1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high-quality educational resources for free. 5 00:00:10,120 --> 00:00:12,660 To make a donation or to view additional materials 6 00:00:12,660 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,850 at osu.mit.edu. 8 00:00:21,560 --> 00:00:24,600 ROBERT FIELD: You know that there is exam tomorrow night. 9 00:00:24,600 --> 00:00:31,210 It's very heavily on the particle 10 00:00:31,210 --> 00:00:33,440 in a box, the harmonic oscillator, 11 00:00:33,440 --> 00:00:35,900 and the time-independent Schrodinger equation. 12 00:00:35,900 --> 00:00:38,610 It's really quite an amazing amount of stuff. 13 00:00:38,610 --> 00:00:39,110 OK. 14 00:00:39,110 --> 00:00:46,160 So last time, we talked about some time-independent 15 00:00:46,160 --> 00:00:47,630 Hamiltonian examples. 16 00:00:47,630 --> 00:00:52,620 And two that I like are the half harmonic oscillator 17 00:00:52,620 --> 00:00:56,960 and the vertical excitation-- the Franck-Condon excitation. 18 00:00:56,960 --> 00:01:00,805 Now, when you're doing time-dependent Hamiltonians-- 19 00:01:00,805 --> 00:01:03,290 or when you're doing time-dependent problems 20 00:01:03,290 --> 00:01:07,370 for a time-independent Hamiltonian-- 21 00:01:07,370 --> 00:01:12,410 You always start with the form of the function at t 22 00:01:12,410 --> 00:01:15,690 equals zero and automatically extend 23 00:01:15,690 --> 00:01:18,470 it using the fact that you have a complete set 24 00:01:18,470 --> 00:01:22,190 of eigenfunctions of the time-independent Hamiltonian 25 00:01:22,190 --> 00:01:25,290 to the time-dependent wave function. 26 00:01:25,290 --> 00:01:27,800 And with the time-dependent wave function, 27 00:01:27,800 --> 00:01:32,870 you're able to calculate almost anything you want. 28 00:01:32,870 --> 00:01:35,759 And I did several examples of things 29 00:01:35,759 --> 00:01:36,800 that you could calculate. 30 00:01:36,800 --> 00:01:41,260 One is the probability density as a function of time. 31 00:01:41,260 --> 00:01:45,540 So the other is the survival probability. 32 00:01:45,540 --> 00:01:51,050 And the survival probability is a really neat thing, 33 00:01:51,050 --> 00:01:55,280 because it says, we've got some object which is particle-like. 34 00:01:55,280 --> 00:01:58,820 And the time evolution makes the wave function 35 00:01:58,820 --> 00:02:01,670 move away from its birthplace. 36 00:02:01,670 --> 00:02:05,990 And that's an easy thing to understand, 37 00:02:05,990 --> 00:02:11,660 but what's surprising is that we think about a wave packet as 38 00:02:11,660 --> 00:02:14,840 localized in position, but it also 39 00:02:14,840 --> 00:02:18,060 has encoded in it momentum information. 40 00:02:18,060 --> 00:02:20,360 And so when the wave packet moves away 41 00:02:20,360 --> 00:02:21,620 from its starting point-- 42 00:02:21,620 --> 00:02:24,920 if it starts at rest-- 43 00:02:24,920 --> 00:02:30,010 The initial fast changes are in momentum. 44 00:02:30,010 --> 00:02:33,790 And the momentum-- the change of the momentum-- 45 00:02:33,790 --> 00:02:37,040 is sampling the gradient of the potential. 46 00:02:37,040 --> 00:02:39,250 And that's something you usually want to know. 47 00:02:39,250 --> 00:02:41,440 The gradient of the potential at a turning 48 00:02:41,440 --> 00:02:43,570 point-- at an energy you know. 49 00:02:43,570 --> 00:02:46,630 And that is a measurable quantity. 50 00:02:46,630 --> 00:02:48,880 And it's really a beautiful example 51 00:02:48,880 --> 00:02:53,140 of how you can find easily observable, or easily 52 00:02:53,140 --> 00:02:56,980 calculable, quantum mechanical things that 53 00:02:56,980 --> 00:03:00,400 reflect classical mechanics. 54 00:03:00,400 --> 00:03:01,750 OK. 55 00:03:01,750 --> 00:03:05,530 I talked about grand rephasings for problems like the harmonic 56 00:03:05,530 --> 00:03:10,240 oscillator, and the rigid rotor, and the particle in a box-- 57 00:03:10,240 --> 00:03:14,200 You have this fantastic property that all of the energy level 58 00:03:14,200 --> 00:03:21,480 differences are integer multiples of a common factor. 59 00:03:21,480 --> 00:03:23,820 And that guarantees that you will 60 00:03:23,820 --> 00:03:26,940 have periodic rephasings at times 61 00:03:26,940 --> 00:03:30,650 related to that common factor. 62 00:03:30,650 --> 00:03:33,730 And so that enables you to observe 63 00:03:33,730 --> 00:03:36,800 the evolution of something for a very long time 64 00:03:36,800 --> 00:03:39,190 and to see whether the rephasings are 65 00:03:39,190 --> 00:03:41,620 perfect or not quite perfect. 66 00:03:41,620 --> 00:03:44,320 And the imperfections are telling you something 67 00:03:44,320 --> 00:03:46,380 beyond the simple model-- 68 00:03:46,380 --> 00:03:50,410 They're telling you about anharmonicity 69 00:03:50,410 --> 00:03:54,250 or some other thing that makes the energy levels not quite 70 00:03:54,250 --> 00:03:57,500 integer multiples of a common factor. 71 00:03:57,500 --> 00:04:01,960 Now, one of the nicest things is the illustration of tunneling. 72 00:04:01,960 --> 00:04:05,650 And we don't observe tunneling. 73 00:04:05,650 --> 00:04:09,190 Tunneling is a quantum mechanical thing. 74 00:04:09,190 --> 00:04:13,540 And it's encoded in what's most easily observed-- the energy 75 00:04:13,540 --> 00:04:15,460 level pattern. 76 00:04:15,460 --> 00:04:18,130 It's encoded as a level staggering. 77 00:04:18,130 --> 00:04:21,680 Now, we talked about this problem, where we 78 00:04:21,680 --> 00:04:24,150 have a barrier in the middle. 79 00:04:24,150 --> 00:04:27,420 And with a barrier in the middle, 80 00:04:27,420 --> 00:04:30,330 half of the energy levels are almost unaffected, 81 00:04:30,330 --> 00:04:32,340 and the other half are affected a lot. 82 00:04:34,930 --> 00:04:37,690 Now, here is a problem that you can understand-- 83 00:04:37,690 --> 00:04:39,750 In the same way, there's no tunneling here, 84 00:04:39,750 --> 00:04:43,200 but there is some extra stuff here. 85 00:04:43,200 --> 00:04:45,900 And the level diagram can tell you the difference 86 00:04:45,900 --> 00:04:47,400 between these two things. 87 00:04:47,400 --> 00:04:49,860 Does anybody want to tell me what's different? 88 00:04:49,860 --> 00:04:52,350 What is the qualitative signature of this? 89 00:04:55,460 --> 00:04:56,720 Yes. 90 00:04:56,720 --> 00:04:58,820 AUDIENCE: The spacings move up a little bit. 91 00:04:58,820 --> 00:05:01,230 The odd spacings are erased in energy, 92 00:05:01,230 --> 00:05:04,280 and the even spacings are just about unaffected. 93 00:05:04,280 --> 00:05:06,384 ROBERT FIELD: Even symmetry levels are shifted up. 94 00:05:06,384 --> 00:05:07,050 AUDIENCE: Right. 95 00:05:07,050 --> 00:05:08,410 Sorry. 96 00:05:08,410 --> 00:05:11,120 ROBERT FIELD: And so now you're ready to answer this one. 97 00:05:18,760 --> 00:05:20,830 If the even symmetry levels are shifted up, 98 00:05:20,830 --> 00:05:24,700 because they feel the barrier, what about the even 99 00:05:24,700 --> 00:05:25,750 symmetry levels here? 100 00:05:28,420 --> 00:05:29,200 Yes? 101 00:05:29,200 --> 00:05:30,220 AUDIENCE: They would be shifted down. 102 00:05:30,220 --> 00:05:31,240 ROBERT FIELD: Right. 