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 --> 00:00:17,250 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,250 --> 00:00:18,210 at ocw.mit.edu. 8 00:00:21,250 --> 00:00:25,120 ROBERT FIELD: Last time, I talked a lot 9 00:00:25,120 --> 00:00:30,970 about the semiclassical method, where we generalize 10 00:00:30,970 --> 00:00:35,830 on this wonderful relationship to say, well, 11 00:00:35,830 --> 00:00:41,180 if the potential is not constant, then we can say, 12 00:00:41,180 --> 00:00:45,550 well, the wavelength changes with position. 13 00:00:45,550 --> 00:00:50,050 And we can say that the momentum changes with position. 14 00:00:50,050 --> 00:00:51,970 But we're using this as the guide. 15 00:00:51,970 --> 00:00:54,550 And so the basis is really just saying, 16 00:00:54,550 --> 00:00:57,730 OK, we're going to take this kind of a relationship 17 00:00:57,730 --> 00:00:59,200 seriously. 18 00:00:59,200 --> 00:01:02,066 Because v is not constant. 19 00:01:02,066 --> 00:01:05,470 We have V of x. 20 00:01:05,470 --> 00:01:12,790 And we also know that the kinetic energy, 21 00:01:12,790 --> 00:01:18,180 which is an operator, is p squared over 2m. 22 00:01:18,180 --> 00:01:24,780 And the energy minus the potential 23 00:01:24,780 --> 00:01:27,940 is the kinetic energy. 24 00:01:27,940 --> 00:01:38,400 And so we can use this to get a classical, mechanical function 25 00:01:38,400 --> 00:01:45,520 that this p of x is going to be 2m E minus V of x. 26 00:01:48,515 --> 00:01:50,640 So why are we doing all this when we can just solve 27 00:01:50,640 --> 00:01:52,840 the differential equations? 28 00:01:52,840 --> 00:01:55,560 And the answer is we want insight. 29 00:01:55,560 --> 00:01:59,130 And we want to build our insight on what we know. 30 00:01:59,130 --> 00:02:02,830 And so we have this momentum function, 31 00:02:02,830 --> 00:02:05,910 which gives us a wavelength function, which tells us 32 00:02:05,910 --> 00:02:08,690 how far apart the waves are. 33 00:02:08,690 --> 00:02:10,699 But we also have another thing, which 34 00:02:10,699 --> 00:02:14,000 was demonstrated by my running across the room, 35 00:02:14,000 --> 00:02:18,020 and that the momentum is related to the velocity, which 36 00:02:18,020 --> 00:02:20,690 is related to the probability of finding the system 37 00:02:20,690 --> 00:02:22,400 at a particular place. 38 00:02:22,400 --> 00:02:29,010 And so we have, if we commit the travesty of saying, OK, 39 00:02:29,010 --> 00:02:32,870 we have a classical function-- it's not the momentum-- 40 00:02:32,870 --> 00:02:37,450 but it's going to somehow encode the classical behavior, 41 00:02:37,450 --> 00:02:42,130 we can determine without solving any differential equation what 42 00:02:42,130 --> 00:02:46,840 the spacing between nodes in the exact wave function is, 43 00:02:46,840 --> 00:02:50,320 and the amplitude in that region. 44 00:02:50,320 --> 00:02:54,220 And that's a lot because then we can 45 00:02:54,220 --> 00:02:59,840 use our knowledge of classical mechanics to say, 46 00:02:59,840 --> 00:03:03,200 oh, this is what we expect in quantum mechanics. 47 00:03:03,200 --> 00:03:05,000 And that's very powerful. 48 00:03:05,000 --> 00:03:10,090 And I keep stressing that you want to draw cartoons. 49 00:03:10,090 --> 00:03:15,480 And this is the way you get into those cartoons. 50 00:03:15,480 --> 00:03:23,190 OK, now, for the free particle, we 51 00:03:23,190 --> 00:03:28,860 have these kinds of wave functions, e to the ikx. 52 00:03:28,860 --> 00:03:33,210 And they're really great because often, we 53 00:03:33,210 --> 00:03:40,250 want to evaluate things like the integral of e to the ikx times 54 00:03:40,250 --> 00:03:43,490 e to the minus ikx dx. 55 00:03:43,490 --> 00:03:47,300 Well, that's one. 56 00:03:47,300 --> 00:03:50,330 And so instead of remembering that we 57 00:03:50,330 --> 00:03:55,460 have to evaluate trigonometric integrals, sine, sine theta, 58 00:03:55,460 --> 00:03:57,440 cosine, we can do this. 59 00:03:57,440 --> 00:04:00,440 And it really simplifies life. 60 00:04:00,440 --> 00:04:08,240 Now, using that insight, I'm asking you, OK, 61 00:04:08,240 --> 00:04:12,200 the expectation value for the momentum could be-- 62 00:04:22,540 --> 00:04:27,640 OK, if this is the expectation value for the momentum, what 63 00:04:27,640 --> 00:04:29,950 is psi of x? 64 00:04:33,400 --> 00:04:36,100 I promised I would ask you this question. 65 00:04:36,100 --> 00:04:38,630 I don't know if anybody really thought about it. 66 00:04:38,630 --> 00:04:44,880 But first of all, we're talking about the free particle. 67 00:04:44,880 --> 00:04:48,500 And this is some sort of eigenfunction 68 00:04:48,500 --> 00:04:51,540 of the Hamiltonian for the free particle. 69 00:04:51,540 --> 00:04:55,305 So what can we say about k? 70 00:05:00,600 --> 00:05:02,940 We have two parts to the wave function. 71 00:05:02,940 --> 00:05:04,580 We have an e to the ikx. 72 00:05:04,580 --> 00:05:06,990 And we have a e to the minus ikx. 73 00:05:06,990 --> 00:05:10,470 And the k is the same for both of them. 74 00:05:10,470 --> 00:05:14,840 So with all those hints, what is this? 75 00:05:14,840 --> 00:05:19,130 We have e to the ikx times something. 76 00:05:19,130 --> 00:05:25,354 And we have plus e to the minus ikx. 77 00:05:25,354 --> 00:05:27,020 So what are the somethings that go here? 78 00:05:34,291 --> 00:05:34,790 Yes? 79 00:05:34,790 --> 00:05:35,540 AUDIENCE: A and B? 80 00:05:35,540 --> 00:05:37,490 ROBERT FIELD: Right. 81 00:05:37,490 --> 00:05:42,080 And that didn't involve very much mental gymnastics 82 00:05:42,080 --> 00:05:44,960 if you really have done a little bit of practicing 83 00:05:44,960 --> 00:05:47,850 of integrals involving these sorts of things. 84 00:05:47,850 --> 00:05:49,110 Now there's two things. 