1 00:00:00,070 --> 00:00:02,410 The following content is provided under a Creative 2 00:00:02,410 --> 00:00:03,830 Commons license. 3 00:00:03,830 --> 00:00:06,040 Your support will help MIT OpenCourseWare 4 00:00:06,040 --> 00:00:10,130 continue to offer high-quality educational resources for free. 5 00:00:10,130 --> 00:00:12,670 To make a donation or to view additional materials 6 00:00:12,670 --> 00:00:16,630 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,630 --> 00:00:17,503 at ocw.mit.edu. 8 00:00:21,890 --> 00:00:27,440 ROBERT FIELD: Well, we're now well underway into quantum 9 00:00:27,440 --> 00:00:30,590 mechanics. 10 00:00:30,590 --> 00:00:38,510 So a lot of the important stuff goes by very fast. 11 00:00:38,510 --> 00:00:42,950 So we represent a quantum mechanical operator 12 00:00:42,950 --> 00:00:46,070 with a little hat, and it means do something 13 00:00:46,070 --> 00:00:48,720 to the thing on its right. 14 00:00:48,720 --> 00:00:50,520 And it has to be a linear operator, 15 00:00:50,520 --> 00:00:53,670 and you want to be sure you know what a linear operator does 16 00:00:53,670 --> 00:00:56,370 and what is not a linear operator. 17 00:00:56,370 --> 00:00:58,630 This is an eigenvalue equation. 18 00:00:58,630 --> 00:01:00,780 So we have some function which, when 19 00:01:00,780 --> 00:01:03,120 the operator operates on it, gives back 20 00:01:03,120 --> 00:01:05,880 a constant times that function. 21 00:01:05,880 --> 00:01:09,120 The constant is the eigenvalue, and the function 22 00:01:09,120 --> 00:01:12,180 is an eigenfunction of the operator that 23 00:01:12,180 --> 00:01:16,860 belongs to this eigenvalue, and all of quantum mechanics 24 00:01:16,860 --> 00:01:20,520 can be expressed in terms of eigenvalue equations. 25 00:01:20,520 --> 00:01:23,670 It's very important, and you sort of take it for granted. 26 00:01:23,670 --> 00:01:28,950 Now, one of the important things about quantum mechanics 27 00:01:28,950 --> 00:01:33,600 is that we have to find a linear operator that 28 00:01:33,600 --> 00:01:37,090 corresponds to the classically observable quantities. 29 00:01:37,090 --> 00:01:40,050 And for x the linear operator is x, 30 00:01:40,050 --> 00:01:42,210 and for the momentum the linear operator 31 00:01:42,210 --> 00:01:45,150 is minus ih bar partial with respect to x. 32 00:01:45,150 --> 00:01:47,260 That should bother you two ways. 33 00:01:47,260 --> 00:01:51,600 One is the i, and the other is the partial derivative. 34 00:01:51,600 --> 00:01:55,380 But when you apply this operator to functions, 35 00:01:55,380 --> 00:02:02,070 you discover that out pops something that has the expected 36 00:02:02,070 --> 00:02:04,920 behavior of momentum, and so this 37 00:02:04,920 --> 00:02:07,590 is in fact the operator that we're 38 00:02:07,590 --> 00:02:10,710 going to use for momentum. 39 00:02:10,710 --> 00:02:14,550 And then there is a commutation rule. 40 00:02:14,550 --> 00:02:22,140 This commutation rule, xp minus px, is equal to this. 41 00:02:22,140 --> 00:02:27,380 This is really the foundation of quantum mechanics, 42 00:02:27,380 --> 00:02:31,900 and as I've said before, many people derive everything 43 00:02:31,900 --> 00:02:34,580 from a few commutation rules. 44 00:02:34,580 --> 00:02:38,140 It's really scary, but you should 45 00:02:38,140 --> 00:02:41,890 be able to work out this commutation 46 00:02:41,890 --> 00:02:47,950 rule by applying xp minus px to some arbitrary function. 47 00:02:47,950 --> 00:02:51,160 And going through the symbolics should take you about 30 48 00:02:51,160 --> 00:02:53,380 seconds, or maybe it shouldn't. 49 00:02:53,380 --> 00:02:56,360 Maybe you can go faster. 50 00:02:56,360 --> 00:02:56,860 OK. 51 00:03:00,020 --> 00:03:03,060 We have an operator, and we often 52 00:03:03,060 --> 00:03:06,660 want to know what is the expectation 53 00:03:06,660 --> 00:03:09,840 value of the particular function, which 54 00:03:09,840 --> 00:03:13,440 we could symbolize here, but it's never done that way. 55 00:03:13,440 --> 00:03:16,500 So we have some function, and we want 56 00:03:16,500 --> 00:03:19,500 to calculate its expectation value of operator A, 57 00:03:19,500 --> 00:03:22,870 and this is it. 58 00:03:22,870 --> 00:03:26,860 And so this is a normalization integral, 59 00:03:26,860 --> 00:03:31,330 and this normalization integral is usually taken for granted, 60 00:03:31,330 --> 00:03:36,520 because we almost always work with sets of functions 61 00:03:36,520 --> 00:03:38,620 which are normalized. 62 00:03:38,620 --> 00:03:41,800 And so if you convince yourself that it is, 63 00:03:41,800 --> 00:03:45,710 in fact, they are normalized, fine, 64 00:03:45,710 --> 00:03:48,940 and then this is the thing that you normally would calculate. 65 00:03:53,840 --> 00:03:56,810 Then, we went to our first problem 66 00:03:56,810 --> 00:04:00,230 in quantum mechanics which is the free particle, 67 00:04:00,230 --> 00:04:05,480 and the free particle has some idiosyncrasies. 68 00:04:05,480 --> 00:04:09,780 The wave function for the free particle 69 00:04:09,780 --> 00:04:18,370 has the form e to the ikx plus e to the minus ikx, 70 00:04:18,370 --> 00:04:27,270 and the Hamiltonian is minus h bar 71 00:04:27,270 --> 00:04:36,900 squared over 2m second partial with respect to x plus v0. 72 00:04:36,900 --> 00:04:48,130 So there's no v0 here, and we have two different exponentials 73 00:04:48,130 --> 00:04:53,680 and so is this really going to be an eigenfunction 74 00:04:53,680 --> 00:04:54,970 of the Hamiltonian? 75 00:04:54,970 --> 00:05:01,060 This is really p squared, and so this 76 00:05:01,060 --> 00:05:03,280 is going to be an eigenfunction of p squared too. 77 00:05:08,950 --> 00:05:11,450 All right, so let's show a little picture here. 78 00:05:11,450 --> 00:05:19,530 Here is energy, and this is v0, and let's say 79 00:05:19,530 --> 00:05:20,850 this is the 0 of energy. 80 00:05:43,040 --> 00:05:46,900 What are the eigenvalues of this? 81 00:05:46,900 --> 00:05:51,290 What does the Hamiltonian do to this function? 82 00:05:55,050 --> 00:06:01,310 Well, in order to do that, you have to calculate something 83 00:06:01,310 --> 00:06:06,320 like where you have to calculate the second derivative of each 84 00:06:06,320 --> 00:06:09,650 of these terms, and the second derivative of this term 85 00:06:09,650 --> 00:06:13,805 brings down a minus k squared. 