103 00:05:31,240 --> 00:05:34,960 And so you get a level staggering where, in this case, 104 00:05:34,960 --> 00:05:40,600 the lowest level is close to the next higher one. 105 00:05:40,600 --> 00:05:44,320 And in this one, the lowest level is shifted way down, 106 00:05:44,320 --> 00:05:45,670 and the next one is not shifted. 107 00:05:45,670 --> 00:05:48,810 And then we get the doubling or the parent. 108 00:05:48,810 --> 00:05:53,110 So that's a kind of intuition that you get just 109 00:05:53,110 --> 00:05:55,520 by looking at these problems. 110 00:05:55,520 --> 00:05:58,690 Now, one thing that is really beautiful 111 00:05:58,690 --> 00:06:02,410 is, when you have a barrier like this, 112 00:06:02,410 --> 00:06:04,750 since this part of the potential problem 113 00:06:04,750 --> 00:06:07,450 is something that is exactly solved, 114 00:06:07,450 --> 00:06:10,480 you propagate the wave function in from the sides 115 00:06:10,480 --> 00:06:13,490 and they have the same phase. 116 00:06:13,490 --> 00:06:20,080 So there is no accumulation of phase under the barrier. 117 00:06:20,080 --> 00:06:23,230 And that means the levels that are trying 118 00:06:23,230 --> 00:06:25,810 to propagate under this barrier are shifted up, 119 00:06:25,810 --> 00:06:28,840 because they have to accumulate enough phase to satisfy 120 00:06:28,840 --> 00:06:31,930 the boundary conditions at the turning points. 121 00:06:31,930 --> 00:06:34,120 Here, you're going to accumulate more phase 122 00:06:34,120 --> 00:06:37,090 in this special region. 123 00:06:37,090 --> 00:06:39,670 Phase is really important. 124 00:06:39,670 --> 00:06:41,530 OK. 125 00:06:41,530 --> 00:06:44,530 So today we're going to talk-- 126 00:06:44,530 --> 00:06:48,880 And this this lecture is basically not on the exam, 127 00:06:48,880 --> 00:06:51,520 although it does connect with topics on the exam 128 00:06:51,520 --> 00:06:55,300 and makes it possible to understand them better. 129 00:06:55,300 --> 00:07:01,300 So instead of learning about the postulates 130 00:07:01,300 --> 00:07:04,660 in a great abstract way at the beginning, before you know what 131 00:07:04,660 --> 00:07:07,720 they're for, now we're going to review 132 00:07:07,720 --> 00:07:09,730 what we understand about them. 133 00:07:09,730 --> 00:07:14,965 And so one thing is, there is a wave function. 134 00:07:18,230 --> 00:07:20,710 And now we're considering not just one dimension, 135 00:07:20,710 --> 00:07:24,780 but any number of dimensions. 136 00:07:24,780 --> 00:07:27,430 And this is the state function that 137 00:07:27,430 --> 00:07:29,110 tells you everything you're allowed 138 00:07:29,110 --> 00:07:33,130 to know about the system. 139 00:07:33,130 --> 00:07:37,030 And if you have this, you can calculate everything. 140 00:07:37,030 --> 00:07:41,290 If you know how observables relate to this, 141 00:07:41,290 --> 00:07:42,040 well, you're fine. 142 00:07:42,040 --> 00:07:45,229 You can then use that to describe the Hamiltonian. 143 00:07:55,020 --> 00:07:57,480 Hermetian operators are the only kind 144 00:07:57,480 --> 00:08:00,570 of operators you can have in quantum mechanics. 145 00:08:00,570 --> 00:08:03,220 And they have the wonderful property 146 00:08:03,220 --> 00:08:04,750 that their eigenvalues-- 147 00:08:07,480 --> 00:08:10,120 all of them-- are real. 148 00:08:10,120 --> 00:08:13,960 Because they correspond to something that's observable. 149 00:08:13,960 --> 00:08:17,380 And when you observe something, it's a real number. 150 00:08:17,380 --> 00:08:18,920 It's not a complex number. 151 00:08:18,920 --> 00:08:20,180 It's not an imaginary number. 152 00:08:25,620 --> 00:08:30,360 So what is Hermetian? 153 00:08:30,360 --> 00:08:34,230 And why does that ensure that you always get a real number? 154 00:08:42,990 --> 00:08:43,490 OK. 155 00:08:43,490 --> 00:08:47,570 You all know, in quantum mechanics, 156 00:08:47,570 --> 00:08:50,630 that if you do an experiment, and the experiment can 157 00:08:50,630 --> 00:08:54,520 be represented by some operator, you 158 00:08:54,520 --> 00:08:58,360 get an eigenvalue of that operator-- 159 00:08:58,360 --> 00:09:02,080 nothing else-- one experiment, one eigenvalue. 160 00:09:02,080 --> 00:09:06,310 100 experiments-- You might get several eigenvalues. 161 00:09:06,310 --> 00:09:10,030 And the relative probabilities of the different eigenvalues-- 162 00:09:10,030 --> 00:09:12,940 I tell you something about, what is this? 163 00:09:12,940 --> 00:09:14,835 What was the initial preparation? 164 00:09:26,960 --> 00:09:30,140 So I've already said something about this. 165 00:09:30,140 --> 00:09:32,780 The expectation value of the wave function 166 00:09:32,780 --> 00:09:42,020 for a particular operator is related to probabilities times 167 00:09:42,020 --> 00:09:42,830 eigenvalues. 168 00:09:46,960 --> 00:09:47,460 OK. 169 00:09:47,460 --> 00:09:50,880 The fifth postulate is the time-dependent Schrodinger 170 00:09:50,880 --> 00:09:52,130 equation. 171 00:09:52,130 --> 00:09:56,970 And I don't need to talk much about that, because it's 172 00:09:56,970 --> 00:09:59,000 all the way around us. 173 00:09:59,000 --> 00:10:05,070 And then I'm going to talk about some neat stuff where we start 174 00:10:05,070 --> 00:10:18,820 using words that are very instructive-- so completeness, 175 00:10:18,820 --> 00:10:23,990 orthogonality, commutators-- 176 00:10:30,260 --> 00:10:37,200 simultaneous eigenfunctions. 177 00:10:37,200 --> 00:10:39,050 And this is really important. 178 00:10:39,050 --> 00:10:42,230 Suppose we have an easy operator, which 179 00:10:42,230 --> 00:10:45,740 commutes with the Hamiltonian. 180 00:10:45,740 --> 00:10:50,520 And it's easy for us to find the eigenvalues of that operator-- 181 00:10:50,520 --> 00:10:53,831 eigenfunctions of that operator. 182 00:10:53,831 --> 00:10:56,080 Those eigenfunctions are automatically eigenfunctions. 183 00:10:56,080 --> 00:10:58,770 It's a hard operator. 184 00:10:58,770 --> 00:11:00,850 So we like them. 185 00:11:00,850 --> 00:11:04,200 And so we are interested in, can we 186 00:11:04,200 --> 00:11:08,470 have simultaneous eigenfunctions of several operators? 187 00:11:08,470 --> 00:11:10,670 And the answer is yes, if the operators can move. 188 00:11:13,480 --> 00:11:23,270 We use the term basis set to describe 189 00:11:23,270 --> 00:11:26,150 the complete set of eigenfunctions 190 00:11:26,150 --> 00:11:29,870 of some operator. 191 00:11:29,870 --> 00:11:43,120 And we use the words mixing coefficients 192 00:11:43,120 --> 00:11:45,820 and mixing fraction. 193 00:11:49,000 --> 00:11:53,200 So here we have a wave function that 194 00:11:53,200 --> 00:11:57,520 is expressed as a linear combination of basis functions. 195 00:11:57,520 --> 00:12:00,400 And the coefficient in front of each one 196 00:12:00,400 --> 00:12:02,270 is the mixing coefficient. 197 00:12:02,270 --> 00:12:04,450 Now, if we're talking about probabilities, 198 00:12:04,450 --> 00:12:07,000 we care about mixing fractions, which 199 00:12:07,000 --> 00:12:10,360 are basically the mixing coefficient squared-- 200 00:12:10,360 --> 00:12:13,390 or [INAUDIBLE] square modulus. 