1 00:00:00,499 --> 00:00:02,920 PROFESSOR: All right, so how do we solve this? 2 00:00:02,920 --> 00:00:05,380 This is a very interesting thing, 3 00:00:05,380 --> 00:00:14,230 and I think it points to all kinds of important things 4 00:00:14,230 --> 00:00:18,620 that people find useful in physics. 5 00:00:18,620 --> 00:00:24,160 So here it is, the way, maybe, we can think about it. 6 00:00:24,160 --> 00:00:27,080 Think variational method. 7 00:00:27,080 --> 00:00:29,410 What is the variational method? 8 00:00:29,410 --> 00:00:32,950 You write a wave function and you 9 00:00:32,950 --> 00:00:37,030 try to see what is the expectation 10 00:00:37,030 --> 00:00:39,430 value of the Hamiltonian and that wave function. 11 00:00:39,430 --> 00:00:42,250 You calculate it, and now you know that the ground state 12 00:00:42,250 --> 00:00:44,890 energy is below that number. 13 00:00:44,890 --> 00:00:47,320 Because for the real ground state 14 00:00:47,320 --> 00:00:49,210 expectation value of the Hamiltonian 15 00:00:49,210 --> 00:00:50,590 is the ground state energy. 16 00:00:50,590 --> 00:00:53,990 For an arbitrary state it's more than that. 17 00:00:53,990 --> 00:00:57,190 So the variational method says, OK, 18 00:00:57,190 --> 00:01:00,820 if you want to figure out what is a good wave 19 00:01:00,820 --> 00:01:05,590 function, compute the expectation value of your wave 20 00:01:05,590 --> 00:01:11,160 function in the Hamiltonian, and then you will see, 21 00:01:11,160 --> 00:01:12,600 you will get some energy. 22 00:01:12,600 --> 00:01:14,430 If you tinker with your wave function 23 00:01:14,430 --> 00:01:17,500 you can get to the right energy. 24 00:01:17,500 --> 00:01:19,320 So this is what we're going to do 25 00:01:19,320 --> 00:01:27,720 we're going to try to take this psi of r, r 26 00:01:27,720 --> 00:01:30,330 and put the total Hamiltonian here. 27 00:01:30,330 --> 00:01:34,110 And now that psi of r, r is going to be this one from now 28 00:01:34,110 --> 00:01:35,550 on. 29 00:01:35,550 --> 00:01:37,832 This is psi of r, r. 30 00:01:37,832 --> 00:01:44,715 We'll put it here, and then I'll put another psi of r, r. 31 00:01:50,170 --> 00:01:54,320 And I have to compute that. 32 00:01:54,320 --> 00:02:00,790 But think of it-- it's actually pretty nice, the situation. 33 00:02:00,790 --> 00:02:03,970 Computing this, you could say how can I 34 00:02:03,970 --> 00:02:09,639 compute that if I don't know the etas, the single eta that 35 00:02:09,639 --> 00:02:10,210 is there? 36 00:02:10,210 --> 00:02:13,750 I know the Hamiltonian but I don't know the eta, 37 00:02:13,750 --> 00:02:16,900 so I can't compute this. 38 00:02:16,900 --> 00:02:21,580 But think of this really, this is an integral, because there's 39 00:02:21,580 --> 00:02:24,790 an [INAUDIBLE] over all the capital R's, 40 00:02:24,790 --> 00:02:28,610 an integral over the little r's, and you can say, 41 00:02:28,610 --> 00:02:34,140 oh, I know the little r's here. 42 00:02:34,140 --> 00:02:35,700 I know them very well. 43 00:02:35,700 --> 00:02:36,720 I found them. 44 00:02:36,720 --> 00:02:39,220 I found this phi r. 45 00:02:39,220 --> 00:02:40,530 I've determined it. 46 00:02:40,530 --> 00:02:43,470 Some way you did. 47 00:02:43,470 --> 00:02:47,690 So if I know that, I know the little r dependence, 48 00:02:47,690 --> 00:02:53,840 and I can do all the little r integrals. 49 00:02:53,840 --> 00:02:57,770 If I can do all the little r integrals, 50 00:02:57,770 --> 00:03:01,700 means that this wave function is a product, 51 00:03:01,700 --> 00:03:04,280 this wave function is a product, this 52 00:03:04,280 --> 00:03:07,800 I can reduce to the following. 