1 00:00:00,990 --> 00:00:04,970 PROFESSOR: So our example is a well-known one. 2 00:00:04,970 --> 00:00:08,700 But it's important to understand it. 3 00:00:08,700 --> 00:00:12,760 And it's a good example. 4 00:00:12,760 --> 00:00:13,980 It's a familiar one. 5 00:00:13,980 --> 00:00:18,380 It's the hydrogen molecule ionized. 6 00:00:20,900 --> 00:00:23,820 And we spoke a little about it last time. 7 00:00:23,820 --> 00:00:27,240 So let's do that now. 8 00:00:32,860 --> 00:00:40,990 So H2 plus, hydrogen ion. 9 00:00:40,990 --> 00:00:44,540 So what is this system? 10 00:00:44,540 --> 00:00:53,125 Well, it is two protons sharing one electron. 11 00:00:58,480 --> 00:01:01,300 That's basically it. 12 00:01:01,300 --> 00:01:10,065 It's the fact that if you have a hydrogen atom-- 13 00:01:13,570 --> 00:01:26,170 this is hydrogen-- you could bring another proton 14 00:01:26,170 --> 00:01:31,220 and form a bound state. 15 00:01:31,220 --> 00:01:37,550 That's an important thing in order to think about it-- 16 00:01:37,550 --> 00:01:41,090 you see, you can ask what is the bound state 17 00:01:41,090 --> 00:01:42,880 energy of your system? 18 00:01:42,880 --> 00:01:47,370 It's the energy you need to dissociate your system. 19 00:01:47,370 --> 00:01:52,220 But this system, if you have two protons-- 20 00:01:52,220 --> 00:01:57,440 proton, proton-- and one electron, 21 00:01:57,440 --> 00:02:00,020 it can be dissociated in many ways. 22 00:02:00,020 --> 00:02:04,190 It can be dissociated by just totally separating 23 00:02:04,190 --> 00:02:06,410 it and destroying it. 24 00:02:06,410 --> 00:02:11,180 But it can be dissociated or liberated 25 00:02:11,180 --> 00:02:13,950 by just removing the proton. 26 00:02:13,950 --> 00:02:15,860 So this is already a bound state. 27 00:02:15,860 --> 00:02:20,450 So if you think of energies, you have 28 00:02:20,450 --> 00:02:27,620 the bound state the energy of hydrogen here, hydrogen itself, 29 00:02:27,620 --> 00:02:30,420 and the H2 ion-- 30 00:02:30,420 --> 00:02:39,050 H2 plus-- must be lower because you could imagine 31 00:02:39,050 --> 00:02:43,670 taking the proton out and somehow this system is 32 00:02:43,670 --> 00:02:47,480 a configuration in which this proton is captured here. 33 00:02:47,480 --> 00:02:53,630 So you should get lower energy than the hydrogen atom energy. 34 00:02:53,630 --> 00:02:56,480 So that's the hydrogen ion. 35 00:02:56,480 --> 00:03:00,620 And that should be still more energy 36 00:03:00,620 --> 00:03:03,200 than if you bring another electron, which 37 00:03:03,200 --> 00:03:06,740 is the H2 molecule. 38 00:03:06,740 --> 00:03:08,000 You bring another electron. 39 00:03:08,000 --> 00:03:10,890 This should bind this and the other electron 40 00:03:10,890 --> 00:03:14,910 and lower its energy even more. 41 00:03:14,910 --> 00:03:19,800 So how does one solve for this hydrogen ion? 42 00:03:19,800 --> 00:03:26,090 Well, as we discussed, you first solve for the electronic wave 43 00:03:26,090 --> 00:03:30,305 function when the protons are separated at distance r. 44 00:03:33,410 --> 00:03:35,960 So we said fix the lattice. 45 00:03:35,960 --> 00:03:36,860 So here it is. 46 00:03:36,860 --> 00:03:38,810 The lattice is fixed. 47 00:03:38,810 --> 00:03:42,860 It's a separation capital R. And now 48 00:03:42,860 --> 00:03:46,290 I'm supposed to find the electron wave function. 