201 00:12:13,390 --> 00:12:18,280 So these are words that are part of the language of someone 202 00:12:18,280 --> 00:12:22,570 who uses quantum mechanics to understand stuff. 203 00:12:22,570 --> 00:12:23,070 OK. 204 00:12:23,070 --> 00:12:26,055 So I'm going to try to develop this in some useful tricks. 205 00:12:31,280 --> 00:12:34,010 So we have a state function. 206 00:12:44,880 --> 00:12:46,760 So we have a state function, which 207 00:12:46,760 --> 00:12:50,510 is a function of coordinates and time. 208 00:12:50,510 --> 00:12:53,810 And this thing is telling you, what 209 00:12:53,810 --> 00:12:58,910 is the probability of finding this system at that coordinate 210 00:12:58,910 --> 00:13:01,820 at the specified time? 211 00:13:01,820 --> 00:13:06,800 And the volume element is dx, dy, and dz. 212 00:13:06,800 --> 00:13:10,730 Now, often, we use an abbreviation-- 213 00:13:10,730 --> 00:13:13,880 d tau-- for the volume element, because we're 214 00:13:13,880 --> 00:13:15,320 going to be dealing with problems 215 00:13:15,320 --> 00:13:18,900 that are not just single particle, but many particles. 216 00:13:18,900 --> 00:13:23,770 And so we use this notation to say, 217 00:13:23,770 --> 00:13:27,240 for the differential associated with every coordinate 218 00:13:27,240 --> 00:13:29,100 associated with the problem, and we're 219 00:13:29,100 --> 00:13:31,100 going to integrate over those sorts of things-- 220 00:13:31,100 --> 00:13:34,350 or at least over some of them. 221 00:13:34,350 --> 00:13:38,010 So this is telling you about a probability within a volume 222 00:13:38,010 --> 00:13:42,030 element at a particular point in space and time. 223 00:13:42,030 --> 00:13:48,770 Now, one of the things that we're told 224 00:13:48,770 --> 00:13:59,860 is, if the wave functions are well behaved. 225 00:13:59,860 --> 00:14:03,130 And that says something about the wave functions 226 00:14:03,130 --> 00:14:07,270 and the derivatives of the wave functions. 227 00:14:07,270 --> 00:14:08,812 So what's well-behaved? 228 00:14:11,650 --> 00:14:13,710 Well, the wave function is normalize-able. 229 00:14:18,020 --> 00:14:20,690 Now, there are two kinds of normalization-- normalization 230 00:14:20,690 --> 00:14:25,850 to one, implying that the system is somewhere within a specified 231 00:14:25,850 --> 00:14:28,540 range of coordinates where there's 232 00:14:28,540 --> 00:14:30,370 one particle in the system. 233 00:14:30,370 --> 00:14:31,690 Whatever. 234 00:14:31,690 --> 00:14:35,860 And there's normalization to number density. 235 00:14:35,860 --> 00:14:38,710 When you have a free particle, the free particle 236 00:14:38,710 --> 00:14:41,200 is not confined. 237 00:14:41,200 --> 00:14:44,720 And so you can't say, I'm normalizing. 238 00:14:44,720 --> 00:14:46,870 So there's one particle somewhere, 239 00:14:46,870 --> 00:14:51,490 because that means there is no particle anywhere. 240 00:14:51,490 --> 00:14:54,850 And so we can extend the concept of normalization to say, 241 00:14:54,850 --> 00:14:56,830 there's one particle in a particular length 242 00:14:56,830 --> 00:14:57,420 of the system. 243 00:15:00,970 --> 00:15:02,930 And that was a problem that got removed 244 00:15:02,930 --> 00:15:06,270 from the draft of the exam. 245 00:15:06,270 --> 00:15:11,540 But one thing that happens is, if I write a beautiful problem, 246 00:15:11,540 --> 00:15:16,940 and it gets bumped from an exam, it might appear in the final. 247 00:15:16,940 --> 00:15:17,620 OK. 248 00:15:17,620 --> 00:15:26,640 So normalize-able is part of well-behaved continuous 249 00:15:26,640 --> 00:15:29,169 and single value-- 250 00:15:29,169 --> 00:15:30,460 we'll talk about all of these-- 251 00:15:33,900 --> 00:15:39,940 and square integrable 252 00:15:39,940 --> 00:15:40,880 OK. 253 00:15:40,880 --> 00:15:44,720 Continuous-- The wave function has 254 00:15:44,720 --> 00:15:49,070 to be continuous everywhere. 255 00:15:49,070 --> 00:15:50,990 The first derivative of the wave function, 256 00:15:50,990 --> 00:15:53,000 with respect to coordinate-- 257 00:15:53,000 --> 00:15:55,730 we already know from the particle in a box 258 00:15:55,730 --> 00:16:00,170 that that is not continuous at an infinite wall. 259 00:16:00,170 --> 00:16:01,760 So an infinite wall-- 260 00:16:01,760 --> 00:16:06,350 not just a vertical wall, but one that goes to infinity-- 261 00:16:06,350 --> 00:16:10,670 guarantees that this guy is not continuous. 262 00:16:10,670 --> 00:16:14,020 But that's a pretty dramatic thing. 263 00:16:14,020 --> 00:16:17,140 The second derivative is not continuous 264 00:16:17,140 --> 00:16:22,460 when you have a vertical step, which is not infinite. 265 00:16:22,460 --> 00:16:26,600 Now, when you have problems where you divide space up 266 00:16:26,600 --> 00:16:28,940 into regions, you're often trying 267 00:16:28,940 --> 00:16:32,810 to establish boundary conditions between the different regions 268 00:16:32,810 --> 00:16:34,640 or at the borders. 269 00:16:34,640 --> 00:16:38,660 And the boundary conditions are usually 270 00:16:38,660 --> 00:16:42,230 expressed in terms of continuity of the wave function 271 00:16:42,230 --> 00:16:45,260 and continuity of the first derivative. 272 00:16:45,260 --> 00:16:48,770 And we don't need this often. 273 00:16:48,770 --> 00:16:51,530 But you're entitled-- if the problem is sufficiently 274 00:16:51,530 --> 00:16:52,130 well-behaved-- 275 00:16:52,130 --> 00:16:54,140 All of these guys are continuous, 276 00:16:54,140 --> 00:16:55,340 and you can use them all. 277 00:16:58,720 --> 00:16:59,860 OK. 278 00:16:59,860 --> 00:17:03,790 So normalize-able-- Well, normalize-able 279 00:17:03,790 --> 00:17:08,260 means it's square integrable, and you don't get infinity 280 00:17:08,260 --> 00:17:13,660 unless we use this other definition of normalize-able. 281 00:17:13,660 --> 00:17:19,119 So one of the things that has to happen 282 00:17:19,119 --> 00:17:26,710 is, at the coordinate plus and minus infinity, 283 00:17:26,710 --> 00:17:28,210 the wave function has to go to zero. 284 00:17:30,850 --> 00:17:39,480 Now, the wave function can be infinite at a very small region 285 00:17:39,480 --> 00:17:41,070 of space. 286 00:17:41,070 --> 00:17:44,650 So there are singularities that can be dealt with. 287 00:17:44,650 --> 00:17:46,890 But normally, you say, the wave function is never 288 00:17:46,890 --> 00:17:49,950 going to be infinite, and it's never 289 00:17:49,950 --> 00:17:52,400 going to be anything but 0 at infinity, 290 00:17:52,400 --> 00:17:53,540 or you're in real trouble. 291 00:18:01,210 --> 00:18:04,990 Now, there is a wonderful example 292 00:18:04,990 --> 00:18:07,690 of a kind of a problem called the delta function. 293 00:18:10,290 --> 00:18:14,280 A delta function is basically an infinite spike-- 294 00:18:14,280 --> 00:18:17,990 infinitely thin, infinitely tall. 295 00:18:17,990 --> 00:18:21,410 And what it does is, it causes this 296 00:18:21,410 --> 00:18:26,150 to be discontinuous by an amount related 297 00:18:26,150 --> 00:18:30,915 to the value of the derivative at the-- 298 00:18:30,915 --> 00:18:33,560 by an amount determined by the value of the wave 299 00:18:33,560 --> 00:18:37,070 function at the delta function. 