1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high-quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,260 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,260 --> 00:00:18,220 at ocw.mit.edu. 8 00:00:21,862 --> 00:00:25,490 ROBERT FIELD: OK, let's get started. 9 00:00:25,490 --> 00:00:34,730 So last time we talked about the one-dimensional wave equation, 10 00:00:34,730 --> 00:00:40,710 which is a second-order partial differential equation. 11 00:00:40,710 --> 00:00:44,220 This is not a math course. 12 00:00:44,220 --> 00:00:47,220 If you have a second-order differential equation, 13 00:00:47,220 --> 00:00:52,940 there will be two linearly independent solutions to it. 14 00:00:52,940 --> 00:00:56,100 And that's important to remember. 15 00:00:56,100 --> 00:01:00,660 Now, there are three steps that we 16 00:01:00,660 --> 00:01:03,727 use in approaching a problem like this. 17 00:01:03,727 --> 00:01:06,060 Does anybody want to tell me what those three steps are? 18 00:01:15,660 --> 00:01:16,415 Yes? 19 00:01:16,415 --> 00:01:20,750 AUDIENCE: [INAUDIBLE] it has separate functions that 20 00:01:20,750 --> 00:01:22,590 take only one variable. 21 00:01:22,590 --> 00:01:25,770 ROBERT FIELD: OK, that's most of it. 22 00:01:25,770 --> 00:01:28,530 You want to solve the general equation, 23 00:01:28,530 --> 00:01:31,560 and one way to solve the general equation 24 00:01:31,560 --> 00:01:33,960 is to try to separate variables. 25 00:01:33,960 --> 00:01:36,390 Always you want to separate variables. 26 00:01:36,390 --> 00:01:39,750 Even if it's not quite legal, you 27 00:01:39,750 --> 00:01:43,560 want to find a way to do that because that breaks the problem 28 00:01:43,560 --> 00:01:45,670 down in a very useful way. 29 00:01:45,670 --> 00:01:53,270 So the first step is the general solution, 30 00:01:53,270 --> 00:02:01,650 and it involves trying something like this 31 00:02:01,650 --> 00:02:06,630 where we say u of x and t is going to be 32 00:02:06,630 --> 00:02:09,060 treated in the separable form. 33 00:02:09,060 --> 00:02:10,949 If it doesn't work, you're going to get 0. 34 00:02:10,949 --> 00:02:14,910 You're going to get the solution. 35 00:02:14,910 --> 00:02:20,980 The only solution with separated variables is nothing happening. 36 00:02:20,980 --> 00:02:23,100 And so that's unfortunate if you do work 37 00:02:23,100 --> 00:02:26,400 and you get nothing for it, but life is complicated. 38 00:02:26,400 --> 00:02:30,060 So then after you do the general solution, what's next? 39 00:02:32,760 --> 00:02:33,260 Yes? 40 00:02:33,260 --> 00:02:34,706 AUDIENCE: You need to set boundary conditions? 41 00:02:34,706 --> 00:02:35,456 ROBERT FIELD: Yes. 42 00:02:44,350 --> 00:02:50,230 Now if it's a second-order differential equation, 43 00:02:50,230 --> 00:02:52,180 you're going to need two boundary conditions. 44 00:02:54,800 --> 00:03:00,930 And when you impose two boundary conditions, 45 00:03:00,930 --> 00:03:05,040 the second one gives some sort of quantization. 46 00:03:05,040 --> 00:03:09,120 It makes the solutions discrete or that there is a discrete set 47 00:03:09,120 --> 00:03:11,250 rather than a continuous set. 48 00:03:11,250 --> 00:03:16,650 The general solution is more or less continuous, or continuous 49 00:03:16,650 --> 00:03:17,860 possibility. 50 00:03:17,860 --> 00:03:20,700 So we have now the boundary conditions, 51 00:03:20,700 --> 00:03:26,910 and that gives us something that we can start to visualize. 52 00:03:26,910 --> 00:03:29,540 So what are the important things that, if you're 53 00:03:29,540 --> 00:03:34,040 going to be drawing pictures rather than actually plotting 54 00:03:34,040 --> 00:03:37,600 some complicated mathematical function, what are 55 00:03:37,600 --> 00:03:39,170 the first questions you ask? 56 00:03:44,260 --> 00:03:44,820 Yes? 57 00:03:44,820 --> 00:03:48,326 AUDIENCE: Is it symmetric or asymmetric? 58 00:03:52,990 --> 00:03:56,680 ROBERT FIELD: Yes, if the problem has symmetry, 59 00:03:56,680 --> 00:03:58,075 the solutions will have symmetry. 60 00:04:01,940 --> 00:04:02,910 But there's more. 61 00:04:02,910 --> 00:04:04,860 I want another, my favorite. 62 00:04:04,860 --> 00:04:06,530 AUDIENCE: Where's it start? 63 00:04:06,530 --> 00:04:06,940 ROBERT FIELD: I'm sorry? 64 00:04:06,940 --> 00:04:08,190 AUDIENCE: Where does it start? 65 00:04:08,190 --> 00:04:11,730 So t equals 0 or x equals 0. 66 00:04:11,730 --> 00:04:14,940 ROBERT FIELD: The initial condition is really 67 00:04:14,940 --> 00:04:18,423 the next thing I'm going to ask you about, 68 00:04:18,423 --> 00:04:19,589 and that's called the pluck. 69 00:04:22,410 --> 00:04:23,870 You're right on target. 70 00:04:23,870 --> 00:04:26,310 But now if you're going to draw a picture, 71 00:04:26,310 --> 00:04:28,650 the best thing you want to do to draw a picture 72 00:04:28,650 --> 00:04:32,360 is have it not move. 73 00:04:32,360 --> 00:04:40,480 So you want to look at the thing in position, 74 00:04:40,480 --> 00:04:45,640 and what are the things about the position function 75 00:04:45,640 --> 00:04:50,460 that you can immediately figure out and use 76 00:04:50,460 --> 00:04:51,855 in drawing a cartoon? 77 00:04:57,593 --> 00:04:58,590 AUDIENCE: Nodes. 78 00:04:58,590 --> 00:04:59,340 ROBERT FIELD: Yep. 79 00:05:04,320 --> 00:05:08,840 So how many nodes and where are they? 80 00:05:08,840 --> 00:05:11,720 Are they equally spaced? 81 00:05:11,720 --> 00:05:16,710 And that's the most important thing in drawing a picture, 82 00:05:16,710 --> 00:05:19,990 the number of zero crossings that a wave function has, 83 00:05:19,990 --> 00:05:24,430 and how are they distributed? 84 00:05:24,430 --> 00:05:29,750 Their spacing of nodes is half the wavelength, 85 00:05:29,750 --> 00:05:31,625 and the wavelength is related to momentum. 86 00:05:34,460 --> 00:05:37,260 And so I'm jumping into quantum mechanics, 87 00:05:37,260 --> 00:05:41,450 but it's still valid for understanding the wave 88 00:05:41,450 --> 00:05:42,450 equation. 89 00:05:42,450 --> 00:05:50,270 So we want the number of nodes for each specific solution 90 00:05:50,270 --> 00:05:54,610 satisfying the boundary conditions and the spacing 91 00:05:54,610 --> 00:05:59,600 and the loops between nodes. 92 00:05:59,600 --> 00:06:02,010 Are they all identical? 93 00:06:02,010 --> 00:06:04,760 Is there some systematic variation 94 00:06:04,760 --> 00:06:11,090 in the magnitude of each of the loops between nodes? 95 00:06:11,090 --> 00:06:15,920 Because if you have just a qualitative sense for how this 96 00:06:15,920 --> 00:06:19,740 works, you can draw the wave function, 97 00:06:19,740 --> 00:06:23,950 and you can begin to make conclusions about it. 98 00:06:23,950 --> 00:06:25,850 But it all starts with nodes. 99 00:06:25,850 --> 00:06:27,780 Nodes are really important. 