clear that C is always correct. 288 00:21:38,120 --> 00:21:42,740 The Lamb shift is actually causing 289 00:21:42,740 --> 00:21:51,030 a deviation from a 1/r potential because-- well, 290 00:21:51,030 --> 00:22:00,220 both the vacuum polarization and the-- well, 291 00:22:00,220 --> 00:22:02,150 you can go ahead and say, the fact 292 00:22:02,150 --> 00:22:03,940 that we have QED, that we have other modes 293 00:22:03,940 --> 00:22:05,920 of the electromagnetic field mean 294 00:22:05,920 --> 00:22:08,940 that there's a deviation from the 1/r potential. 295 00:22:08,940 --> 00:22:12,960 The interesting question is the Darwin term. 296 00:22:12,960 --> 00:22:16,840 And the people who clicked D included the Darwin term. 297 00:22:16,840 --> 00:22:20,600 That's a little bit trickier because I explained the Darwin 298 00:22:20,600 --> 00:22:23,970 term as Zitterbewegung, as this trembling motion 299 00:22:23,970 --> 00:22:28,470 of the electron which smears out the 1/r potential. 300 00:22:28,470 --> 00:22:32,250 So you would think coming from the non-relativistic 301 00:22:32,250 --> 00:22:35,810 Schrodinger equation, that there is an effect which 302 00:22:35,810 --> 00:22:38,350 is smearing out the 1/r potential. 303 00:22:38,350 --> 00:22:40,650 On the other hand, the Zitterbewegung, 304 00:22:40,650 --> 00:22:43,490 the Darwin term, is just one term 305 00:22:43,490 --> 00:22:46,890 which is included in the Dirac equation. 306 00:22:46,890 --> 00:22:50,680 And the Dirac equation, which includes fine structure 307 00:22:50,680 --> 00:22:54,830 and relativistic energy corrections and the Darwin term 308 00:22:54,830 --> 00:22:57,910 is an exact relativistic formulation 309 00:22:57,910 --> 00:23:00,960 of the 1/r potential. 310 00:23:00,960 --> 00:23:15,470 So in other words, I would say the correct answer is E. The 311 00:23:15,470 --> 00:23:18,970 people who included the Darwin term, 312 00:23:18,970 --> 00:23:24,210 I would say the Darwin term is not 313 00:23:24,210 --> 00:23:27,730 a deviation of the 1/r potential because it's simply a way 314 00:23:27,730 --> 00:23:32,070 to explain what is the result of the Dirac equation. 315 00:23:32,070 --> 00:23:36,650 The Dirac equation uses exactly the 1/r potential 316 00:23:36,650 --> 00:23:38,180 without any corrections. 317 00:23:38,180 --> 00:23:41,340 So you can say that if you want to understand 318 00:23:41,340 --> 00:23:45,010 the relativistic solution to the 1/r problem, 319 00:23:45,010 --> 00:23:50,150 you include a term which in the non-relativistic equation 320 00:23:50,150 --> 00:23:53,820 slightly changes the Coulomb potential. 321 00:23:53,820 --> 00:23:54,970 Questions about that? 322 00:23:57,665 --> 00:23:58,165 OK. 323 00:24:04,820 --> 00:24:05,820 Fine structure. 324 00:24:10,680 --> 00:24:13,810 The fine structure affects only states with L 325 00:24:13,810 --> 00:24:16,235 equals 0 through a coupling term L 326 00:24:16,235 --> 00:24:23,965 dot S. Is this statement, the way how it is written, 327 00:24:23,965 --> 00:24:24,960 true of false? 328 00:24:47,940 --> 00:24:54,420 I would say it's false because the fine structure has 329 00:24:54,420 --> 00:24:56,700 three contributions-- the Darwin term, 330 00:24:56,700 --> 00:25:03,621 the relativistic kinetic energy contribution, and this L dot S 331 00:25:03,621 --> 00:25:04,120 term. 332 00:25:07,120 --> 00:25:11,920 And it effects all states, also the S states, 333 00:25:11,920 --> 00:25:14,590 through the Darwin term and the relativistic energy 334 00:25:14,590 --> 00:25:16,120 contribution. 335 00:25:16,120 --> 00:25:20,659 So the fine structure is more than just an L dot S term. 336 00:25:25,880 --> 00:25:32,490 Next question is for L equals 0, the orbiting electron 337 00:25:32,490 --> 00:25:34,390 creates a magnetic field. 338 00:25:34,390 --> 00:25:38,140 And spin orbit interaction can be simply regarded 339 00:25:38,140 --> 00:25:44,460 as the energy of the electron's spin in this magnetic field. 340 00:25:44,460 --> 00:25:47,810 Would you say that this sentence is true or false? 341 00:26:11,700 --> 00:26:14,210 I thought it's true, but maybe people 342 00:26:14,210 --> 00:26:16,430 want to tell me what is false about the statement? 343 00:26:24,200 --> 00:26:26,790 Maybe the first sentence people did-- tell me, 344 00:26:26,790 --> 00:26:29,340 the orbiting electron creates a magnetic field. 345 00:26:32,440 --> 00:26:34,148 Yes. 346 00:26:34,148 --> 00:26:38,594 AUDIENCE: I said false for this because I normally 347 00:26:38,594 --> 00:26:41,558 would picture that from the electron's frame of reference, 348 00:26:41,558 --> 00:26:43,534 the nucleus creating a magnetic field 349 00:26:43,534 --> 00:26:45,487 is the magnetic field [INAUDIBLE]. 350 00:26:45,487 --> 00:26:48,070 PROFESSOR: OK, the second part, that there is a magnetic field 351 00:26:48,070 --> 00:26:49,569 and it's been orbiting the action is 352 00:26:49,569 --> 00:26:52,410 the energy of the electron spin in this magnetic field 353 00:26:52,410 --> 00:26:54,210 is probably generally accepted. 354 00:26:54,210 --> 00:26:56,860 But the first question is, does the orbiting electron 355 00:26:56,860 --> 00:27:00,050 create the magnetic field? 356 00:27:00,050 --> 00:27:02,510 Well, we have the two options. 357 00:27:02,510 --> 00:27:06,650 We can say the electron moves and in its own frame 358 00:27:06,650 --> 00:27:08,670 there is a v cross e term. 359 00:27:08,670 --> 00:27:10,870 And therefore, magnetic field. 360 00:27:10,870 --> 00:27:14,010 So we can say that the electron's motion creates 361 00:27:14,010 --> 00:27:15,930 a magnetic field in its own frame 362 00:27:15,930 --> 00:27:18,850 for the relativistic transformation. 363 00:27:18,850 --> 00:27:20,600 So in that sense, it is correct. 364 00:27:20,600 --> 00:27:22,870 but I would also side with you that there 365 00:27:22,870 --> 00:27:26,800 is an alternative view of saying in the electron's flame, 366 00:27:26,800 --> 00:27:29,250 the nucleus rotates around the electron. 367 00:27:29,250 --> 00:27:32,102 And it's the nucleus which creates the magnetic field. 368 00:27:32,102 --> 00:27:34,310 In the end, it's the relative motion between the two. 369 00:27:39,300 --> 00:27:41,110 Well, isn't it a good thing that we are not 370 00:27:41,110 --> 00:27:42,310 giving scores on that? 371 00:27:42,310 --> 00:27:46,110 So yes, if you want, you everybody 372 00:27:46,110 --> 00:27:49,320 can feel that you have given the right answer. 373 00:27:52,550 --> 00:27:53,210 Oh, yeah. 374 00:27:53,210 --> 00:27:55,710 In Dirac-- that should be easy, but it's just 375 00:27:55,710 --> 00:27:59,030 a warm-up question for the next one-- which states 376 00:27:59,030 --> 00:28:00,850 are degenerate in Dirac theory? 377 00:28:03,766 --> 00:28:04,890 And you have a few choices. 