300 00:18:37,070 --> 00:18:41,190 And delta functions are computationally wonderful, 301 00:18:41,190 --> 00:18:46,770 because they enable you to treat certain kinds of problems 302 00:18:46,770 --> 00:18:49,120 in a trivial way. 303 00:18:49,120 --> 00:18:50,750 Like, if you have a barrier-- 304 00:18:50,750 --> 00:18:51,550 Yes. 305 00:18:51,550 --> 00:18:54,790 AUDIENCE: Does it relate at all to the integral of the spike? 306 00:18:54,790 --> 00:18:55,540 ROBERT FIELD: Yes. 307 00:18:59,530 --> 00:19:03,960 So we have an integral of the delta function 308 00:19:03,960 --> 00:19:10,480 at x i times the wave function at x, dx. 309 00:19:10,480 --> 00:19:13,606 And that gives you the wave function at x i. 310 00:19:16,730 --> 00:19:19,070 OK. 311 00:19:19,070 --> 00:19:22,530 I haven't really talked much about delta functions. 312 00:19:22,530 --> 00:19:25,790 But because it acts like a barrier 313 00:19:25,790 --> 00:19:28,520 and is a trivial barrier, it enables 314 00:19:28,520 --> 00:19:32,630 you to solve barrier problems or at least understand them 315 00:19:32,630 --> 00:19:35,900 in a very quick and easy way. 316 00:19:35,900 --> 00:19:40,080 And vertical steps are also not physical, 317 00:19:40,080 --> 00:19:42,020 but we like vertical steps because it's 318 00:19:42,020 --> 00:19:45,140 easy to apply boundary conditions. 319 00:19:45,140 --> 00:19:48,820 And so these are all just computationally tricky things 320 00:19:48,820 --> 00:19:50,470 that are wonderful. 321 00:19:50,470 --> 00:19:54,700 And we don't worry about, is there 322 00:19:54,700 --> 00:19:57,760 a real system that acts like a delta function 323 00:19:57,760 --> 00:19:58,910 or a vertical step? 324 00:19:58,910 --> 00:19:59,410 No. 325 00:19:59,410 --> 00:20:00,700 There isn't. 326 00:20:00,700 --> 00:20:03,400 But everything you get easily, mathematically, 327 00:20:03,400 --> 00:20:06,040 from these simple things is great. 328 00:20:08,860 --> 00:20:10,700 OK. 329 00:20:10,700 --> 00:20:12,385 Did I satisfy you on the-- 330 00:20:12,385 --> 00:20:13,010 AUDIENCE: Sure. 331 00:20:13,010 --> 00:20:14,570 I'll read into it. 332 00:20:14,570 --> 00:20:15,450 ROBERT FIELD: OK. 333 00:20:15,450 --> 00:20:18,300 And there's also a notation where 334 00:20:18,300 --> 00:20:26,880 you have x, xi, or x minus x i, and they're basically 335 00:20:26,880 --> 00:20:28,340 all the same sort of thing. 336 00:20:30,850 --> 00:20:31,860 If you have this-- 337 00:20:31,860 --> 00:20:34,260 the argument-- when the argument is 0, 338 00:20:34,260 --> 00:20:36,170 you get the infinite spike. 339 00:20:36,170 --> 00:20:38,771 And there's just lots of things. 340 00:20:38,771 --> 00:20:39,270 OK. 341 00:20:43,880 --> 00:20:54,240 So for every classical mechanical observable, 342 00:20:54,240 --> 00:20:58,313 there is a quantum mechanical operator, which is Hermetian. 343 00:21:04,470 --> 00:21:08,090 And the main point of Hermetian, as I said, 344 00:21:08,090 --> 00:21:11,570 is that its eigenvalues are real. 345 00:21:11,570 --> 00:21:13,850 And so what is the thing that assures 346 00:21:13,850 --> 00:21:15,950 that we get real eigenvalues? 347 00:21:15,950 --> 00:21:20,930 Well, here is the definition in a peculiar form. 348 00:21:20,930 --> 00:21:25,010 So we have an integral from minus infinity to infinity 349 00:21:25,010 --> 00:21:28,400 of some function-- complex conjugate-- 350 00:21:28,400 --> 00:21:32,675 a times some other function dx. 351 00:21:32,675 --> 00:21:35,300 So this could be a wave function and a different wave function. 352 00:21:38,990 --> 00:21:46,130 And the definition of Hermetian is this abstract definition. 353 00:21:46,130 --> 00:21:48,770 We have, say, interval from minus infinity 354 00:21:48,770 --> 00:21:53,840 to infinity g a-- 355 00:21:53,840 --> 00:21:56,830 Let's put a hat on it. 356 00:21:56,830 --> 00:22:01,280 Star, f-star, dx. 357 00:22:04,530 --> 00:22:06,770 Well, this is kind of fancy. 358 00:22:06,770 --> 00:22:07,820 So we have an operator. 359 00:22:07,820 --> 00:22:10,130 We can take the complex conjugate of the operator. 360 00:22:10,130 --> 00:22:11,780 We have functions. 361 00:22:11,780 --> 00:22:15,110 We can take the complex conjugates of the function. 362 00:22:15,110 --> 00:22:17,930 But here, what we're seeing is, the operator 363 00:22:17,930 --> 00:22:21,680 is operating on the g function-- the function 364 00:22:21,680 --> 00:22:25,190 that started out on the right. 365 00:22:25,190 --> 00:22:29,930 And here, what we have is this operator operating 366 00:22:29,930 --> 00:22:35,630 on the f function, which was initially on the left. 367 00:22:35,630 --> 00:22:38,830 And so this is prescription for operating on the left. 368 00:22:42,040 --> 00:22:46,870 And it's also an invitation to use 369 00:22:46,870 --> 00:22:50,140 a really convenient and compact notation. 370 00:22:50,140 --> 00:22:52,590 And that is this-- 371 00:22:52,590 --> 00:22:56,190 Put two subscripts on a. 372 00:22:56,190 --> 00:23:00,700 A subscript says the first guy is the function over here, 373 00:23:00,700 --> 00:23:02,810 which is complex conjugated. 374 00:23:02,810 --> 00:23:04,740 And the second one is the function 375 00:23:04,740 --> 00:23:07,710 over here, which is not complex conjugated. 376 00:23:07,710 --> 00:23:19,200 And so this equation reduces to a g f star, where, now, this 377 00:23:19,200 --> 00:23:22,470 is a wonderful shorthand. 378 00:23:22,470 --> 00:23:27,471 And this is another way of saying that A 379 00:23:27,471 --> 00:23:29,565 has to be equal to A dagger. 380 00:23:32,370 --> 00:23:34,830 Where now we're talking about operators, 381 00:23:34,830 --> 00:23:38,130 and matrix representations of operators. 382 00:23:38,130 --> 00:23:42,620 Because here we have a number with two indices, 383 00:23:42,620 --> 00:23:45,010 and that's how we represent elements of a matrix. 384 00:23:45,010 --> 00:23:47,730 And we're soon going to be playing with linear algebra, 385 00:23:47,730 --> 00:23:50,460 and talking about matrices. 386 00:23:50,460 --> 00:23:53,910 And so this is just saying, well, we can take one matrix, 387 00:23:53,910 --> 00:23:56,820 and it's equal to the complex conjugate 388 00:23:56,820 --> 00:23:58,530 of every term in the matrix. 389 00:23:58,530 --> 00:24:00,075 And the order switched. 390 00:24:03,090 --> 00:24:07,420 So this is a warning that we're going 391 00:24:07,420 --> 00:24:10,630 to be using a notation, which is way simpler than taking 392 00:24:10,630 --> 00:24:12,920 these integrals. 393 00:24:12,920 --> 00:24:17,037 And so once you recognize what this symbol means, 394 00:24:17,037 --> 00:24:18,620 you're never going to want to go back. 395 00:24:24,760 --> 00:24:25,260 OK. 396 00:24:33,870 --> 00:24:35,370 Why real? 397 00:24:40,700 --> 00:24:43,100 So let's look at a specific example, 398 00:24:43,100 --> 00:24:47,390 where instead of having two different functions, let's just 399 00:24:47,390 --> 00:24:48,770 look at one. 400 00:24:48,770 --> 00:24:55,595 So we have this integral f star A f d x. 401 00:24:59,490 --> 00:25:00,600 So that's a f f. 402 00:25:03,940 --> 00:25:09,170 And the definition says, well, we're going to get the-- 403 00:25:17,100 --> 00:25:25,080 So it says, replace the original thing by moving this-- 404 00:25:25,080 --> 00:25:27,330 anyway, yes. 405 00:25:27,330 --> 00:25:32,760 And this is a f f star. 406 00:25:32,760 --> 00:25:37,230 If you know how this notation translates into-- 407 00:25:37,230 --> 00:25:39,690 now you can see, oh, well, what is this? 408 00:25:39,690 --> 00:25:43,750 It's just taking the complex conjugate. 