85 00:05:49,110 --> 00:05:53,900 One is when you have the same k in e to the ikx 86 00:05:53,900 --> 00:05:55,340 and e to the minus ikx. 87 00:05:55,340 --> 00:05:58,150 The product is one. 88 00:05:58,150 --> 00:06:03,000 If you have different k's and you're 89 00:06:03,000 --> 00:06:08,970 integrating over all space, or over some cleverly chosen 90 00:06:08,970 --> 00:06:11,165 region, that integral is zero. 91 00:06:14,080 --> 00:06:17,830 Because these are eigenfunctions of-- 92 00:06:17,830 --> 00:06:20,050 these are different eigenfunctions 93 00:06:20,050 --> 00:06:24,200 of an operator belonging to different eigenvalues. 94 00:06:24,200 --> 00:06:27,110 And you always can count on those things being zero. 95 00:06:27,110 --> 00:06:31,270 Now, quantum mechanics is full of integrals. 96 00:06:31,270 --> 00:06:34,410 Basically, there's an infinite number of them. 97 00:06:34,410 --> 00:06:37,140 And most of them are zero. 98 00:06:37,140 --> 00:06:39,540 And you want to be able to look at an integral 99 00:06:39,540 --> 00:06:42,035 and say, oh, I don't need to evaluate that. 100 00:06:42,035 --> 00:06:44,160 And often, you want to look at an integral and say, 101 00:06:44,160 --> 00:06:46,140 I do know how to evaluate that. 102 00:06:46,140 --> 00:06:49,900 And I know an infinite number of those like that. 103 00:06:49,900 --> 00:06:53,460 And all of a sudden, it starts to be transparent again. 104 00:06:53,460 --> 00:06:59,250 Because the barrier between insight and quantum 105 00:06:59,250 --> 00:07:03,110 mechanics is usually a whole bunch of integrals. 106 00:07:03,110 --> 00:07:05,280 And they're all yours. 107 00:07:05,280 --> 00:07:12,000 And so we like problems where the wave functions 108 00:07:12,000 --> 00:07:14,260 have simple forms. 109 00:07:14,260 --> 00:07:16,710 And this is true for free particle. 110 00:07:16,710 --> 00:07:18,455 It's true for the particle in a box. 111 00:07:21,114 --> 00:07:22,530 We're going to start talking about 112 00:07:22,530 --> 00:07:24,880 their harmonic oscillator. 113 00:07:24,880 --> 00:07:30,200 And it seems like those integrals are not simple, 114 00:07:30,200 --> 00:07:31,580 but they are. 115 00:07:31,580 --> 00:07:33,690 I have to teach you why they're not simple. 116 00:07:33,690 --> 00:07:39,740 So today, we're starting on the classical mechanical treatment 117 00:07:39,740 --> 00:07:41,570 of the harmonic oscillator. 118 00:07:41,570 --> 00:07:46,020 Then we'll do the traditional quantum mechanical treatment. 119 00:07:46,020 --> 00:07:49,430 And then, we'll come back and use these creation 120 00:07:49,430 --> 00:07:51,600 and annililation operators, which 121 00:07:51,600 --> 00:07:54,270 are the magic decoders for essentially evaluating 122 00:07:54,270 --> 00:07:57,940 all the integrals trivially. 123 00:07:57,940 --> 00:08:00,640 And then, with all that in hand, we're 124 00:08:00,640 --> 00:08:03,370 going to make our first step into time-dependent quantum 125 00:08:03,370 --> 00:08:05,420 mechanics. 126 00:08:05,420 --> 00:08:07,400 And we're going to use time-dependent quantum 127 00:08:07,400 --> 00:08:08,280 mechanics. 128 00:08:08,280 --> 00:08:11,860 Well, we're going to use our facility with integrals 129 00:08:11,860 --> 00:08:15,700 to describe the properties of some particle-like state we 130 00:08:15,700 --> 00:08:17,350 can construct. 131 00:08:17,350 --> 00:08:21,160 And these constructions are really simple 132 00:08:21,160 --> 00:08:24,970 for the particle in a box or the harmonic oscillator, 133 00:08:24,970 --> 00:08:28,020 depending on which properties you want. 134 00:08:28,020 --> 00:08:33,450 So your ability to draw cartoons and to use classical insights 135 00:08:33,450 --> 00:08:37,620 for the particle in a box and the harmonic oscillator 136 00:08:37,620 --> 00:08:41,100 will be incredibly valuable once we take our first step 137 00:08:41,100 --> 00:08:43,799 into the reality of time-dependent quantum 138 00:08:43,799 --> 00:08:46,270 mechanics. 139 00:08:46,270 --> 00:08:50,380 Now, one of the things that's going to happen 140 00:08:50,380 --> 00:08:56,260 is that we're going to describe real situations, 141 00:08:56,260 --> 00:08:59,580 real situations that are not one of the standard solved 142 00:08:59,580 --> 00:09:00,880 problems. 143 00:09:00,880 --> 00:09:09,440 The standard solved problems are the particle in a box, 144 00:09:09,440 --> 00:09:13,970 or a particle in the infinite box, the harmonic oscillator, 145 00:09:13,970 --> 00:09:19,970 the hydrogen atom, and the rigid rotor. 146 00:09:19,970 --> 00:09:22,520 And the particle in a box-- 147 00:09:22,520 --> 00:09:25,770 so the potential for each of these is different. 148 00:09:25,770 --> 00:09:27,220 It looks like this. 149 00:09:27,220 --> 00:09:28,860 It looks like this. 150 00:09:28,860 --> 00:09:31,110 It looks like this. 151 00:09:31,110 --> 00:09:34,520 And it looks like that. 152 00:09:34,520 --> 00:09:38,230 So there is no stretching in a rigid rotor. 153 00:09:38,230 --> 00:09:42,250 And so all of the complexity is in the kinetic energy, not 154 00:09:42,250 --> 00:09:43,240 the potential. 155 00:09:43,240 --> 00:09:47,350 But each of these has a different potential energy. 156 00:09:47,350 --> 00:09:51,486 And the energy levels for the particle in a box 157 00:09:51,486 --> 00:09:52,735 are proportional to n squared. 158 00:09:57,390 --> 00:09:59,170 For the harmonic oscillator, they're 159 00:09:59,170 --> 00:10:03,280 proportional to n plus 1/2. 160 00:10:03,280 --> 00:10:06,310 For the hydrogen atom, they are proportional to 1 161 00:10:06,310 --> 00:10:09,470 over n squared. 162 00:10:09,470 --> 00:10:12,650 And for the rigid rotor, they're proportional to n times n 163 00:10:12,650 --> 00:10:14,390 plus 1. 164 00:10:14,390 --> 00:10:16,640 So one of the valuable things you 165 00:10:16,640 --> 00:10:21,020 get from looking at these exactly soluble problems 166 00:10:21,020 --> 00:10:24,350 is that the energy level patterns for each of them 167 00:10:24,350 --> 00:10:26,710 are slightly different. 