100 00:06:27,780 --> 00:06:33,260 And quantum mechanics, the wave functions have nodes. 101 00:06:33,260 --> 00:06:37,650 You can't do better than focus on the nodes. 102 00:06:37,650 --> 00:06:42,240 And then I like to call it pluck, 103 00:06:42,240 --> 00:06:52,270 but it's a superposition of eigenstates. 104 00:06:52,270 --> 00:06:57,070 And those superpositions for this problem can move. 105 00:07:00,510 --> 00:07:06,190 And so-- you're missing a great-- 106 00:07:06,190 --> 00:07:07,360 anyway. 107 00:07:07,360 --> 00:07:11,890 And so you want to know, what is the kind of motion 108 00:07:11,890 --> 00:07:14,270 that this thing can have? 109 00:07:14,270 --> 00:07:17,020 And so one of the things-- 110 00:07:17,020 --> 00:07:19,840 this is really the three steps you go through 111 00:07:19,840 --> 00:07:22,540 in order to make a picture in which you hang up 112 00:07:22,540 --> 00:07:24,070 your insights. 113 00:07:24,070 --> 00:07:35,900 And so if there's one state or two states, 114 00:07:35,900 --> 00:07:40,370 one state is just going to be standing waves. 115 00:07:40,370 --> 00:07:45,740 Two states is all the complexity you're ever going to need. 116 00:07:45,740 --> 00:07:49,630 And if two states have different frequencies, 117 00:07:49,630 --> 00:07:52,030 there will be motion, and the motion 118 00:07:52,030 --> 00:07:57,550 can be side-to-side motion or sort of breathing motion 119 00:07:57,550 --> 00:08:01,790 where amplitude moves in from the turning-point region 120 00:08:01,790 --> 00:08:05,000 to the middle and back out again. 121 00:08:05,000 --> 00:08:09,040 And so you can be able to classify 122 00:08:09,040 --> 00:08:15,670 what you can understand and to imagine doing experiments 123 00:08:15,670 --> 00:08:19,975 based on this simplified version of the flux. 124 00:08:22,980 --> 00:08:25,200 In my opinion, the most important thing 125 00:08:25,200 --> 00:08:30,480 you can do as a professional quantum machinist 126 00:08:30,480 --> 00:08:33,299 and in preparation for exams is to be 127 00:08:33,299 --> 00:08:37,595 able to draw these cartoons quickly, really quickly. 128 00:08:37,595 --> 00:08:39,719 That means you have to think about them in advance. 129 00:08:44,460 --> 00:08:48,570 And so this recipe is how you're going 130 00:08:48,570 --> 00:08:52,880 to understand quantum mechanical problems too. 131 00:08:52,880 --> 00:08:55,457 And this differential equation is actually 132 00:08:55,457 --> 00:08:57,290 a little more complicated than the first few 133 00:08:57,290 --> 00:09:01,220 that we're going to encounter because the first few problems 134 00:09:01,220 --> 00:09:04,590 we're going to face are not time dependent. 135 00:09:04,590 --> 00:09:08,590 There may still be a separation of variables situation 136 00:09:08,590 --> 00:09:11,620 and imposing boundary conditions and so on, 137 00:09:11,620 --> 00:09:15,880 but there is no motion. 138 00:09:15,880 --> 00:09:18,970 But eventually we'll get motion because our real world has 139 00:09:18,970 --> 00:09:22,450 motion, and quantum mechanics has to reproduce everything 140 00:09:22,450 --> 00:09:24,010 that our real world does. 141 00:09:26,680 --> 00:09:30,470 So, we're going to begin quantum mechanics, 142 00:09:30,470 --> 00:09:33,860 and first of all I will describe some 143 00:09:33,860 --> 00:09:37,020 of the rules we have to obey in building 144 00:09:37,020 --> 00:09:39,440 a quantum-mechanical picture. 145 00:09:39,440 --> 00:09:42,470 And then I'll approach two of the easiest 146 00:09:42,470 --> 00:09:46,310 problems, the free particle and the particle 147 00:09:46,310 --> 00:09:47,230 in an infinite box. 148 00:10:05,730 --> 00:10:14,450 So we have the one-dimensional Schrodinger equation, 149 00:10:14,450 --> 00:10:17,540 and the one-dimensional Schrodinger equation looks 150 00:10:17,540 --> 00:10:19,340 like the wave equation. 151 00:10:19,340 --> 00:10:20,750 And why? 152 00:10:20,750 --> 00:10:25,700 Because waves interfere with each other. 153 00:10:25,700 --> 00:10:28,880 We can have constructive and destructive interference. 154 00:10:28,880 --> 00:10:33,620 Almost everything that is wonderful about quantum 155 00:10:33,620 --> 00:10:38,390 mechanics is the solutions to this Schrodinger equation 156 00:10:38,390 --> 00:10:42,780 also exhibit constructive and destructive interference. 157 00:10:42,780 --> 00:10:46,370 And that's essential to our understanding of how quantum 158 00:10:46,370 --> 00:10:50,150 mechanics describes the world. 159 00:10:50,150 --> 00:10:52,250 The next thing I want to do is talk a little bit 160 00:10:52,250 --> 00:10:53,160 about postulates. 161 00:10:56,780 --> 00:11:00,410 Now I'm going to be introducing the quantum-mechanic postulates 162 00:11:00,410 --> 00:11:02,750 as we need them as opposed to just 163 00:11:02,750 --> 00:11:08,510 a dry lecture of these strange and wonderful things 164 00:11:08,510 --> 00:11:11,630 before we're ready for them. 165 00:11:11,630 --> 00:11:15,520 But a postulate is something that can't be proven right. 166 00:11:15,520 --> 00:11:18,140 It can be proven wrong. 167 00:11:18,140 --> 00:11:24,610 And we build a system of logic based on these postulates. 168 00:11:24,610 --> 00:11:27,490 Now one of the great experiences in my life 169 00:11:27,490 --> 00:11:33,760 was one time when I visited the Exploratorium in San Francisco 170 00:11:33,760 --> 00:11:37,510 where there are rather crude, or at least when I visited 171 00:11:37,510 --> 00:11:40,300 almost 50 years ago, there were rather 172 00:11:40,300 --> 00:11:42,940 crude interactive experiments where 173 00:11:42,940 --> 00:11:47,410 people can turn knobs and push buttons and make things happen. 174 00:11:47,410 --> 00:11:52,300 And the most wonderful thing was really young kids 175 00:11:52,300 --> 00:11:55,840 trying to break these exhibits. 176 00:11:55,840 --> 00:11:58,540 And what they did is by trying to break them 177 00:11:58,540 --> 00:12:01,430 they discovered patterns, some of them, 178 00:12:01,430 --> 00:12:02,920 and that's what we're going to do. 179 00:12:02,920 --> 00:12:06,820 We're going to try to break or think about breaking postulates 180 00:12:06,820 --> 00:12:12,040 and then see what we learn. 181 00:12:12,040 --> 00:12:14,140 So, let's begin. 182 00:12:17,630 --> 00:12:19,570 We have operators in quantum mechanics. 183 00:12:23,010 --> 00:12:31,940 And we denote them either with a hat or as a boldface object. 184 00:12:31,940 --> 00:12:33,870 We start using this kind of notation when 185 00:12:33,870 --> 00:12:37,260 we do matrix mechanics, which we will do, 186 00:12:37,260 --> 00:12:41,960 but this is just a general symbol for an operator. 187 00:12:41,960 --> 00:12:46,460 And an operator operates on a function 188 00:12:46,460 --> 00:12:50,050 and gives a different function. 