378 00:28:24,310 --> 00:28:25,150 That should be easy. 379 00:28:34,380 --> 00:28:38,700 Yeah, it's the S 1/2 and P 1/2. 380 00:28:38,700 --> 00:28:42,950 Dirac theory does not lift the degeneracy between states 381 00:28:42,950 --> 00:28:45,910 with the same J, 1/2 and 1/2. 382 00:28:45,910 --> 00:28:50,118 But between 1/2 and 3/2 states, there is, actually, 383 00:28:50,118 --> 00:28:52,493 the fine structure splitting, which we've just discussed. 384 00:28:55,470 --> 00:29:00,380 OK, the next question is, what effects lift now 385 00:29:00,380 --> 00:29:05,866 the degeneracy between the 2 S 1/2 and the 2 P 1/2 term? 386 00:29:36,470 --> 00:29:39,770 OK, we have three candidates-- the Lamb shift-- well, the Lamb 387 00:29:39,770 --> 00:29:43,050 shift is famous and the Lamb shift was discovered because it 388 00:29:43,050 --> 00:29:45,620 splits the degeneracy between the two. 389 00:29:45,620 --> 00:29:52,000 QED corrections have different effects on S 1/2 and P 1/2. 390 00:29:52,000 --> 00:29:56,130 The size of the proton does also shift it, 391 00:29:56,130 --> 00:29:59,520 because the size of the proton-- the volume effect 392 00:29:59,520 --> 00:30:03,630 is more important for S states than for P states. 393 00:30:03,630 --> 00:30:07,760 Maybe the question is, does the mass of the proton 394 00:30:07,760 --> 00:30:10,030 lift the degeneracy? 395 00:30:10,030 --> 00:30:11,660 No, it doesn't. 396 00:30:11,660 --> 00:30:15,650 It would just mean if your nucleus has a finite mass, 397 00:30:15,650 --> 00:30:17,790 you simply have a two-body problem 398 00:30:17,790 --> 00:30:21,005 with a reduced mass, which is different from the bare mass 399 00:30:21,005 --> 00:30:22,160 of the electron. 400 00:30:22,160 --> 00:30:24,510 But nothing else is changed, no degeneracies. 401 00:30:27,670 --> 00:30:30,190 It's as if the electron has a different mass, which 402 00:30:30,190 --> 00:30:31,880 is the effective mass. 403 00:30:31,880 --> 00:30:40,250 So the correct answer here is D. 404 00:30:40,250 --> 00:30:42,300 OK, four more questions. 405 00:30:42,300 --> 00:30:43,935 And this is about hyperfine structure. 406 00:30:52,230 --> 00:31:03,810 So the question is, what-- well, hydrogen in the ground state 407 00:31:03,810 --> 00:31:05,280 has four states. 408 00:31:05,280 --> 00:31:08,560 Because the electron has a spin up and down and the proton 409 00:31:08,560 --> 00:31:09,570 has a spin up and down. 410 00:31:09,570 --> 00:31:11,600 And 2 times 2 is 4. 411 00:31:11,600 --> 00:31:15,930 So we're talking about multiplicity of 4. 412 00:31:15,930 --> 00:31:20,770 And I'm asking you now about the limits of high and low field. 413 00:31:20,770 --> 00:31:23,180 First at high fields, then at low fields. 414 00:31:23,180 --> 00:31:26,680 And the question is, what are the magnetic moments 415 00:31:26,680 --> 00:31:28,550 of those hyperfine states? 416 00:31:28,550 --> 00:31:31,650 And we neglect the nuclear magneton 417 00:31:31,650 --> 00:31:35,090 compared to the Bohr magneton. 418 00:31:35,090 --> 00:31:39,100 So what are t magnetic moments of those hyperfine states 419 00:31:39,100 --> 00:31:40,780 in units of the Bohr magneton? 420 00:32:24,940 --> 00:32:27,110 Oh, yeah. 421 00:32:27,110 --> 00:32:30,470 What happens at high magnetic fields? 422 00:32:30,470 --> 00:32:33,800 Remember, at high magnetic fields, 423 00:32:33,800 --> 00:32:35,760 this is actually the simpler case. 424 00:32:35,760 --> 00:32:37,960 Often, you think the low magnetic field is simpler 425 00:32:37,960 --> 00:32:41,590 because it connects more with the isolated atom. 426 00:32:41,590 --> 00:32:43,520 But you should take away the message 427 00:32:43,520 --> 00:32:45,640 that high magnetic fields are simple. 428 00:32:45,640 --> 00:32:48,830 Because in high magnetic field, each spin 429 00:32:48,830 --> 00:32:52,220 couplets to the magnetic field by itself 430 00:32:52,220 --> 00:32:55,830 because the coupling to the strong magnetic field-- that's 431 00:32:55,830 --> 00:32:57,780 the definition of a strong magnetic field-- 432 00:32:57,780 --> 00:33:01,580 is stronger than the coupling of the two spins with each other. 433 00:33:01,580 --> 00:33:05,240 So the problem I'm giving you is that you have an electron 434 00:33:05,240 --> 00:33:07,950 spin which can be up and down and it 435 00:33:07,950 --> 00:33:09,930 couples to the magnetic field. 436 00:33:09,930 --> 00:33:11,830 And then we have the nucleus spin, 437 00:33:11,830 --> 00:33:13,725 but the magnetic moment of the nucleus 438 00:33:13,725 --> 00:33:17,920 is so small that we neglect it. 439 00:33:17,920 --> 00:33:21,520 So what are the possibilities now? 440 00:33:21,520 --> 00:33:23,310 Well, we have four states of hydrogen 441 00:33:23,310 --> 00:33:24,750 at high magnetic field. 442 00:33:24,750 --> 00:33:28,780 Two have the electron spin up, nucleus spin up/down. 443 00:33:28,780 --> 00:33:31,000 Two have the electron spin down. 444 00:33:31,000 --> 00:33:33,400 And then when the nucleus spin is up or down 445 00:33:33,400 --> 00:33:34,810 in those two states. 446 00:33:34,810 --> 00:33:37,250 So all the states at high magnetic fields 447 00:33:37,250 --> 00:33:41,890 have either the electron spin up or the electron spin down. 448 00:33:41,890 --> 00:33:43,580 So therefore, the correct answer is 449 00:33:43,580 --> 00:33:46,930 A. We have two states where the electron spin is up 450 00:33:46,930 --> 00:33:50,600 and two states where the electron spin is down. 451 00:33:50,600 --> 00:33:53,340 It's just a complicated way of asking you, 452 00:33:53,340 --> 00:33:55,260 what are the possible energy levels 453 00:33:55,260 --> 00:33:58,000 of an electron in a magnetic field? 454 00:33:58,000 --> 00:34:02,150 And the answer is, well, plus-minus 1 Bohr magneton 455 00:34:02,150 --> 00:34:04,950 times the magnetic field. 456 00:34:04,950 --> 00:34:05,880 Questions about it? 457 00:34:11,150 --> 00:34:12,389 OK. 458 00:34:12,389 --> 00:34:18,209 Now, we go to the more complicated case, 459 00:34:18,209 --> 00:34:20,310 to low magnetic fields. 460 00:34:20,310 --> 00:34:22,007 And again, same question. 461 00:34:22,007 --> 00:34:24,340 What are the magnetic moments of those hyperfine states? 462 00:34:28,080 --> 00:34:29,870 So you have four states. 463 00:34:29,870 --> 00:34:31,870 The number of states, of course, doesn't change. 464 00:34:31,870 --> 00:34:34,760 That's the dimension of our Hilbert space. 465 00:34:34,760 --> 00:34:36,900 But now we are at low magnetic field, 466 00:34:36,900 --> 00:34:39,750 and what is the magnetic moment, which 467 00:34:39,750 --> 00:34:43,850 is nothing else than the derivative of the energy 468 00:34:43,850 --> 00:34:45,370 with respect to the magnetic field? 469 00:35:04,250 --> 00:35:09,310 Yes, the correct answer is B. We have two manifolds, 470 00:35:09,310 --> 00:35:11,620 one is F equals 1, where one slope is 0 471 00:35:11,620 --> 00:35:14,080 and one slope is plus-minus 1. 