86 00:06:13,805 --> 00:06:16,240 Now, the second derivative of this term 87 00:06:16,240 --> 00:06:22,970 brings down a minus k squared, and so the energy eigenvalues 88 00:06:22,970 --> 00:06:32,270 are going to be given by h bar squared a squared over 2m plus 89 00:06:32,270 --> 00:06:33,720 v0. 90 00:06:33,720 --> 00:06:38,430 So these are the eigenvalues, eigenfunctions, 91 00:06:38,430 --> 00:06:44,870 the energies of a free particle, and they're not quantized. 92 00:06:44,870 --> 00:06:48,780 Now this v0 is something that will often trip you up, 93 00:06:48,780 --> 00:06:52,630 because it's hidden here. 94 00:06:52,630 --> 00:06:53,340 It's not in here. 95 00:06:55,770 --> 00:06:56,270 OK. 96 00:07:00,300 --> 00:07:03,030 I'm going to torture you with something. 97 00:07:03,030 --> 00:07:10,000 So why are these two k's the same? 98 00:07:10,000 --> 00:07:14,990 What would happen if the k for the positive term 99 00:07:14,990 --> 00:07:17,080 were different from the k for the negative term? 100 00:07:19,720 --> 00:07:22,051 Simple answer. 101 00:07:22,051 --> 00:07:22,550 Yeah? 102 00:07:22,550 --> 00:07:24,476 AUDIENCE: There'd be two eigenvalues? 103 00:07:24,476 --> 00:07:25,600 ROBERT FIELD: That's right. 104 00:07:25,600 --> 00:07:29,355 It wouldn't be an eigenvalue. 105 00:07:29,355 --> 00:07:31,480 It wouldn't be an eigenfunction of the Hamiltonian. 106 00:07:31,480 --> 00:07:33,920 It's a mixture of two eigenvalues, 107 00:07:33,920 --> 00:07:36,670 and so that's simple. 108 00:07:36,670 --> 00:07:42,430 But often we might be dealing with a potential that's 109 00:07:42,430 --> 00:07:47,440 not simple like this but has got complexity. 110 00:07:50,500 --> 00:07:54,605 So suppose we had a potential that did this. 111 00:07:59,270 --> 00:08:05,380 The potential is constant piecewise, 112 00:08:05,380 --> 00:08:06,270 and so what do we do? 113 00:08:09,311 --> 00:08:09,810 Yes? 114 00:08:09,810 --> 00:08:11,610 AUDIENCE: Break down the function into pieces 115 00:08:11,610 --> 00:08:12,901 for each in certain boundaries? 116 00:08:12,901 --> 00:08:16,710 ROBERT FIELD: Yes, and that's exactly right. 117 00:08:16,710 --> 00:08:19,700 You do want to break it up, but one 118 00:08:19,700 --> 00:08:21,800 of the things I'm stressing here is 119 00:08:21,800 --> 00:08:24,110 that you want to be able to draw cartoons. 120 00:08:26,800 --> 00:08:32,820 And so we know that if we choose an energy here, 121 00:08:32,820 --> 00:08:39,860 there is a certain momentum, or a certain kinetic energy here, 122 00:08:39,860 --> 00:08:45,280 and a different kinetic energy here, and so somehow, 123 00:08:45,280 --> 00:08:50,680 what you write for the wave function will reflect that. 124 00:08:50,680 --> 00:08:53,650 But now qualitatively, pictorially, 125 00:08:53,650 --> 00:08:59,830 if we have a wave function in this region which 126 00:08:59,830 --> 00:09:03,010 is oscillating like this, and it'll 127 00:09:03,010 --> 00:09:06,790 be oscillating at the same spatial frequency over here, 128 00:09:06,790 --> 00:09:08,930 well what's going to be happening here? 129 00:09:08,930 --> 00:09:10,930 Is it going to be oscillating faster or slower? 130 00:09:14,157 --> 00:09:16,605 AUDIENCE: Faster. 131 00:09:16,605 --> 00:09:18,520 Faster. 132 00:09:18,520 --> 00:09:21,120 ROBERT FIELD: Absolutely, and is it 133 00:09:21,120 --> 00:09:25,325 going to have amplitude smaller or larger than here? 134 00:09:29,600 --> 00:09:31,170 You're going to answer, yes? 135 00:09:33,740 --> 00:09:38,040 Well, let me do a thought experiment. 136 00:09:38,040 --> 00:09:41,240 So I'm going to walk from one side of the blackboard 137 00:09:41,240 --> 00:09:44,680 to the other, and I'm going to walk at a constant velocity. 138 00:09:44,680 --> 00:09:47,705 Then, I'm going to walk faster and then back 139 00:09:47,705 --> 00:09:51,310 to this original velocity. 140 00:09:51,310 --> 00:09:54,040 So what's the probability of seeing me 141 00:09:54,040 --> 00:09:58,750 in the middle region relative to the edge regions? 142 00:09:58,750 --> 00:10:00,800 Yes? 143 00:10:00,800 --> 00:10:02,646 AUDIENCE: It's less. 144 00:10:02,646 --> 00:10:03,770 ROBERT FIELD: That's right. 145 00:10:03,770 --> 00:10:06,650 The probability, local probability, 146 00:10:06,650 --> 00:10:10,330 is proportional to 1 over the velocity, 147 00:10:10,330 --> 00:10:12,980 and the wave function is proportional to 1 148 00:10:12,980 --> 00:10:15,890 over the square root of the velocity. 149 00:10:15,890 --> 00:10:17,970 And the velocity is related to the momentum, 150 00:10:17,970 --> 00:10:19,680 and so we have everything. 151 00:10:19,680 --> 00:10:26,070 So we know that the wave here will be oscillating faster 152 00:10:26,070 --> 00:10:28,740 and with lower amplitude. 153 00:10:28,740 --> 00:10:32,220 This is what I want you to know, and you'll 154 00:10:32,220 --> 00:10:34,860 be able to use that cartoon to solve problems. 155 00:10:37,420 --> 00:10:40,580 If you understand what's going on here, 156 00:10:40,580 --> 00:10:43,090 these pictures will be equivalent 157 00:10:43,090 --> 00:10:46,720 to global understanding, and these pictures 158 00:10:46,720 --> 00:10:57,480 are also part of semi-classical quantum mechanics. 159 00:10:57,480 --> 00:11:01,980 I believe you all know classical mechanics at least a little, 160 00:11:01,980 --> 00:11:03,960 enough to be useful. 161 00:11:03,960 --> 00:11:07,220 And what we want to be able to do in order to draw pictures 162 00:11:07,220 --> 00:11:11,130 and to understand stuff is to insert just enough quantum 163 00:11:11,130 --> 00:11:14,870 mechanics into classical mechanics so that it's correct. 164 00:11:17,640 --> 00:11:21,650 Then, all of a sudden, it starts to make a lot more sense. 165 00:11:21,650 --> 00:11:22,150 OK. 166 00:11:36,110 --> 00:11:44,190 So the particle in a box, well, we have this sort of situation, 167 00:11:44,190 --> 00:11:49,830 and we have 0 and a. 168 00:11:49,830 --> 00:11:56,730 So the length of the box is a, and the bottom of the box v0 169 00:11:56,730 --> 00:12:00,392 is 0 for this picture. 170 00:12:00,392 --> 00:12:02,600 Now, one of the things that I want you to think about 171 00:12:02,600 --> 00:12:05,100 is, OK, I understand. 172 00:12:05,100 --> 00:12:06,920 I've solved this problem. 173 00:12:06,920 --> 00:12:08,870 I know how to solve this problem. 