49 00:03:49,150 --> 00:03:50,010 Easy? 50 00:03:50,010 --> 00:03:51,720 No. 51 00:03:51,720 --> 00:03:53,150 It already is hard. 52 00:03:53,150 --> 00:03:57,920 We spent several lectures-- you've been studying this three 53 00:03:57,920 --> 00:04:01,760 times already, maybe, in quantum mechanics, the hydrogen atom, 54 00:04:01,760 --> 00:04:03,980 one proton, one electron. 55 00:04:03,980 --> 00:04:07,590 To protons, one electron is much harder. 56 00:04:07,590 --> 00:04:12,530 I don't think there's a simple analytical way to solve it. 57 00:04:12,530 --> 00:04:16,490 So already at this step we have trouble solving it. 58 00:04:16,490 --> 00:04:20,959 What is the electron, Hamiltonian in this case? 59 00:04:20,959 --> 00:04:24,410 The electron Hamiltonian is just one Laplacian here 60 00:04:24,410 --> 00:04:34,710 minus h squared over 2m for the electron variable, 61 00:04:34,710 --> 00:04:41,270 and then the potential that is the interactions 62 00:04:41,270 --> 00:04:43,830 of the electron with the two nuclei. 63 00:04:43,830 --> 00:04:52,250 So if we call this distance r1 and this distance r2, 64 00:04:52,250 --> 00:04:54,860 that is the whole electron Hamiltonian. 65 00:04:54,860 --> 00:04:58,100 Now, don't think of r1 and r2 as you have 66 00:04:58,100 --> 00:05:00,260 two variables, two positions. 67 00:05:00,260 --> 00:05:03,680 No, there is just one position. 68 00:05:03,680 --> 00:05:07,340 The dynamical variables for this electron, this p 69 00:05:07,340 --> 00:05:10,820 and r for this electron-- 70 00:05:10,820 --> 00:05:12,570 r is the position of the electron. 71 00:05:12,570 --> 00:05:15,770 It happens to be that r is equal to r1. 72 00:05:15,770 --> 00:05:18,350 And if you know-- so r1 is really 73 00:05:18,350 --> 00:05:22,310 the length of r, the position of the electron. 74 00:05:22,310 --> 00:05:26,870 And r2 is the length of capital R, 75 00:05:26,870 --> 00:05:32,720 if we wish, all the way here, minus r. 76 00:05:35,961 --> 00:05:38,145 r is the vector here, as well. 77 00:05:42,920 --> 00:05:45,230 So there is r1 and r2, but it's not 78 00:05:45,230 --> 00:05:47,750 two electrons or two variables. 79 00:05:47,750 --> 00:05:51,860 It's just two distances that depend on the single position 80 00:05:51,860 --> 00:05:53,030 of the electron. 81 00:05:53,030 --> 00:05:55,385 And this is the momentum operator of the electron. 82 00:05:58,520 --> 00:06:05,360 OK, so you can't solve this. 83 00:06:05,360 --> 00:06:06,250 That's life. 84 00:06:06,250 --> 00:06:07,960 It can't be solved. 85 00:06:07,960 --> 00:06:12,430 But we can do something and we can do something variational. 86 00:06:12,430 --> 00:06:17,560 We can try to find some kind of approximation 87 00:06:17,560 --> 00:06:20,900 for the state of electrons and use that. 88 00:06:32,740 --> 00:06:34,750 So there is an [INAUDIBLE] that you 89 00:06:34,750 --> 00:06:37,990 could try, a variational wave function, 90 00:06:37,990 --> 00:06:39,430 for even the electron. 91 00:06:39,430 --> 00:06:42,310 You see, in our argument in this lecture, 92 00:06:42,310 --> 00:06:44,260 we did the variation and approximation 93 00:06:44,260 --> 00:06:47,470 for the nuclear degrees of freedom or for that thing. 