472 00:35:14,080 --> 00:35:16,340 And then we have an F equals 0 state. 473 00:35:16,340 --> 00:35:19,930 So it is 1, minus 1, and 0, 0. 474 00:35:23,820 --> 00:35:24,320 OK. 475 00:35:27,750 --> 00:35:30,550 Let's now make it more interesting. 476 00:35:30,550 --> 00:35:35,920 Let's replace the proton by a positron, 477 00:35:35,920 --> 00:35:38,580 the anti-particle of the electron. 478 00:35:38,580 --> 00:35:42,610 So now we have a similar situation, but what happens 479 00:35:42,610 --> 00:35:45,260 now is, of course, the contribution 480 00:35:45,260 --> 00:35:48,440 to the magnetic moment form the nucleus, which is now 481 00:35:48,440 --> 00:35:50,975 the positron, is as important as the contribution 482 00:35:50,975 --> 00:35:52,510 of the electron. 483 00:35:52,510 --> 00:35:56,320 So you have two spin 1/2's coupled now. 484 00:35:56,320 --> 00:35:58,620 One is positive, one is negative. 485 00:35:58,620 --> 00:36:06,460 And you should figure out again, what are the energies? 486 00:36:06,460 --> 00:36:09,930 But before we talk about the energies, let's first talk 487 00:36:09,930 --> 00:36:12,860 about, how many hyperfine states do 488 00:36:12,860 --> 00:36:16,130 we have in the ground state-- 1, 2, 3, or 4? 489 00:36:41,790 --> 00:36:43,830 Yes, we have four states because we 490 00:36:43,830 --> 00:36:46,670 have two particles-- positron, electron. 491 00:36:46,670 --> 00:36:48,600 Each of them has spin up, spin down. 492 00:36:48,600 --> 00:36:51,290 2 times 2 is 4. 493 00:36:51,290 --> 00:36:56,460 And therefore, we have now four states. 494 00:36:56,460 --> 00:37:01,710 And the question is again, at high and low magnetic fields, 495 00:37:01,710 --> 00:37:03,710 what are the magnetic moments of those states? 496 00:37:06,320 --> 00:37:08,370 So we have four states-- spin up, 497 00:37:08,370 --> 00:37:12,600 spin down-- of the electron and the positron. 498 00:37:12,600 --> 00:37:15,320 And the question is, what are the magnetic moments 499 00:37:15,320 --> 00:37:16,510 of those hyperfine states? 500 00:37:43,490 --> 00:37:45,312 D is correct. 501 00:37:45,312 --> 00:37:50,040 We have 1/2, spin 1/2. 502 00:37:50,040 --> 00:37:53,210 If the two couple like up-up and down-down, 503 00:37:53,210 --> 00:37:55,500 we have the maximum spin. 504 00:37:55,500 --> 00:37:59,800 But since one particle is positive, one is negative, 505 00:37:59,800 --> 00:38:03,280 when the spins are aligned, the angular momenta 506 00:38:03,280 --> 00:38:04,900 are anti-aligned. 507 00:38:04,900 --> 00:38:08,730 And therefore, the magnetic moment is 0. 508 00:38:08,730 --> 00:38:12,440 So when they couple parallel, the magnetic moment is 0. 509 00:38:12,440 --> 00:38:15,980 When they couple anti-parallel, the two magnetic moments 510 00:38:15,980 --> 00:38:18,520 of one Bohr magneton each add up, 511 00:38:18,520 --> 00:38:22,007 and we have either 2 or minus 2 as the magnetic moment. 512 00:38:25,577 --> 00:38:26,160 Any questions? 513 00:38:29,280 --> 00:38:33,020 Then finally, the last question. 514 00:38:33,020 --> 00:38:38,680 Same situation, positronium, but now at low magnetic fields. 515 00:38:46,050 --> 00:38:50,270 What are the magnetic moments of the four hyperfine states 516 00:38:50,270 --> 00:38:52,750 of positronium at low magnetic field? 517 00:39:21,170 --> 00:39:21,770 All right. 518 00:39:24,740 --> 00:39:28,210 Let's discuss it. 519 00:39:28,210 --> 00:39:31,910 What is the structure of the ground 520 00:39:31,910 --> 00:39:33,101 state at low magnetic field? 521 00:39:33,101 --> 00:39:35,350 What is the good quantum number at low magnetic field? 522 00:39:39,823 --> 00:39:40,820 AUDIENCE: F. 523 00:39:40,820 --> 00:39:42,489 PROFESSOR: F. It's hydrogen. 524 00:39:42,489 --> 00:39:43,280 It's like hydrogen. 525 00:39:43,280 --> 00:39:44,230 1/2 and 1/2. 526 00:39:46,775 --> 00:39:50,850 If we have an S of 1/2 of the electron, the I of the positron 527 00:39:50,850 --> 00:39:52,000 is also 1/2. 528 00:39:52,000 --> 00:39:57,960 And 1/2 and 1/2 couple to F. And what are the values for F? 529 00:39:57,960 --> 00:39:59,990 F equals 1 and F equals 0. 530 00:40:03,980 --> 00:40:08,570 OK, what is the magnetic moment of the F equals 1 state? 531 00:40:18,030 --> 00:40:22,700 In order to get F equals 1 out of 1/2 and 1/2, 532 00:40:22,700 --> 00:40:26,450 you have to align the spin of the electron with the positron. 533 00:40:26,450 --> 00:40:29,660 So the F equals 1/2 state is the state where 534 00:40:29,660 --> 00:40:32,440 the two spins are aligned. 535 00:40:32,440 --> 00:40:35,111 What is the magnetic moment or this state? 536 00:40:35,111 --> 00:40:35,610 AUDIENCE: 0. 537 00:40:35,610 --> 00:40:36,290 PROFESSOR: 0. 538 00:40:36,290 --> 00:40:38,920 How many states are in the F equals 1 manifold? 539 00:40:44,240 --> 00:40:46,630 What's a multiplicity of F equals 1? 540 00:40:46,630 --> 00:40:47,520 3. 541 00:40:47,520 --> 00:40:49,360 Plus, minus 1 and 0. 542 00:40:49,360 --> 00:40:51,930 So we have an F equals 1 state which 543 00:40:51,930 --> 00:40:54,560 has angular momentum but no magnetic moment, 544 00:40:54,560 --> 00:40:56,480 and it has a multiplicity of 3. 545 00:40:56,480 --> 00:41:01,230 So three states have 0, 0 magnetic moment. 546 00:41:01,230 --> 00:41:05,180 In other words, you would expect an F equals 547 00:41:05,180 --> 00:41:13,280 1 state to have this kind of Zeeman structure. 548 00:41:13,280 --> 00:41:17,260 But because of the special situation in positronium, 549 00:41:17,260 --> 00:41:19,580 the Zeeman structure is like this. 550 00:41:19,580 --> 00:41:21,010 There is no linear effect. 551 00:41:21,010 --> 00:41:22,380 It's a quadratic effect. 552 00:41:22,380 --> 00:41:26,710 All three states start out with 0 slope 553 00:41:26,710 --> 00:41:30,450 because as long as the spins couple to F equals 1 554 00:41:30,450 --> 00:41:32,610 and we don't have a magnetic field messing up 555 00:41:32,610 --> 00:41:36,660 with the coupling, the magnetic moment is 0. 556 00:41:36,660 --> 00:41:40,000 OK, now what happens in the fourth state, which is F 557 00:41:40,000 --> 00:41:41,670 equals 0? 558 00:41:41,670 --> 00:41:45,240 In the fourth state, which is F equals 0, 559 00:41:45,240 --> 00:41:50,540 the two spins couple in an anti-parallel way. 560 00:41:50,540 --> 00:41:52,340 So now, what is the magnetic moment 561 00:41:52,340 --> 00:41:54,540 when the two spins couple in an anti-parallel way? 562 00:41:58,430 --> 00:42:00,950 The spins subtract. 563 00:42:00,950 --> 00:42:04,200 But because of the different charge, plus and minus, 564 00:42:04,200 --> 00:42:06,690 the magnetic moments would add up. 565 00:42:06,690 --> 00:42:09,290 That's what we just discussed in the high field case. 566 00:42:09,290 --> 00:42:13,860 So you would think the F equals 0 state has a magnetic moment. 