409 00:25:43,750 --> 00:25:46,270 And these two guys are equal. 410 00:25:46,270 --> 00:25:50,400 And so a number is real if it's equal to its complex conjugate. 411 00:25:53,100 --> 00:25:54,960 And this is just a special case. 412 00:25:54,960 --> 00:25:58,490 It's-- the Hermitian property is a little bit more powerful than 413 00:25:58,490 --> 00:26:03,440 that, but the important thing is that it guarantees that if you 414 00:26:03,440 --> 00:26:08,960 calculate the expectation value of an observable quantity, 415 00:26:08,960 --> 00:26:11,470 you're going to get a real number, 416 00:26:11,470 --> 00:26:13,580 if the operator is Hermitian. 417 00:26:13,580 --> 00:26:16,299 And it has to be Hermitian, if it's observable. 418 00:26:19,233 --> 00:26:27,010 Now, it's often useful if you have 419 00:26:27,010 --> 00:26:30,550 a classical mechanical operator, and you translate it 420 00:26:30,550 --> 00:26:33,730 into a quantum mechanical operator 421 00:26:33,730 --> 00:26:37,125 by doing the usual replacement of x with x, and p with i 422 00:26:37,125 --> 00:26:40,540 h bar, the derivative with respect to x. 423 00:26:40,540 --> 00:26:44,040 If you do all that sort of stuff, you might be unlucky 424 00:26:44,040 --> 00:26:47,390 and you might get a non-Hermitian operator. 425 00:26:47,390 --> 00:26:50,020 And so you can generate a Hermitian operator 426 00:26:50,020 --> 00:26:52,990 if you write-- 427 00:26:57,590 --> 00:27:01,510 that's guaranteed to be Hermitian. 428 00:27:01,510 --> 00:27:05,470 So you take classic mechanics, you generate something 429 00:27:05,470 --> 00:27:08,230 following some simple rules, and you have bad luck, 430 00:27:08,230 --> 00:27:09,730 it doesn't come out to be Hermitian. 431 00:27:09,730 --> 00:27:11,831 This is the way we make it Hermitian. 432 00:27:19,060 --> 00:27:24,180 So if this is not Hermitian, and this is not Hermitian, 433 00:27:24,180 --> 00:27:27,560 but that we're defining something that is Hermitian, 434 00:27:27,560 --> 00:27:30,790 so let's just put a little twiddle over A, 435 00:27:30,790 --> 00:27:31,896 that's Hermitian. 436 00:27:36,360 --> 00:27:43,420 OK then we're now talking about the third postulate, 437 00:27:43,420 --> 00:27:55,470 and each measurement of a gives an eigenvalue of a. 438 00:27:58,560 --> 00:28:01,360 We've talked about this enough. 439 00:28:01,360 --> 00:28:07,640 But your first encounter of this was the Two-Slit experiment. 440 00:28:07,640 --> 00:28:13,560 In the Two-Slit experiment, this experiment is an operator. 441 00:28:13,560 --> 00:28:19,125 And you have some initial photons entering this operator 442 00:28:19,125 --> 00:28:23,600 , and you get dots, on the screen, 443 00:28:23,600 --> 00:28:27,350 those are eigenvalues of this operator. 444 00:28:27,350 --> 00:28:34,970 Now it may be that the operator has continuous eigenvalues, 445 00:28:34,970 --> 00:28:37,760 but they don't have uniform probabilities. 446 00:28:37,760 --> 00:28:40,880 And so what you observe is a whole distribution of dots that 447 00:28:40,880 --> 00:28:43,460 doesn't look like anything special, 448 00:28:43,460 --> 00:28:46,430 and it's not reproducible from one experiment to the other, 449 00:28:46,430 --> 00:28:49,310 but you have this periodic structure that's appearing, 450 00:28:49,310 --> 00:28:53,580 which is related to the properties of the operator. 451 00:28:53,580 --> 00:28:56,810 And so there is not uniform probabilities 452 00:28:56,810 --> 00:29:00,470 of each eigenvalues, and so you get that. 453 00:29:00,470 --> 00:29:04,360 OK, these are simple, but really beautiful. 454 00:29:11,090 --> 00:29:26,170 OK, if we have a normalized state then we can say, OK, 455 00:29:26,170 --> 00:29:28,440 and we never use this notation. 456 00:29:28,440 --> 00:29:32,370 But whenever you see this symbol, 457 00:29:32,370 --> 00:29:36,300 it means the expectation value of an operator for some wave 458 00:29:36,300 --> 00:29:38,940 functions, so we could actually symbolically put 459 00:29:38,940 --> 00:29:42,480 that wave function down here, or some symbols saying, OK, 460 00:29:42,480 --> 00:29:45,120 which one? 461 00:29:45,120 --> 00:29:55,900 And that this is equal to psi star A psi d tau, or dx. 462 00:29:55,900 --> 00:29:57,600 And if the wave function is normalized, 463 00:29:57,600 --> 00:30:01,070 we don't need to divide by a normalization integral. 464 00:30:01,070 --> 00:30:05,960 If it's not normalized, like if it's a free particle, 465 00:30:05,960 --> 00:30:08,670 we divide by some kind of free particle integral. 466 00:30:18,330 --> 00:30:24,450 So now the next topic, which is related to this, 467 00:30:24,450 --> 00:30:36,567 is completeness and orthogonality. 468 00:30:45,340 --> 00:30:48,830 So we have a particular operator. 469 00:30:48,830 --> 00:30:53,510 There exists some complete set of eigenfunctions 470 00:30:53,510 --> 00:30:55,490 of that operator. 471 00:30:55,490 --> 00:30:59,500 Usually that complete set is infinite, 472 00:30:59,500 --> 00:31:03,220 but they're related to each other in a simple way. 473 00:31:03,220 --> 00:31:05,140 You have some class of functions, 474 00:31:05,140 --> 00:31:07,210 and you change an integer to get a new function. 475 00:31:09,760 --> 00:31:15,370 And orthogonality is, well, if you have all the eigenfunctions 476 00:31:15,370 --> 00:31:20,830 of an operator, if they belong to different eigenvalues, 477 00:31:20,830 --> 00:31:25,230 they're guaranteed to be orthogonal. 478 00:31:25,230 --> 00:31:27,990 Which is convenient, because that means you get lots of 0s 479 00:31:27,990 --> 00:31:30,600 from integrals, and we like that, because we 480 00:31:30,600 --> 00:31:32,880 don't have to worry about them. 481 00:31:32,880 --> 00:31:36,664 And you want to be able to recognize the zero integrals, 482 00:31:36,664 --> 00:31:38,830 so that you can move very quickly through a problem. 483 00:31:43,510 --> 00:31:48,670 Completeness means, take any function defined 484 00:31:48,670 --> 00:31:51,520 on the space of the operator you're interested in. 485 00:31:51,520 --> 00:31:52,900 You might have an operator that's 486 00:31:52,900 --> 00:31:58,390 only operating on a particular coordinate of a many electron 487 00:31:58,390 --> 00:32:01,000 atom or molecule. 488 00:32:01,000 --> 00:32:04,450 There's lots of ways of saying, it's not over all space, 489 00:32:04,450 --> 00:32:07,900 but for each operator you know what space the operator 490 00:32:07,900 --> 00:32:10,290 in question is dealing with. 491 00:32:10,290 --> 00:32:12,730 And then it's always possible to write. 