189 00:12:50,050 --> 00:12:56,130 It operates to the right, or at least we 190 00:12:56,130 --> 00:12:58,716 like to think about it as operating to the right. 191 00:12:58,716 --> 00:13:00,090 If we let it operate to the left, 192 00:13:00,090 --> 00:13:02,230 we have to figure out what the rules are, 193 00:13:02,230 --> 00:13:04,260 and I'm not ready to tell you about that. 194 00:13:04,260 --> 00:13:06,700 So this operator has to be linear. 195 00:13:09,530 --> 00:13:14,590 And so if we have an operator operating 196 00:13:14,590 --> 00:13:24,855 on a function, A f plus bg, It has to do this. 197 00:13:30,010 --> 00:13:33,430 Now you'd think, well, that's pretty simple. 198 00:13:33,430 --> 00:13:35,490 Anything should do that. 199 00:13:35,490 --> 00:13:37,460 So taking the derivative does that. 200 00:13:37,460 --> 00:13:41,796 Doing an integral does that, but taking the square root doesn't. 201 00:13:41,796 --> 00:13:44,220 So taking the square root-- an operator says 202 00:13:44,220 --> 00:13:48,520 take the square root, well, that's not a linear operator. 203 00:13:48,520 --> 00:13:50,700 Now the only operators in quantum mechanics 204 00:13:50,700 --> 00:13:51,979 are linear operators. 205 00:13:58,400 --> 00:14:00,410 We have eigenvalue equations. 206 00:14:08,430 --> 00:14:13,090 So we have an operator operating in some function. 207 00:14:13,090 --> 00:14:18,485 It gives a number and the function back again. 208 00:14:18,485 --> 00:14:20,490 And this is called the eigenvalues, 209 00:14:20,490 --> 00:14:21,740 and this is the eigenfunction. 210 00:14:21,740 --> 00:14:22,900 AUDIENCE: Dr. Field? 211 00:14:22,900 --> 00:14:23,730 ROBERT FIELD: Yes? 212 00:14:23,730 --> 00:14:25,140 AUDIENCE: In the [INAUDIBLE] that 213 00:14:25,140 --> 00:14:28,228 should be b times A hat [INAUDIBLE].. 214 00:14:31,210 --> 00:14:32,780 ROBERT FIELD: What did I do? 215 00:14:32,780 --> 00:14:35,490 Well, it should just be-- 216 00:14:35,490 --> 00:14:38,060 sorry about that. 217 00:14:38,060 --> 00:14:41,910 I'm going to make mistakes like this. 218 00:14:41,910 --> 00:14:43,870 The TAs are going to catch me on it, 219 00:14:43,870 --> 00:14:45,150 and you're going to do it too. 220 00:14:48,230 --> 00:14:51,890 All right, so now here we have an operator operating 221 00:14:51,890 --> 00:14:53,180 on some function. 222 00:14:53,180 --> 00:14:55,790 And this function is special because when 223 00:14:55,790 --> 00:14:57,710 the operator operates on it, it returns 224 00:14:57,710 --> 00:15:01,130 the function times a number, the eigenvalue 225 00:15:01,130 --> 00:15:03,590 and the eigenfunction. 226 00:15:03,590 --> 00:15:04,520 We like these. 227 00:15:04,520 --> 00:15:07,460 Almost all quantum mechanics is expressed in terms 228 00:15:07,460 --> 00:15:09,448 of eigenvalue equations. 229 00:15:17,420 --> 00:15:25,430 Operators in quantum mechanics-- 230 00:15:25,430 --> 00:15:32,430 so for every physical quantity in non-quantum-mechanical life 231 00:15:32,430 --> 00:15:35,620 there corresponds an operator in quantum mechanics. 232 00:15:35,620 --> 00:15:40,960 So for the coordinate, the operator 233 00:15:40,960 --> 00:15:43,530 is just the coordinate. 234 00:15:43,530 --> 00:15:46,550 For the momentum, the operator is minus 235 00:15:46,550 --> 00:15:51,270 ih bar partial with respect to x or derivative with respect 236 00:15:51,270 --> 00:15:52,650 to x. 237 00:15:52,650 --> 00:15:54,570 Now this is not too surprising, but this 238 00:15:54,570 --> 00:16:00,610 is really puzzling because why is there an imaginary number? 239 00:16:00,610 --> 00:16:04,680 This is the square root of minus 1, which we call i. 240 00:16:04,680 --> 00:16:08,370 Why is that there and why is the operator a derivative 241 00:16:08,370 --> 00:16:09,930 rather than just some simple thing? 242 00:16:13,350 --> 00:16:16,620 Another operator is the kinetic energy, 243 00:16:16,620 --> 00:16:20,070 and the kinetic energy is p squared over 2m. 244 00:16:24,760 --> 00:16:30,570 And so that comes out to be minus h bar squared 245 00:16:30,570 --> 00:16:36,570 over 2m second partial with respect to x. 246 00:16:43,240 --> 00:16:45,800 Well, it's nice that I don't have to memorize this 247 00:16:45,800 --> 00:16:49,940 because I can just square this and this pops out, 248 00:16:49,940 --> 00:16:52,400 but you have to be aware of how to operate 249 00:16:52,400 --> 00:16:56,060 with complex and imaginary numbers. 250 00:16:56,060 --> 00:17:00,050 And there are so many exercises on the problem set, 251 00:17:00,050 --> 00:17:03,440 so you should be up to date on that. 252 00:17:03,440 --> 00:17:12,540 And now the potential is just the potential. 253 00:17:12,540 --> 00:17:14,220 And now the most important operator, 254 00:17:14,220 --> 00:17:16,680 at least when we start out, is the Hamiltonian, 255 00:17:16,680 --> 00:17:20,040 which is the operator that corresponds to energy, 256 00:17:20,040 --> 00:17:26,236 and that is kinetic energy plus potential energy. 257 00:17:26,236 --> 00:17:28,369 And this is called the Hamiltonian, 258 00:17:28,369 --> 00:17:32,010 and we're going to be focusing a lot on that. 259 00:17:32,010 --> 00:17:35,650 So these are the operators you're going to care about. 260 00:17:35,650 --> 00:17:42,470 The next thing we talk about is commutation rules 261 00:17:42,470 --> 00:17:44,800 or commutators. 262 00:17:44,800 --> 00:17:48,520 And one really important commutator 263 00:17:48,520 --> 00:17:50,830 is the commutator of the coordinate 264 00:17:50,830 --> 00:17:53,470 with the conjugate momentum, conjugate 265 00:17:53,470 --> 00:17:55,390 meaning in the same direction. 266 00:17:55,390 --> 00:18:02,350 And that is defined as xp minus px. 267 00:18:02,350 --> 00:18:08,350 And the obvious thing is that this commutator would be 0. 268 00:18:08,350 --> 00:18:11,490 Why does it matter which order you write things? 269 00:18:11,490 --> 00:18:13,160 But it does matter. 270 00:18:13,160 --> 00:18:17,260 And, in fact, one approach to quantum mechanics 271 00:18:17,260 --> 00:18:20,290 is to start not with the postulates 272 00:18:20,290 --> 00:18:24,880 that you normally deal with but a set of commutation rules, 273 00:18:24,880 --> 00:18:29,360 and everything can be derived from the commutation rules. 274 00:18:29,360 --> 00:18:32,150 It's a much more abstract approach, 275 00:18:32,150 --> 00:18:35,790 but it's a very powerful approach. 