567 00:42:13,860 --> 00:42:17,840 But in an F equals 0 state, it cannot point anywhere 568 00:42:17,840 --> 00:42:20,560 because the angular momentum is 0. 569 00:42:20,560 --> 00:42:24,990 And therefore, in a most trivial way, this is hydrogen 570 00:42:24,990 --> 00:42:27,360 and this is positronium. 571 00:42:27,360 --> 00:42:31,700 So positronium has four hyperfine states. 572 00:42:31,700 --> 00:42:36,210 And the slope of all four, for the reasons discussed, 573 00:42:36,210 --> 00:42:37,080 is all 0. 574 00:42:39,710 --> 00:42:44,790 So sorry, A is the correct answer, without any ambiguity 575 00:42:44,790 --> 00:42:45,530 this time. 576 00:42:57,635 --> 00:42:58,135 OK. 577 00:43:12,877 --> 00:43:13,460 Any questions? 578 00:43:19,440 --> 00:43:24,010 OK, then let's talk about atoms in electric field. 579 00:43:28,450 --> 00:43:34,220 We start out in-- we put the atoms 580 00:43:34,220 --> 00:43:40,590 in a uniform electric field. 581 00:43:40,590 --> 00:43:43,870 Again, we assume that it points in the z-direction 582 00:43:43,870 --> 00:43:46,790 and its magnitude is epsilon. 583 00:43:46,790 --> 00:43:56,750 And we want to ask, what is the electrostatic energy 584 00:43:56,750 --> 00:43:59,770 in this electric field? 585 00:43:59,770 --> 00:44:07,980 And we are using the fact that electrostatic energy can 586 00:44:07,980 --> 00:44:11,265 be expanded in a multi-pole expansion. 587 00:44:13,800 --> 00:44:24,760 We have a monopole term, we have a dipole term, 588 00:44:24,760 --> 00:44:27,470 and we have a quadratic term. 589 00:44:33,960 --> 00:44:44,180 So the charge, of course, is-- the atom, itself, 590 00:44:44,180 --> 00:44:45,020 is a neutral atom. 591 00:44:45,020 --> 00:44:47,620 So there is no monopole term. 592 00:44:47,620 --> 00:44:51,970 The linear term in the electric field 593 00:44:51,970 --> 00:44:58,260 would correspond to a permanent dipole moment. 594 00:44:58,260 --> 00:45:01,940 And I will remind you in a moment that this is 0. 595 00:45:01,940 --> 00:45:07,030 And then the term which provides us with a stark effect, 596 00:45:07,030 --> 00:45:11,950 with the energy shift of atoms in electric field will 597 00:45:11,950 --> 00:45:18,260 be the third term here, which is characterized 598 00:45:18,260 --> 00:45:21,520 by the polarizability alpha. 599 00:45:24,820 --> 00:45:30,620 And it corresponds to an induced dipole moment. 600 00:45:30,620 --> 00:45:33,770 That there is an induced dipole moment, 601 00:45:33,770 --> 00:45:35,940 which is alpha times epsilon. 602 00:45:35,940 --> 00:45:39,420 And then the induced dipole moment 603 00:45:39,420 --> 00:45:41,400 interacts with the electric field. 604 00:45:41,400 --> 00:45:43,800 And that gives then, epsilon times epsilon-- epsilon 605 00:45:43,800 --> 00:45:44,300 squared. 606 00:45:49,190 --> 00:45:53,090 So this would be a classical multi-pole expansion. 607 00:45:53,090 --> 00:46:00,190 And we will now derive results quantum mechanically. 608 00:46:00,190 --> 00:46:09,230 The perturbation operator for us is the dipole operator. 609 00:46:09,230 --> 00:46:10,870 And that could, in principle, include 610 00:46:10,870 --> 00:46:13,240 a permanent or an induced dipole moment. 611 00:46:13,240 --> 00:46:16,100 So it would take care of the second and third term, 612 00:46:16,100 --> 00:46:19,910 the dipole operator and its projection on the z-axis. 613 00:46:25,410 --> 00:46:27,380 So the dipole operator is the charge 614 00:46:27,380 --> 00:46:28,960 of the electron times the position. 615 00:46:36,060 --> 00:46:40,150 And as long as the polarizability 616 00:46:40,150 --> 00:46:41,620 and the situation is isotropic. 617 00:46:46,090 --> 00:46:46,940 A minus sign. 618 00:46:46,940 --> 00:46:49,270 Minus E is the charge. 619 00:46:49,270 --> 00:46:51,910 If you apply an electric field in the z-direction, 620 00:46:51,910 --> 00:46:55,050 all the relevant dipole moments are in the z-direction. 621 00:46:55,050 --> 00:46:58,290 For anisotropic materials, you could have an electric field 622 00:46:58,290 --> 00:47:00,850 in the z-direction and the dipole moment points 623 00:47:00,850 --> 00:47:05,055 at an angle, but we do not have such a situation for our atoms. 624 00:47:07,670 --> 00:47:13,190 OK, so the operator is then simply 625 00:47:13,190 --> 00:47:16,170 charge of the electron, the z-coordinate times 626 00:47:16,170 --> 00:47:17,010 the electric field. 627 00:47:21,890 --> 00:47:24,760 And this has o parity. 628 00:47:24,760 --> 00:47:27,280 And that leads us immediate to the result 629 00:47:27,280 --> 00:47:32,740 when we have an atom in an eigenstate n 630 00:47:32,740 --> 00:47:38,800 and we ask, what is the expectation value of H prime? 631 00:47:38,800 --> 00:47:42,580 It is 0 because of parity. 632 00:47:47,530 --> 00:47:50,390 So the answer is, we have no permanent dipole 633 00:47:50,390 --> 00:47:54,620 moment until we have degenerate energy levels. 634 00:47:54,620 --> 00:47:58,140 If n is a non-degenerate level, this matrix element 635 00:47:58,140 --> 00:48:00,260 is 0 by the parity selection rule. 636 00:48:05,606 --> 00:48:09,640 OK, now we want to do perturbation theory. 637 00:48:12,240 --> 00:48:17,154 So our perturbation operator is this. 638 00:48:35,440 --> 00:48:37,600 And since we have the clickers, I just 639 00:48:37,600 --> 00:48:41,100 want to ask you two quick questions. 640 00:48:41,100 --> 00:48:42,740 I will do the perturbation theory 641 00:48:42,740 --> 00:48:48,840 and I will explain everything, but maybe you 642 00:48:48,840 --> 00:48:51,280 want to predict the result, which 643 00:48:51,280 --> 00:48:53,900 I want to derive in the next 10 minutes. 644 00:48:53,900 --> 00:49:01,650 And the question is, what will we actually 645 00:49:01,650 --> 00:49:10,730 get for the expectation value of H prime? 646 00:49:10,730 --> 00:49:21,990 Will we get the expectation value of the dipole operator 647 00:49:21,990 --> 00:49:25,280 times the electric field or will we 648 00:49:25,280 --> 00:49:30,080 get the expectation value of the dipole operator 649 00:49:30,080 --> 00:49:31,520 times the electric field over 2? 650 00:49:35,530 --> 00:49:39,590 And the next question would be the same, 651 00:49:39,590 --> 00:49:42,130 but what do we get for the total Hamiltonian? 652 00:49:44,770 --> 00:49:46,240 So these are the questions. 653 00:49:46,240 --> 00:49:49,200 I want to discuss with you in the next 10 minutes, 654 00:49:49,200 --> 00:49:53,020 simply using perturbation theory, expectation values. 655 00:49:53,020 --> 00:49:57,030 Expectation values of the total energy each, not plus H prime. 656 00:49:57,030 --> 00:50:01,170 But also, expectation values of the electrostatic energy, 657 00:50:01,170 --> 00:50:03,680 which is H prime. 