168 00:10:26,710 --> 00:10:30,160 And you can tell what you've got from the energy level pattern. 169 00:10:30,160 --> 00:10:33,210 And so when you take a spectrum, so often you want to know 170 00:10:33,210 --> 00:10:34,460 what kind of spectrum is this? 171 00:10:34,460 --> 00:10:38,170 Sometimes you can tell just by what frequency region it is. 172 00:10:38,170 --> 00:10:40,210 But usually, in a spectrum, there's 173 00:10:40,210 --> 00:10:42,040 a pattern of energy levels. 174 00:10:42,040 --> 00:10:47,140 And that gets you focused on well, what kind of problem-- 175 00:10:47,140 --> 00:10:51,339 what is the nature of the building 176 00:10:51,339 --> 00:10:52,630 blocks that we're going to use? 177 00:10:56,660 --> 00:10:58,420 So this is wonderful. 178 00:10:58,420 --> 00:11:03,040 Now, we're also going to find that when 179 00:11:03,040 --> 00:11:05,950 we solve these problems in quantum mechanics, 180 00:11:05,950 --> 00:11:11,680 we get an infinite number of eigenfunctions and eigenvalues. 181 00:11:11,680 --> 00:11:16,450 And often, we get presented to us a lot of integrals 182 00:11:16,450 --> 00:11:23,230 involving the operators between different wave functions. 183 00:11:23,230 --> 00:11:24,790 And one of the beautiful things is 184 00:11:24,790 --> 00:11:27,670 when the theory gives you an infinite number 185 00:11:27,670 --> 00:11:29,230 of those integrals. 186 00:11:29,230 --> 00:11:31,750 So now we would have a collection 187 00:11:31,750 --> 00:11:33,340 of all sorts of integrals that are 188 00:11:33,340 --> 00:11:39,360 evaluated for us in a simple dependence on quantum numbers. 189 00:11:39,360 --> 00:11:41,350 So what do we do with them? 190 00:11:41,350 --> 00:11:44,780 Well, there's two main things we do with these infinite numbers 191 00:11:44,780 --> 00:11:46,310 of integrals. 192 00:11:46,310 --> 00:11:50,360 One is we say, well, this problem is not 193 00:11:50,360 --> 00:11:51,800 the standard problem. 194 00:11:51,800 --> 00:11:53,150 There are defects. 195 00:11:53,150 --> 00:11:56,230 Instead of having the potential for the harmonic oscillator 196 00:11:56,230 --> 00:11:59,450 that looked like this, it might look like that. 197 00:11:59,450 --> 00:12:02,950 There might be some anharmonicity. 198 00:12:02,950 --> 00:12:07,560 Or there can be all sorts of defects. 199 00:12:07,560 --> 00:12:10,160 And these defects can be expressed by integrals 200 00:12:10,160 --> 00:12:12,310 of some other operator. 201 00:12:12,310 --> 00:12:15,530 And that goes into perturbation theory. 202 00:12:15,530 --> 00:12:18,470 Or if we're going to want to look at wavepackets, 203 00:12:18,470 --> 00:12:20,810 creating a particle-like state and asking 204 00:12:20,810 --> 00:12:26,310 how it propagates in time, those integrals are useful. 205 00:12:26,310 --> 00:12:29,180 So there's all sorts of fantastic stuff, 206 00:12:29,180 --> 00:12:32,690 once we kill the standard problems. 207 00:12:32,690 --> 00:12:36,350 It's not just some thing you have to study, 208 00:12:36,350 --> 00:12:38,250 and it's over with. 209 00:12:38,250 --> 00:12:39,620 You're going to use this. 210 00:12:39,620 --> 00:12:42,440 As long as you're involved in physical chemistry, 211 00:12:42,440 --> 00:12:45,170 you're going to be using perturbation theory 212 00:12:45,170 --> 00:12:50,130 to understand what's going on beyond the simple stuff. 213 00:12:50,130 --> 00:12:56,595 OK, so let's get started. 214 00:12:59,480 --> 00:13:02,680 So why is the harmonic oscillator so special? 215 00:13:02,680 --> 00:13:06,910 If we look at any potential energy curve, any one 216 00:13:06,910 --> 00:13:11,200 dimensional problem, it's typically 217 00:13:11,200 --> 00:13:13,960 harmonic at the bottom, no matter what it does. 218 00:13:13,960 --> 00:13:18,330 I mean, this is a typical molecular diatomic molecule 219 00:13:18,330 --> 00:13:19,420 potential. 220 00:13:19,420 --> 00:13:22,000 We have a hard inner wall. 221 00:13:22,000 --> 00:13:25,570 And we have bond breaking at the outer wall. 222 00:13:25,570 --> 00:13:28,040 And so this is an anharmonic potential. 223 00:13:28,040 --> 00:13:33,120 But the bottom part is harmonic. 224 00:13:33,120 --> 00:13:37,800 And so we can use everything we learn 225 00:13:37,800 --> 00:13:39,660 about the harmonic oscillator to begin 226 00:13:39,660 --> 00:13:44,780 to draw a picture of arbitrary potentials. 227 00:13:44,780 --> 00:13:54,695 So this is hard wall bond breaking. 228 00:13:57,530 --> 00:13:59,300 Now, we're mostly chemists. 229 00:13:59,300 --> 00:14:02,540 And breaking a bond is-- 230 00:14:02,540 --> 00:14:04,440 how much energy does it take to break a bond? 231 00:14:04,440 --> 00:14:06,023 Well, is that encoded in the spectrum? 232 00:14:06,023 --> 00:14:07,650 Yeah. 233 00:14:07,650 --> 00:14:10,860 So this is the sort of thing we would care about. 234 00:14:10,860 --> 00:14:18,580 And now let me put some notation here. 235 00:14:18,580 --> 00:14:21,360 So this is the potential. 236 00:14:21,360 --> 00:14:26,440 And this is the potential of-- the internuclear resistance, 237 00:14:26,440 --> 00:14:30,000 which is traditionally called R. We're going to switch notations 238 00:14:30,000 --> 00:14:32,160 really quickly from R to x. 239 00:14:32,160 --> 00:14:36,460 But this, the minimum of the potential, is the equilibrium 240 00:14:36,460 --> 00:14:38,834 internuclear distance R sub e. 241 00:14:42,810 --> 00:14:52,680 So for R near R sub e the potential V of R 242 00:14:52,680 --> 00:14:59,730 looks simply like k over 2, a forced constant, R minus R sub 243 00:14:59,730 --> 00:15:00,345 e squared. 244 00:15:05,980 --> 00:15:09,580 So this is harmonic oscillator. 245 00:15:09,580 --> 00:15:12,920 And now we're going to change notation a little bit. 246 00:15:12,920 --> 00:15:18,010 We're going to use lowercase x to be R minus Re. 247 00:15:18,010 --> 00:15:21,040 In other words, it's the distortion away 248 00:15:21,040 --> 00:15:23,550 from equilibrium. 