94 00:06:47,470 --> 00:06:51,310 But even for the electron wave function, I can't solve them. 95 00:06:51,310 --> 00:06:53,240 So I have to do a variational method. 96 00:06:53,240 --> 00:06:55,510 Now, these variational methods have 97 00:06:55,510 --> 00:06:58,690 become extremely sophisticated. 98 00:06:58,690 --> 00:07:03,670 People do this with a series of Eigenfunctions 99 00:07:03,670 --> 00:07:09,190 and find answers that converge with 15 digits accuracy. 100 00:07:09,190 --> 00:07:12,610 It's just unbelievable what people can do by now with this. 101 00:07:12,610 --> 00:07:17,320 It's a very nice and developed field. 102 00:07:17,320 --> 00:07:22,030 But we'll do a baby version of it. 103 00:07:22,030 --> 00:07:24,610 So this baby version is going to stay. 104 00:07:24,610 --> 00:07:28,030 My wave function for the electron 105 00:07:28,030 --> 00:07:33,840 as a function of position and as a function of the separation 106 00:07:33,840 --> 00:07:35,660 is going to be simple. 107 00:07:35,660 --> 00:07:38,355 It's going to be a sum of ground state wave functions. 108 00:07:38,355 --> 00:07:40,270 It's going to be a number-- 109 00:07:40,270 --> 00:07:42,670 a, that's just a number-- 110 00:07:42,670 --> 00:07:46,960 times the ground state wave function 111 00:07:46,960 --> 00:07:52,430 based on the first proton plus the ground state wave 112 00:07:52,430 --> 00:07:55,910 function based on the second proton. 113 00:07:55,910 --> 00:08:04,100 So these psis, or psi0 of r is the ground state wave function 114 00:08:04,100 --> 00:08:12,140 of hydrogen pi a 0 cubed e to the minus r over a0. 115 00:08:19,130 --> 00:08:23,450 This is called-- people have given it a name, 116 00:08:23,450 --> 00:08:24,930 even though it's-- 117 00:08:24,930 --> 00:08:28,130 you would say it's not that original to put some wave 118 00:08:28,130 --> 00:08:29,240 function like that. 119 00:08:29,240 --> 00:08:33,350 This is a simple approximation. 120 00:08:33,350 --> 00:08:40,700 And the technique is called LCAO technique. 121 00:08:44,480 --> 00:08:56,140 And this calls for Linear Combination of Atomic Orbitals. 122 00:09:01,540 --> 00:09:04,800 A big name for a rather simple thing. 123 00:09:09,080 --> 00:09:15,150 It has some nice things about this wave function. 124 00:09:15,150 --> 00:09:20,630 This system is invariant. 125 00:09:20,630 --> 00:09:25,790 This molecule is invariant. 126 00:09:25,790 --> 00:09:31,880 And they're taking a reflection around this thing changing 127 00:09:31,880 --> 00:09:35,270 the first proton and the second proton. 128 00:09:35,270 --> 00:09:38,900 So that's the symmetry of your Hamiltonian. 129 00:09:38,900 --> 00:09:42,700 And it's the symmetry of [INAUDIBLE] 130 00:09:42,700 --> 00:09:43,970 with the Hamiltonian. 131 00:09:43,970 --> 00:09:48,290 So you can demand that your wave functions have that symmetry. 132 00:09:48,290 --> 00:09:52,905 And this is nice here because if you change r1 and r2, 133 00:09:52,905 --> 00:09:55,560 the wave function is invariant. 134 00:09:55,560 --> 00:10:03,720 So that's a very nice thing about this wave function. 135 00:10:03,720 --> 00:10:07,110 It shows-- you see, you can put in the variational method 136 00:10:07,110 --> 00:10:09,960 anything and it will still give you some answer. 