276 00:18:35,790 --> 00:18:41,350 So this commutator is not zero. 277 00:18:41,350 --> 00:18:44,080 And how do you find out what a commutator is? 278 00:18:44,080 --> 00:18:52,180 Well, you do xp minus px, operate on some function, 279 00:18:52,180 --> 00:18:52,990 and you find out. 280 00:18:55,770 --> 00:18:58,140 And you could do that. 281 00:18:58,140 --> 00:18:59,310 I could do that. 282 00:18:59,310 --> 00:19:03,860 But the commutator is going to be equal to ih bar. 283 00:19:07,850 --> 00:19:10,880 Now there is a little bit of trickiness 284 00:19:10,880 --> 00:19:17,360 because the commutator xp is ih bar and px is minus ih bar. 285 00:19:17,360 --> 00:19:20,870 And so I don't recommend memorizing it. 286 00:19:20,870 --> 00:19:26,900 I recommend being able to do this operation 287 00:19:26,900 --> 00:19:30,470 at the speed of light so you know whether it's 288 00:19:30,470 --> 00:19:33,690 plus ih bar or minus ih bar because you 289 00:19:33,690 --> 00:19:36,120 get into a whole lot of trouble if you get it wrong. 290 00:19:39,390 --> 00:19:44,310 So this is really where it all begins, and this 291 00:19:44,310 --> 00:19:48,480 is why you can't make simultaneously 292 00:19:48,480 --> 00:19:53,160 precise measurements of position and momentum, 293 00:19:53,160 --> 00:19:54,510 and lots of other good things. 294 00:19:57,250 --> 00:19:58,690 And then we have wave functions. 295 00:20:06,200 --> 00:20:16,480 So wave function for when the time independent Hamiltonian is 296 00:20:16,480 --> 00:20:21,340 a function of one variable, and it contains everything we could 297 00:20:21,340 --> 00:20:25,890 possibly know about the system. 298 00:20:25,890 --> 00:20:28,410 But this strange and wonderful thing, 299 00:20:28,410 --> 00:20:32,680 which leads to all sorts of philosophical debates, 300 00:20:32,680 --> 00:20:37,360 is that this guy, which contains everything that we can know, 301 00:20:37,360 --> 00:20:40,920 can never be directly measured. 302 00:20:40,920 --> 00:20:43,350 You can only measure what happens 303 00:20:43,350 --> 00:20:46,950 when you act on something with a given wave function. 304 00:20:46,950 --> 00:20:49,320 You cannot observe the wave function. 305 00:20:49,320 --> 00:20:57,490 And for a subject area where the central thing is unobservable 306 00:20:57,490 --> 00:20:59,540 is rather spooky. 307 00:20:59,540 --> 00:21:02,310 And a lot of people don't like that approach 308 00:21:02,310 --> 00:21:07,680 because it says we've got this thing that we're relying on, 309 00:21:07,680 --> 00:21:08,790 but we can't observe it. 310 00:21:08,790 --> 00:21:13,260 We can only observe what we do when we act on it. 311 00:21:13,260 --> 00:21:16,540 And usually the action is destructive. 312 00:21:16,540 --> 00:21:18,690 It's destructive of the state of the system. 313 00:21:18,690 --> 00:21:21,420 It causes the state of the system 314 00:21:21,420 --> 00:21:26,580 to give you a set of possible answers, 315 00:21:26,580 --> 00:21:30,680 and not the same thing each time. 316 00:21:30,680 --> 00:21:31,880 So it's really weird. 317 00:21:34,780 --> 00:21:37,370 So we have wave functions. 318 00:21:37,370 --> 00:21:42,530 And we can use the wave function to find 319 00:21:42,530 --> 00:21:51,640 the probability of the system at x with a range of x, x 320 00:21:51,640 --> 00:21:53,730 to x plus dx. 321 00:21:53,730 --> 00:21:58,645 And that's psi star x psi of x dx. 322 00:22:01,420 --> 00:22:04,270 So you notice we have two wave functions, the product of two, 323 00:22:04,270 --> 00:22:07,550 and this star means takes a complex conjugate. 324 00:22:07,550 --> 00:22:16,270 So if you have a complex number z is equal to x plus iy-- 325 00:22:16,270 --> 00:22:18,860 real part, imaginary part-- 326 00:22:18,860 --> 00:22:22,900 and if we take z star, that's x minus iy. 327 00:22:27,690 --> 00:22:32,340 So these wave functions are complex functions 328 00:22:32,340 --> 00:22:35,110 of a real variable. 329 00:22:35,110 --> 00:22:39,170 And so we do things like take the complex conjugate, 330 00:22:39,170 --> 00:22:43,220 and you have to become familiar with that. 331 00:22:43,220 --> 00:22:56,620 Now we have what we call the expectation value 332 00:22:56,620 --> 00:23:05,090 or the average value, and we denote this as A. 333 00:23:05,090 --> 00:23:08,020 So for the state function psi, we 334 00:23:08,020 --> 00:23:13,820 want the average value of the operator A. Now in most life, 335 00:23:13,820 --> 00:23:20,960 that symbol is not included just because people assume 336 00:23:20,960 --> 00:23:22,340 you know what you're doing. 337 00:23:22,340 --> 00:23:36,130 And this is psi star A hat psi dx over psi star psi dx. 338 00:23:36,130 --> 00:23:39,640 And this is integral from minus infinity to infinity. 339 00:23:42,330 --> 00:23:48,360 So this down here is a normalization integral. 340 00:23:48,360 --> 00:23:53,690 Now we normally deal with state functions 341 00:23:53,690 --> 00:23:56,840 which are normalized to 1, meaning the particle 342 00:23:56,840 --> 00:23:59,300 is somewhere. 343 00:23:59,300 --> 00:24:02,320 But if the particle can go anywhere, 344 00:24:02,320 --> 00:24:08,600 then normalization to 1 means it's approximately nowhere. 345 00:24:08,600 --> 00:24:11,380 And so we have to think a little bit about what 346 00:24:11,380 --> 00:24:14,920 do we mean by normalization, but this 347 00:24:14,920 --> 00:24:19,300 is how we define the average value or the expectation 348 00:24:19,300 --> 00:24:23,120 value of the quantity A for the state psi. 349 00:24:28,350 --> 00:24:30,510 So this is just a little bit of a warning 350 00:24:30,510 --> 00:24:33,280 that, yeah, you would think this is all you need, 351 00:24:33,280 --> 00:24:35,790 but you also need to at least think about this. 352 00:24:39,549 --> 00:24:40,090 That's great. 353 00:24:40,090 --> 00:24:44,610 I'm at the top of the board and we're now at the beginning. 354 00:24:44,610 --> 00:24:48,070 So the Schrodinger equation is the last thing, 355 00:24:48,070 --> 00:24:51,850 and that's the Hamiltonian operating on the function 356 00:24:51,850 --> 00:24:56,350 and gives an energy times that function. 357 00:24:56,350 --> 00:24:57,820 And if it's an eigenvalue, then we 358 00:24:57,820 --> 00:25:00,050 have this eigenvalue equation. 359 00:25:00,050 --> 00:25:03,110 We have these symbols here. 360 00:25:03,110 --> 00:25:07,820 So that's the energy associated with the psi n function. 361 00:25:12,070 --> 00:25:15,910 So now we're ready to start playing games 362 00:25:15,910 --> 00:25:18,710 with this strange new world. 363 00:25:18,710 --> 00:25:20,850 And so let's start out with the free particle. 364 00:25:29,190 --> 00:25:33,510 Now because the free particle has a complicated feature 365 00:25:33,510 --> 00:25:36,630 about how do we normalize it, it really 366 00:25:36,630 --> 00:25:39,630 shouldn't be the first thing we talk about. 367 00:25:39,630 --> 00:25:44,400 But it seems like the simplest problem, so we will. 368 00:25:44,400 --> 00:25:47,314 So what's the Hamiltonian? 369 00:25:47,314 --> 00:25:52,460 The Hamiltonian is the kinetic energy, minus h bar squared, 370 00:25:52,460 --> 00:25:58,830 or 2m second derivative with respect to x 371 00:25:58,830 --> 00:26:02,346 plus the potential energy, V0. 