174 00:12:08,870 --> 00:12:10,762 I know how to get the eigenvalues, 175 00:12:10,762 --> 00:12:12,470 and I know how to get the eigenfunctions, 176 00:12:12,470 --> 00:12:14,970 and I know how to normalize them. 177 00:12:14,970 --> 00:12:19,550 Well, suppose I move the box to the side. 178 00:12:19,550 --> 00:12:25,970 So I move it from say b to a plus b. 179 00:12:25,970 --> 00:12:29,406 So it's the same width, but it's just in a different place. 180 00:12:29,406 --> 00:12:30,630 Well, did anything change? 181 00:12:37,830 --> 00:12:39,960 The only thing that changes is the wave function, 182 00:12:39,960 --> 00:12:43,690 because you have to shift the coordinates. 183 00:12:43,690 --> 00:12:47,570 What happens if I raise the box or lower the box? 184 00:12:47,570 --> 00:12:49,861 Will anything change? 185 00:12:49,861 --> 00:12:51,690 AUDIENCE: [INAUDIBLE] 186 00:12:51,690 --> 00:12:52,850 ROBERT FIELD: You're hot. 187 00:12:52,850 --> 00:12:55,010 AUDIENCE: [INAUDIBLE] 188 00:12:55,256 --> 00:12:56,256 ROBERT FIELD: I'm sorry? 189 00:12:56,256 --> 00:12:58,050 AUDIENCE: [INAUDIBLE] 190 00:13:00,119 --> 00:13:00,910 ROBERT FIELD: Yeah. 191 00:13:00,910 --> 00:13:06,680 If I move the box so that v0 is not 0, but v0 is 10. 192 00:13:06,680 --> 00:13:10,490 AUDIENCE: Then, the weight function will oscillate slower. 193 00:13:10,490 --> 00:13:12,095 ROBERT FIELD: No. 194 00:13:12,095 --> 00:13:14,280 AUDIENCE: [INAUDIBLE] 195 00:13:14,280 --> 00:13:20,600 ROBERT FIELD: So if you move the box up in energy, 196 00:13:20,600 --> 00:13:24,321 the wave function is going to look exactly the same, 197 00:13:24,321 --> 00:13:26,820 but the energies are going to be different by the amount you 198 00:13:26,820 --> 00:13:30,840 move the box up or down, and this is really important. 199 00:13:30,840 --> 00:13:35,580 It may seem trivial to some of you and really obscure 200 00:13:35,580 --> 00:13:38,670 to others, but you really want to be 201 00:13:38,670 --> 00:13:41,370 able to take these things apart. 202 00:13:41,370 --> 00:13:44,070 Because that will enable you to understand them 203 00:13:44,070 --> 00:13:49,400 in a permanent way, and the cartoons are really important. 204 00:13:49,400 --> 00:13:53,480 So if you have the solution to the particle in a box, 205 00:13:53,480 --> 00:13:56,010 then it doesn't matter where the box is. 206 00:13:56,010 --> 00:13:58,515 You know the solution to any particle in a box. 207 00:14:04,950 --> 00:14:05,940 OK. 208 00:14:05,940 --> 00:14:10,560 There is something that I meant to talk about briefly, 209 00:14:10,560 --> 00:14:18,350 but when we write these solutions-- where 210 00:14:18,350 --> 00:14:22,210 did the other blackboard go? 211 00:14:22,210 --> 00:14:25,220 All right, well, I've hidden it-- 212 00:14:25,220 --> 00:14:29,460 so when we have solutions like e to the ikx 213 00:14:29,460 --> 00:14:36,920 and e to the minus ikx, so we have say a here and b here. 214 00:14:36,920 --> 00:14:39,903 When we go to normalize a function like this-- 215 00:14:39,903 --> 00:14:42,050 let's put the plus in here-- 216 00:14:42,050 --> 00:14:49,220 then we write psi star psi dx. 217 00:14:49,220 --> 00:14:52,430 So psi star would make this go a star 218 00:14:52,430 --> 00:14:55,160 and this go to e to the minus ikx, 219 00:14:55,160 --> 00:15:00,960 and this go to b star e to the plus ikx. 220 00:15:00,960 --> 00:15:02,870 So now, we multiply things together. 221 00:15:02,870 --> 00:15:06,530 We get an a, a star which is the square modulus of a, 222 00:15:06,530 --> 00:15:09,680 and we get e to the ikx and e to the minus ikx. 223 00:15:09,680 --> 00:15:11,480 It's 1. 224 00:15:11,480 --> 00:15:13,780 This is why we use this form. 225 00:15:13,780 --> 00:15:18,560 The integrals for things involving e to the ikx 226 00:15:18,560 --> 00:15:20,850 are either 1 or 0. 227 00:15:23,530 --> 00:15:28,650 So if you took e to the ikx, this term, 228 00:15:28,650 --> 00:15:30,780 and multiplied it by this term, you'd get an a, 229 00:15:30,780 --> 00:15:35,330 b star e to the 2 ikx integrated over a finite region. 230 00:15:35,330 --> 00:15:35,850 That's 0. 231 00:15:38,880 --> 00:15:42,060 So we really like this exponential notation, 232 00:15:42,060 --> 00:15:45,180 even if you've been brought up on sines and cosines, 233 00:15:45,180 --> 00:15:47,880 and you use the sines and cosines to impose the boundary 234 00:15:47,880 --> 00:15:48,995 conditions. 235 00:16:01,890 --> 00:16:06,670 OK, another challenge. 236 00:16:06,670 --> 00:16:16,230 So this is v0, and the only problem is this v0-- 237 00:16:16,230 --> 00:16:21,310 well, it looks like this. 238 00:16:21,310 --> 00:16:25,760 So this is v1. 239 00:16:25,760 --> 00:16:31,880 OK, so we have now a particle in this straight. 240 00:16:31,880 --> 00:16:35,570 It's a hybrid between the free particle 241 00:16:35,570 --> 00:16:36,605 and a particle in a box. 242 00:16:40,290 --> 00:16:42,150 So suppose we're at an energy like this. 243 00:16:45,000 --> 00:16:47,160 What's going to happen? 244 00:16:47,160 --> 00:16:50,610 Well, everything that's outside-- 245 00:16:50,610 --> 00:16:54,120 everything that's in the classically-allowed region, 246 00:16:54,120 --> 00:16:54,990 we understand. 247 00:16:54,990 --> 00:17:00,860 We know how to deal with it, but in here, well, that's OK too. 248 00:17:00,860 --> 00:17:04,819 But inside this classically-forbidden region, 249 00:17:04,819 --> 00:17:06,980 the wave function is going to behave differently. 250 00:17:06,980 --> 00:17:11,280 Now, I'm going to assert something. 251 00:17:11,280 --> 00:17:12,900 It doesn't have nodes. 252 00:17:12,900 --> 00:17:14,520 It doesn't oscillate. 253 00:17:14,520 --> 00:17:16,319 It's either exponentially decreasing 254 00:17:16,319 --> 00:17:24,750 or exponentially increasing, and it will never cross 0, never. 255 00:17:24,750 --> 00:17:25,670 OK. 256 00:17:25,670 --> 00:17:32,060 So now, if we're solving a problem involving 257 00:17:32,060 --> 00:17:42,300 any kind of 1D potential, number of nodes. 258 00:17:58,610 --> 00:18:07,190 So for 2D-bound problems, the number of nodes starts with 0, 259 00:18:07,190 --> 00:18:10,070 and it corresponds to the lowest energy state. 260 00:18:10,070 --> 00:18:15,450 The next state up has 1 node, and the next state has 2 nodes. 261 00:18:15,450 --> 00:18:16,940 So by counting the nodes, you would 262 00:18:16,940 --> 00:18:21,890 know what the energy order is of these eigenvalues which is also 263 00:18:21,890 --> 00:18:24,080 an extremely useful thing. 