492 00:32:19,920 --> 00:32:22,020 So this is some general function, 493 00:32:22,020 --> 00:32:24,670 defined in the space of the operator, 494 00:32:24,670 --> 00:32:27,300 and this is the equation that says, well, completeness 495 00:32:27,300 --> 00:32:31,350 tells us that we can take the sum over all 496 00:32:31,350 --> 00:32:35,850 of the eigenfunctions with mixing coefficients. 497 00:32:35,850 --> 00:32:38,560 c j. 498 00:32:38,560 --> 00:32:41,290 And this set of all of the eigenfunctions 499 00:32:41,290 --> 00:32:43,510 is called the basis set. 500 00:32:43,510 --> 00:32:45,040 It's a complete basis set. 501 00:32:48,070 --> 00:32:50,300 So it's always true, you can do this. 502 00:32:50,300 --> 00:32:53,090 So you know from other problems, that if you 503 00:32:53,090 --> 00:32:59,630 have a finite region of space, you can represent anything 504 00:32:59,630 --> 00:33:06,770 within that finite region via sum over Fourier components. 505 00:33:06,770 --> 00:33:08,930 That's a discrete sum. 506 00:33:08,930 --> 00:33:10,430 If you have an infinite space, you 507 00:33:10,430 --> 00:33:13,490 have to do a Fourier integral, but it's basically 508 00:33:13,490 --> 00:33:14,120 the same thing. 509 00:33:14,120 --> 00:33:18,170 You're expressing a function, anything you want, 510 00:33:18,170 --> 00:33:21,920 in terms of simple, manipulable objects. 511 00:33:24,730 --> 00:33:29,410 So sometimes the these things are Fourier components, 512 00:33:29,410 --> 00:33:33,028 and sometimes they're just simple wave functions. 513 00:33:36,020 --> 00:33:41,380 OK, now suppose you have two operators, a and b. 514 00:33:45,130 --> 00:33:47,780 If they operate over the same space, 515 00:33:47,780 --> 00:33:55,140 the question is, can we take eigenfunctions of one 516 00:33:55,140 --> 00:33:57,450 and be sure that they're eigenfunctions of the other? 517 00:34:09,840 --> 00:34:13,719 OK, but let's deal with something simpler. 518 00:34:13,719 --> 00:34:37,650 So suppose we have psi i and psi j, both belong to a sub i. 519 00:34:37,650 --> 00:34:39,784 So they both have the same eigenvalue. 520 00:34:43,659 --> 00:34:48,310 Well, in that case, we cannot be sure that these two functions 521 00:34:48,310 --> 00:34:50,850 are orthogonal. 522 00:34:50,850 --> 00:34:53,960 So there is a handy dandy procedure 523 00:34:53,960 --> 00:35:01,580 called Schmidt Orthogonalization that 524 00:35:01,580 --> 00:35:08,000 says, take any two functions, and let's 525 00:35:08,000 --> 00:35:11,650 construct an orthogonal pair. 526 00:35:11,650 --> 00:35:15,010 This is amazingly useful when you're 527 00:35:15,010 --> 00:35:19,330 trying to understand a problem using 528 00:35:19,330 --> 00:35:21,520 a complete orthonormal basis set. 529 00:35:24,470 --> 00:35:27,890 We know, I'm not going to prove it, 530 00:35:27,890 --> 00:35:30,950 because I don't know where it is in my notes, what 531 00:35:30,950 --> 00:35:35,040 the sequence is, I'm just going to forget it-- 532 00:35:35,040 --> 00:35:37,100 I think I'm going to do it. 533 00:35:37,100 --> 00:35:41,300 If you have functions belonging to different eigenvalues, 534 00:35:41,300 --> 00:35:43,010 they are automatically orthogonal. 535 00:35:43,010 --> 00:35:48,140 That's also really valuable, because you have to check. 536 00:35:48,140 --> 00:35:50,480 So you have harmonic oscillator functions, 537 00:35:50,480 --> 00:35:53,690 and they're all orthogonal to each other. 538 00:35:53,690 --> 00:35:57,110 You have, perhaps, one harmonic oscillator, 539 00:35:57,110 --> 00:35:59,250 and a different harmonic oscillator, 540 00:35:59,250 --> 00:36:02,044 and the functions for these two different harmonic oscillators 541 00:36:02,044 --> 00:36:03,210 don't know about each other. 542 00:36:03,210 --> 00:36:05,960 They're not guaranteed to be orthogonal. 543 00:36:05,960 --> 00:36:11,075 But any two eigenvalues of this guy are orthogonal, 544 00:36:11,075 --> 00:36:12,770 and any of those are orthogonal. 545 00:36:12,770 --> 00:36:14,680 But not here. 546 00:36:14,680 --> 00:36:21,840 OK, so let's just say we have now two 547 00:36:21,840 --> 00:36:24,390 eigenfunctions of an operator that 548 00:36:24,390 --> 00:36:28,250 belong to the same eigenvalues. 549 00:36:28,250 --> 00:36:29,280 And that happens. 550 00:36:29,280 --> 00:36:33,630 There are often very many, very high degeneracies. 551 00:36:33,630 --> 00:36:35,880 But we want to make sure that we've got two. 552 00:36:35,880 --> 00:36:40,770 So let's say, here is a number, the overlap integral 553 00:36:40,770 --> 00:36:45,600 between psi 1 and psi 2 dx. 554 00:36:45,600 --> 00:36:47,580 This is a calculable number. 555 00:36:47,580 --> 00:36:50,220 So you have two original functions, 556 00:36:50,220 --> 00:36:52,920 which are not guaranteed to be orthogonal, 557 00:36:52,920 --> 00:36:55,590 because they belong to the same eigenvalue, 558 00:36:55,590 --> 00:36:58,290 and you can calculate this number. 559 00:36:58,290 --> 00:37:02,760 And then we can say, let us now construct something up psi 560 00:37:02,760 --> 00:37:08,610 2, which is guaranteed to be a psi 2 prime which is guaranteed 561 00:37:08,610 --> 00:37:10,680 to be orthogonal to psi 1. 562 00:37:10,680 --> 00:37:11,430 How do we do that? 563 00:37:14,300 --> 00:37:19,310 Well, we define psi 2 to be a normalization 564 00:37:19,310 --> 00:37:22,700 factor, times psi 2 the original, 565 00:37:22,700 --> 00:37:28,250 plus some constant times psi 1. 566 00:37:28,250 --> 00:37:32,900 And then we say, let us calculate the overlap integral 567 00:37:32,900 --> 00:37:36,305 between psi 1 and psi 2 prime. 568 00:37:39,780 --> 00:37:42,240 OK, so we do that. 569 00:37:42,240 --> 00:37:49,980 And so we have the integral, and we have psi 1 star times psi 2 570 00:37:49,980 --> 00:37:54,320 plus a psi 1 dx. 571 00:37:54,320 --> 00:37:57,910 Psi 1 on psi 1 is 1. 572 00:37:57,910 --> 00:37:59,250 So we get an a. 573 00:37:59,250 --> 00:38:05,310 And psi 1 star times psi 2 gives s. 574 00:38:05,310 --> 00:38:10,110 So this integral, which is supposed 575 00:38:10,110 --> 00:38:14,970 to be 0, because we want psi 1 to be 576 00:38:14,970 --> 00:38:21,270 orthogonal to psi 2 prime, is going to be n times s plus a. 577 00:38:23,970 --> 00:38:25,230 Well, how do we satisfy this? 578 00:38:25,230 --> 00:38:29,770 We just make a to be minus s. 579 00:38:29,770 --> 00:38:30,414 Guaranteed. 580 00:38:30,414 --> 00:38:32,080 Now this is one of the tricks that I use 581 00:38:32,080 --> 00:38:35,230 the most when I do derivations. 582 00:38:35,230 --> 00:38:38,020 I want orthogonal functions, and this little Schmidt 583 00:38:38,020 --> 00:38:41,740 Orthogonalization enables me to take my simple idea 584 00:38:41,740 --> 00:38:44,180 and propagate it into a complicated problem. 585 00:38:44,180 --> 00:38:45,460 It's very valuable. 586 00:38:45,460 --> 00:38:49,150 You'll probably never use it, but I love it. 587 00:38:49,150 --> 00:38:53,890 So if a is equal to minus s, you've got orthogonality. 588 00:38:53,890 --> 00:38:59,830 And the general formula is psi 2 prime is equal 589 00:38:59,830 --> 00:39:07,770 to 1 minus x squared, the square root of that, 590 00:39:07,770 --> 00:39:11,810 times psi 2 minus s psi 1. 591 00:39:15,460 --> 00:39:20,140 So this is a normalized function which is orthogonal to psi 1. 