249 00:15:23,550 --> 00:15:28,830 And we can take any potential and write it as a power series. 250 00:15:51,262 --> 00:15:51,762 Sorry. 251 00:16:00,670 --> 00:16:01,290 And so on. 252 00:16:04,320 --> 00:16:10,035 So if we know these derivatives, we know what to do with them. 253 00:16:10,035 --> 00:16:10,535 OK. 254 00:16:20,900 --> 00:16:27,870 So one standard potential is called the Morse potential. 255 00:16:27,870 --> 00:16:29,710 Because it looks like this. 256 00:16:29,710 --> 00:16:33,530 It looks like what you need for a diatomic molecule. 257 00:16:33,530 --> 00:16:36,200 And the Morse potential has an analytic form, 258 00:16:36,200 --> 00:16:41,400 where the potential v of x is equal to D sub 259 00:16:41,400 --> 00:16:46,185 e, the dissociation energy, 1 minus e to the minus ax. 260 00:16:49,050 --> 00:16:52,640 Well, that doesn't look like polynomial of x. 261 00:16:52,640 --> 00:16:56,930 But we power series expand this, and we get a polynomial x. 262 00:16:56,930 --> 00:16:59,610 OK, now, what is this D sub e? 263 00:16:59,610 --> 00:17:03,890 So if x is equal to 0-- 264 00:17:03,890 --> 00:17:05,810 this is 1 0. 265 00:17:05,810 --> 00:17:08,490 And so we get the energy. 266 00:17:08,490 --> 00:17:12,859 The potential at x equals 0 is 0. 267 00:17:12,859 --> 00:17:17,359 And if x is equal to infinity, v of x 268 00:17:17,359 --> 00:17:21,630 equals D sub e, so basically, this 269 00:17:21,630 --> 00:17:26,099 is the energy between the bottom of the well 270 00:17:26,099 --> 00:17:27,490 and where it breaks. 271 00:17:27,490 --> 00:17:30,050 This is D sub e. 272 00:17:30,050 --> 00:17:30,550 OK? 273 00:17:30,550 --> 00:17:34,890 So this is well interpreted. 274 00:17:34,890 --> 00:17:36,630 All the rest of the action is in here. 275 00:17:36,630 --> 00:17:41,300 AUDIENCE: [INAUDIBLE] 276 00:17:41,300 --> 00:17:42,300 ROBERT FIELD: I'm sorry? 277 00:17:42,300 --> 00:17:44,170 AUDIENCE: Do you just square the x? 278 00:17:44,170 --> 00:17:46,400 ROBERT FIELD: Yep! 279 00:17:46,400 --> 00:17:47,000 Thank you. 280 00:17:50,150 --> 00:17:51,580 It's an innocent factor. 281 00:17:51,580 --> 00:17:54,010 But it turns out to be very important. 282 00:17:54,010 --> 00:17:57,300 OK, so let's square this. 283 00:17:57,300 --> 00:18:02,740 We have v of x is equal to De times 1 284 00:18:02,740 --> 00:18:12,245 minus 2e to the minus ax plus e to the minus 2ax. 285 00:18:16,190 --> 00:18:19,640 OK, so we're going to do power series expansions of these. 286 00:18:19,640 --> 00:18:21,910 And you can do that. 287 00:18:21,910 --> 00:18:32,950 And so we have plus some function 288 00:18:32,950 --> 00:18:43,660 of x plus some function of x squared, et cetera. 289 00:18:43,660 --> 00:18:47,270 Now, what is the function of x? 290 00:18:47,270 --> 00:18:53,320 We're taking our derivatives at the equilibrium, at x equals 0. 291 00:18:53,320 --> 00:18:58,110 And we have a potential, which has a minimum. 292 00:18:58,110 --> 00:19:02,050 So what's the derivative of the potential at equilibrium? 293 00:19:02,050 --> 00:19:02,550 Yes? 294 00:19:02,550 --> 00:19:03,150 AUDIENCE: Zero. 295 00:19:03,150 --> 00:19:03,983 ROBERT FIELD: Right. 296 00:19:03,983 --> 00:19:06,650 And so the x term goes away. 297 00:19:06,650 --> 00:19:08,730 And so we have a constant term, which we usually 298 00:19:08,730 --> 00:19:11,630 choose to be zero. 299 00:19:11,630 --> 00:19:16,420 And we have this quadratic term. 300 00:19:16,420 --> 00:19:18,440 Looks like a harmonic oscillator. 301 00:19:18,440 --> 00:19:19,960 And then there are other terms which 302 00:19:19,960 --> 00:19:24,310 express the personality of the potential. 303 00:19:24,310 --> 00:19:25,240 This is universal. 304 00:19:25,240 --> 00:19:30,290 And the rest becomes a special case. 305 00:19:30,290 --> 00:19:39,850 So we have v of x is De a squared 306 00:19:39,850 --> 00:19:44,770 x squared plus other stuff. 307 00:19:44,770 --> 00:19:46,530 OK, we're going to call this 1/2k. 308 00:19:49,350 --> 00:19:49,850 why? 309 00:19:49,850 --> 00:19:53,210 Because we'd like it to look like a harmonic oscillator. 310 00:19:53,210 --> 00:19:55,430 I mean, we know that the potential 311 00:19:55,430 --> 00:19:59,930 for a harmonic oscillator is described in this way. 312 00:20:02,810 --> 00:20:07,510 So let's just draw a picture. 313 00:20:15,840 --> 00:20:19,740 So we have a spring and a mass. 314 00:20:19,740 --> 00:20:24,960 And I should have drawn this up a little higher. 315 00:20:24,960 --> 00:20:29,340 OK, so at equilibrium there is no force. 316 00:20:29,340 --> 00:20:33,850 If the mass is down here, there is a force pulling it up. 317 00:20:33,850 --> 00:20:36,740 And if it's up here, there's a force pushing it down. 318 00:20:36,740 --> 00:20:48,390 Hooke's Law says that the force is 319 00:20:48,390 --> 00:20:56,380 equal to minus k x minus x 0. 320 00:20:56,380 --> 00:21:01,420 OK, and now we're going to switch to just the lowercase x 321 00:21:01,420 --> 00:21:02,710 in a second. 322 00:21:02,710 --> 00:21:09,860 But now, the force according to Newton 323 00:21:09,860 --> 00:21:17,530 is minus gradient of the potential. 324 00:21:17,530 --> 00:21:20,970 So the potential for this problem 325 00:21:20,970 --> 00:21:28,382 is v of little x is equal to 1/2k x squared. 326 00:21:32,380 --> 00:21:39,310 OK, so we have what we expect for a harmonic oscillator. 327 00:21:39,310 --> 00:21:44,700 And we're going to say the small displacement part of the Morse 328 00:21:44,700 --> 00:21:46,950 oscillator looks like 1/2k. 329 00:21:46,950 --> 00:21:48,910 x squared looks harmonic. 330 00:21:48,910 --> 00:21:50,550 And so what do we do now? 331 00:21:54,970 --> 00:22:02,250 Well, one of Newton's laws, force 332 00:22:02,250 --> 00:22:04,770 is equal to the mass times the acceleration. 333 00:22:04,770 --> 00:22:14,490 And so we can say, oh, well, the acceleration 334 00:22:14,490 --> 00:22:22,190 is the second derivative of x with respect to t. 335 00:22:22,190 --> 00:22:29,850 And the force is a minus gradient of the potential. 