264 00:18:24,080 --> 00:18:26,420 If you're thinking about it or telling your computer 265 00:18:26,420 --> 00:18:29,750 to find the 33rd eigenvalue of something, 266 00:18:29,750 --> 00:18:34,550 because you just run a calculation that solves 267 00:18:34,550 --> 00:18:38,090 for an approximate wave function, and the 33rd, 268 00:18:38,090 --> 00:18:43,040 it needs 32 nodes. 269 00:18:43,040 --> 00:18:47,150 And so the computer says, oh, thank you, master, 270 00:18:47,150 --> 00:18:49,572 and here is your wave function, but you 271 00:18:49,572 --> 00:18:50,780 have to find the right thing. 272 00:18:50,780 --> 00:18:51,500 OK. 273 00:18:51,500 --> 00:19:01,040 Now, here is the picture that you use to remember everything 274 00:19:01,040 --> 00:19:02,240 about a particle in a box. 275 00:19:05,610 --> 00:19:08,030 And the wave function looks like this, 276 00:19:08,030 --> 00:19:10,565 and the next wave function looks like that, 277 00:19:10,565 --> 00:19:16,270 and the next wave function looks like this. 278 00:19:16,270 --> 00:19:24,280 And so no nodes, 1 node, 2 nodes, the nodes 279 00:19:24,280 --> 00:19:29,530 are symmetrically arranged in the space available. 280 00:19:29,530 --> 00:19:34,120 And the lobes on one side of the node and the other side 281 00:19:34,120 --> 00:19:38,360 have the same amplitude, different sine, 282 00:19:38,360 --> 00:19:40,550 and they're all normalized. 283 00:19:40,550 --> 00:19:43,790 And so the maximum value for each of these guys 284 00:19:43,790 --> 00:19:48,920 is 2 over a square root, where this is 0 to a. 285 00:19:51,950 --> 00:19:54,570 So that's a fantastic simplification, 286 00:19:54,570 --> 00:19:59,060 and it also reminds you of Mr. DeBroglie. 287 00:19:59,060 --> 00:20:01,040 He said, you have to have an integer 288 00:20:01,040 --> 00:20:03,770 number of half wavelengths-- 289 00:20:03,770 --> 00:20:06,140 well, for the hydrogen-- an integer number 290 00:20:06,140 --> 00:20:09,150 of wavelengths around a path. 291 00:20:09,150 --> 00:20:11,220 And for here, you need an integer number 292 00:20:11,220 --> 00:20:12,910 of wavelengths for that round trip 293 00:20:12,910 --> 00:20:14,760 which is the same thing or an integer 294 00:20:14,760 --> 00:20:18,880 number of half wavelengths. 295 00:20:18,880 --> 00:20:22,850 That's DeBroglie's idea, and it enables 296 00:20:22,850 --> 00:20:25,580 you to say, oh well, let's see if we 297 00:20:25,580 --> 00:20:28,640 can use this concept of wavelength 298 00:20:28,640 --> 00:20:30,530 to approach general problems. 299 00:20:38,650 --> 00:20:39,160 OK. 300 00:20:39,160 --> 00:20:45,520 Well, if you do something to the potential 301 00:20:45,520 --> 00:20:52,010 by putting a little thing in it, well, the wave function 302 00:20:52,010 --> 00:20:55,660 will oscillate more slowly in that region, 303 00:20:55,660 --> 00:21:02,960 and that causes it to be at a higher or lower energy? 304 00:21:02,960 --> 00:21:05,520 If it's oscillating more slowly here, 305 00:21:05,520 --> 00:21:09,740 it has to make it to an integer number of half wavelengths, 306 00:21:09,740 --> 00:21:12,170 and so that means it pushes it up. 307 00:21:12,170 --> 00:21:16,120 And if you do this, it'll push it down, 308 00:21:16,120 --> 00:21:18,490 and you can do terrible things. 309 00:21:18,490 --> 00:21:21,280 You can put a delta function there, 310 00:21:21,280 --> 00:21:24,370 and now you know everything qualitatively 311 00:21:24,370 --> 00:21:26,065 that can happen in a 1D box. 312 00:21:30,661 --> 00:21:31,160 OK. 313 00:21:31,160 --> 00:21:33,620 One of the things that bothers people a lot 314 00:21:33,620 --> 00:21:37,930 is, OK, so we have some wave function, 315 00:21:37,930 --> 00:21:42,590 it's got lots of nodes, and the particle starts out over here. 316 00:21:42,590 --> 00:21:45,006 How did it get across the node? 317 00:21:45,006 --> 00:21:46,380 How does it move across the node? 318 00:21:48,980 --> 00:21:51,740 Well, the answer is it's not moving. 319 00:21:51,740 --> 00:21:52,300 It's here. 320 00:21:52,300 --> 00:21:52,690 It's here. 321 00:21:52,690 --> 00:21:53,189 It's here. 322 00:21:53,189 --> 00:21:56,340 It's everywhere, and this is just the probability amplitude. 323 00:21:56,340 --> 00:22:02,200 There is no motion through a node, no motion at all. 324 00:22:02,200 --> 00:22:04,000 We are going to do time-dependent quantum 325 00:22:04,000 --> 00:22:07,150 mechanics before too long, and then there will be motion, 326 00:22:07,150 --> 00:22:11,710 but that motion is encoded in a different way. 327 00:22:14,220 --> 00:22:14,720 OK. 328 00:22:14,720 --> 00:22:21,110 Another thing, suppose you have a particle in a box, 329 00:22:21,110 --> 00:22:24,320 and it's in some state, and I'm going to draw something 330 00:22:24,320 --> 00:22:26,750 like this again. 331 00:22:26,750 --> 00:22:29,300 OK, first of all, one, two, three, four, five, 332 00:22:29,300 --> 00:22:31,730 which state is that? 333 00:22:39,010 --> 00:22:43,081 I got-- the hands are right, six, it's the sixth eigenstate. 334 00:22:43,081 --> 00:22:43,580 OK. 335 00:22:43,580 --> 00:22:45,770 Now, suppose-- nothing is moving. 336 00:22:45,770 --> 00:22:46,610 Right? 337 00:22:46,610 --> 00:22:49,250 This is a stationary state. 338 00:22:49,250 --> 00:22:52,940 How would you experimentally, in principle, 339 00:22:52,940 --> 00:23:00,710 determine that the particle is in this n equals 6 state? 340 00:23:03,840 --> 00:23:07,530 Now, this can be a completely fanciful experiment, which 341 00:23:07,530 --> 00:23:10,740 you would never do, but you could still 342 00:23:10,740 --> 00:23:14,490 describe how you would do it and what it would tell you. 343 00:23:14,490 --> 00:23:17,420 And so, yes. 344 00:23:17,420 --> 00:23:19,580 AUDIENCE: Try to find the n equals 6 to n 345 00:23:19,580 --> 00:23:22,802 equals 7 transition by irradiating it or something? 346 00:23:22,802 --> 00:23:23,510 ROBERT FIELD: OK. 347 00:23:23,510 --> 00:23:27,260 That's the quantum mechanical-- 348 00:23:27,260 --> 00:23:30,320 I agree, spectroscopy wins always. 349 00:23:30,320 --> 00:23:37,780 But if you want to observe the wave function or something 350 00:23:37,780 --> 00:23:42,460 related to the wave function, like the number of nodes, 351 00:23:42,460 --> 00:23:44,270 what would you do? 352 00:23:44,270 --> 00:23:48,310 And the reason I'm being very apologetic about this 353 00:23:48,310 --> 00:23:50,830 is because it's a crazy idea, But this 354 00:23:50,830 --> 00:23:52,575 is a one-dimensional system. 