592 00:39:23,040 --> 00:39:24,290 And it doesn't take much work. 593 00:39:24,290 --> 00:39:27,880 You calculate one interval, and it's done. 594 00:39:27,880 --> 00:39:30,160 Now later in the course, we're going 595 00:39:30,160 --> 00:39:34,060 to talk we're going to talk about a secular determinant 596 00:39:34,060 --> 00:39:37,570 that you use to solve for all the eigenvalues 597 00:39:37,570 --> 00:39:40,480 and eigenfunctions of a complicated problem. 598 00:39:40,480 --> 00:39:44,410 And often, when you do that, you use convenient functions 599 00:39:44,410 --> 00:39:47,020 which are not guaranteed to be orthogonal. 600 00:39:47,020 --> 00:39:52,570 And there is a procedure you apply to this secular matrix, 601 00:39:52,570 --> 00:39:55,240 which orthogonalizes it first. 602 00:39:55,240 --> 00:39:57,760 Then you diagonalize a simple thing, 603 00:39:57,760 --> 00:40:00,170 and then you undiagonalize, if you need to. 604 00:40:00,170 --> 00:40:03,640 Anyway, this is terribly valuable, especially 605 00:40:03,640 --> 00:40:05,710 when you're doing quantum chemistry, which you're 606 00:40:05,710 --> 00:40:08,511 going to see towards the end. 607 00:40:08,511 --> 00:40:09,010 OK. 608 00:40:17,630 --> 00:40:22,750 Often, we would like to express a particular eigenfunction 609 00:40:22,750 --> 00:40:29,890 expectation value as a sum over P i a i. 610 00:40:29,890 --> 00:40:32,720 So this is a probability. 611 00:40:32,720 --> 00:40:35,260 And so how do we do that? 612 00:40:35,260 --> 00:40:41,320 Certainly, the average of this operator over psi 613 00:40:41,320 --> 00:40:46,270 can be written as the eigenvalues of a times 614 00:40:46,270 --> 00:40:48,281 the probability of each operator. 615 00:40:48,281 --> 00:40:49,030 And what are they? 616 00:40:49,030 --> 00:40:52,480 Well, you can show that this is equal to c i-- 617 00:40:56,230 --> 00:40:58,892 I better be careful here. 618 00:40:58,892 --> 00:40:59,850 Well, let's just do it. 619 00:41:04,030 --> 00:41:05,320 Well, I'll just write it. 620 00:41:09,110 --> 00:41:11,300 Where this is the mixing coefficient 621 00:41:11,300 --> 00:41:17,750 of the eigenfunction of a in the original function. 622 00:41:17,750 --> 00:41:22,130 So we get probabilities mixing fractions, 623 00:41:22,130 --> 00:41:24,950 and we have mixing coefficients. 624 00:41:24,950 --> 00:41:26,550 And this is the language we're going 625 00:41:26,550 --> 00:41:29,336 to use to describe many things. 626 00:41:29,336 --> 00:41:30,520 AUDIENCE: [INAUDIBLE] 627 00:41:30,520 --> 00:41:31,520 ROBERT FIELD: I'm sorry? 628 00:41:31,520 --> 00:41:32,895 AUDIENCE: The probability doesn't 629 00:41:32,895 --> 00:41:35,502 have an a i, the average does. 630 00:41:35,502 --> 00:41:39,950 You just remove the a i. 631 00:41:39,950 --> 00:41:41,615 ROBERT FIELD: So the probability-- 632 00:41:41,615 --> 00:41:42,740 AUDIENCE: The probability-- 633 00:41:42,740 --> 00:41:44,750 ROBERT FIELD: Oh, yes. 634 00:41:44,750 --> 00:41:49,390 You see, when I start lecturing from what's 635 00:41:49,390 --> 00:41:50,395 in my addled brain-- 636 00:41:52,900 --> 00:41:55,210 OK, thank you. 637 00:42:02,640 --> 00:42:04,440 OK. 638 00:42:04,440 --> 00:42:06,150 Now let's do some really neat things. 639 00:42:19,930 --> 00:42:23,800 So we have a commutator of two operators. 640 00:42:26,730 --> 00:42:34,490 If that commutator is 0, then all non-degenerate 641 00:42:34,490 --> 00:42:45,770 eigenfunctions of A are eigenfunctions of B. 642 00:42:45,770 --> 00:42:52,670 If this is not equal to 0, then we can say something about 643 00:42:52,670 --> 00:43:02,130 the variances of A and B. So these quantities are greater 644 00:43:02,130 --> 00:43:11,450 than or equal to minus 1/4 times the integral-- 645 00:43:11,450 --> 00:43:13,620 I better write this on the board below. 646 00:43:34,530 --> 00:43:37,820 And this is greater than 0, and real. 647 00:43:37,820 --> 00:43:39,705 This is the uncertainty principle. 648 00:43:42,620 --> 00:43:48,740 So it's possible to prove this, and it's really strange, 649 00:43:48,740 --> 00:43:50,990 because we have a square of a number here. 650 00:43:54,010 --> 00:43:58,230 We think the square of a number is going to be real, 651 00:43:58,230 --> 00:44:01,200 but not if it's imaginary. 652 00:44:01,200 --> 00:44:07,740 Most non-zero commutators are imaginary. 653 00:44:07,740 --> 00:44:11,340 And so this thing is negative, and it's canceled. 654 00:44:11,340 --> 00:44:16,710 So the joint uncertainty is related to the expectation 655 00:44:16,710 --> 00:44:19,590 value of a commutator. 656 00:44:19,590 --> 00:44:26,850 And this is all traced back to x P x is equal to ih bar. 657 00:44:30,360 --> 00:44:35,090 And this commutator is imaginary. 658 00:44:35,090 --> 00:44:40,350 And everything that appears in here, 659 00:44:40,350 --> 00:44:42,840 it comes from the non-commutation 660 00:44:42,840 --> 00:44:45,360 of coordinate and momentum. 661 00:44:45,360 --> 00:44:50,070 And this is why this commutator is often 662 00:44:50,070 --> 00:44:55,840 regarded as the foundation of quantum mechanics. 663 00:44:55,840 --> 00:44:58,610 Because all of the strangeness comes from it. 664 00:45:03,740 --> 00:45:06,500 So yes, this is surprising. 665 00:45:06,500 --> 00:45:09,110 It's saying that, when we have a non-zero commutator, 666 00:45:09,110 --> 00:45:12,500 this is what determines the joint uncertainty of two 667 00:45:12,500 --> 00:45:13,220 properties. 668 00:45:16,220 --> 00:45:18,050 This commutator is always imaginary, 669 00:45:18,050 --> 00:45:21,830 that is a big, big surprise. 670 00:45:21,830 --> 00:45:27,390 And as a result the joint uncertainty is greater than 0, 671 00:45:27,390 --> 00:45:29,580 If the two operators don't commute, 672 00:45:29,580 --> 00:45:31,230 it's because x and P don't commute. 673 00:45:33,870 --> 00:45:34,820 It's really scary. 674 00:45:39,100 --> 00:45:40,440 OK, what time is it? 675 00:45:40,440 --> 00:45:45,680 We've got five minutes left, and I can do one more thing. 676 00:45:45,680 --> 00:45:49,700 Well, I guess I'm going to be just talking about-- 677 00:45:49,700 --> 00:45:53,730 So the uncertainty principle. 678 00:46:01,820 --> 00:46:12,900 If we know operator A and B, we can calculate their commutator. 679 00:46:12,900 --> 00:46:15,150 This is a property of the operators. 680 00:46:15,150 --> 00:46:16,890 It doesn't have to do with constructing 681 00:46:16,890 --> 00:46:19,560 some clever experiment, where you tried to measure 682 00:46:19,560 --> 00:46:23,170 two things simultaneously. 683 00:46:23,170 --> 00:46:29,980 It says, all of the problems with simultaneous observations 684 00:46:29,980 --> 00:46:33,940 of two operators, two things, comes from the structure 685 00:46:33,940 --> 00:46:34,780 of the operators. 686 00:46:34,780 --> 00:46:37,060 Comes from their commutation rule. 687 00:46:37,060 --> 00:46:49,030 Which traces all the way back to the commutator between x and P. 688 00:46:49,030 --> 00:46:52,330 So at the beginning, I said I don't like these experiments, 689 00:46:52,330 --> 00:46:59,620 where we try to confine the coordinate of the photon 690 00:46:59,620 --> 00:47:04,060 or the electron, and it results in uncertainty 691 00:47:04,060 --> 00:47:06,410 in the measurement of the conjugate property, 692 00:47:06,410 --> 00:47:09,550 the momentum, or something like that. 693 00:47:09,550 --> 00:47:13,771 These experiments depend on your cleverness, but this doesn't. 694 00:47:13,771 --> 00:47:16,970 This Is fundamental. 