53 00:03:07,800 --> 00:03:16,470 Eta of r, because this one I don't know, and then eta of r, 54 00:03:16,470 --> 00:03:19,010 and I claim that all the part that 55 00:03:19,010 --> 00:03:23,060 had to do with the phi of r, we can take care, 56 00:03:23,060 --> 00:03:27,920 and then there's going to be some Hamiltonian here left. 57 00:03:27,920 --> 00:03:33,760 And this we're going to call h effective of the r 58 00:03:33,760 --> 00:03:34,810 degrees of freedom. 59 00:03:42,240 --> 00:03:45,530 So a lot of physics in that step. 60 00:03:45,530 --> 00:03:50,630 Seems like we did nothing, but look, 61 00:03:50,630 --> 00:03:55,790 I'm supposed to minimize this by adjusting my trial wave 62 00:03:55,790 --> 00:03:57,320 functions. 63 00:03:57,320 --> 00:04:00,380 OK, but I don't have room to adjust the phi, 64 00:04:00,380 --> 00:04:03,050 because I determined that it's a good one. 65 00:04:03,050 --> 00:04:06,180 So I put it in and I calculate this. 66 00:04:06,180 --> 00:04:09,490 Now I have to minimize this. 67 00:04:09,490 --> 00:04:15,030 But if I have to minimize this for just v nuclear wave 68 00:04:15,030 --> 00:04:19,850 function, I have found the effective Hamiltonian 69 00:04:19,850 --> 00:04:29,380 for the nuclei, in which the electron has essentially 70 00:04:29,380 --> 00:04:32,020 disappeared from this interaction. 71 00:04:32,020 --> 00:04:36,130 It just doesn't play a role anymore. 72 00:04:36,130 --> 00:04:38,890 This is the idea of effective Hamiltonians 73 00:04:38,890 --> 00:04:42,520 for slow degrees of freedom when you integrate 74 00:04:42,520 --> 00:04:44,320 fast degrees of freedom. 75 00:04:44,320 --> 00:04:46,550 The fast degrees of freedom are your electrons. 76 00:04:46,550 --> 00:04:49,720 We're getting rid of them, we're integrating them. 77 00:04:49,720 --> 00:04:51,760 We're going to try to do that. 78 00:04:51,760 --> 00:04:56,290 So this will be an effective Hamiltonian 79 00:04:56,290 --> 00:05:00,830 for the nuclear degrees of freedom. 80 00:05:00,830 --> 00:05:03,600 So let me do one term in here. 81 00:05:03,600 --> 00:05:08,500 One term, you can play trying to do other terms, 82 00:05:08,500 --> 00:05:12,550 and the notes will deal with that. 83 00:05:12,550 --> 00:05:17,470 So let's consider the term He. 84 00:05:17,470 --> 00:05:24,790 This Hamiltonian is some terms plus He. 85 00:05:24,790 --> 00:05:26,750 It's over there. 86 00:05:26,750 --> 00:05:31,000 So let's calculate what He does. 87 00:05:31,000 --> 00:05:38,350 So I now have to put effect of He. 88 00:05:38,350 --> 00:05:41,940 So I have to put here eta of r-- 89 00:05:41,940 --> 00:05:44,050 so let me put the whole thing. 90 00:05:44,050 --> 00:05:48,550 It's integral over all the R's-- 91 00:05:48,550 --> 00:05:53,065 R1, R2, R3-- integral over all the little r's. 92 00:05:53,065 --> 00:05:53,815 That's an overlap. 93 00:05:56,390 --> 00:06:02,620 Eta star of all the R's-- that's the [INAUDIBLE],, 94 00:06:02,620 --> 00:06:24,310 phi star R of little r, He, then etas of capital R, phi R of r. 95 00:06:27,520 --> 00:06:28,540 Everybody happy? 96 00:06:28,540 --> 00:06:32,180 That's my term. 97 00:06:32,180 --> 00:06:36,710 That's the contribution of He to the left hand side. 98 00:06:44,534 --> 00:06:47,470 Good? 99 00:06:47,470 --> 00:06:54,410 Now He is an operator and looks at this thing and says, 100 00:06:54,410 --> 00:06:59,920 well, I don't think derivatives with respect to capital R, 101 00:06:59,920 --> 00:07:01,810 so I'm there. 102 00:07:01,810 --> 00:07:04,300 I take the derivatives with respect to little r, 103 00:07:04,300 --> 00:07:07,780 I have a multiplicative factor, another multiplicative factor. 104 00:07:07,780 --> 00:07:12,700 So I basically act just directly on this. 105 00:07:12,700 --> 00:07:18,100 But He already acting on that gives you 106 00:07:18,100 --> 00:07:20,120 the electronic energy. 107 00:07:20,120 --> 00:07:28,030 So this would be the electronic energy in the ground state. 108 00:07:28,030 --> 00:07:36,760 That depends on R times phi R. And lots of vectors. 