336 00:22:29,850 --> 00:22:32,300 So it's minus 1/2kx squared. 337 00:22:37,620 --> 00:22:38,370 Minus kx. 338 00:22:41,680 --> 00:22:45,900 OK, so this is the gradient of the potential. 339 00:22:45,900 --> 00:22:48,790 This is the mass-- 340 00:22:48,790 --> 00:22:50,570 well, I need the mass-- 341 00:22:50,570 --> 00:22:52,116 times the acceleration. 342 00:22:52,116 --> 00:22:53,990 And so this is the equation we have to solve. 343 00:23:00,590 --> 00:23:05,680 And so we want to find the solution x of t. 344 00:23:05,680 --> 00:23:08,080 And this is a second order differential equation 345 00:23:08,080 --> 00:23:10,760 because we have a second derivative. 346 00:23:10,760 --> 00:23:12,830 So there's going to be two terms. 347 00:23:12,830 --> 00:23:21,530 And they'll be-- we'll have some sine and some cosine function. 348 00:23:21,530 --> 00:23:24,940 And we're going to want a derivative, 349 00:23:24,940 --> 00:23:30,460 a second derivative, that brings down a constant k. 350 00:23:30,460 --> 00:23:32,780 And so we know that these are going 351 00:23:32,780 --> 00:23:45,500 to be things that have the form a sine kx k or m square root 352 00:23:45,500 --> 00:23:56,840 x plus b cosine k m x. 353 00:23:56,840 --> 00:23:58,620 So this is the solution. 354 00:23:58,620 --> 00:24:00,560 And we have to find A and B. 355 00:24:00,560 --> 00:24:02,870 Now, why do we have k over m? 356 00:24:02,870 --> 00:24:05,550 Well, you can look at this differential equation. 357 00:24:05,550 --> 00:24:07,385 And you can see that we have an m here. 358 00:24:07,385 --> 00:24:08,510 We have a k here. 359 00:24:08,510 --> 00:24:11,180 And that's what you need in order to solve it. 360 00:24:11,180 --> 00:24:14,340 So now our job is simply to find A and B. 361 00:24:14,340 --> 00:24:19,740 AUDIENCE: Should these be functions of t and not x? 362 00:24:19,740 --> 00:24:21,560 ROBERT FIELD: You know, sometimes 363 00:24:21,560 --> 00:24:27,480 I go onto automatic pilot because it's so familiar to me. 364 00:24:27,480 --> 00:24:31,030 I'm just writing what comes up in my subconscious. 365 00:24:31,030 --> 00:24:34,860 But yes, that's a good point. 366 00:24:34,860 --> 00:24:37,350 All right, so we want x of t. 367 00:24:37,350 --> 00:24:40,470 And it is this combination. 368 00:24:40,470 --> 00:24:43,635 OK, so now we put in some insights. 369 00:24:52,920 --> 00:24:56,340 We want to know what the period of oscillation is. 370 00:24:56,340 --> 00:25:04,860 And so x t plus tau has to be equal to x of t. 371 00:25:07,520 --> 00:25:12,210 And so when we do that, we discover-- 372 00:25:12,210 --> 00:25:14,700 and I'm just going to skip a lot of steps-- 373 00:25:14,700 --> 00:25:26,210 that k over m square root times tau has to be equal to 2 pi 374 00:25:26,210 --> 00:25:27,350 in order to satisfy that. 375 00:25:35,490 --> 00:25:42,230 We call k over m square root omega, just 376 00:25:42,230 --> 00:25:47,090 to simplify the equations. 377 00:25:47,090 --> 00:25:49,610 But we also discover that this actually 378 00:25:49,610 --> 00:25:51,650 is an angular frequency. 379 00:25:55,980 --> 00:26:04,390 So if we say omega tau is equal to 2 pi, 380 00:26:04,390 --> 00:26:09,690 then tau equals 2 pi over omega, which is 381 00:26:09,690 --> 00:26:13,140 equal to 1 over the frequency. 382 00:26:13,140 --> 00:26:14,680 Just exactly what we expect. 383 00:26:19,150 --> 00:26:26,430 So we have the beginning of a solution and omega, tau, 384 00:26:26,430 --> 00:26:32,190 and frequency make sense. 385 00:26:35,640 --> 00:26:38,790 Everything is what we sort of expect. 386 00:26:38,790 --> 00:26:41,460 OK, so now the next step is to determine 387 00:26:41,460 --> 00:26:43,560 the values of the constants. 388 00:26:48,950 --> 00:26:52,810 So normally, when we have a differential equation, 389 00:26:52,810 --> 00:27:01,530 after we find the general form, we apply boundary conditions. 390 00:27:01,530 --> 00:27:06,550 And so we're going to apply some boundary conditions. 391 00:27:06,550 --> 00:27:09,130 So here we have the potential. 392 00:27:09,130 --> 00:27:26,450 And-- so this is the potential as a function of coordinate. 393 00:27:26,450 --> 00:27:30,740 And this is the turning point, this 394 00:27:30,740 --> 00:27:33,160 is the inner turning point at energy E. 395 00:27:33,160 --> 00:27:36,860 This is the outer turning point. 396 00:27:36,860 --> 00:27:39,380 Well, what's true at the turning point? 397 00:27:39,380 --> 00:27:42,020 The turning point, the potential is 398 00:27:42,020 --> 00:27:49,100 equal to the energy at the two turning points. 399 00:27:49,100 --> 00:27:58,790 So 1/2k x plus or minus of E squared. 400 00:27:58,790 --> 00:28:02,900 So if we know E, we know where the turning points are. 401 00:28:02,900 --> 00:28:12,030 And so x of plus minus of E, we can solve this. 402 00:28:12,030 --> 00:28:19,740 And so we have 2E over k. 403 00:28:19,740 --> 00:28:20,240 Yeah. 404 00:28:27,200 --> 00:28:28,602 So this is the turning points. 405 00:28:28,602 --> 00:28:29,810 These are the turning points. 406 00:28:32,750 --> 00:28:36,810 Now, this is a fairly frequent exercise in quantum mechanics. 407 00:28:36,810 --> 00:28:41,180 You're going to want to know where are the turning points. 408 00:28:41,180 --> 00:28:44,420 Because this is how you impose boundary conditions easily. 409 00:28:44,420 --> 00:28:47,780 And so knowing that a turning point corresponds 410 00:28:47,780 --> 00:28:50,570 to where the potential is equal to the total energy 411 00:28:50,570 --> 00:28:54,820 is enough to be able to solve for this. 412 00:28:54,820 --> 00:29:07,600 OK, so suppose we start at x equals x plus. 413 00:29:07,600 --> 00:29:14,620 And so x of 0 is equal to x plus. 414 00:29:14,620 --> 00:29:19,730 And that determines one of the coefficients. 