372 00:26:02,346 --> 00:26:04,935 Free particle, the potential is constant. 373 00:26:07,530 --> 00:26:11,520 We normally think of it as the potential is zero, 374 00:26:11,520 --> 00:26:15,620 but there is no absolute scale of a zero of energy, 375 00:26:15,620 --> 00:26:17,340 so we just need to specify this. 376 00:26:20,900 --> 00:26:24,770 And so we want to write the Schrodinger equation, 377 00:26:24,770 --> 00:26:30,380 and we want to arrange it in a form that is easy to solve. 378 00:26:30,380 --> 00:26:32,270 There is two steps to the rearrangement, 379 00:26:32,270 --> 00:26:37,650 and I'll just write the final thing. 380 00:26:37,650 --> 00:26:40,010 So the second derivative of psi is 381 00:26:40,010 --> 00:26:51,380 equal to minus 2m over h bar squared times E minus V0 psi. 382 00:26:51,380 --> 00:26:55,260 So this is the differential equation that we have to solve. 383 00:26:55,260 --> 00:26:57,260 So there was a little bit of rearrangement here, 384 00:26:57,260 --> 00:27:00,320 but you can do that. 385 00:27:00,320 --> 00:27:03,170 So the second derivative of some function 386 00:27:03,170 --> 00:27:06,740 is equal to some constant times that function. 387 00:27:06,740 --> 00:27:09,590 We've seen that problem before. 388 00:27:09,590 --> 00:27:12,260 It makes a lot of difference whether that constant 389 00:27:12,260 --> 00:27:17,380 is positive or negative, and it better, 390 00:27:17,380 --> 00:27:20,830 because if we have a potential V0 391 00:27:20,830 --> 00:27:24,370 and we have an energy up here, well, 392 00:27:24,370 --> 00:27:25,540 that's perfectly reasonable. 393 00:27:25,540 --> 00:27:28,210 The particle can be there, classically. 394 00:27:28,210 --> 00:27:29,770 But suppose the energy is down here. 395 00:27:32,420 --> 00:27:36,720 If the zero of energy is here, you can't go below it. 396 00:27:39,940 --> 00:27:42,340 That's a classically forbidden situation. 397 00:27:45,030 --> 00:27:48,930 And so for the classically allowed situation, 398 00:27:48,930 --> 00:27:53,910 the quantity, this constant, is negative. 399 00:27:53,910 --> 00:27:56,400 For the classically forbidden situation, 400 00:27:56,400 --> 00:27:58,800 this constant is positive. 401 00:27:58,800 --> 00:28:00,630 You've already seen the big difference 402 00:28:00,630 --> 00:28:06,090 in the way a second derivative, this kind of equation, 403 00:28:06,090 --> 00:28:12,700 works when the constant is positive or negative. 404 00:28:12,700 --> 00:28:17,900 When this constant is negative, you get oscillation. 405 00:28:17,900 --> 00:28:22,130 When this constant is positive, you'll get exponential. 406 00:28:27,780 --> 00:28:35,440 Now we're interested in a free particle, 407 00:28:35,440 --> 00:28:44,980 so free particle can be anywhere. 408 00:28:44,980 --> 00:28:48,790 And we insist that the solution to 409 00:28:48,790 --> 00:28:55,390 our quantum-mechanical problem, the wave function 410 00:28:55,390 --> 00:28:56,925 is what we say well behaved. 411 00:28:59,960 --> 00:29:02,200 So well behaved has many meanings, but one of them 412 00:29:02,200 --> 00:29:04,170 is it never goes to infinity. 413 00:29:07,170 --> 00:29:09,400 Another is that when you go to infinity, 414 00:29:09,400 --> 00:29:11,175 the wave function should go to zero. 415 00:29:14,680 --> 00:29:20,130 But there's also things about continuity and continuity 416 00:29:20,130 --> 00:29:22,200 of first derivatives and continuity 417 00:29:22,200 --> 00:29:24,180 of second derivatives. 418 00:29:24,180 --> 00:29:28,410 We'll get into those, but you know immediately 419 00:29:28,410 --> 00:29:32,380 that if this constant is positive, 420 00:29:32,380 --> 00:29:37,300 you get an exponential behavior, and you get the e 421 00:29:37,300 --> 00:29:40,120 to the ikx and e to the-- 422 00:29:40,120 --> 00:29:44,980 not ik-- e to the kx and e to the minus kx. 423 00:29:44,980 --> 00:29:50,590 And one of those blows up at either positive infinity 424 00:29:50,590 --> 00:29:52,670 or negative infinity. 425 00:29:52,670 --> 00:29:57,400 So it's telling you that in agreement 426 00:29:57,400 --> 00:30:02,240 with what you expect for the classical world, 427 00:30:02,240 --> 00:30:07,740 an energy below the constant potential is illegal. 428 00:30:10,480 --> 00:30:16,360 It's illegal when this situation persists to infinity. 429 00:30:16,360 --> 00:30:19,570 But we'll discover that it is legal 430 00:30:19,570 --> 00:30:26,680 if the range of coordinate for which the energy is less 431 00:30:26,680 --> 00:30:28,570 than the potential is finite. 432 00:30:28,570 --> 00:30:31,120 And that's called tunneling, and tunneling 433 00:30:31,120 --> 00:30:32,800 is a quantum-mechanical phenomenon. 434 00:30:32,800 --> 00:30:36,090 We will encounter that. 435 00:30:36,090 --> 00:30:40,980 So we know from our experience with this kind of differential 436 00:30:40,980 --> 00:30:49,350 equation that the solutions will have the form sine 437 00:30:49,350 --> 00:30:52,890 kx and cosine kx. 438 00:30:55,570 --> 00:31:01,690 But we choose to use instead e to the ikx 439 00:31:01,690 --> 00:31:12,480 and e to the minus ikx because this cosine kx 440 00:31:12,480 --> 00:31:18,570 is 1/2 e to the ikx plus e to the minus ikx. 441 00:31:18,570 --> 00:31:21,040 And so we can use these functions 442 00:31:21,040 --> 00:31:23,750 because they're more convenient, more memorable. 443 00:31:23,750 --> 00:31:28,270 All the integrals and derivatives are trivial. 444 00:31:28,270 --> 00:31:31,350 And so we do that. 445 00:31:34,550 --> 00:31:39,488 So the differential equation-- 446 00:31:58,940 --> 00:32:04,250 and we saw before that we already have what k is. 447 00:32:04,250 --> 00:32:11,120 So minus k squared is minus 2m over h bar squared-- 448 00:32:11,120 --> 00:32:16,260 minus 2m over h bar squared E minus V0. 449 00:32:26,840 --> 00:32:29,375 We take the derivative of this function. 450 00:32:35,180 --> 00:32:41,550 This is the function, and this is the eigenvalue. 451 00:32:41,550 --> 00:32:43,740 We take the second derivative with respect to x. 452 00:32:43,740 --> 00:32:47,190 We get an ik from this term and then 453 00:32:47,190 --> 00:32:52,050 another ik, which makes minus k squared. 454 00:32:52,050 --> 00:32:56,070 And we get a minus ik and another minus ik, 455 00:32:56,070 --> 00:33:03,790 and that gives a minus k squared. 456 00:33:03,790 --> 00:33:09,700 And so, in fact, this is an eigenvalue equation. 457 00:33:09,700 --> 00:33:15,490 We have the form where this equation is an eigenfunction. 458 00:33:15,490 --> 00:33:17,830 With this, we have everything. 459 00:33:17,830 --> 00:33:30,600 So the energies for the free particle, h bar k over h bar k 460 00:33:30,600 --> 00:33:46,920 squared over 2m plus V0, so this is an eigenfunction, 461 00:33:46,920 --> 00:33:50,760 and this is the eigenvalue associated with that function. 