109 00:07:43,001 --> 00:07:43,500 OK. 110 00:07:52,750 --> 00:08:07,920 So this integral now becomes integral dR eta star of R. 111 00:08:07,920 --> 00:08:14,880 Then I'll put the other integral, integral dr, phi 112 00:08:14,880 --> 00:08:19,950 of r, star of r. 113 00:08:19,950 --> 00:08:44,750 And then I have just this number times eta of R. 114 00:08:44,750 --> 00:08:47,210 All right, so what is this integral? 115 00:08:47,210 --> 00:08:49,650 Happily, this is just a number. 116 00:08:49,650 --> 00:08:52,980 It goes out of this, it's an orthonormal state. 117 00:08:52,980 --> 00:09:10,440 So this whole thing is dR eta star of R, Ee of R, eta of R, 118 00:09:10,440 --> 00:09:21,020 which you could say is eta of R, the number Ee of R times 119 00:09:21,020 --> 00:09:27,140 eta of R, and therefore you've confirmed 120 00:09:27,140 --> 00:09:32,360 that the contribution to the effective Hamiltonian of all 121 00:09:32,360 --> 00:09:40,520 this electron cloud is just this number, the electronic energy 122 00:09:40,520 --> 00:09:41,030 like that. 123 00:09:41,030 --> 00:09:46,190 So in the effective Hamiltonian, h effective, 124 00:09:46,190 --> 00:09:49,130 there will be some terms and there will 125 00:09:49,130 --> 00:09:56,750 be Ee of R. That's one term. 126 00:10:02,880 --> 00:10:09,070 OK, so what is the whole answer for the effective Hamiltonian? 127 00:10:12,400 --> 00:10:16,840 No way to guess it, I think, unless you're-- 128 00:10:16,840 --> 00:10:19,330 well, you have incredible intuition. 129 00:10:24,030 --> 00:10:26,500 It's going to be a little complicated. 130 00:10:26,500 --> 00:10:29,280 If you think about it, why? 131 00:10:29,280 --> 00:10:34,800 Suppose this term is going to create no trouble whatsoever. 132 00:10:34,800 --> 00:10:36,840 It's just going to copy itself. 133 00:10:36,840 --> 00:10:40,470 This thing doesn't act on the electronic wave function, 134 00:10:40,470 --> 00:10:42,600 so the electronic wave function's going to cancel, 135 00:10:42,600 --> 00:10:45,210 is just going to contribute by itself. 136 00:10:45,210 --> 00:10:47,670 So this term we've done. 137 00:10:47,670 --> 00:10:50,240 This term is easy. 138 00:10:50,240 --> 00:10:52,950 It just gives itself. 139 00:10:52,950 --> 00:10:59,460 But this term is tough, because this term takes derivatives 140 00:10:59,460 --> 00:11:04,410 with respect to capital R, and your wave function 141 00:11:04,410 --> 00:11:12,370 depends on capital R. So it's going to be interesting. 142 00:11:12,370 --> 00:11:13,920 So what is the answer? 143 00:11:13,920 --> 00:11:18,660 I want to give you the answer rather than 144 00:11:18,660 --> 00:11:19,910 do that calculation. 145 00:11:22,570 --> 00:11:29,670 And it's almost like sometimes physics 146 00:11:29,670 --> 00:11:35,130 seems to have just a limited bag of tricks, 147 00:11:35,130 --> 00:11:41,980 and things come out in some way or another in a simple way. 148 00:11:41,980 --> 00:11:46,410 So what is the effective Hamiltonian? 149 00:11:46,410 --> 00:11:50,580 You could say, OK, it's going to be the sum of kinetic terms, 150 00:11:50,580 --> 00:11:58,680 2m alpha over alpha, and you have p alpha squared. 151 00:11:58,680 --> 00:12:01,420 Well, it's not going to be just p alpha squared. 152 00:12:01,420 --> 00:12:03,090 There's going to be a little more. 153 00:12:03,090 --> 00:12:07,020 And then there's going to be a potential, a great potential 154 00:12:07,020 --> 00:12:12,840 of R, and that great potential of R 155 00:12:12,840 --> 00:12:17,720 is going to include the nuclear nuclear R potential. 156 00:12:17,720 --> 00:12:22,290 We said that this nuclear nuclear term would just 157 00:12:22,290 --> 00:12:23,820 copy along. 