695 00:47:16,970 --> 00:47:18,160 So I like that a lot better. 696 00:47:21,930 --> 00:47:24,340 OK, the last thing I want to talk about, 697 00:47:24,340 --> 00:47:27,070 which I have just barely enough time to do, 698 00:47:27,070 --> 00:47:41,770 is suppose we have a wave function. 699 00:47:41,770 --> 00:47:49,510 Let's call it psi 2 in quotes of x for the particle in a box. 700 00:47:49,510 --> 00:47:52,140 Particle in a infinite box. 701 00:47:52,140 --> 00:47:56,290 This is a wave function, which is not the eigenfunction, 702 00:47:56,290 --> 00:47:59,790 but it is constructed to look like the eigenfunction. 703 00:47:59,790 --> 00:48:04,350 Mainly because it has the right number of nodes. 704 00:48:04,350 --> 00:48:07,890 And so suppose we call this thing 705 00:48:07,890 --> 00:48:19,050 some normalization factor x x minus a and x minus a over 2. 706 00:48:19,050 --> 00:48:23,610 This guarantees you have a node at x equals 0. 707 00:48:23,610 --> 00:48:27,990 This guarantees you have a node at x equals a. 708 00:48:27,990 --> 00:48:32,400 This guarantees you have a node at x equals a over 2. 709 00:48:32,400 --> 00:48:36,360 So this is that the generic property 710 00:48:36,360 --> 00:48:40,320 of the second eigenfunction for the particle in a box. 711 00:48:43,140 --> 00:48:47,080 And this is a very clever guess, and often you 712 00:48:47,080 --> 00:48:49,030 want to make a guess. 713 00:48:49,030 --> 00:48:53,140 And so, how well you do with a guess? 714 00:48:53,140 --> 00:49:01,470 And so this function, this part of it 715 00:49:01,470 --> 00:49:08,620 looks sort of like this between x and a, and this part of it 716 00:49:08,620 --> 00:49:11,110 looks like this. 717 00:49:11,110 --> 00:49:13,390 And we multiply these two together 718 00:49:13,390 --> 00:49:19,220 and we get something that looks like this. 719 00:49:19,220 --> 00:49:22,600 Which is, at least a sketch, looks 720 00:49:22,600 --> 00:49:26,689 like the n equals 2 eigenfunction 721 00:49:26,689 --> 00:49:27,730 with a particle in a box. 722 00:49:27,730 --> 00:49:33,970 So let's just go through and see how well we can do. 723 00:49:33,970 --> 00:49:37,390 So first of all, we have to determine n. 724 00:49:37,390 --> 00:49:40,625 So we do the normalization integral. 725 00:49:40,625 --> 00:49:49,310 Psi 2 star psi 2 dx. 726 00:49:49,310 --> 00:49:50,400 That has to be 1. 727 00:49:55,590 --> 00:49:59,490 So what we do, now this is kind of a cheat, 728 00:49:59,490 --> 00:50:03,270 because we do this because we don't know an eigenfunction. 729 00:50:03,270 --> 00:50:04,980 But we do know these eigenfunctions, 730 00:50:04,980 --> 00:50:06,840 so we can expand these functions, 731 00:50:06,840 --> 00:50:13,890 in terms of the particle in a box eigenfunctions. 732 00:50:13,890 --> 00:50:19,920 So we use these things, which you know very well. 733 00:50:28,090 --> 00:50:31,130 Now we have a sine function here. 734 00:50:31,130 --> 00:50:33,250 That's because I've chosen the box that 735 00:50:33,250 --> 00:50:39,310 doesn't have 0 left edge, it's symmetric about x equals 0. 736 00:50:39,310 --> 00:50:41,700 And that would be appropriate for this kind of function. 737 00:50:46,610 --> 00:50:51,560 So anyway, when we do this, we find the mixing coefficients. 738 00:50:51,560 --> 00:50:55,630 And I know I just said something wrong 739 00:50:55,630 --> 00:50:58,870 and that's, we don't have time to correct it. 740 00:50:58,870 --> 00:51:03,210 Because I said it the wave function is 0 at x equals 0, 741 00:51:03,210 --> 00:51:04,192 and at x equals a. 742 00:51:04,192 --> 00:51:05,650 And I'm now saying, all of a sudden 743 00:51:05,650 --> 00:51:09,400 I'm using symmetric box-- 744 00:51:09,400 --> 00:51:12,520 this does not matter, because the calculation 745 00:51:12,520 --> 00:51:13,900 is done correctly. 746 00:51:13,900 --> 00:51:19,270 And what we end up getting, is that the mixing 747 00:51:19,270 --> 00:51:27,690 coefficients for these functions of the general form, 800-- 748 00:51:27,690 --> 00:51:34,640 I'm sorry, 840-- this is algebra!-- 749 00:51:34,640 --> 00:51:44,520 Square root a to the minus 7/2 2 over n-- 750 00:51:44,520 --> 00:51:47,112 2 over a. 751 00:51:47,112 --> 00:51:47,820 I think that's a. 752 00:51:51,850 --> 00:51:53,890 Well, I'm not sure whether that's a or n, 753 00:51:53,890 --> 00:52:00,190 but let's just say it's 2 over n square root-- 754 00:52:00,190 --> 00:52:02,320 oh, it's going to be 2 over a. 755 00:52:02,320 --> 00:52:12,340 And times the integral-- anyway, so we 756 00:52:12,340 --> 00:52:29,830 get c 2n is equal to 1680 square root over 6 over 2n pi cubed. 757 00:52:33,270 --> 00:52:43,230 And that becomes equal to 0.9914n to the minus 3. 758 00:52:43,230 --> 00:52:46,950 So this is a general formula which you can derive. 759 00:52:46,950 --> 00:52:48,420 I don't recommend it, and I don't 760 00:52:48,420 --> 00:52:49,650 think it's really important. 761 00:52:49,650 --> 00:52:52,540 The important thing is what I'm about to say. 762 00:52:52,540 --> 00:52:54,570 And I have no time. 763 00:52:54,570 --> 00:52:55,710 This is almost 1. 764 00:53:01,140 --> 00:53:10,450 So when we calculate the energy using these functions, 765 00:53:10,450 --> 00:53:17,290 we get that the energy of this 2n function 766 00:53:17,290 --> 00:53:31,160 is equal to 4E1 times 0.983 integral from n 767 00:53:31,160 --> 00:53:39,900 equals 1 to infinity times n to the minus 4. 768 00:53:43,180 --> 00:53:47,050 OK, the first term and this is 1. 769 00:53:47,050 --> 00:53:54,040 And so we have something that looks like 4 times E1. 770 00:53:54,040 --> 00:53:58,740 Now the sum is larger than one, and the product of these two 771 00:53:58,740 --> 00:54:01,140 things is larger than 1. 772 00:54:01,140 --> 00:54:10,080 And so what we get is that, E2n is larger, but only slightly 773 00:54:10,080 --> 00:54:17,660 larger, than the exact results. 774 00:54:17,660 --> 00:54:22,630 So this is sort of a taste of a variation calculation. 775 00:54:22,630 --> 00:54:30,060 We can solve for the form of a function 776 00:54:30,060 --> 00:54:36,780 by doing a minimization of the energy of that function. 777 00:54:36,780 --> 00:54:41,040 And that function will look like the true function, 778 00:54:41,040 --> 00:54:45,570 but its energy will always be larger than the true function. 779 00:54:48,450 --> 00:54:51,460 But it's great, because the bigger the calculation, 780 00:54:51,460 --> 00:54:52,750 the better you do. 781 00:54:52,750 --> 00:54:55,390 And that's how most of the money, the computer 782 00:54:55,390 --> 00:54:58,330 time in the world, is expended. 783 00:54:58,330 --> 00:55:00,970 Doing large variational calculations 784 00:55:00,970 --> 00:55:06,110 to find eigenfunctions of complicated problems. 785 00:55:06,110 --> 00:55:08,720 OK, good luck on the exam. 786 00:55:08,720 --> 00:55:14,130 I hope you find it fun, and I meant it to be fun.