355 00:23:52,575 --> 00:23:53,075 Right? 356 00:23:53,075 --> 00:23:57,260 It's in the blackboard, and so you could stand out here 357 00:23:57,260 --> 00:24:01,580 and shoot particles at it from the perpendicular direction 358 00:24:01,580 --> 00:24:04,070 and collect the number of times you have a hit. 359 00:24:07,600 --> 00:24:11,020 And so you would discover that you 360 00:24:11,020 --> 00:24:16,180 would measure a probability distribution which 361 00:24:16,180 --> 00:24:19,465 had the form-- 362 00:24:25,060 --> 00:24:27,720 well, I can't draw this properly. 363 00:24:27,720 --> 00:24:31,380 It's going to have one, two, three, four, five, six, 364 00:24:31,380 --> 00:24:37,200 six regions separated by a gap, and what it's measuring 365 00:24:37,200 --> 00:24:39,500 is psi 6 squared. 366 00:24:42,020 --> 00:24:47,930 Well, you can't measure, you cannot observe a wave function, 367 00:24:47,930 --> 00:24:51,850 but you can observe a probability distribution wave 368 00:24:51,850 --> 00:24:52,750 function squared. 369 00:24:55,520 --> 00:24:57,840 You can also do a spectroscopic experiment 370 00:24:57,840 --> 00:25:01,184 and find out what is the nature of the Hamiltonian. 371 00:25:01,184 --> 00:25:03,100 And if you know the nature of the Hamiltonian, 372 00:25:03,100 --> 00:25:07,039 you can calculate the wave function, 373 00:25:07,039 --> 00:25:08,080 but you can't observe it. 374 00:25:13,830 --> 00:25:14,330 OK. 375 00:25:14,330 --> 00:25:19,330 Another thing, this harmonic oscillator-- this particle 376 00:25:19,330 --> 00:25:22,784 in a box has a minimum energy which is not 377 00:25:22,784 --> 00:25:23,825 at the bottom of the box. 378 00:25:33,530 --> 00:25:36,355 Well, we have something called the uncertainty principle. 379 00:25:42,300 --> 00:25:44,980 Now, I'm just pulling this out of my pocket, 380 00:25:44,980 --> 00:25:52,450 but I know that x, p is equal to i h bar not 0, 381 00:25:52,450 --> 00:25:57,880 and one can derive some uncertainty principle by doing 382 00:25:57,880 --> 00:25:59,830 a little bit more mathematics. 383 00:25:59,830 --> 00:26:02,860 But basically that uncertainty principle 384 00:26:02,860 --> 00:26:20,350 is where sigma x is expectation value 385 00:26:20,350 --> 00:26:27,270 of x squared minus expectation value of x squared square root. 386 00:26:27,270 --> 00:26:29,800 So if we can calculate this and calculate that, 387 00:26:29,800 --> 00:26:33,750 we can calculate the variance in x, 388 00:26:33,750 --> 00:26:35,600 and you can calculate the variance in p. 389 00:26:35,600 --> 00:26:38,680 That's exact, and that's what you 390 00:26:38,680 --> 00:26:48,130 can derive from the computation rule, but for our purposes, 391 00:26:48,130 --> 00:26:49,510 we can be really crude. 392 00:26:49,510 --> 00:26:55,400 And so if I'm in this state, what is delta x? 393 00:26:59,590 --> 00:27:02,752 What is the range of possibilities for x? 394 00:27:02,752 --> 00:27:04,210 AUDIENCE: The box link? 395 00:27:04,210 --> 00:27:06,700 ROBERT FIELD: Yeah, a. 396 00:27:06,700 --> 00:27:12,800 OK, and what is the possibility-- 397 00:27:12,800 --> 00:27:15,640 what is the uncertainty in p sub x? 398 00:27:24,000 --> 00:27:29,340 In an eigenstate, we've got equal amplitudes going this way 399 00:27:29,340 --> 00:27:30,180 and going that way. 400 00:27:33,770 --> 00:27:44,510 So we could just say p sub x positive minus p sub x negative 401 00:27:44,510 --> 00:27:45,485 which is 2p. 402 00:27:48,550 --> 00:27:51,490 That's the uncertainty. 403 00:27:51,490 --> 00:27:55,120 And if we know what quantum number we're in we 404 00:27:55,120 --> 00:28:00,670 know what the expectation value for p of the momentum is, 405 00:28:00,670 --> 00:28:10,810 and what we derive is that delta x delta P is equal to hn. 406 00:28:15,290 --> 00:28:18,490 You can do that, and maybe I should ask you to really 407 00:28:18,490 --> 00:28:19,490 be sure you can do that. 408 00:28:25,250 --> 00:28:29,620 In seconds, because you really know 409 00:28:29,620 --> 00:28:35,020 what the possible values of momentum or momentum squared 410 00:28:35,020 --> 00:28:36,775 are for a product of a box. 411 00:28:39,811 --> 00:28:40,310 OK. 412 00:28:43,190 --> 00:28:47,690 So why is there zero-point energy, because if you said, 413 00:28:47,690 --> 00:28:51,140 I had a level at the bottom of the box, 414 00:28:51,140 --> 00:28:55,170 we would have the momentum 0. 415 00:28:55,170 --> 00:28:58,370 The uncertainty and the momentum is 0, 416 00:28:58,370 --> 00:29:00,067 and the product of the uncertainty 417 00:29:00,067 --> 00:29:02,150 of the moment times the product of the uncertainty 418 00:29:02,150 --> 00:29:06,540 in the position has to be some finite number, 419 00:29:06,540 --> 00:29:08,580 and you can't do that here. 420 00:29:08,580 --> 00:29:11,430 And so this is a simple illustration of the uncertainty 421 00:29:11,430 --> 00:29:16,820 principle that you have to have a non-zero zero-point energy. 422 00:29:16,820 --> 00:29:18,770 That's true for all one-dimensional problems. 423 00:29:23,290 --> 00:29:24,096 OK. 424 00:29:24,096 --> 00:29:25,680 We've got lots of time. 425 00:29:33,830 --> 00:29:37,620 One of the beautiful things about quantum mechanics 426 00:29:37,620 --> 00:29:41,990 is that if you solved one problem, 427 00:29:41,990 --> 00:29:44,110 you could solve a whole bunch of problems, 428 00:29:44,110 --> 00:29:47,950 and so to illustrate that, let's consider 429 00:29:47,950 --> 00:29:50,893 the 3D particle in a box. 430 00:29:53,660 --> 00:29:58,760 So for the 3D particle in a box, the Hamiltonian 431 00:29:58,760 --> 00:30:03,050 can be written as a little Hamiltonian for the x degree 432 00:30:03,050 --> 00:30:06,960 of freedom y and z. 433 00:30:06,960 --> 00:30:08,270 OK. 434 00:30:08,270 --> 00:30:12,950 So we have three independent motions of the particle. 435 00:30:12,950 --> 00:30:15,590 They're not coupled. 436 00:30:15,590 --> 00:30:18,604 They could be, and we're interested in letting 437 00:30:18,604 --> 00:30:19,270 them be coupled. 438 00:30:19,270 --> 00:30:24,210 But that's where we start asking questions about reality, 439 00:30:24,210 --> 00:30:27,120 and that's where we bring in perturbation theory. 440 00:30:27,120 --> 00:30:30,270 But for this, oh, that's fantastic, 441 00:30:30,270 --> 00:30:33,460 because I know the eigenvalues of this operator 442 00:30:33,460 --> 00:30:36,290 and of this operator, eigenfunctions, 443 00:30:36,290 --> 00:30:37,800 and of this operator. 