658 00:50:03,680 --> 00:50:05,630 And the question is-- I mean, you 659 00:50:05,630 --> 00:50:07,920 can say for dimensionless units, what 660 00:50:07,920 --> 00:50:12,360 we get is a dipole moment times an electric field. 661 00:50:12,360 --> 00:50:17,130 And this is one of the situation where factors of 1/2 662 00:50:17,130 --> 00:50:18,920 are not just bookkeeping. 663 00:50:18,920 --> 00:50:24,340 Factors of 1/2 really reflect interesting physics. 664 00:50:24,340 --> 00:50:27,210 And I want to sort of highlight it by asking you, 665 00:50:27,210 --> 00:50:29,970 what would you expect what we get 666 00:50:29,970 --> 00:50:38,550 for those expectation values when we solve for atomic energy 667 00:50:38,550 --> 00:50:44,140 levels in electric fields? 668 00:50:44,140 --> 00:50:46,350 So we're discussing first question 1. 669 00:50:57,580 --> 00:50:59,707 OK, let's go to question 2. 670 00:51:31,020 --> 00:51:33,200 OK. 671 00:51:33,200 --> 00:51:35,460 Anyway, now I know I'm not boring you 672 00:51:35,460 --> 00:51:38,050 with the derivation I want to give you in the next 10 673 00:51:38,050 --> 00:51:38,890 minutes. 674 00:51:38,890 --> 00:51:41,330 I want to give you the answer right away 675 00:51:41,330 --> 00:51:44,370 by drawing up another problem where 676 00:51:44,370 --> 00:51:46,860 maybe the answer is more intuitive. 677 00:51:46,860 --> 00:51:54,920 And this is we have a mass on a spring with spring constant k. 678 00:51:54,920 --> 00:51:59,040 And now the equivalent to the electric field 679 00:51:59,040 --> 00:52:03,070 which we switch on is-- we switch on gravity. 680 00:52:03,070 --> 00:52:10,800 And due to gravity, the object zags by an amount delta z. 681 00:52:10,800 --> 00:52:14,250 So the question is, what is-- and delta z 682 00:52:14,250 --> 00:52:16,160 is like the dipole moment. 683 00:52:16,160 --> 00:52:22,830 What is the gravitational energy gained by the object 684 00:52:22,830 --> 00:52:24,590 because it has fallen down? 685 00:52:24,590 --> 00:52:28,160 It is zagging down due to gravity. 686 00:52:28,160 --> 00:52:32,260 Well, I think you would agree that the answer is, 687 00:52:32,260 --> 00:52:34,590 it is mg times delta z. 688 00:52:34,590 --> 00:52:38,680 This is the work done by gravity with a minus sign. 689 00:52:38,680 --> 00:52:43,320 So the expectation value of the perturbation operator 690 00:52:43,320 --> 00:52:45,340 is minus mg times delta z. 691 00:52:45,340 --> 00:52:47,540 Or in electrostatic units, it's simply 692 00:52:47,540 --> 00:52:51,500 the dipole moment times the electric field. 693 00:52:51,500 --> 00:52:58,800 But what happens is the-- so this is gravitational energy. 694 00:52:58,800 --> 00:53:05,900 How much is the total energy affected 695 00:53:05,900 --> 00:53:08,396 when we switch on the gravitational field? 696 00:53:14,620 --> 00:53:17,620 1/2 of it, because the negative energy 697 00:53:17,620 --> 00:53:22,820 which is gained in the gravitational field, 1/2 of it 698 00:53:22,820 --> 00:53:25,790 is used to stretch the spring. 699 00:53:25,790 --> 00:53:29,900 1/2 of it goes into the internal energy of the system. 700 00:53:35,020 --> 00:53:41,820 So therefore, the electrostatic energy H prime-- H prime 701 00:53:41,820 --> 00:53:44,520 is the operator of the electrostatic energy. 702 00:53:44,520 --> 00:53:48,700 The answer here is A. But the total energy 703 00:53:48,700 --> 00:53:52,090 is B because part of the energy is 704 00:53:52,090 --> 00:53:56,610 needed to stretch the spring. 705 00:53:56,610 --> 00:54:00,260 And as I want to show you, stretching the spring 706 00:54:00,260 --> 00:54:04,190 is-- we admix to the ground state some excited state. 707 00:54:04,190 --> 00:54:07,330 This costs energy, like stretching the spring costs 708 00:54:07,330 --> 00:54:09,150 energy. 709 00:54:09,150 --> 00:54:12,300 And this is responsible for the fact of which is exactly 1/2. 710 00:54:27,540 --> 00:54:29,590 Well, I could stop here. 711 00:54:29,590 --> 00:54:33,670 I think I've explained it all, but let's follow 712 00:54:33,670 --> 00:54:38,900 the usual-- the standard approach. 713 00:54:38,900 --> 00:54:41,140 And let's do second-order perturbation theory 714 00:54:41,140 --> 00:54:45,380 and calculate the energy, calculate the dipole moment, 715 00:54:45,380 --> 00:54:49,246 and see that everything is as we expect now. 716 00:54:55,200 --> 00:54:57,500 So we want to do second-order perturbation theory. 717 00:55:00,530 --> 00:55:03,210 We know already the first-order term is 0. 718 00:55:03,210 --> 00:55:06,090 This was a discussion about parity. 719 00:55:06,090 --> 00:55:12,040 And in second-order perturbation theory, 720 00:55:12,040 --> 00:55:17,700 the state n has an energy, which is the unperturbed energy. 721 00:55:17,700 --> 00:55:23,320 And then in second-order, we have the matrix element 722 00:55:23,320 --> 00:55:25,450 to all other states. 723 00:55:25,450 --> 00:55:27,880 We square it. 724 00:55:27,880 --> 00:55:33,960 We divide by the energy denominator. 725 00:55:33,960 --> 00:55:38,930 We sum over all states m, but I make a prime here. 726 00:55:38,930 --> 00:55:41,240 Of course, we are not summing over 727 00:55:41,240 --> 00:55:44,390 the state-- we exclude n from the summation. 728 00:55:51,236 --> 00:55:57,900 And the prefactor here is electron charge 729 00:55:57,900 --> 00:56:01,910 squared times electric field squared. 730 00:56:01,910 --> 00:56:04,730 OK, pretty much that's the resulting 731 00:56:04,730 --> 00:56:07,270 second-order perturbation theory. 732 00:56:07,270 --> 00:56:09,690 So this is the energy and we want 733 00:56:09,690 --> 00:56:15,270 to relate the energy to the dipole moment. 734 00:56:15,270 --> 00:56:19,080 So the next step is now we calculate d. 735 00:56:19,080 --> 00:56:26,180 and we calculate d from the first-order wave function 736 00:56:26,180 --> 00:56:29,350 because we already get an effect in first order and everything 737 00:56:29,350 --> 00:56:32,360 here is about leading order. 738 00:56:32,360 --> 00:56:46,580 So the expectation value of the dipole operator-- 739 00:56:46,580 --> 00:56:51,040 so we take the expectation value of the dipole operator and we 740 00:56:51,040 --> 00:56:58,600 use the 0-th order, the unperturbed state, 741 00:56:58,600 --> 00:57:00,660 plus the first-order correction. 742 00:57:05,910 --> 00:57:09,990 And we know already that the diagonal terms do not 743 00:57:09,990 --> 00:57:10,880 contribute. 744 00:57:10,880 --> 00:57:14,930 This is a parity selection rule. 745 00:57:14,930 --> 00:57:21,110 So we get contributions from the course term, 746 00:57:21,110 --> 00:57:26,845 which is n0 the dipole operator with n1. 747 00:57:32,300 --> 00:57:36,166 So let's just suppress vectorial notation. 748 00:57:36,166 --> 00:57:37,790 We know everything is along the z-axis. 749 00:57:42,810 --> 00:57:47,620 So we have the 0-th order wave function. 