137 00:10:09,960 --> 00:10:13,180 But if you put something that mimics the real wave 138 00:10:13,180 --> 00:10:16,170 functions and the real wave function of this system 139 00:10:16,170 --> 00:10:19,140 is going to have a symmetry under the exchange of the two 140 00:10:19,140 --> 00:10:19,830 protons. 141 00:10:19,830 --> 00:10:21,310 They're identical. 142 00:10:21,310 --> 00:10:24,850 So this lattice that you've created has a symmetry. 143 00:10:24,850 --> 00:10:30,100 So the electron configuration has to respect that symmetry. 144 00:10:30,100 --> 00:10:31,410 So that's very nice. 145 00:10:31,410 --> 00:10:35,340 Now, this is electronic wave function. 146 00:10:35,340 --> 00:10:41,050 So I ask you, is this wave function better 147 00:10:41,050 --> 00:10:45,000 where the protons are far away or when the protons 148 00:10:45,000 --> 00:10:47,220 are very close to each other? 149 00:10:47,220 --> 00:10:49,680 Give you a minute to think about it. 150 00:10:49,680 --> 00:10:53,370 When is this approximation or this wave function better? 151 00:10:56,090 --> 00:10:58,070 If the distance between the protons 152 00:10:58,070 --> 00:11:01,310 is very little or the distance between the protons 153 00:11:01,310 --> 00:11:02,210 is very large? 154 00:11:06,960 --> 00:11:10,470 You're right in that when the two things collapse, 155 00:11:10,470 --> 00:11:14,850 this looks like a true wave function 156 00:11:14,850 --> 00:11:17,430 because this is a solution when there's 157 00:11:17,430 --> 00:11:20,070 a single nucleus and a single electron. 158 00:11:20,070 --> 00:11:23,150 But this is the solution for hydrogen. 159 00:11:23,150 --> 00:11:25,830 And it gives you the ground state of hydrogen. 160 00:11:25,830 --> 00:11:27,900 But when the two protons collapse, 161 00:11:27,900 --> 00:11:30,630 it has become helium nucleus. 162 00:11:30,630 --> 00:11:32,160 It has two protons. 163 00:11:32,160 --> 00:11:36,780 So it doesn't have the right decay rate. 164 00:11:36,780 --> 00:11:39,480 It doesn't have the right Born radius. 165 00:11:39,480 --> 00:11:43,640 On the other hand, when they're far apart, 166 00:11:43,640 --> 00:11:45,080 it does have the right thing. 167 00:11:45,080 --> 00:11:47,240 So this is very good. 168 00:11:47,240 --> 00:11:50,750 It's excellent when the things are far away. 169 00:11:50,750 --> 00:11:55,500 But it's not great when they're close together. 170 00:11:55,500 --> 00:11:58,040 So here you go. 171 00:11:58,040 --> 00:12:00,590 You have to normalize this wave function. 172 00:12:00,590 --> 00:12:03,320 Even that takes some effort. 173 00:12:03,320 --> 00:12:08,360 It turns out that the value of a for normalization-- 174 00:12:11,600 --> 00:12:21,360 the value of a is 1 over 2 1 plus a constant i. 175 00:12:21,360 --> 00:12:27,350 And that constant i is e to the minus capital 176 00:12:27,350 --> 00:12:35,360 R over a not times 1 plus r over a not plus 1/3 177 00:12:35,360 --> 00:12:38,330 of r over a not squared. 178 00:12:38,330 --> 00:12:42,140 Wow, so very funny. 179 00:12:42,140 --> 00:12:47,120 It's not so easy to even calculate the normalization 180 00:12:47,120 --> 00:12:50,180 of this thing. 181 00:12:50,180 --> 00:12:51,830 But that's the normalization. 