158 00:12:23,820 --> 00:12:30,960 The He we calculated, and that gives us the Ee of R. 159 00:12:30,960 --> 00:12:32,640 And you say, well, maybe that's it, 160 00:12:32,640 --> 00:12:35,500 but no, it's not that, that's it. 161 00:12:35,500 --> 00:12:38,520 So let's see what it gives you. 162 00:12:38,520 --> 00:12:42,730 OK, so here it comes. 163 00:12:42,730 --> 00:12:45,100 You remember electromagnetism, you 164 00:12:45,100 --> 00:12:46,990 could change this thing here? 165 00:12:46,990 --> 00:12:56,350 Well, it gets changed, and it will become p minus a alpha. 166 00:12:56,350 --> 00:12:59,740 Like if there would be a connection. 167 00:12:59,740 --> 00:13:02,360 Like if there would be an electromagnetic field. 168 00:13:02,360 --> 00:13:06,510 There's no electromagnetic field here, nothing whatsoever. 169 00:13:06,510 --> 00:13:10,390 But there is one thing, and what is it? 170 00:13:10,390 --> 00:13:16,330 As we'll write here, it turns out to be a Berry connection. 171 00:13:16,330 --> 00:13:18,430 Because you have the adiabatic thing 172 00:13:18,430 --> 00:13:20,770 and you have your states that depend on R, 173 00:13:20,770 --> 00:13:23,380 the Berry connection shows up. 174 00:13:23,380 --> 00:13:27,730 So it shows up like this here. 175 00:13:27,730 --> 00:13:33,010 And here it shows up in a couple of funny terms. 176 00:13:33,010 --> 00:13:37,630 A term of the form alpha h squared over 177 00:13:37,630 --> 00:13:45,970 2m alpha, integral vr, gradient sub r 178 00:13:45,970 --> 00:13:56,470 alpha of phi r squared, minus sum over alpha 179 00:13:56,470 --> 00:14:01,645 a alpha squared over 2m alpha. 180 00:14:04,250 --> 00:14:09,160 So that's the whole thing, but I haven't defined the a. 181 00:14:19,492 --> 00:14:20,970 What is the a? 182 00:14:20,970 --> 00:14:24,060 Well, it's sort of a Berry connection. 183 00:14:24,060 --> 00:14:26,430 So you know what a Berry connection is. 184 00:14:26,430 --> 00:14:31,290 It's a v with a R in configuration space of a wave 185 00:14:31,290 --> 00:14:37,560 function with the star wave function, integrated. 186 00:14:37,560 --> 00:14:39,660 That was kind of a Berry connection. 187 00:14:39,660 --> 00:14:42,716 So it turns out to be very similar here. 188 00:14:42,716 --> 00:14:51,180 The a alpha vector of R is ih bar, 189 00:14:51,180 --> 00:14:55,710 the integral over the little space-- the electron space. 190 00:14:55,710 --> 00:14:58,170 That's where you integrate things-- 191 00:14:58,170 --> 00:15:02,550 of the electric wave function star times 192 00:15:02,550 --> 00:15:08,850 the gradient with respect to R alpha of the electron wave 193 00:15:08,850 --> 00:15:14,130 function of R. 194 00:15:14,130 --> 00:15:16,290 That is the Berry connection. 195 00:15:16,290 --> 00:15:21,080 If you remember, it's an overlap of wave function, wave function 196 00:15:21,080 --> 00:15:26,860 and the gradient with respect to the configuration space. 197 00:15:26,860 --> 00:15:36,690 So that's a good Hamiltonian for a molecule 198 00:15:36,690 --> 00:15:38,750 in the adiabatic approximation. 199 00:15:38,750 --> 00:15:41,230 If this looks complicated, it's already 200 00:15:41,230 --> 00:15:45,670 much simpler than what the real problem is, because 201 00:15:45,670 --> 00:15:47,650 of the adiabatic approximation. 202 00:15:47,650 --> 00:15:50,920 If you have these wave functions, 203 00:15:50,920 --> 00:15:56,800 these electronic wave functions you can calculate these a's. 204 00:15:56,800 --> 00:16:01,180 And this plays the role of the effective Hamiltonian 205 00:16:01,180 --> 00:16:06,180 for the nuclei in this approximation. 206 00:16:06,180 --> 00:16:09,750 So the variational principle essentially told us 207 00:16:09,750 --> 00:16:15,150 what is the right way to describe 208 00:16:15,150 --> 00:16:20,640 the interactions of the nuclei, given that the electrons have 209 00:16:20,640 --> 00:16:27,210 already been fixed into a class of wave functions. 