750 00:57:47,620 --> 00:57:50,790 Our operator is z. 751 00:57:50,790 --> 00:57:54,540 And now we have to write down the first-order correction 752 00:57:54,540 --> 00:57:55,970 to the wave function. 753 00:57:55,970 --> 00:57:58,860 And the first-order correction is 754 00:57:58,860 --> 00:58:02,940 the sum over all other states. 755 00:58:02,940 --> 00:58:06,440 We make an admixture of the state m, 756 00:58:06,440 --> 00:58:13,890 and this admixture uses a matrix element. 757 00:58:13,890 --> 00:58:15,740 And here, we have the energy denominator. 758 00:58:20,960 --> 00:58:28,070 So what we obtain is-- we have the electron charge here 759 00:58:28,070 --> 00:58:29,160 from the dipole moment. 760 00:58:29,160 --> 00:58:33,790 We have the electron charge due to the perturbation operator. 761 00:58:33,790 --> 00:58:36,430 So it's electron squared. 762 00:58:36,430 --> 00:58:39,370 We have the electric field. 763 00:58:39,370 --> 00:58:50,030 And then, this is due to the admixture of the wave 764 00:58:50,030 --> 00:58:52,120 function with the dipole operator. 765 00:58:52,120 --> 00:58:55,640 And now because we take the matrix element 766 00:58:55,640 --> 00:58:58,950 of the dipole operator, we get another occurrence 767 00:58:58,950 --> 00:59:00,690 of the dipole operator. 768 00:59:00,690 --> 00:59:03,840 So therefore, we do first-order perturbation theory, 769 00:59:03,840 --> 00:59:06,860 but we take the first-order result 770 00:59:06,860 --> 00:59:10,150 and ask, what is the expectation value for the dipole moment? 771 00:59:10,150 --> 00:59:12,450 And that means the dipole operator, 772 00:59:12,450 --> 00:59:16,140 or the perturbation operator appears twice. 773 00:59:16,140 --> 00:59:18,230 And our result is as expected. 774 00:59:21,550 --> 00:59:26,330 Quadratic in the matrix element and it 775 00:59:26,330 --> 00:59:29,650 has this energy denominator. 776 00:59:29,650 --> 00:59:36,410 So the definition is that a dipole moment is alpha 777 00:59:36,410 --> 00:59:38,460 times the electric field. 778 00:59:41,000 --> 00:59:44,170 So therefore, all that equals alpha. 779 00:59:46,680 --> 00:59:52,360 And if you compare now the result for the dipole moment 780 00:59:52,360 --> 00:59:54,340 with the second-order perturbation 781 00:59:54,340 --> 00:59:59,200 theory for the electric field, for the energy, 782 00:59:59,200 --> 01:00:05,620 we find-- here's a factor of 2, but there is no factor of 2 783 01:00:05,620 --> 01:00:06,880 up there. 784 01:00:06,880 --> 01:00:23,530 We find that the energy or the energy shift delta En, 785 01:00:23,530 --> 01:00:26,960 it has exactly the same matrix element as the polarizability. 786 01:00:33,170 --> 01:00:36,560 It is-- yes. 787 01:00:39,885 --> 01:00:45,402 It is this, 1/2 alpha epsilon squared. 788 01:00:57,690 --> 01:01:00,800 Since the perturbation operator, I'm just writing it down here, 789 01:01:00,800 --> 01:01:03,320 was dipole moment times electron, 790 01:01:03,320 --> 01:01:08,120 that means that the energy shift is-- and this 791 01:01:08,120 --> 01:01:12,820 is what we expected now, is 1/2 times 792 01:01:12,820 --> 01:01:15,350 the expectation value of the dipole moment 793 01:01:15,350 --> 01:01:17,237 times the electric field. 794 01:01:23,730 --> 01:01:28,220 So now we have obtained with a quantum mechanical calculation 795 01:01:28,220 --> 01:01:28,780 the result. 796 01:01:28,780 --> 01:01:36,200 I told you that the energy shift of the energy levels 797 01:01:36,200 --> 01:01:39,158 is 1/2 the dipole moment times the electric field. 798 01:01:44,540 --> 01:01:51,280 Let me just redo the calculation in a way I like. 799 01:01:51,280 --> 01:01:56,790 And this is I want to determine now the total energy, 800 01:01:56,790 --> 01:02:01,470 but sort the terms in a little bit different way. 801 01:02:01,470 --> 01:02:06,660 So I want to know, what is the energy in our result? 802 01:02:06,660 --> 01:02:13,370 And what we do is we are calculating the energy 803 01:02:13,370 --> 01:02:14,620 using our wave functions. 804 01:02:21,400 --> 01:02:34,890 We take the total Hamiltonian and take the wave function. 805 01:02:42,090 --> 01:02:48,120 So this leads us to three terms. 806 01:02:48,120 --> 01:02:52,470 One is the unperturbed energy. 807 01:03:05,080 --> 01:03:13,060 The unperturbed energy, the energy contribution 808 01:03:13,060 --> 01:03:14,460 of the first-order correction. 809 01:03:20,470 --> 01:03:24,170 This is the part due to H0. 810 01:03:24,170 --> 01:03:34,690 And the part due to H prime is simply 811 01:03:34,690 --> 01:03:37,790 the dipole moment times the electric field. 812 01:03:40,580 --> 01:03:48,420 So the first part is, of course, simply the energy E0 813 01:03:48,420 --> 01:03:52,760 times the norm of the wave function n0. 814 01:03:58,170 --> 01:04:10,320 For the second term, we use the first-order perturbation theory 815 01:04:10,320 --> 01:04:14,040 for n1. 816 01:04:14,040 --> 01:04:15,250 This is our sum over m. 817 01:04:21,304 --> 01:04:24,130 Em minus E0. 818 01:04:24,130 --> 01:04:26,565 m H prime 0. 819 01:04:30,200 --> 01:04:33,410 Because n1 is on either side, this 820 01:04:33,410 --> 01:04:36,360 is the amplitude of the state n1. 821 01:04:36,360 --> 01:04:37,860 We have to square it. 822 01:04:37,860 --> 01:04:44,490 And since we calculate what is the expectation value of H0, 823 01:04:44,490 --> 01:04:46,420 we multiply with the energy Em. 824 01:04:52,650 --> 01:04:54,630 OK, so we are done. 825 01:04:54,630 --> 01:04:56,700 We calculate the total energy. 826 01:04:56,700 --> 01:04:58,230 We get three terms. 827 01:04:58,230 --> 01:05:00,770 One is the unperturbed energy, one 828 01:05:00,770 --> 01:05:03,760 is the dipole energy in the electric field, 829 01:05:03,760 --> 01:05:07,100 and one is the extra term, which I want to discuss. 830 01:05:07,100 --> 01:05:11,120 Actually, this term is the internal energy, 831 01:05:11,120 --> 01:05:13,015 which would correspond to the stretching 832 01:05:13,015 --> 01:05:16,020 of the spring in Hooke's law. 833 01:05:16,020 --> 01:05:21,890 Now, in order to show it to you explicitly, 834 01:05:21,890 --> 01:05:27,890 I want to use E0 equal 0 for the energy. 835 01:05:27,890 --> 01:05:31,210 Because then this term is 0. 836 01:05:31,210 --> 01:05:33,520 I can neglect this. 837 01:05:33,520 --> 01:05:39,520 And one of the squares, 1 over Em squared, 838 01:05:39,520 --> 01:05:40,650 cancels with the Em. 839 01:05:51,396 --> 01:05:53,145 It confused me for a while. 840 01:05:53,145 --> 01:05:57,410 If I don't set E0 to 0, the result looks different. 841 01:05:57,410 --> 01:06:01,950 But what happens is, if you do perturbation theory, 842 01:06:01,950 --> 01:06:05,100 there are certain issues with the normalization of the wave 843 01:06:05,100 --> 01:06:06,300 function. 