182 00:12:54,440 --> 00:12:59,810 And then recall that what we have to do 183 00:12:59,810 --> 00:13:07,370 is just put the electronic Hamiltonian inside this wave 184 00:13:07,370 --> 00:13:11,550 function, this phi of r. 185 00:13:11,550 --> 00:13:13,940 And this is going to give us an energy which 186 00:13:13,940 --> 00:13:22,080 is approximately the electronic energy as a function of r. 187 00:13:22,080 --> 00:13:28,010 So you have to evaluate the Hamiltonian-- 188 00:13:28,010 --> 00:13:32,400 I'm sorry, that Hamiltonian there-- 189 00:13:32,400 --> 00:13:34,280 should have a square here. 190 00:13:40,550 --> 00:13:43,790 And now we evaluate this. 191 00:13:43,790 --> 00:13:46,250 We have to go even more into gross 192 00:13:46,250 --> 00:13:53,900 and get what is the potential energy contributed 193 00:13:53,900 --> 00:13:56,060 by the electronic cloud. 194 00:14:03,038 --> 00:14:08,580 So here is the function that you get. 195 00:14:08,580 --> 00:14:16,230 So our plot here I'll call-- 196 00:14:16,230 --> 00:14:19,800 this calls for a variable x, which 197 00:14:19,800 --> 00:14:24,990 is going to be the separation divided by a not. 198 00:14:24,990 --> 00:14:29,100 This is the separation between the protons divided by a not. 199 00:14:29,100 --> 00:14:33,660 So here is going to be the electronic energy 200 00:14:33,660 --> 00:14:35,160 as a function of x. 201 00:14:35,160 --> 00:14:36,225 And here is x. 202 00:14:38,760 --> 00:14:44,210 And all right. 203 00:14:44,210 --> 00:14:51,520 So as x goes to 0, means the nuclei 204 00:14:51,520 --> 00:14:53,480 are going on top of each other. 205 00:14:53,480 --> 00:14:58,760 That's the place where the wave function is not all that great. 206 00:14:58,760 --> 00:15:01,040 And that's not so good. 207 00:15:01,040 --> 00:15:07,980 It turns out that here you get minus 3 Rydbergs. 208 00:15:07,980 --> 00:15:14,360 Remember, the Rydberg was e squared over 2 a not. 209 00:15:14,360 --> 00:15:16,890 And it's about 13.6 ev. 210 00:15:16,890 --> 00:15:19,620 It's just the ground state energy of the hydrogen. 211 00:15:19,620 --> 00:15:29,130 You do get that the energy due to the electronic configuration 212 00:15:29,130 --> 00:15:32,980 is negative and it goes to this value. 213 00:15:32,980 --> 00:15:37,670 And then it starts to grow quadratically and goes up. 214 00:15:37,670 --> 00:15:44,010 And I want to know, in your opinion, what's 215 00:15:44,010 --> 00:15:47,910 going to be the next asymptote? 216 00:15:47,910 --> 00:15:52,560 So what is it going to asymptote as x goes to infinity? 217 00:15:55,460 --> 00:15:56,880 Let let's make the analogy. 218 00:15:56,880 --> 00:16:00,870 The right answer is it actually stops here. 219 00:16:03,770 --> 00:16:10,450 And you'll remember this problem that you've solved many times, 220 00:16:10,450 --> 00:16:12,160 probably. 221 00:16:12,160 --> 00:16:13,340 If you have a-- 222 00:16:13,340 --> 00:16:15,400 suppose a square well. 223 00:16:15,400 --> 00:16:18,460 You have a wave function that is like that. 224 00:16:18,460 --> 00:16:20,670 Now, if you have two square well-- and it 225 00:16:20,670 --> 00:16:23,080 has a ground state energy. 226 00:16:23,080 --> 00:16:28,960 If you have two square wells like that and you ask, 227 00:16:28,960 --> 00:16:32,740 what is the ground state energy? 