210 00:16:27,210 --> 00:16:32,940 Now in general, this is the complete solution, 211 00:16:32,940 --> 00:16:40,050 but in general it's hard to do it, still complicated. 212 00:16:40,050 --> 00:16:43,280 So many times, people simplify. 213 00:16:43,280 --> 00:16:45,460 Well, it might happen, for example, 214 00:16:45,460 --> 00:16:49,420 if you have a simple enough molecule that your wave 215 00:16:49,420 --> 00:16:54,120 functions phi r are real. 216 00:16:54,120 --> 00:16:57,510 Now you remember that the Berry phase vanishes when 217 00:16:57,510 --> 00:16:59,490 the wave functions are real. 218 00:16:59,490 --> 00:17:02,250 These things have to be imaginary. 219 00:17:02,250 --> 00:17:04,890 With another i they have to be real. 220 00:17:04,890 --> 00:17:13,310 Connections are always real operators in the sense of r. 221 00:17:13,310 --> 00:17:17,970 So if the electronic wave functions are real, forget it, 222 00:17:17,970 --> 00:17:20,550 there's no Berry connection. 223 00:17:20,550 --> 00:17:24,510 So many times the Berry connections are not there. 224 00:17:24,510 --> 00:17:26,970 Even if there are no Berry connections, 225 00:17:26,970 --> 00:17:30,310 this term might be there. 226 00:17:30,310 --> 00:17:33,300 This is the gradient of this wave function with respect 227 00:17:33,300 --> 00:17:37,470 to the R. And this term can be important, 228 00:17:37,470 --> 00:17:41,290 even if there is no Berry connection. 229 00:17:41,290 --> 00:17:54,380 So in some cases, the lowest order approximation of this-- 230 00:17:54,380 --> 00:17:56,580 I think that's what this classically called 231 00:17:56,580 --> 00:17:59,320 the [INAUDIBLE] Oppenheimer-- 232 00:18:02,830 --> 00:18:09,770 ignores the Berry phase and ignores this extra term. 233 00:18:09,770 --> 00:18:14,500 So in that approximation, the lowest order approximation, 234 00:18:14,500 --> 00:18:18,310 the effective Hamiltonian, is just 235 00:18:18,310 --> 00:18:22,090 the sum over alpha minus h squared 236 00:18:22,090 --> 00:18:30,240 over 2m alpha, Laplacian R alpha plus the nuclear nuclear 237 00:18:30,240 --> 00:18:37,420 potential, plus the electronic contribution to the potential. 238 00:18:37,420 --> 00:18:43,240 So you keep-- basically, forget about Berry. 239 00:18:43,240 --> 00:18:48,040 That's higher approximation, higher order approximation, 240 00:18:48,040 --> 00:18:49,690 higher order approximation. 241 00:18:49,690 --> 00:18:53,690 Keep just the p, keep the nuclear nuclear, 242 00:18:53,690 --> 00:18:55,960 keep the electronic. 243 00:18:55,960 --> 00:18:59,720 And people have done a lot with this. 244 00:18:59,720 --> 00:19:07,220 So that's how our treatment of molecules. 245 00:19:07,220 --> 00:19:10,060 A pretty interesting thing. 246 00:19:10,060 --> 00:19:13,330 And perhaps the most important lesson of all of this 247 00:19:13,330 --> 00:19:20,470 is your way in which you can see how light degrees of freedom 248 00:19:20,470 --> 00:19:24,040 impose dynamics in slow degrees of freedom. 249 00:19:24,040 --> 00:19:27,460 And that's something you do in physics, in quantum field 250 00:19:27,460 --> 00:19:29,290 theory, all the time. 251 00:19:29,290 --> 00:19:33,880 You have the slow-moving degrees of freedom 252 00:19:33,880 --> 00:19:35,630 and the fast-moving degrees of freedom. 253 00:19:35,630 --> 00:19:38,530 And you integrate the fast-moving degrees of freedom, 254 00:19:38,530 --> 00:19:43,180 and you find an effective Hamiltonian for the big things. 255 00:19:43,180 --> 00:19:46,700 You may have quarks and all kinds of complicated things. 256 00:19:46,700 --> 00:19:50,020 But at the end of the day, they form protons and neutrons, 257 00:19:50,020 --> 00:19:53,230 and you can integrate the interactions of the quarks 258 00:19:53,230 --> 00:19:57,860 to sort of find the dynamics of the particle that you observe. 259 00:19:57,860 --> 00:20:02,740 So that step in that top blackboard is pretty important.