444 00:30:37,800 --> 00:30:39,480 So the problem is basically solved 445 00:30:39,480 --> 00:30:42,840 once you solve the 1D box. 446 00:30:42,840 --> 00:30:49,240 Now, one proviso, what you can do this separation completely 447 00:30:49,240 --> 00:30:59,140 formally as long as hx, hy commute, 448 00:30:59,140 --> 00:31:02,000 and basically we say the x, y, and z directions 449 00:31:02,000 --> 00:31:06,740 don't interact with each other. 450 00:31:06,740 --> 00:31:08,940 The particle is free inside the box. 451 00:31:08,940 --> 00:31:10,520 It's just encountering walls. 452 00:31:10,520 --> 00:31:16,490 There are no springs or anything expressing the number 453 00:31:16,490 --> 00:31:17,960 of degrees of freedom. 454 00:31:17,960 --> 00:31:19,670 OK. 455 00:31:19,670 --> 00:31:24,380 So we have now a wave function, which is a function of x, y, 456 00:31:24,380 --> 00:31:26,750 and z, but we can always write it 457 00:31:26,750 --> 00:31:36,890 as psi x of x, psi y of y, psi z of z. 458 00:31:36,890 --> 00:31:41,490 So it's a product of three wave functions that we know, 459 00:31:41,490 --> 00:31:47,920 and the energy is going to be expressed 460 00:31:47,920 --> 00:32:25,400 as a function of three quantum numbers, where the box is 461 00:32:25,400 --> 00:32:26,840 edge lengths a, b, and c. 462 00:32:30,110 --> 00:32:32,850 You didn't see me looking at my notes. 463 00:32:32,850 --> 00:32:38,380 I'm just taking the solution to 1D box, and I'm multiplying it. 464 00:32:38,380 --> 00:32:43,970 And so now we have the particle in a 3D box, 465 00:32:43,970 --> 00:32:49,020 and this is where the ideal gas law comes from, but not 466 00:32:49,020 --> 00:32:51,090 in this course. 467 00:32:51,090 --> 00:32:53,480 So anyway, this is a simple thing, 468 00:32:53,480 --> 00:32:55,850 and the wave functions are simple as well, 469 00:32:55,850 --> 00:32:59,600 and you can do all these fantastic things. 470 00:33:05,740 --> 00:33:11,220 So there are many problems like a polyatomic molecule. 471 00:33:11,220 --> 00:33:15,010 In a polyatomic molecule, if you have n atoms, 472 00:33:15,010 --> 00:33:20,280 you have 3n minus 6 vibrational modes. 473 00:33:20,280 --> 00:33:23,250 You might ask, what is a vibrational mode? 474 00:33:23,250 --> 00:33:26,640 Well, are they're independent motions 475 00:33:26,640 --> 00:33:33,210 of the atoms that satisfy the harmonic oscillator 476 00:33:33,210 --> 00:33:37,410 Hamiltonian, which we'll come to next time. 477 00:33:37,410 --> 00:33:44,490 And so we have 3n minus 6 exactly solved problems 478 00:33:44,490 --> 00:33:48,540 all cohabiting in one Hamiltonian. 479 00:33:48,540 --> 00:33:51,300 And then we can say, oh yeah, we got these oscillators, 480 00:33:51,300 --> 00:33:53,990 and if I stretch a particular bond, 481 00:33:53,990 --> 00:33:58,680 it might affect the force constant for the bending. 482 00:33:58,680 --> 00:34:03,720 So we can introduce couplings between the oscillators, 483 00:34:03,720 --> 00:34:06,320 and in fact, that's what we do with perturbation theory. 484 00:34:06,320 --> 00:34:08,030 That's the whole purpose. 485 00:34:08,030 --> 00:34:12,110 And with that, we can describe both the spectrum 486 00:34:12,110 --> 00:34:16,370 and how the spectrum encodes the couplings between the modes. 487 00:34:16,370 --> 00:34:18,800 And also, we can describe what's called intramolecular 488 00:34:18,800 --> 00:34:21,500 vibrational redistribution, which 489 00:34:21,500 --> 00:34:24,210 happens when you have a very high density 490 00:34:24,210 --> 00:34:25,760 of vibrational state. 491 00:34:25,760 --> 00:34:29,210 Energy moves around, because all the modes are coupled, 492 00:34:29,210 --> 00:34:31,610 and so even if you've plucked one, 493 00:34:31,610 --> 00:34:33,320 the excitation would go to others. 494 00:34:33,320 --> 00:34:37,340 And we can understand that all using the same formalism 495 00:34:37,340 --> 00:34:39,170 that we're about to develop. 496 00:34:52,070 --> 00:34:54,670 All right. 497 00:34:54,670 --> 00:34:56,760 I'm not using my notes this time, 498 00:34:56,760 --> 00:34:59,600 because I think there's just so much insight, 499 00:34:59,600 --> 00:35:02,105 so I have to keep checking to see what I've skipped. 500 00:35:06,881 --> 00:35:07,380 All right. 501 00:35:07,380 --> 00:35:21,510 So what I've been saying is whenever the Hamiltonian can 502 00:35:21,510 --> 00:35:32,500 be expressed as a sum of individual Hamiltonians, 503 00:35:32,500 --> 00:35:36,790 whenever we can write the Hamiltonian this way, 504 00:35:36,790 --> 00:35:45,790 we can write the wave function as a product of wave 505 00:35:45,790 --> 00:35:59,870 functions for coordinates, xi i1, 2 N. 506 00:35:59,870 --> 00:36:11,660 And the energies will be the sum Ein, 507 00:36:11,660 --> 00:36:16,085 i equals little ei n sub i. 508 00:36:16,085 --> 00:36:18,740 I equals 1 to N. 509 00:36:18,740 --> 00:36:19,970 So this is really easy. 510 00:36:19,970 --> 00:36:26,780 If we have simply a Hamiltonian, which 511 00:36:26,780 --> 00:36:30,500 is a sum of individual particle Hamiltonians, 512 00:36:30,500 --> 00:36:32,330 we don't even have to stop to think. 513 00:36:32,330 --> 00:36:34,663 We know the wave functions and the eigenvalues. 514 00:36:38,310 --> 00:36:38,810 OK. 515 00:36:44,370 --> 00:36:56,140 Now, suppose the Hamiltonian is this plus that. 516 00:36:56,140 --> 00:37:00,370 So here, we have a Hamiltonian, and this is this simply the 517 00:37:00,370 --> 00:37:01,750 uncoupled Hamiltonian. 518 00:37:01,750 --> 00:37:06,340 This is what we'd like nature to be, but nature isn't so kind, 519 00:37:06,340 --> 00:37:08,770 and there are some coupling terms. 520 00:37:08,770 --> 00:37:13,960 And so we know the eigenfunctions and eigenvalues 521 00:37:13,960 --> 00:37:15,100 for this Hamiltonian. 522 00:37:15,100 --> 00:37:20,830 We call them the basis functions and the zero-order energies, 523 00:37:20,830 --> 00:37:22,390 and then there is this thing that 524 00:37:22,390 --> 00:37:24,909 couples them and leads to complications, 525 00:37:24,909 --> 00:37:26,200 and that's perturbation theory. 526 00:37:26,200 --> 00:37:27,730 We're going to do that. 527 00:37:27,730 --> 00:37:28,830 OK. 528 00:37:28,830 --> 00:37:40,850 So now, let me just say, on page nine of your notes, 529 00:37:40,850 --> 00:37:48,420 there's the words next time, and those are going 530 00:37:48,420 --> 00:37:50,495 to be replaced by you should. 