415 00:29:19,730 --> 00:29:22,690 So x of 0-- 416 00:29:22,690 --> 00:29:29,500 so we have at t equals 0 the sine term is 0 417 00:29:29,500 --> 00:29:32,580 and the cosine term is 1. 418 00:29:32,580 --> 00:29:39,440 And so the first thing we get at t 419 00:29:39,440 --> 00:29:48,307 equals 0 is that B is equal to x plus. 420 00:29:48,307 --> 00:29:48,806 Right? 421 00:29:52,070 --> 00:29:57,224 The next step is to find a. 422 00:29:57,224 --> 00:29:58,640 There are several ways to do this. 423 00:29:58,640 --> 00:30:03,220 But it's useful to draw a little picture. 424 00:30:03,220 --> 00:30:06,550 So x plus is here. 425 00:30:06,550 --> 00:30:12,190 And that occurs at t equals 0. 426 00:30:12,190 --> 00:30:17,880 And x minus occurs at tau over 2. 427 00:30:17,880 --> 00:30:20,020 And in the middle we have tau over 4. 428 00:30:22,710 --> 00:30:29,190 So let's ask for, what is the value of the wave function 429 00:30:29,190 --> 00:30:33,150 when x is equal to 0? 430 00:30:33,150 --> 00:30:38,010 So how do we make the x be 0 at tau over 4? 431 00:30:38,010 --> 00:30:45,750 And that determines the value of B. I mean, the value of A. 432 00:30:45,750 --> 00:30:49,906 So we have everything we need. 433 00:30:49,906 --> 00:30:54,480 And now, before just rushing on, let me just say, 434 00:30:54,480 --> 00:31:02,160 well, this just gives A is equal to zero. 435 00:31:02,160 --> 00:31:05,650 OK, there's a different approach. 436 00:31:05,650 --> 00:31:10,230 When we have a sine plus a cosine term, 437 00:31:10,230 --> 00:31:14,100 we can always re-express it as x of t 438 00:31:14,100 --> 00:31:22,550 is equal to some other constant times sine omega t plus phi. 439 00:31:28,620 --> 00:31:31,480 And so the same sort of analysis gives 440 00:31:31,480 --> 00:31:39,330 c is equal to E over k square root. 441 00:31:39,330 --> 00:31:44,900 And phi is equal to minus pi over 2. 442 00:31:44,900 --> 00:31:46,377 OK, I'm just writing this. 443 00:31:46,377 --> 00:31:47,210 You want to do that. 444 00:31:54,740 --> 00:32:05,000 OK, so now we're going to be preparing 445 00:32:05,000 --> 00:32:09,830 to do quantum mechanics, the quantum mechanical solution 446 00:32:09,830 --> 00:32:12,300 of the harmonic oscillator. 447 00:32:12,300 --> 00:32:15,020 And so there are going to be other things that we care 448 00:32:15,020 --> 00:32:18,570 about, and one is the kinetic energy 449 00:32:18,570 --> 00:32:20,400 and one is the potential energy. 450 00:32:23,420 --> 00:32:27,230 And in particular, we'd like to know the kinetic energy, which 451 00:32:27,230 --> 00:32:28,670 we call t. 452 00:32:28,670 --> 00:32:33,700 And we'd like to know the expectation value of t 453 00:32:33,700 --> 00:32:37,280 as a function of time, or just T bar of t. 454 00:32:40,120 --> 00:32:46,940 And similarly, we'd like to know the expectation 455 00:32:46,940 --> 00:32:51,450 value of the potential energy as a function of time. 456 00:32:51,450 --> 00:32:58,190 And that's going to be V bar of t. 457 00:32:58,190 --> 00:33:00,680 And so from classical mechanics, we 458 00:33:00,680 --> 00:33:03,560 should be able to determine what these average values 459 00:33:03,560 --> 00:33:06,420 of the kinetic and potential energy. 460 00:33:06,420 --> 00:33:10,870 So what do we know? 461 00:33:10,870 --> 00:33:17,580 We know that the frequency is omega over 2 pi. 462 00:33:17,580 --> 00:33:22,360 We know the period is 1 over omega. 463 00:33:22,360 --> 00:33:35,280 OK, and so T of t is 1/2 mv squared of t, 464 00:33:35,280 --> 00:33:37,800 or p squared over 2m. 465 00:33:43,550 --> 00:33:50,120 But all right, v is the derivative 466 00:33:50,120 --> 00:33:52,670 of x with respect to t. 467 00:33:52,670 --> 00:33:55,500 And we have the solutions over here. 468 00:33:55,500 --> 00:34:07,480 And so we know that we can write x of t and v of t 469 00:34:07,480 --> 00:34:08,800 just by taking derivatives. 470 00:34:08,800 --> 00:34:17,710 And so we have x of t is 2e over k 471 00:34:17,710 --> 00:34:23,934 square root sine omega t plus phi. 472 00:34:26,780 --> 00:34:41,370 And v of t is omega times 2e over k square root cosine omega 473 00:34:41,370 --> 00:34:42,604 t plus phi. 474 00:34:45,510 --> 00:34:48,790 So we want v squared to be able to calculate 475 00:34:48,790 --> 00:34:50,550 the kinetic energy. 476 00:34:50,550 --> 00:34:51,659 And so we do that. 477 00:34:54,250 --> 00:35:01,860 And so the kinetic energy T of t is 478 00:35:01,860 --> 00:35:16,170 1/2m m omega squared 2e over k cosine squared 479 00:35:16,170 --> 00:35:18,430 omega t plus phi. 480 00:35:22,290 --> 00:35:27,780 This here, omega is k over m, the square root of k over m. 481 00:35:27,780 --> 00:35:30,740 S this is m times k over m. 482 00:35:30,740 --> 00:35:32,100 So we just get k out there. 483 00:35:40,070 --> 00:35:43,840 And so now we would like to know the average value 484 00:35:43,840 --> 00:35:45,460 of the kinetic energy. 485 00:35:55,210 --> 00:36:00,380 OK, so we have m omega squared. 486 00:36:00,380 --> 00:36:05,190 And so that's k over k. 487 00:36:05,190 --> 00:36:12,980 And so we just get E integral from 0 488 00:36:12,980 --> 00:36:27,470 to tau bt of cosine square omega t plus phi over tau. 489 00:36:27,470 --> 00:36:29,160 We want the time average. 490 00:36:29,160 --> 00:36:32,670 And so we calculate this integral and we divide by tau. 491 00:36:32,670 --> 00:36:36,000 That's how we take an average. 492 00:36:36,000 --> 00:36:47,870 And what we discover is that this integral is 493 00:36:47,870 --> 00:36:55,010 the numerator is 1/2 tau times E. No, I'm sorry. 494 00:36:55,010 --> 00:36:58,340 1/2 tau, we have E times 1/2 tau divided by tau. 495 00:36:58,340 --> 00:37:06,560 And this becomes E over 2, an important result. 496 00:37:06,560 --> 00:37:11,000 And we're going to discover that the average value 497 00:37:11,000 --> 00:37:13,820 of the momentum for a harmonic oscillator 498 00:37:13,820 --> 00:37:17,360 is E over 2 in quantum mechanics. 