462 00:33:53,412 --> 00:33:54,480 We're done. 463 00:33:54,480 --> 00:33:55,530 That was an easy problem. 464 00:33:55,530 --> 00:33:58,100 I skipped some steps because it's an easy problem 465 00:33:58,100 --> 00:34:01,400 and I want you to go over it and make sure 466 00:34:01,400 --> 00:34:05,060 that you understand the logic and can 467 00:34:05,060 --> 00:34:06,363 come to the same solution. 468 00:34:11,170 --> 00:34:14,120 Let's take a little side issue. 469 00:34:14,120 --> 00:34:20,370 Suppose we have psi of x is e to the ikx. 470 00:34:24,870 --> 00:34:32,719 Well, we're going to find that this is an eigenfunction of p, 471 00:34:32,719 --> 00:34:38,570 and the eigenvalue or the expectation value of p 472 00:34:38,570 --> 00:34:42,650 is h bar k. 473 00:34:42,650 --> 00:34:48,620 And if we had minus e to the minus ikx, then what we'd get 474 00:34:48,620 --> 00:34:49,880 is minus h bar k. 475 00:34:52,750 --> 00:35:00,480 So we have this relationship between p, expectation value, 476 00:35:00,480 --> 00:35:02,238 and h bar k. 477 00:35:04,830 --> 00:35:07,830 So this corresponds to the particle going 478 00:35:07,830 --> 00:35:10,750 in the positive x direction, and this corresponds 479 00:35:10,750 --> 00:35:15,180 to the particle going in the negative x direction. 480 00:35:15,180 --> 00:35:18,180 Everything is perfectly reasonable. 481 00:35:18,180 --> 00:35:21,420 We have solutions to the Schrodinger equation 482 00:35:21,420 --> 00:35:23,160 for the free particle. 483 00:35:23,160 --> 00:35:25,590 The solutions to the free particle 484 00:35:25,590 --> 00:35:30,600 are also solutions to the eigenvalue equation 485 00:35:30,600 --> 00:35:32,270 for momentum. 486 00:35:32,270 --> 00:35:36,498 And the two possible eigenvalues for a given k 487 00:35:36,498 --> 00:35:41,080 are plus h bar k minus h bar k. 488 00:35:41,080 --> 00:35:45,540 Now that's fine. 489 00:35:45,540 --> 00:35:48,000 So everything works out. 490 00:35:48,000 --> 00:35:50,250 We're getting things, although we 491 00:35:50,250 --> 00:35:55,980 have the definition of the momentum having a minus 1, 492 00:35:55,980 --> 00:36:00,510 an i factor, and a derivative factor. 493 00:36:00,510 --> 00:36:02,540 Everything works. 494 00:36:02,540 --> 00:36:04,240 Everything is as you would expect. 495 00:36:07,760 --> 00:36:16,060 And the general solution to the Schrodinger equation 496 00:36:16,060 --> 00:36:20,980 can have two different values, the superposition of these two. 497 00:36:24,750 --> 00:36:36,460 Right now, this wave function is the localized overall space. 498 00:36:36,460 --> 00:36:39,020 Now if we want to normalize it, we'd 499 00:36:39,020 --> 00:36:44,150 like to calculate integral minus infinity to infinity 500 00:36:44,150 --> 00:36:52,334 of psi star x psi of x dx. 501 00:36:55,680 --> 00:37:00,390 This is why we like this notation 502 00:37:00,390 --> 00:37:08,640 because suppose we have a function like this, psi star-- 503 00:37:08,640 --> 00:37:10,280 well, actually like this. 504 00:37:10,280 --> 00:37:28,250 Psi star is equal to a star e to the minus ikx plus b star 505 00:37:28,250 --> 00:37:39,200 e to the ikx, and psi is a e to the plus ikx. 506 00:37:39,200 --> 00:37:44,770 This would be e to the minus ikx. 507 00:37:44,770 --> 00:37:47,620 And so when we write this integral, what we get 508 00:37:47,620 --> 00:37:55,840 is integral of psi star psi dx is 509 00:37:55,840 --> 00:38:06,200 equal to a squared integral minus infinity to infinity 510 00:38:06,200 --> 00:38:12,720 of a squared plus b squared dx. 511 00:38:15,350 --> 00:38:18,200 So we have two constants which are real numbers 512 00:38:18,200 --> 00:38:20,750 because they're square modulus. 513 00:38:20,750 --> 00:38:22,630 They're additive. 514 00:38:22,630 --> 00:38:27,040 And we're integrating this constant from minus infinity 515 00:38:27,040 --> 00:38:29,770 to infinity. 516 00:38:29,770 --> 00:38:31,010 We'll get infinity. 517 00:38:31,010 --> 00:38:32,510 We can't make this equal to 1. 518 00:38:39,480 --> 00:38:41,330 So we have to put this in our head 519 00:38:41,330 --> 00:38:43,400 and say, well, there's a problem when 520 00:38:43,400 --> 00:38:46,640 you have a wave function that extends over all space. 521 00:38:46,640 --> 00:38:49,070 It can't be normalized to 1, but it 522 00:38:49,070 --> 00:38:55,740 can be normalized so that for a given distance in real space, 523 00:38:55,740 --> 00:38:59,910 it's got a probability of 1 in that distance. 524 00:38:59,910 --> 00:39:03,060 So we have a different form of normalization. 525 00:39:03,060 --> 00:39:07,960 But when we actually calculate expectation values, 526 00:39:07,960 --> 00:39:12,550 we can still use this naive idea of the normalization interval 527 00:39:12,550 --> 00:39:14,980 and we get the right answer, even 528 00:39:14,980 --> 00:39:18,400 though because both the numerator and denominator 529 00:39:18,400 --> 00:39:21,560 go to infinity and those infinities cancel 530 00:39:21,560 --> 00:39:22,850 and everything works out. 531 00:39:22,850 --> 00:39:27,450 This is why we don't do this first usually 532 00:39:27,450 --> 00:39:28,950 because there's all of these things 533 00:39:28,950 --> 00:39:30,840 that you have to convince yourself are OK. 534 00:39:33,400 --> 00:39:36,140 And they are and you should. 535 00:39:36,140 --> 00:39:43,850 But now let's go to the famous particle in a box. 536 00:39:43,850 --> 00:39:50,540 It's so famous that we always use this notation. 537 00:39:50,540 --> 00:39:53,570 This is particle in an infinite box, 538 00:39:53,570 --> 00:39:57,650 and that means the particle is in a box like this 539 00:39:57,650 --> 00:40:00,660 where the walls go to infinity. 540 00:40:00,660 --> 00:40:06,620 And so we normally locate this box 541 00:40:06,620 --> 00:40:09,550 at a place where this is the x coordinate 542 00:40:09,550 --> 00:40:13,520 and this is the potential energy, 543 00:40:13,520 --> 00:40:16,871 and the width of the box is a. 544 00:40:16,871 --> 00:40:22,790 And we normally put the left edge of the box at zero 545 00:40:22,790 --> 00:40:24,950 because that problem is a little easier 546 00:40:24,950 --> 00:40:27,740 to solve than the more logical thing 547 00:40:27,740 --> 00:40:33,300 where you say, OK, this box is centered about zero. 548 00:40:33,300 --> 00:40:35,660 And that should bother you because anytime 549 00:40:35,660 --> 00:40:39,522 you're interested in asking about the symmetry of things 550 00:40:39,522 --> 00:40:41,480 you'll want to choose a coordinate system which 551 00:40:41,480 --> 00:40:44,360 reflects that. 552 00:40:44,360 --> 00:40:46,400 Don't worry. 553 00:40:46,400 --> 00:40:48,800 I am going to ask you about symmetry, 554 00:40:48,800 --> 00:40:52,490 and it's a simple thing to take the solution for this problem 555 00:40:52,490 --> 00:40:54,110 and move it to the left by a over 2. 