844 01:06:06,300 --> 01:06:11,720 And the wave function n0 has to be-- 845 01:06:11,720 --> 01:06:14,420 the contribution if you look at the wave 846 01:06:14,420 --> 01:06:17,470 function in perturbation theory of a state, 847 01:06:17,470 --> 01:06:21,960 the 0-th order wave function has an amplitude of 1. 848 01:06:21,960 --> 01:06:26,230 And this amplitude of 1 only changes in second order. 849 01:06:26,230 --> 01:06:29,670 So since I'm doing a second-order calculation here, 850 01:06:29,670 --> 01:06:32,290 I have to include those non-standard terms. 851 01:06:32,290 --> 01:06:36,640 But I can also bypass it by setting E0 to 0, 852 01:06:36,640 --> 01:06:39,720 then the second-order term in the norm doesn't matter. 853 01:06:39,720 --> 01:06:42,440 So in other words, if you set E0 to 0, 854 01:06:42,440 --> 01:06:43,920 you make your life easier. 855 01:06:43,920 --> 01:06:46,652 If you do not set E0 to 0, you have 856 01:06:46,652 --> 01:06:48,610 to include some more terms in your calculation. 857 01:06:52,720 --> 01:07:04,344 But the result is-- just one second. 858 01:07:10,990 --> 01:07:14,270 But yeah, the result which I wanted to emphasize 859 01:07:14,270 --> 01:07:15,820 is this one here. 860 01:07:15,820 --> 01:07:18,770 It is a positive energy. 861 01:07:18,770 --> 01:07:23,910 You can immediately inspect that t positive energy 862 01:07:23,910 --> 01:07:29,170 is the dipole moment times the electric field over 2. 863 01:07:31,840 --> 01:07:45,900 This is exactly analogous to the energy 864 01:07:45,900 --> 01:07:48,270 of the spring in the gravitational problem. 865 01:07:50,800 --> 01:07:55,360 So in other words, this is the energy, internal energy, 866 01:07:55,360 --> 01:07:58,050 because we admix excited states to the ground state. 867 01:07:58,050 --> 01:08:00,400 This crosses energy and it exactly 868 01:08:00,400 --> 01:08:07,638 accounts for the occurrence of the factors of 1/2. 869 01:08:10,890 --> 01:08:13,580 Anyway, this is just the standard theory 870 01:08:13,580 --> 01:08:17,600 of the DC stark effect of the atomic polarizability, 871 01:08:17,600 --> 01:08:19,260 but I put a little bit of emphasis 872 01:08:19,260 --> 01:08:21,439 on those factors of 1/2 and tried 873 01:08:21,439 --> 01:08:25,130 to explain in greater detail the contributions 874 01:08:25,130 --> 01:08:28,010 to the AC stark effect-- to the DC stark effect 875 01:08:28,010 --> 01:08:30,339 which come from the electrostatic energy 876 01:08:30,339 --> 01:08:34,000 and which come from the internal energy. 877 01:08:34,000 --> 01:08:34,500 Questions? 878 01:08:38,955 --> 01:08:39,980 Yes. 879 01:08:39,980 --> 01:08:40,604 AUDIENCE: Sure. 880 01:08:40,604 --> 01:08:42,915 I have a study question. 881 01:08:42,915 --> 01:08:49,350 What is allowing you to use non-degenerate perturbation 882 01:08:49,350 --> 01:08:51,825 theory? 883 01:08:51,825 --> 01:08:55,330 What's the operator that [INAUDIBLE]? 884 01:08:55,330 --> 01:08:57,296 PROFESSOR: Well, I'm looking-- what 885 01:08:57,296 --> 01:08:59,420 allows me to do non-degenerate perturbation theory. 886 01:08:59,420 --> 01:09:02,250 Well, I assume we don't have degeneracies. 887 01:09:02,250 --> 01:09:05,200 If you would go to very high Rydberg states-- and actually, 888 01:09:05,200 --> 01:09:10,229 we do that not today, but on Monday-- we 889 01:09:10,229 --> 01:09:15,474 are looking at a situation where the splitting between states 890 01:09:15,474 --> 01:09:20,720 of different L become so small that the electric field mixes 891 01:09:20,720 --> 01:09:21,920 them. 892 01:09:21,920 --> 01:09:26,290 Then, we have to do degenerate perturbation theory. 893 01:09:26,290 --> 01:09:29,240 And that means we get now a linear term, 894 01:09:29,240 --> 01:09:32,529 linear stark effect, not a quadratic stark effect. 895 01:09:32,529 --> 01:09:35,505 Here, I would say we are doing perturbation 896 01:09:35,505 --> 01:09:37,020 theory of the ground state. 897 01:09:37,020 --> 01:09:37,970 It's an S state. 898 01:09:37,970 --> 01:09:38,803 It's not degenerate. 899 01:09:41,410 --> 01:09:44,410 Maybe your question is also addressing, 900 01:09:44,410 --> 01:09:46,569 but we have multiple ground states. 901 01:09:46,569 --> 01:09:48,490 We have hyperfine structure. 902 01:09:48,490 --> 01:09:52,240 However, the electronics-- the stark effect, 903 01:09:52,240 --> 01:09:56,040 the electric field does not couple to the spin at all. 904 01:09:56,040 --> 01:09:58,460 So therefore, all the magnetic energies-- 905 01:09:58,460 --> 01:10:02,420 the hyperfine energies are completely unaffected. 906 01:10:02,420 --> 01:10:05,690 And also, if we apply an electric field, 907 01:10:05,690 --> 01:10:12,280 all the hyperfine states experience the same shift 908 01:10:12,280 --> 01:10:14,850 and there is no coupling between them. 909 01:10:14,850 --> 01:10:16,910 So also, we have multiple ground states. 910 01:10:16,910 --> 01:10:18,660 We have hyperfine structure. 911 01:10:18,660 --> 01:10:20,995 It's a non-degenerate problem because there 912 01:10:20,995 --> 01:10:24,550 is no coupling between the different hyperfine states. 913 01:10:24,550 --> 01:10:28,610 In other words, the theory or the discussion of the DC 914 01:10:28,610 --> 01:10:31,820 stark shift is you have an S state, 915 01:10:31,820 --> 01:10:33,370 you couple with an electric field, 916 01:10:33,370 --> 01:10:38,022 and there is no degeneracy in the S state. 917 01:10:38,022 --> 01:10:38,970 Other questions? 918 01:10:42,300 --> 01:10:48,450 Well, then we've talked about alpha. 919 01:10:48,450 --> 01:10:55,310 The only parameter which comes out of this treatment is alpha. 920 01:10:55,310 --> 01:10:58,440 And now we want to discuss, how big is alpha? 921 01:10:58,440 --> 01:11:00,730 Or first, what are the units of alpha? 922 01:11:03,430 --> 01:11:12,280 Well, the units of alpha were-- you can go back 923 01:11:12,280 --> 01:11:14,230 to the second-order perturbation result. 924 01:11:14,230 --> 01:11:18,630 But the units of alpha were the charge, time [INAUDIBLE]. 925 01:11:18,630 --> 01:11:20,765 This was the dipole operator. 926 01:11:20,765 --> 01:11:23,330 It was squared. 927 01:11:23,330 --> 01:11:26,790 And in perturbation theory, we divided by energy 928 01:11:26,790 --> 01:11:28,490 because we had an energy denominator. 929 01:11:34,270 --> 01:11:46,190 Well, we can write that as q squared over l times l cubed. 930 01:11:46,190 --> 01:11:49,740 But q squared over l is the Coulomb energy. 931 01:11:49,740 --> 01:11:53,180 And therefore, when I'm interested in the units, 932 01:11:53,180 --> 01:11:55,270 the units cancel. 933 01:11:55,270 --> 01:12:05,020 So therefore, we find that the unit of the polarizability, 934 01:12:05,020 --> 01:12:07,720 at least in [INAUDIBLE] unit or atomic units, which 935 01:12:07,720 --> 01:12:11,750 I've chosen here, is simply the volume. 936 01:12:11,750 --> 01:12:13,460 The question is, what volume? 