228 00:16:32,740 --> 00:16:36,550 The ground state energy is roughly equal to the ground 229 00:16:36,550 --> 00:16:40,180 state energy for a single square well because what happens 230 00:16:40,180 --> 00:16:46,240 is that the wave function goes like that and then like-- 231 00:16:46,240 --> 00:16:51,060 well, actually, like that. 232 00:16:51,060 --> 00:16:54,210 So yes, the wave function spends equal time. 233 00:16:54,210 --> 00:16:56,730 But it's the same energy as if there 234 00:16:56,730 --> 00:16:58,950 would be a single square well. 235 00:16:58,950 --> 00:17:00,090 So here it is. 236 00:17:00,090 --> 00:17:04,511 The protons are very far away in the symmetric state, the ground 237 00:17:04,511 --> 00:17:05,010 state. 238 00:17:05,010 --> 00:17:09,480 The electron is half the times here, half the times there. 239 00:17:09,480 --> 00:17:11,940 But the energy is the same energy 240 00:17:11,940 --> 00:17:15,400 as if it would be in either one. 241 00:17:15,400 --> 00:17:20,760 So this is an intuition that you may have from 804 or 805. 242 00:17:20,760 --> 00:17:21,599 So here it is. 243 00:17:21,599 --> 00:17:23,940 That's what it does. 244 00:17:23,940 --> 00:17:25,589 That's good. 245 00:17:25,589 --> 00:17:34,410 In fact, if you ride the ee of x over Rydbergs, 246 00:17:34,410 --> 00:17:39,390 it behaves here like minus 3 plus x 247 00:17:39,390 --> 00:17:41,710 squared plus dot, dot, dot. 248 00:17:41,710 --> 00:17:44,520 So it starts growing and then asymptotes 249 00:17:44,520 --> 00:17:48,080 with an exponential there. 250 00:17:48,080 --> 00:17:54,260 OK, but what did we say was the Born-Oppenheimer approximation 251 00:17:54,260 --> 00:18:03,255 was the idea that then you have dynamics of the nuclei based 252 00:18:03,255 --> 00:18:03,755 on-- 253 00:18:07,440 --> 00:18:13,470 so Born-Oppenheimer tells you that the Hamiltonian 254 00:18:13,470 --> 00:18:20,010 for the nuclear degrees of freedom is given by h. 255 00:18:20,010 --> 00:18:26,160 This h effective is capital P over 2m squared 256 00:18:26,160 --> 00:18:32,440 plus the nuclear nuclear potential, 257 00:18:32,440 --> 00:18:36,610 plus this electronic energy. 258 00:18:36,610 --> 00:18:40,750 So this electronic energy we already calculated. 259 00:18:43,410 --> 00:18:46,530 The nuclear nuclear potential, in this case, 260 00:18:46,530 --> 00:18:55,530 is the proton proton potential, v m n is e squared over r. 261 00:18:55,530 --> 00:18:58,740 It is repulsive. 262 00:18:58,740 --> 00:19:02,370 You see, this-- if you wanted to minimize the electronic energy, 263 00:19:02,370 --> 00:19:04,650 you still don't get anything. 264 00:19:04,650 --> 00:19:09,420 You get a system that collapses to zero. 265 00:19:09,420 --> 00:19:12,450 That's certainly not the molecule. 266 00:19:12,450 --> 00:19:21,240 But this n n is this, so we can write it as e squared over 2a0 267 00:19:21,240 --> 00:19:27,810 and then put a 2 over r over a 0 to get 268 00:19:27,810 --> 00:19:29,890 all these numbers nicely. 269 00:19:29,890 --> 00:19:36,720 So the nvv is a Rydberg times 2 over x. 270 00:19:36,720 --> 00:19:46,680 So in terms of x there, vnn is a 2 over x-- 271 00:19:46,680 --> 00:19:54,560 oops-- a 2 over x potential like that. 272 00:19:54,560 --> 00:19:59,720 So now you have the possibility of getting a stable grounding, 273 00:19:59,720 --> 00:20:03,260 so the potential for the nucleons. 274 00:20:03,260 --> 00:20:07,210 So the total potential for the nucleons 275 00:20:07,210 --> 00:20:09,520 is the sum of these two potentials. 