556 00:40:56,630 --> 00:41:05,630 So we have basically a problem where the potential is 557 00:41:05,630 --> 00:41:14,910 equal to 0 for 0 is less than or equal to x less than 558 00:41:14,910 --> 00:41:23,130 or equal to a, and it's equal to infinity when x is less than 0 559 00:41:23,130 --> 00:41:24,060 or greater than a. 560 00:41:28,420 --> 00:41:34,540 So inside the box it looks like a free particle, 561 00:41:34,540 --> 00:41:36,550 but it can't be a free particle because there's 562 00:41:36,550 --> 00:41:40,800 got to be nodes at the walls. 563 00:41:40,800 --> 00:41:44,640 We know that outside the box, the wave function 564 00:41:44,640 --> 00:41:50,280 has to be 0 everywhere because it's classically forbidden, 565 00:41:50,280 --> 00:41:51,830 strongly forbidden. 566 00:41:56,060 --> 00:42:03,900 We know that the wave function psi is continuous. 567 00:42:03,900 --> 00:42:08,610 So if it's at 0 outside, it's going to be 0 at the wall. 568 00:42:08,610 --> 00:42:13,390 And so the wave functions have boundary conditions 569 00:42:13,390 --> 00:42:17,140 where, at the wall, the wave function goes to 0. 570 00:42:24,140 --> 00:42:28,110 So now we go and we solve this problem. 571 00:42:28,110 --> 00:42:38,450 And so the Schrodinger equation for the particle in the box 572 00:42:38,450 --> 00:42:41,120 where V of x is 0. 573 00:42:41,120 --> 00:42:42,410 Well, we don't need it. 574 00:42:42,410 --> 00:42:44,435 We just have the kinetic-energy term, 575 00:42:44,435 --> 00:42:51,860 h bar squared over 2m second derivative with respect to x. 576 00:42:51,860 --> 00:42:56,700 Psi is equal to e psi. 577 00:42:56,700 --> 00:42:58,980 Again, we rearrange it. 578 00:42:58,980 --> 00:43:03,090 And so we put the derivative outside, 579 00:43:03,090 --> 00:43:10,547 and we have minus 2me over h bar squared psi. 580 00:43:14,300 --> 00:43:16,670 And this is a number. 581 00:43:16,670 --> 00:43:21,760 And so we just call it minus k squared psi. 582 00:43:24,390 --> 00:43:28,620 We know what that k is as long as we know what the energy is, 583 00:43:28,620 --> 00:43:35,330 and k squared is equal to 2me over h bar squared. 584 00:43:39,430 --> 00:43:45,160 Now we have this thing which is equal to a negative number 585 00:43:45,160 --> 00:43:47,140 times a wave function, and we already 586 00:43:47,140 --> 00:43:49,285 know we have exponential behavior. 587 00:43:51,790 --> 00:43:54,820 But in this case, we use sines and cosines 588 00:43:54,820 --> 00:43:57,920 because it's more convenient. 589 00:43:57,920 --> 00:44:08,970 So psi of x is going to be written as A sine kx plus B 590 00:44:08,970 --> 00:44:12,670 cosine kx. 591 00:44:12,670 --> 00:44:19,680 This is the general solution for this differential equation 592 00:44:19,680 --> 00:44:25,700 where we have a negative constant times the function. 593 00:44:25,700 --> 00:44:30,526 So the boundary condition, psi of 0-- 594 00:44:30,526 --> 00:44:35,180 well, psi of 0, this is 0, but this part is 1. 595 00:44:35,180 --> 00:44:41,150 So that means that psi of 0 has to be 0, 596 00:44:41,150 --> 00:44:45,050 so B has to be equal to 0. 597 00:44:45,050 --> 00:44:48,530 And here, the other boundary condition, 598 00:44:48,530 --> 00:44:57,290 that also has to be equal to 0, and that has to be A sine ka. 599 00:44:57,290 --> 00:45:02,750 And ka has to be equal to n times pi in order 600 00:45:02,750 --> 00:45:03,845 to satisfy this equation. 601 00:45:07,850 --> 00:45:14,943 Sine is 0 at 0, pi, 2 pi, 3 pi, et cetera. 602 00:45:20,870 --> 00:45:25,310 So k is equal to n pi over a. 603 00:45:28,214 --> 00:45:30,790 And so we have the solutions. 604 00:45:30,790 --> 00:45:43,340 Psi of x is equal to A sine np a over x. 605 00:45:43,340 --> 00:45:47,609 And so we could put a little n here. 606 00:45:47,609 --> 00:45:49,150 And this is starting to make you feel 607 00:45:49,150 --> 00:45:55,470 really good because for all positive integers 608 00:45:55,470 --> 00:45:57,890 there is a solution. 609 00:45:57,890 --> 00:46:00,930 There's an infinite number of solutions. 610 00:46:00,930 --> 00:46:04,530 And their scaling with quantum number is trivial. 611 00:46:04,530 --> 00:46:08,670 And it's really great when you solve an equation 612 00:46:08,670 --> 00:46:11,640 and you are given an infinite number of solutions. 613 00:46:15,317 --> 00:46:17,150 Well, there's one thing more you have to do. 614 00:46:17,150 --> 00:46:20,510 You have to find out what the normalization constant is, 615 00:46:20,510 --> 00:46:22,860 so you do the normalization integral. 616 00:46:22,860 --> 00:46:25,880 And when you do that, you discover 617 00:46:25,880 --> 00:46:37,880 that this is equal to 2 over a square root sine n pi over a. 618 00:46:37,880 --> 00:46:41,660 So these are all the solutions for the particle 619 00:46:41,660 --> 00:46:44,436 in an infinite box, all of them. 620 00:46:44,436 --> 00:46:54,300 And the energies you can write as n squared times h squared 621 00:46:54,300 --> 00:47:02,590 over 8 m a squared or n squared times E1. 622 00:47:02,590 --> 00:47:03,970 There's another thing. 623 00:47:03,970 --> 00:47:07,060 n equals 0. 624 00:47:07,060 --> 00:47:12,760 If n is equal to 0, the wave function corresponding to n 625 00:47:12,760 --> 00:47:14,890 equals 0 is 0 everywhere. 626 00:47:14,890 --> 00:47:17,130 The particle isn't in the box. 627 00:47:17,130 --> 00:47:21,260 So n equals 0 is not a solution. 628 00:47:21,260 --> 00:47:22,960 So the solutions we have-- 629 00:47:22,960 --> 00:47:27,580 n equals 1 2, et cetera up to infinity, 630 00:47:27,580 --> 00:47:33,700 and the energies are integer square multiples 631 00:47:33,700 --> 00:47:34,915 of a common factor. 632 00:47:39,350 --> 00:47:43,770 This is wonderful because basically we have a problem. 633 00:47:43,770 --> 00:47:45,830 Maybe it's not that interesting now 634 00:47:45,830 --> 00:47:49,820 because why do we have infinite boxes and stuff like that? 635 00:47:49,820 --> 00:47:55,920 But if you ask, what about the ideal gas law? 636 00:47:55,920 --> 00:47:59,740 We have particles that don't interact with each other inside 637 00:47:59,740 --> 00:48:02,940 a container which has infinite walls. 638 00:48:02,940 --> 00:48:05,520 And I can tell you that in 5.62, there's 639 00:48:05,520 --> 00:48:07,500 a three-line derivation of the ideal gas 640 00:48:07,500 --> 00:48:11,960 law based on solutions to the particle in a box. 641 00:48:11,960 --> 00:48:14,690 Also, we often have situations where 642 00:48:14,690 --> 00:48:17,690 you have molecules where there's conjugation 643 00:48:17,690 --> 00:48:22,250 so that the molecule looks like a not quite flat bottom 644 00:48:22,250 --> 00:48:25,000 box with walls. 