499 00:37:17,360 --> 00:37:21,400 We do the same thing for the potential. 500 00:37:21,400 --> 00:37:24,290 And we discover that it is also E over 2. 501 00:37:29,890 --> 00:37:40,240 So we know that E is equal to T of t plus V of t. 502 00:37:40,240 --> 00:37:44,470 But it's also true that the average is equal-- 503 00:37:44,470 --> 00:38:00,870 so T bar is equal to V bar which is equal to E over 2. 504 00:38:00,870 --> 00:38:04,080 So now we have an important interpretation 505 00:38:04,080 --> 00:38:06,130 of the harmonic oscillator. 506 00:38:06,130 --> 00:38:09,090 The harmonic oscillator is moving 507 00:38:09,090 --> 00:38:12,630 from turning point to the middle to the other turning point. 508 00:38:12,630 --> 00:38:15,690 And what's happening is energy is being exchanged 509 00:38:15,690 --> 00:38:21,570 between all potential energy at a turning point 510 00:38:21,570 --> 00:38:24,750 to all kinetic energy in the middle. 511 00:38:24,750 --> 00:38:27,420 So energy is going back and forth 512 00:38:27,420 --> 00:38:33,750 between kinetic energy and potential energy. 513 00:38:33,750 --> 00:38:40,490 We can solve for the relationship between T of t 514 00:38:40,490 --> 00:38:41,240 and V of t. 515 00:38:41,240 --> 00:38:53,770 And we can find that V of t is equal to T minus pi over 4. 516 00:38:58,730 --> 00:38:59,610 tau over 4. 517 00:39:04,170 --> 00:39:08,040 So this is telling us just what I said before, 518 00:39:08,040 --> 00:39:11,790 that energy is being exchanged between potential and kinetic 519 00:39:11,790 --> 00:39:12,630 energy. 520 00:39:12,630 --> 00:39:16,470 And that the potential energy is lagging by tau over 4 521 00:39:16,470 --> 00:39:20,220 behind the momentum. 522 00:39:24,734 --> 00:39:25,650 This is all very fast. 523 00:39:25,650 --> 00:39:29,210 But it's all classical mechanics, which you know. 524 00:39:29,210 --> 00:39:31,100 And we're going to be rediscovering all 525 00:39:31,100 --> 00:39:33,840 of this in quantum mechanics. 526 00:39:33,840 --> 00:39:36,080 And so we have to know what are we 527 00:39:36,080 --> 00:39:39,270 aiming for in quantum mechanics? 528 00:39:39,270 --> 00:39:41,270 So that we can completely say, yes, it's 529 00:39:41,270 --> 00:39:45,450 consistent with classical mechanics. 530 00:39:45,450 --> 00:39:46,870 And there's some really-- 531 00:39:46,870 --> 00:39:48,510 now, there's another really neat thing. 532 00:39:52,030 --> 00:40:07,480 So if we look at X of t, X of t is oscillating. 533 00:40:07,480 --> 00:40:13,929 And suppose we start out at a turning 534 00:40:13,929 --> 00:40:15,220 point, the outer turning point. 535 00:40:19,450 --> 00:40:23,970 So we can tell from this, the derivative of x with respect 536 00:40:23,970 --> 00:40:29,786 to t at x equals x plus is going to be 0. 537 00:40:33,440 --> 00:40:35,510 So at a turning point, the particle 538 00:40:35,510 --> 00:40:38,090 is hardly moving, not moving. 539 00:40:41,080 --> 00:40:44,470 And so what about the momentum? 540 00:40:44,470 --> 00:40:49,245 Well, the momentum is going to look like this. 541 00:40:53,270 --> 00:40:54,810 Yeah, it's going to look like that. 542 00:40:57,820 --> 00:41:07,020 And so at the time that we are starting at a turning point, 543 00:41:07,020 --> 00:41:08,910 the time derivative of the momentum 544 00:41:08,910 --> 00:41:10,260 is at its maximum value. 545 00:41:13,570 --> 00:41:17,310 So this is going to be really important when 546 00:41:17,310 --> 00:41:21,030 we start looking at properties of time-evolving wave 547 00:41:21,030 --> 00:41:21,540 functions. 548 00:41:21,540 --> 00:41:24,090 Because what we're going to discover is, 549 00:41:24,090 --> 00:41:30,420 suppose we start our system here, at a turning point. 550 00:41:30,420 --> 00:41:33,750 And that's actually something that we can do very easily 551 00:41:33,750 --> 00:41:35,670 in an experiment. 552 00:41:35,670 --> 00:41:43,830 Because when we excite from one electronic state to another, 553 00:41:43,830 --> 00:41:47,760 you automatically create a wave packet, 554 00:41:47,760 --> 00:41:50,220 which is localized at a particular internuclear 555 00:41:50,220 --> 00:41:50,720 distance. 556 00:41:53,460 --> 00:41:57,400 And so you go typically, to a turning point. 557 00:41:57,400 --> 00:42:00,630 So then, well, what's going to happen? 558 00:42:00,630 --> 00:42:02,910 Well, there is a thing in quantum mechanics, which 559 00:42:02,910 --> 00:42:05,880 you will become familiar with, called the autocorrelation 560 00:42:05,880 --> 00:42:07,320 function. 561 00:42:07,320 --> 00:42:07,910 No, it's not. 562 00:42:07,910 --> 00:42:09,570 It's called the survival probability. 563 00:42:18,550 --> 00:42:23,400 And that's going to be the product 564 00:42:23,400 --> 00:42:32,760 of the time-dependent wave function at x and T, 565 00:42:32,760 --> 00:42:37,380 the time-dependent wave function at x and 0. 566 00:42:37,380 --> 00:42:40,620 So this is expressing somehow how 567 00:42:40,620 --> 00:42:43,620 the wave function that is created at t 568 00:42:43,620 --> 00:42:47,740 equals 0 gets away from itself. 569 00:42:47,740 --> 00:42:49,440 It's a very important idea. 570 00:42:49,440 --> 00:42:51,450 Because you make something. 571 00:42:51,450 --> 00:42:53,580 And it evolves. 572 00:42:53,580 --> 00:42:57,300 And for a harmonic oscillator, if you make it at a turning 573 00:42:57,300 --> 00:43:00,360 point, this thing changes. 574 00:43:00,360 --> 00:43:03,320 Because the momentum changes. 575 00:43:03,320 --> 00:43:08,670 And the contribution of the coordinate 576 00:43:08,670 --> 00:43:15,120 to the decay of the survival probability is-- 577 00:43:15,120 --> 00:43:19,950 it's all due to the momentum, and not the coordinate change. 578 00:43:19,950 --> 00:43:22,380 That's actually, a very important insight. 579 00:43:22,380 --> 00:43:27,210 Because the momentum, the time rate of change of the momentum, 580 00:43:27,210 --> 00:43:30,610 is minus the gradient of the potential. 