531 00:37:54,930 --> 00:37:57,790 There's a whole bunch of things that I want you to consider, 532 00:37:57,790 --> 00:38:02,130 and I was planning on talking about them, 533 00:38:02,130 --> 00:38:03,434 but they're all pretty trivial. 534 00:38:03,434 --> 00:38:05,100 And so there are a whole bunch of things 535 00:38:05,100 --> 00:38:10,697 you should study, because I will ask you questions about them. 536 00:38:13,620 --> 00:38:18,000 And of greatest importance is the ability 537 00:38:18,000 --> 00:38:21,351 to calculate things like that. 538 00:38:25,050 --> 00:38:25,550 OK. 539 00:38:25,550 --> 00:38:29,810 Now, I'm going to give you a whole bunch of facts 540 00:38:29,810 --> 00:38:32,630 which I may not have derived. 541 00:38:32,630 --> 00:38:37,620 But you're going to live with them, 542 00:38:37,620 --> 00:38:39,870 and you can ask me questions. 543 00:38:39,870 --> 00:38:42,300 Some of these things are theorems that we can prove, 544 00:38:42,300 --> 00:38:45,930 but the proof of the theorem is really boring. 545 00:38:45,930 --> 00:38:49,330 Understanding what it is is really wonderful. 546 00:38:49,330 --> 00:39:01,070 So all eigenfunctions that belong 547 00:39:01,070 --> 00:39:05,090 to different eigenvalues-- 548 00:39:09,440 --> 00:39:13,930 of whatever operator we want, the Hamiltonian, 549 00:39:13,930 --> 00:39:17,620 some other operator-- 550 00:39:17,620 --> 00:39:20,155 are orthogonal. 551 00:39:23,340 --> 00:39:27,060 That's a fantastic simplification. 552 00:39:27,060 --> 00:39:29,310 So if you have two eigenfunctions 553 00:39:29,310 --> 00:39:36,390 of the Hamiltonian, of the position operator, anything, 554 00:39:36,390 --> 00:39:39,730 those eigenfunctions are orthogonal. 555 00:39:39,730 --> 00:39:46,450 Their integral is 0, very, very useful. 556 00:39:46,450 --> 00:39:53,410 Then, one of the initial postulates 557 00:39:53,410 --> 00:39:57,670 about quantum mechanics is this idea 558 00:39:57,670 --> 00:40:00,810 that the wave functions are well-behaved. 559 00:40:04,310 --> 00:40:08,870 Well, if I were to state it at the beginning, 560 00:40:08,870 --> 00:40:11,980 you wouldn't know what's well-behaved and ill-behaved, 561 00:40:11,980 --> 00:40:12,950 but now I can tell you. 562 00:40:20,240 --> 00:40:23,360 One of the things is that the wave function is continuous, 563 00:40:23,360 --> 00:40:25,000 no matter what the potential does. 564 00:40:27,680 --> 00:40:39,640 The derivative is continuous, except at an infinite barrier. 565 00:40:39,640 --> 00:40:42,070 So you come along, and you hit an infinite barrier, 566 00:40:42,070 --> 00:40:49,400 and you've already seen that with the particle in a box. 567 00:40:49,400 --> 00:40:52,330 The wave function is continuous at the edge of the box, 568 00:40:52,330 --> 00:40:54,420 but the derivative is discontinuous, 569 00:40:54,420 --> 00:40:58,830 and it's because it's an infinite wall. 570 00:40:58,830 --> 00:41:01,950 That's a pretty violent thing to make the first derivative be 571 00:41:01,950 --> 00:41:03,360 discontinuous. 572 00:41:03,360 --> 00:41:09,080 The secondary derivative is continuous, 573 00:41:09,080 --> 00:41:16,424 except at any sudden change in the potential. 574 00:41:20,140 --> 00:41:24,480 So when you're solving 1D problems, 575 00:41:24,480 --> 00:41:28,830 and you've got a solution that works in the various regions, 576 00:41:28,830 --> 00:41:31,440 and you want to connect them together, 577 00:41:31,440 --> 00:41:35,790 these provide some rules about the boundary conditions. 578 00:41:40,150 --> 00:41:46,780 So now, most real systems don't have infinite walls 579 00:41:46,780 --> 00:41:52,020 or infinitely sharp steps. 580 00:41:52,020 --> 00:41:56,164 So for calculation of physically reasonable things, 581 00:41:56,164 --> 00:41:58,580 wave function for the first derivative, second derivative, 582 00:41:58,580 --> 00:42:00,080 they are continuous. 583 00:42:00,080 --> 00:42:05,340 But for solving a problem, we like these steps, 584 00:42:05,340 --> 00:42:09,210 because then we know how to impose boundary conditions, 585 00:42:09,210 --> 00:42:14,420 and that gets us a much easier thing to calculate. 586 00:42:14,420 --> 00:42:15,020 OK. 587 00:42:15,020 --> 00:42:20,070 Now-- oh, that's where it went, OK-- 588 00:42:26,680 --> 00:42:33,200 semi classical quantum mechanics. 589 00:42:39,360 --> 00:42:42,420 We know that the energy classically 590 00:42:42,420 --> 00:42:44,646 is p squared over 2m. 591 00:42:44,646 --> 00:42:47,030 Right? 592 00:42:47,030 --> 00:42:48,980 It's 1/2 mv squared, but that's the same thing 593 00:42:48,980 --> 00:42:50,450 as p squared over 2m. 594 00:42:50,450 --> 00:42:53,340 In quantum mechanics, when we talk about Hamiltonians, 595 00:42:53,340 --> 00:42:58,180 the variables are x and p not x and v. 596 00:42:58,180 --> 00:43:01,820 So that seems like a picky thing, 597 00:43:01,820 --> 00:43:04,280 but it turns out to be very important. 598 00:43:04,280 --> 00:43:13,680 And so we can say, well, the momentum 599 00:43:13,680 --> 00:43:17,344 can be a function of x, classically. 600 00:43:24,610 --> 00:43:32,950 So I just solved this problem, and if the potential is not 601 00:43:32,950 --> 00:43:37,480 constant, then the momentum, classical momentum, 602 00:43:37,480 --> 00:43:39,000 is not constant. 603 00:43:39,000 --> 00:43:43,420 But we know what it is everywhere, 604 00:43:43,420 --> 00:43:50,880 and we also know that the wavelength is 605 00:43:50,880 --> 00:43:52,290 equal to h over p. 606 00:43:54,970 --> 00:43:59,680 So we could make a step into the unknown saying, 607 00:43:59,680 --> 00:44:03,280 well, the wavelength for a non-constant potential 608 00:44:03,280 --> 00:44:05,860 is a function of coordinate, and it's 609 00:44:05,860 --> 00:44:10,370 going to be equal to h over p of x. 610 00:44:10,370 --> 00:44:14,180 That's semi-classical quantum mechanics, everything 611 00:44:14,180 --> 00:44:16,230 you would possibly want. 612 00:44:16,230 --> 00:44:18,110 Now, for one-dimensional problems, 613 00:44:18,110 --> 00:44:25,960 you can solve in terms of this coordinate dependent wavelength 614 00:44:25,960 --> 00:44:29,900 which is related to the local momentum. 