645 00:48:25,000 --> 00:48:29,080 And this equation enables you to learn something 646 00:48:29,080 --> 00:48:31,600 about the electronic energy levels 647 00:48:31,600 --> 00:48:35,270 for linear conjugated molecules. 648 00:48:35,270 --> 00:48:37,730 And this leads to a lot of qualitative insight 649 00:48:37,730 --> 00:48:42,870 into problems in photochemistry. 650 00:48:45,530 --> 00:48:53,900 Now the most important thing, in my opinion, 651 00:48:53,900 --> 00:48:58,300 is being able to draw cartoons, and these cartoons 652 00:48:58,300 --> 00:49:03,310 for the solutions the particle in a box look like this. 653 00:49:03,310 --> 00:49:07,990 So what you frequently do is draw the potential, 654 00:49:07,990 --> 00:49:11,600 and then you draw the energy levels and wave functions. 655 00:49:17,710 --> 00:49:19,490 I have to cheat a little bit. 656 00:49:24,580 --> 00:49:27,430 So number of nodes-- 657 00:49:27,430 --> 00:49:29,770 number of nodes, internal nodes. 658 00:49:29,770 --> 00:49:32,020 We don't count zeros at the walls. 659 00:49:32,020 --> 00:49:35,560 The number of nodes is n minus 1. 660 00:49:38,110 --> 00:49:46,270 The maximum of the wave function is always 2 over a. 661 00:49:50,330 --> 00:49:53,780 So here 2 over a, here 2 over a, here minus 2 662 00:49:53,780 --> 00:49:57,320 over a i square root of 2 over a. 663 00:49:57,320 --> 00:50:00,209 This slope is identical to this slope. 664 00:50:00,209 --> 00:50:02,000 This slope is identical to this slope which 665 00:50:02,000 --> 00:50:04,980 is identical to that slope. 666 00:50:04,980 --> 00:50:07,490 So there's a tremendous amount that you 667 00:50:07,490 --> 00:50:12,576 can get by understanding how the wave function looks 668 00:50:12,576 --> 00:50:13,700 and drawing these cartoons. 669 00:50:18,790 --> 00:50:27,250 And so now if instead we were looking at problems where, 670 00:50:27,250 --> 00:50:30,700 instead of a particle in a box like this, 671 00:50:30,700 --> 00:50:33,460 we have a little dimple in the bottom of the box 672 00:50:33,460 --> 00:50:38,020 or we have something at the bottom or the bottom of the box 673 00:50:38,020 --> 00:50:41,270 is slanted. 674 00:50:41,270 --> 00:50:45,290 You should be able to intuit what these things do 675 00:50:45,290 --> 00:50:48,050 to the energy levels, at least have 676 00:50:48,050 --> 00:50:52,290 the beginning of an intuition. 677 00:50:52,290 --> 00:50:59,730 So, we have an infinite number of oscillating solutions. 678 00:50:59,730 --> 00:51:02,970 That means that we could solve the problem 679 00:51:02,970 --> 00:51:07,740 for any kind of a box as long as it has vertical walls, 680 00:51:07,740 --> 00:51:09,710 and that's called a Fourier series. 681 00:51:12,890 --> 00:51:17,270 So for a finite range, we can describe the solution 682 00:51:17,270 --> 00:51:20,240 to any quantum-mechanical infinite 683 00:51:20,240 --> 00:51:24,140 box with a terrible bottom, in principle, 684 00:51:24,140 --> 00:51:28,019 by a superposition of our basis functions. 685 00:51:28,019 --> 00:51:29,060 That's what we call them. 686 00:51:32,090 --> 00:51:35,800 Now, there are several methods for doing 687 00:51:35,800 --> 00:51:42,220 the solution of a problem like this efficiently. 688 00:51:42,220 --> 00:51:46,620 And you're going to see perturbation theory. 689 00:51:46,620 --> 00:51:48,120 And at the end of the course, you're 690 00:51:48,120 --> 00:51:50,620 going to see something that will really knock your socks off 691 00:51:50,620 --> 00:51:55,110 which is called the discrete variable representation. 692 00:51:55,110 --> 00:51:58,080 And that enables you to say, yeah, well, 693 00:51:58,080 --> 00:52:00,120 potential does terrible things. 694 00:52:00,120 --> 00:52:03,210 I solved the problem by not doing any calculation at all 695 00:52:03,210 --> 00:52:06,630 because it's already been done. 696 00:52:06,630 --> 00:52:11,160 So these things are fantastic that we 697 00:52:11,160 --> 00:52:15,360 have an infinite number of solutions to a simple problem. 698 00:52:15,360 --> 00:52:20,880 We're always looking for a way to describe a simple problem 699 00:52:20,880 --> 00:52:23,820 or maybe not so simple problem with an infinite number 700 00:52:23,820 --> 00:52:28,230 of solutions where the energies for the solutions and the wave 701 00:52:28,230 --> 00:52:32,771 functions behave in a simple n-scaled, quantum-number-scaled 702 00:52:32,771 --> 00:52:33,270 way. 703 00:52:35,800 --> 00:52:40,450 And this provides us with a way of looking at what 704 00:52:40,450 --> 00:52:43,730 these things do in real life. 705 00:52:43,730 --> 00:52:46,180 You do an experiment on an eigenfunction 706 00:52:46,180 --> 00:52:50,470 of a box like this, and it will have certain characteristics. 707 00:52:50,470 --> 00:52:52,960 And it tells you, oh, if I measured the energy 708 00:52:52,960 --> 00:53:00,580 levels of a pathological box, the quantum-number dependence 709 00:53:00,580 --> 00:53:03,590 of the energy levels has a certain form, 710 00:53:03,590 --> 00:53:09,350 and each of the constants in that special form 711 00:53:09,350 --> 00:53:14,530 sample a particular feature of the pathological potential. 712 00:53:14,530 --> 00:53:16,360 And that's what we do as spectroscopists. 713 00:53:16,360 --> 00:53:20,770 We find an efficient way to fit the observables in order 714 00:53:20,770 --> 00:53:27,340 to characterize what's going on inside something we can't see. 715 00:53:27,340 --> 00:53:29,440 And that's the game in quantum mechanics. 716 00:53:29,440 --> 00:53:32,650 We can't see the wave function ever. 717 00:53:32,650 --> 00:53:34,900 We know there are eigenstates. 718 00:53:34,900 --> 00:53:39,310 We can observe energy levels and transition probabilities, 719 00:53:39,310 --> 00:53:45,100 and between those two things, we can determine quantitatively 720 00:53:45,100 --> 00:53:50,470 all of the internal structure of objects that we can't see. 721 00:53:50,470 --> 00:53:52,960 And this is what I do as a spectroscopist. 722 00:53:52,960 --> 00:53:57,130 And I'm really a little bit crazy about it 723 00:53:57,130 --> 00:54:02,880 because most people instead of saying 724 00:54:02,880 --> 00:54:05,990 let's try to understand based on something simple, 725 00:54:05,990 --> 00:54:08,180 they will just solve the Schrodinger equations 726 00:54:08,180 --> 00:54:13,430 numerically and get a bunch of small results 727 00:54:13,430 --> 00:54:16,730 and no intuition, no cartoons, and no ability 728 00:54:16,730 --> 00:54:19,610 to do dynamics except another picture 729 00:54:19,610 --> 00:54:23,950 where you have to work things out in a complicated way. 730 00:54:23,950 --> 00:54:27,460 But I'm giving you the standard problems from which 731 00:54:27,460 --> 00:54:30,620 you can solve almost anything. 732 00:54:30,620 --> 00:54:33,970 And this should sound like fun, I hope. 733 00:54:33,970 --> 00:54:36,790 OK, so have a great weekend.