276 00:20:15,270 --> 00:20:18,240 And how does it look? 277 00:20:18,240 --> 00:20:22,050 Well, some of the two potential-- 278 00:20:22,050 --> 00:20:33,370 so ee of x plus vnn of x. 279 00:20:33,370 --> 00:20:41,310 Let's divide by Rydberg as a function of x, goes-- 280 00:20:41,310 --> 00:20:42,950 here is minus 1. 281 00:20:46,070 --> 00:20:47,990 And it goes more or less like-- 282 00:20:47,990 --> 00:20:48,660 OK. 283 00:20:48,660 --> 00:20:49,910 Let me try to get this right. 284 00:20:49,910 --> 00:20:58,490 1, 2, 3, 4, 5. 285 00:20:58,490 --> 00:21:00,620 OK. 286 00:21:00,620 --> 00:21:15,630 It goes up here, down, crosses the minus 1 line, 287 00:21:15,630 --> 00:21:22,571 and moves to a minimum here. 288 00:21:22,571 --> 00:21:26,680 And then it goes like that. 289 00:21:26,680 --> 00:21:28,250 Pretty much it's something like that. 290 00:21:28,250 --> 00:21:33,820 It's a rather little quadratic minimum. 291 00:21:33,820 --> 00:21:36,800 It's rather flat. 292 00:21:36,800 --> 00:21:39,970 And these are the numbers you care for. 293 00:21:39,970 --> 00:21:41,920 This is the minimum. 294 00:21:41,920 --> 00:21:45,350 Maybe it doesn't look like that in my graph too well. 295 00:21:45,350 --> 00:21:49,000 But it's a very flat minimum. 296 00:21:49,000 --> 00:21:59,340 And r over a0 at the minimum is, in fact, 2.49. 297 00:21:59,340 --> 00:22:19,680 And e over Rydberg at the minimum is minus 1.297. 298 00:22:19,680 --> 00:22:21,540 Those are results. 299 00:22:21,540 --> 00:22:23,220 Now, you have a potential. 300 00:22:23,220 --> 00:22:25,710 You could calculate the quadratic term 301 00:22:25,710 --> 00:22:26,790 around the potential. 302 00:22:26,790 --> 00:22:31,020 And then you get an approximate oscillation Hamiltonian 303 00:22:31,020 --> 00:22:34,050 that those would be the nuclear vibrations. 304 00:22:34,050 --> 00:22:35,760 You could calculate the frequency 305 00:22:35,760 --> 00:22:38,580 of the nuclear vibrations and the energy 306 00:22:38,580 --> 00:22:40,170 of the nuclear vibration. 307 00:22:40,170 --> 00:22:43,080 It's a very simple but nice model. 308 00:22:43,080 --> 00:22:48,180 You can calculate everything pretty much about this molecule 309 00:22:48,180 --> 00:22:49,920 and compare with experiments. 310 00:22:49,920 --> 00:22:53,450 So how does it do the comparison with experiment? 311 00:22:53,450 --> 00:22:55,070 It does OK. 312 00:22:55,070 --> 00:22:57,690 It doesn't do very well. 313 00:22:57,690 --> 00:23:00,670 So let me tell you what's happening. 314 00:23:00,670 --> 00:23:06,630 So again, we do get this system. 315 00:23:06,630 --> 00:23:10,060 Happily, the minimum was below minus 1. 316 00:23:10,060 --> 00:23:15,270 The minus 1 was the situation in which you have a hydrogen 317 00:23:15,270 --> 00:23:17,520 atom and a free proton. 318 00:23:17,520 --> 00:23:21,960 So there is a bound state energy but the bounds state energy 319 00:23:21,960 --> 00:23:28,450 is not conceptually right to think of it as this big energy. 