937 01:12:18,110 --> 01:12:35,570 Well, if you would calculate the polarizability for hydrogen, 938 01:12:35,570 --> 01:12:39,230 and simply make the assumption that the only important matrix 939 01:12:39,230 --> 01:12:42,490 element goes from the S to the P state, 940 01:12:42,490 --> 01:12:46,130 then we have a matrix element which is on the order of 941 01:12:46,130 --> 01:12:47,400 [INAUDIBLE]. 942 01:12:47,400 --> 01:12:51,270 And the energy splitting between the first ground state 943 01:12:51,270 --> 01:12:54,807 and first excited state is three quarter 944 01:12:54,807 --> 01:12:55,807 of the Rydberg constant. 945 01:13:00,972 --> 01:13:14,470 So for hydrogen in the 1s state, if you only use the coupling 946 01:13:14,470 --> 01:13:22,410 to the 2p state, we find that alpha is the Bohr radius cubed. 947 01:13:22,410 --> 01:13:27,380 And the prefactor is 2.96. 948 01:13:27,380 --> 01:13:29,820 If you do the summation over all states, 949 01:13:29,820 --> 01:13:33,430 the prefactor would be 4.5 because there 950 01:13:33,430 --> 01:13:36,620 are higher states, especially continuum states, which 951 01:13:36,620 --> 01:13:37,720 contribute to the sum. 952 01:13:56,680 --> 01:13:59,400 We have only five minutes left, but that 953 01:13:59,400 --> 01:14:09,320 allows me to show you that this is not a coincidence that we 954 01:14:09,320 --> 01:14:13,400 obtain-- here, what we obtain is the Bohr radius cubed, which 955 01:14:13,400 --> 01:14:15,590 is pretty much the volume of the hydrogen atom. 956 01:14:18,490 --> 01:14:21,580 But we can now do an approximation. 957 01:14:21,580 --> 01:14:23,980 It's not really relevant, but it has an historic name-- 958 01:14:23,980 --> 01:14:25,840 [INAUDIBLE] approximation. 959 01:14:25,840 --> 01:14:27,975 It's just nice to show how things work out. 960 01:14:34,390 --> 01:14:36,105 We have a second-order matrix element, 961 01:14:36,105 --> 01:14:41,360 so we couple the state n with the operator z to a state m. 962 01:14:41,360 --> 01:14:47,720 But if we assume that all energy denominators can be taken out 963 01:14:47,720 --> 01:14:51,420 of the summation by assuming that we have 964 01:14:51,420 --> 01:14:57,170 some kind of average excited energy, 965 01:14:57,170 --> 01:15:00,590 then the sum of-- maybe I should have 966 01:15:00,590 --> 01:15:11,670 said it the sum m z n, which we sum over m 967 01:15:11,670 --> 01:15:15,020 and it just cancels out. 968 01:15:15,020 --> 01:15:19,270 So what we have is, if you take the energy, 969 01:15:19,270 --> 01:15:23,170 an average energy denominator out of the summation, 970 01:15:23,170 --> 01:15:27,530 what we find is that what matters 971 01:15:27,530 --> 01:15:29,560 is the matrix element z squared. 972 01:15:32,360 --> 01:15:39,340 And we can even assume that in the energy denominator, 973 01:15:39,340 --> 01:15:42,370 the excited state energy is negligible. 974 01:15:42,370 --> 01:15:45,790 The hydrogen atom has a binding energy of 1 Rydberg 975 01:15:45,790 --> 01:15:48,910 and the first excited state has a quarter Rydberg. 976 01:15:48,910 --> 01:15:53,590 So at the 25% level, we can set that to 0. 977 01:15:53,590 --> 01:16:07,160 So I'm waving all my hands, but I'm getting a simple expression 978 01:16:07,160 --> 01:16:11,090 for the polarizability in the ground state. 979 01:16:11,090 --> 01:16:15,330 And this goes as follows-- the ground state energy is-- 980 01:16:15,330 --> 01:16:18,110 we have discussed Coulomb energy, Virial's 981 01:16:18,110 --> 01:16:20,870 theorem, and all that. 982 01:16:20,870 --> 01:16:24,360 We need just 1/r in the ground state. 983 01:16:24,360 --> 01:16:30,010 And for the z squared matrix element, 984 01:16:30,010 --> 01:16:35,290 we can simply say for an S state that it 985 01:16:35,290 --> 01:16:38,350 is x squared y squared z squared. 986 01:16:38,350 --> 01:16:42,090 It is 1/2 of r squared in the ground state. 987 01:16:42,090 --> 01:16:47,530 So therefore, continuously waving our hands 988 01:16:47,530 --> 01:16:53,940 and making approximations, we find that the polarizability 989 01:16:53,940 --> 01:17:00,500 is r squared expectation value divided by an r 990 01:17:00,500 --> 01:17:01,800 to the minus expectation value. 991 01:17:05,720 --> 01:17:10,720 So this is some r cubed expectation 992 01:17:10,720 --> 01:17:13,916 value which is an atomic value. 993 01:17:17,080 --> 01:17:20,710 So you see the nature of the perturbation expression 994 01:17:20,710 --> 01:17:27,350 suggests that cannot be anything else than the atomic volume. 995 01:17:27,350 --> 01:17:30,940 I sort of like that because when people discuss, for instance 996 01:17:30,940 --> 01:17:34,340 in my group, does lithium or rubidium 997 01:17:34,340 --> 01:17:36,600 have a bigger polarizability? 998 01:17:36,600 --> 01:17:40,170 Well, the bigger atom has a bigger volume 999 01:17:40,170 --> 01:17:43,410 and the more fluffier atoms have the larger polarizability. 1000 01:17:43,410 --> 01:17:46,412 And that's pretty much based on that result. 1001 01:17:50,700 --> 01:17:56,790 Now, let me finally do a comparison. 1002 01:17:56,790 --> 01:17:59,230 There is another system for which 1003 01:17:59,230 --> 01:18:01,320 you have done calculations of the dipole moment. 1004 01:18:04,170 --> 01:18:11,862 And this is in classical E and M for conducting sphere. 1005 01:18:19,590 --> 01:18:24,070 For conducting [INAUDIBLE] electric field, 1006 01:18:24,070 --> 01:18:27,320 you can exactly solve the boundary conditions, 1007 01:18:27,320 --> 01:18:30,360 the boundary value problem, get the electric field, 1008 01:18:30,360 --> 01:18:31,605 and find the dipole moment. 1009 01:18:37,930 --> 01:18:44,580 And the exact result is that the dipole moment 1010 01:18:44,580 --> 01:18:52,900 is the electric field times the cube of the sphere. 1011 01:18:52,900 --> 01:18:56,960 So in other words, the dipole moment, or the polarizability 1012 01:18:56,960 --> 01:19:00,170 of this sphere, is-- and neglecting 1013 01:19:00,170 --> 01:19:02,590 factors, which are only factors of unity. 1014 01:19:02,590 --> 01:19:08,180 The dipole moment-- sorry, the polarizability of a conducting 1015 01:19:08,180 --> 01:19:11,890 sphere is the volume of the sphere. 1016 01:19:11,890 --> 01:19:15,040 The dipole moment of a hydrogen atom, 1017 01:19:15,040 --> 01:19:17,250 or using [INAUDIBLE] approximation 1018 01:19:17,250 --> 01:19:21,030 for all simple atoms is the volume of the atom. 1019 01:19:21,030 --> 01:19:22,910 So I find it sort of interesting that when 1020 01:19:22,910 --> 01:19:25,710 it comes to dipole moments and to polarizability, 1021 01:19:25,710 --> 01:19:29,230 that atoms pretty much behave like 1022 01:19:29,230 --> 01:19:31,960 metallic-conducting spheres of the same volume. 1023 01:19:45,277 --> 01:19:45,860 Any questions? 1024 01:19:49,980 --> 01:19:53,310 OK, then let's stop here and we meet again on Monday.