615 00:44:29,900 --> 00:44:34,340 And so it doesn't matter how complicated the problem is, 616 00:44:34,340 --> 00:44:37,610 you know that you can calculate the spatial modulation 617 00:44:37,610 --> 00:44:40,790 frequency, you could calculate the amplitude, 618 00:44:40,790 --> 00:44:45,860 is it big or small, based on these ideas 619 00:44:45,860 --> 00:44:50,090 of the classical momentum function. 620 00:44:50,090 --> 00:44:52,670 So this demonstration of my walking 621 00:44:52,670 --> 00:44:59,060 across the room slow, fast, slow tells you about probability. 622 00:44:59,060 --> 00:45:05,390 So if you use this formula, you know the node spacings, 623 00:45:05,390 --> 00:45:09,340 and you know the amplitudes. 624 00:45:09,340 --> 00:45:13,770 Now, what you don't know is where are the nodes? 625 00:45:13,770 --> 00:45:17,520 You know how far they are apart, but I 626 00:45:17,520 --> 00:45:19,230 have to be humble about this. 627 00:45:19,230 --> 00:45:21,030 In order to calculate where the nodes are, 628 00:45:21,030 --> 00:45:25,980 I have to do a little bit more in order to pin them down. 629 00:45:25,980 --> 00:45:28,800 But mostly, when you're trying to understand how something 630 00:45:28,800 --> 00:45:33,350 works, you want to know the amplitude of the envelope, 631 00:45:33,350 --> 00:45:36,180 and that's a probability, and so it's 632 00:45:36,180 --> 00:45:40,320 related to 1 over the square root of the momentum. 633 00:45:40,320 --> 00:45:42,540 I'm sorry, the amplitude is 1 over the square root 634 00:45:42,540 --> 00:45:44,371 of the momentum, and the nodes spacings, 635 00:45:44,371 --> 00:45:45,870 those are the things you want to do, 636 00:45:45,870 --> 00:45:47,775 and you want to know them immediately. 637 00:45:53,230 --> 00:45:56,460 And a couple other facts that I told you earlier, but I think I 638 00:45:56,460 --> 00:45:58,360 want to emphasize them-- 639 00:45:58,360 --> 00:46:08,920 the energy order, number of nodes, 640 00:46:08,920 --> 00:46:10,090 number of internal nodes. 641 00:46:14,200 --> 00:46:17,410 For 1D problems, you never skip a number of-- 642 00:46:17,410 --> 00:46:22,626 so you can't say, there is no wave function with 13 nodes, 643 00:46:22,626 --> 00:46:26,310 even if you don't like being unlucky. 644 00:46:26,310 --> 00:46:33,470 And it's there, and so if you want the 13th energy level, 645 00:46:33,470 --> 00:46:38,170 you want something with 12 nodes, 646 00:46:38,170 --> 00:46:40,160 and that also focuses things. 647 00:46:40,160 --> 00:46:42,800 So these are amazingly wonderful things, 648 00:46:42,800 --> 00:46:44,930 because you can get them from what 649 00:46:44,930 --> 00:46:48,280 you know about classical mechanics, 650 00:46:48,280 --> 00:46:53,750 and it's easy to embed them into a kind of half quantum 651 00:46:53,750 --> 00:46:56,080 mechanics. 652 00:46:56,080 --> 00:46:58,360 And since, I told you, this course 653 00:46:58,360 --> 00:47:03,340 is for use and insight, not admiration of philosophy 654 00:47:03,340 --> 00:47:07,160 or historical development, and this is what you want to do. 655 00:47:07,160 --> 00:47:09,580 You look at the problem and sketch 656 00:47:09,580 --> 00:47:12,460 how is the wave function going to behave 657 00:47:12,460 --> 00:47:15,010 and perhaps how a particular thing 658 00:47:15,010 --> 00:47:16,630 at some place in the potential is 659 00:47:16,630 --> 00:47:19,840 going to affect the energy levels or any other observable 660 00:47:19,840 --> 00:47:21,080 property. 661 00:47:21,080 --> 00:47:25,660 And so the cartoons are really your guide to getting things 662 00:47:25,660 --> 00:47:28,770 right, but you really have to invest 663 00:47:28,770 --> 00:47:35,491 in developing the sense of how to build these cartoons. 664 00:47:35,491 --> 00:47:35,990 OK. 665 00:47:35,990 --> 00:47:37,830 I'm finished early again. 666 00:47:37,830 --> 00:47:39,260 Does anybody have any questions? 667 00:47:42,020 --> 00:47:42,520 OK. 668 00:47:42,520 --> 00:47:46,190 We start the harmonic oscillator next time. 669 00:47:46,190 --> 00:47:48,870 OK, so I can say a couple of things. 670 00:47:48,870 --> 00:47:52,320 The wave functions, the solution to the 1D 671 00:47:52,320 --> 00:47:56,570 box and the free particle, they're really simple. 672 00:47:56,570 --> 00:47:58,280 The solution to the harmonic oscillator 673 00:47:58,280 --> 00:48:01,740 involves a complicated differential equation 674 00:48:01,740 --> 00:48:04,385 which the mathematicians have solved and worked 675 00:48:04,385 --> 00:48:07,210 out all the properties. 676 00:48:07,210 --> 00:48:12,610 But there is a really important simplification 677 00:48:12,610 --> 00:48:17,590 that enables us to proceed with even greater velocity 678 00:48:17,590 --> 00:48:19,120 in the harmonic oscillator than we 679 00:48:19,120 --> 00:48:20,920 would in a particle in a box. 680 00:48:20,920 --> 00:48:26,320 And they are these things called a and a 681 00:48:26,320 --> 00:48:30,370 dagger, creation and annihilation operators, where 682 00:48:30,370 --> 00:48:37,450 when we operate on psi with this creation operator, 683 00:48:37,450 --> 00:48:48,460 it converts it to square root of v plus 1, psi v plus 1 further. 684 00:48:48,460 --> 00:48:51,520 These things, we don't ever have to do an integral. 685 00:48:51,520 --> 00:48:53,530 Once you're in harmonic oscillator land, 686 00:48:53,530 --> 00:49:00,490 everything you need comes from these wonderful operators. 687 00:49:00,490 --> 00:49:02,650 And so even though the differential equation 688 00:49:02,650 --> 00:49:07,070 is a little bit scary for chemists, 689 00:49:07,070 --> 00:49:10,200 these things make everything trivial. 690 00:49:10,200 --> 00:49:14,460 And so we use the harmonic oscillator, 691 00:49:14,460 --> 00:49:16,910 and the particle in a box to illustrate 692 00:49:16,910 --> 00:49:19,440 time-dependent quantum mechanics. 693 00:49:19,440 --> 00:49:24,840 They each have their own special advantages for simplifications, 694 00:49:24,840 --> 00:49:30,390 but it's wonderful, because we can use something we barely 695 00:49:30,390 --> 00:49:32,550 understand for the first time. 696 00:49:32,550 --> 00:49:34,590 And actually reach that level of, yeah, I 697 00:49:34,590 --> 00:49:38,030 can understand macroscopic behaviors too 698 00:49:38,030 --> 00:49:42,180 and how they relate to quantum mechanical behavior 699 00:49:42,180 --> 00:49:43,760 of simple systems. 700 00:49:43,760 --> 00:49:45,115 OK, so that's where we're going. 701 00:49:45,115 --> 00:49:46,823 We're going to have two or three lectures 702 00:49:46,823 --> 00:49:49,460 on harmonic oscillator.