320 00:23:28,450 --> 00:23:32,550 It's just this little part here in which 321 00:23:32,550 --> 00:23:34,800 this is the extra binding that you 322 00:23:34,800 --> 00:23:39,520 can have when you bring a proton near to the hydrogen atom. 323 00:23:39,520 --> 00:23:42,330 So this extra part-- 324 00:23:42,330 --> 00:23:43,965 so the bound state energy-- 325 00:23:47,370 --> 00:23:51,930 I'll write a couple of words here and we'll stop. 326 00:23:58,090 --> 00:24:03,250 The bound state energy is E. This E that we determined here 327 00:24:03,250 --> 00:24:07,300 at the minimum is minus 1 Rydberg 328 00:24:07,300 --> 00:24:21,580 minus 0.1297 Rydbergs which is minus 1 Rydberg minus 1.76 ev. 329 00:24:21,580 --> 00:24:24,340 So this is what you would call the bound state 330 00:24:24,340 --> 00:24:30,250 energy of this system because it's the least energy you need 331 00:24:30,250 --> 00:24:32,410 to start to dissociate it-- 332 00:24:32,410 --> 00:24:37,120 not to dissociate it completely, but to start to dissociate it. 333 00:24:37,120 --> 00:24:40,240 1.76. 334 00:24:40,240 --> 00:24:51,070 True value is 2.8 ev, so not great, 335 00:24:51,070 --> 00:24:54,760 but sort of order of magnitude. 336 00:24:54,760 --> 00:24:57,560 Another thing that you can ask is, 337 00:24:57,560 --> 00:25:03,100 what is the separation between the nuclei in this ion? 338 00:25:03,100 --> 00:25:05,110 How far away are they? 339 00:25:05,110 --> 00:25:11,220 And this gives a prediction that it's about 2.49 a0. 340 00:25:11,220 --> 00:25:24,430 So separation is r equal 2.49 a0, which is 2.49 times 341 00:25:24,430 --> 00:25:28,910 0.529 angstroms. 342 00:25:28,910 --> 00:25:33,880 It's about 1.32 angstroms. 343 00:25:33,880 --> 00:25:37,600 And through value's 2.8. 344 00:25:37,600 --> 00:25:45,600 Experimental value, the value of r, our experiment 345 00:25:45,600 --> 00:25:48,550 is not that far. 346 00:25:48,550 --> 00:25:51,590 It's 106 angstroms. 347 00:25:51,590 --> 00:25:58,360 So it's 20% there, 25% there. 348 00:25:58,360 --> 00:26:01,750 But it gives you a way to think about this system. 349 00:26:01,750 --> 00:26:03,190 You get all the physics. 350 00:26:03,190 --> 00:26:04,990 You understand the physics. 351 00:26:04,990 --> 00:26:09,220 The Born-Oppenheimer physics is that the electron 352 00:26:09,220 --> 00:26:16,770 energy in the fixed lattice is a potential term for the nuclei. 353 00:26:16,770 --> 00:26:19,150 Added to the nucleus nucleus repulsion 354 00:26:19,150 --> 00:26:23,080 gives you the total potential term for the nuclear degrees 355 00:26:23,080 --> 00:26:23,620 of freedom. 356 00:26:23,620 --> 00:26:25,810 You can find the stable situation. 357 00:26:25,810 --> 00:26:28,900 And if you want to do quantum mechanics and vibrations 358 00:26:28,900 --> 00:26:33,430 of the nuclei, well, do the harmonic oscillator 359 00:26:33,430 --> 00:26:36,460 associated with this thing. 360 00:26:36,460 --> 00:26:38,500 The nice thing is you can complicate 361 00:26:38,500 --> 00:26:40,300 these wave functions a little and get 362 00:26:40,300 --> 00:26:41,980 better and better answers. 363 00:26:41,980 --> 00:26:43,660 And it's fun. 364 00:26:43,660 --> 00:26:46,570 It's things that can be done numerically. 365 00:26:46,570 --> 00:26:51,010 And you have a remarkably powerful tool 366 00:26:51,010 --> 00:26:53,050 to understand these things. 367 00:26:53,050 --> 00:26:56,590 So with this, we'll close our whole chapter 368 00:26:56,590 --> 00:26:58,910 in adiabatic physics. 369 00:26:58,910 --> 00:27:02,220 And next time, we will begin with scattering.