581 00:43:30,610 --> 00:43:34,380 So this is one way we learn about the potential 582 00:43:34,380 --> 00:43:37,350 simply by starting at a turning point 583 00:43:37,350 --> 00:43:40,380 and knowing that this thing, which we can measure, 584 00:43:40,380 --> 00:43:43,980 is measuring the thing we want to know. 585 00:43:43,980 --> 00:43:45,981 Now, we will get to this very soon. 586 00:43:45,981 --> 00:43:48,480 Because I haven't even told you about time-dependent quantum 587 00:43:48,480 --> 00:43:49,410 mechanics. 588 00:43:49,410 --> 00:43:52,664 But those are the things that we expect to encounter. 589 00:43:58,100 --> 00:44:02,440 OK, now I want to give you-- 590 00:44:02,440 --> 00:44:04,280 I'm going to throw at you-- 591 00:44:04,280 --> 00:44:09,770 some useful stuff, which turns out to be really easy. 592 00:44:09,770 --> 00:44:14,450 Suppose we want to know the expectation 593 00:44:14,450 --> 00:44:20,030 values, or the average values, of x, x squared, 594 00:44:20,030 --> 00:44:21,620 p, and p squared. 595 00:44:25,310 --> 00:44:29,490 OK, so we have a harmonic oscillator. 596 00:44:29,490 --> 00:44:34,400 And do we know what this is? 597 00:44:34,400 --> 00:44:37,868 Do we know the expectation value of the coordinate? 598 00:44:37,868 --> 00:44:38,740 AUDIENCE: 0. 599 00:44:38,740 --> 00:44:39,740 ROBERT FIELD: We have 0. 600 00:44:39,740 --> 00:44:41,330 Why is it 0? 601 00:44:41,330 --> 00:44:44,130 There's two ways of answering that question. 602 00:44:44,130 --> 00:44:47,660 But these are easy questions, which on an exam, 603 00:44:47,660 --> 00:44:49,310 you don't want to evaluate an integral. 604 00:44:49,310 --> 00:44:51,530 You want to know why is it 0. 605 00:44:51,530 --> 00:44:53,030 And there's two answers. 606 00:44:53,030 --> 00:44:54,950 You said 0, right? 607 00:44:54,950 --> 00:44:56,328 Why did you say 0? 608 00:44:56,328 --> 00:44:58,240 AUDIENCE: Because it's symmetric. 609 00:44:58,240 --> 00:44:59,430 ROBERT FIELD: Yes. 610 00:44:59,430 --> 00:45:03,100 OK, so there is a symmetry argument. 611 00:45:03,100 --> 00:45:06,930 Another is well, is the potential moving? 612 00:45:06,930 --> 00:45:08,440 The particle is in a potential. 613 00:45:08,440 --> 00:45:09,523 It's going back and forth. 614 00:45:09,523 --> 00:45:11,390 The potential is stationary. 615 00:45:11,390 --> 00:45:13,530 So there's no way that x could move. 616 00:45:13,530 --> 00:45:14,910 X could be time dependent. 617 00:45:14,910 --> 00:45:19,110 So we know this is 0. 618 00:45:19,110 --> 00:45:21,510 What about p? 619 00:45:21,510 --> 00:45:27,540 Same thing, whether you use symmetry or just 620 00:45:27,540 --> 00:45:29,784 physical insight. 621 00:45:29,784 --> 00:45:30,825 But what about x squared? 622 00:45:36,150 --> 00:45:44,050 Well, x squared is equal to V of x over k over 2. 623 00:45:48,650 --> 00:45:52,370 So expectation value of x squared 624 00:45:52,370 --> 00:46:00,800 is equal to the expectation value of V over k over 2. 625 00:46:00,800 --> 00:46:04,850 But we know the expectation value of V. It's E over 2. 626 00:46:13,920 --> 00:46:17,880 So we know without doing any integrals what 627 00:46:17,880 --> 00:46:21,300 the expectation value of x squared is, and similarly, 628 00:46:21,300 --> 00:46:22,050 for p squared. 629 00:46:26,980 --> 00:46:32,300 And that's just going to be m times E. OK, so why 630 00:46:32,300 --> 00:46:33,990 do I care about these things? 631 00:46:33,990 --> 00:46:37,360 Well, we have a little thing called the uncertainty 632 00:46:37,360 --> 00:46:39,190 principle. 633 00:46:39,190 --> 00:46:44,920 And we'd like to know the uncertainty in the coordinate 634 00:46:44,920 --> 00:46:46,090 and the momentum. 635 00:46:46,090 --> 00:46:48,230 And this is our classical view of it. 636 00:46:48,230 --> 00:46:52,510 But it's going to remind us of what 637 00:46:52,510 --> 00:46:54,400 we find quantum mechanically. 638 00:46:54,400 --> 00:46:59,530 So the uncertainty in x can be defined 639 00:46:59,530 --> 00:47:08,650 as the average value of x squared minus the average value 640 00:47:08,650 --> 00:47:11,910 of x squared square root. 641 00:47:11,910 --> 00:47:14,460 That's just the variance. 642 00:47:14,460 --> 00:47:17,085 And well, this is 0. 643 00:47:17,085 --> 00:47:19,090 And we know what this is. 644 00:47:19,090 --> 00:47:27,670 So we know that the uncertainty in x is E over k square root. 645 00:47:27,670 --> 00:47:30,400 And the uncertainty in p is going 646 00:47:30,400 --> 00:47:36,970 to be p squared average value minus p squared. 647 00:47:36,970 --> 00:47:38,330 This is still 0. 648 00:47:38,330 --> 00:47:44,060 And this one is then m times E square root. 649 00:47:47,010 --> 00:47:55,310 So delta x delta p, it looks like the uncertainty principle. 650 00:47:55,310 --> 00:48:01,235 This is classic mechanics, is just E over omega. 651 00:48:01,235 --> 00:48:02,710 But what's E over omega? 652 00:48:06,034 --> 00:48:06,534 AUDIENCE: h? 653 00:48:09,890 --> 00:48:10,950 h bar. 654 00:48:10,950 --> 00:48:12,450 ROBERT FIELD: Right. 655 00:48:12,450 --> 00:48:16,470 So it's all going to come around. 656 00:48:16,470 --> 00:48:21,360 This is related to the uncertainty principle 657 00:48:21,360 --> 00:48:22,500 in quantum mechanics. 658 00:48:22,500 --> 00:48:26,340 There is a minimum joint uncertainty between x and p. 659 00:48:26,340 --> 00:48:29,620 And it's just related to this constant. 660 00:48:29,620 --> 00:48:32,080 Now, this doesn't say the uncertainty grows 661 00:48:32,080 --> 00:48:34,510 as you go to higher and higher energy, 662 00:48:34,510 --> 00:48:36,280 as it will in quantum mechanics. 663 00:48:36,280 --> 00:48:39,080 But this is really a neat thing to see. 664 00:48:39,080 --> 00:48:41,020 OK, I've got two minutes left. 665 00:48:44,010 --> 00:48:49,760 So if we wanted to know the probability of finding 666 00:48:49,760 --> 00:49:03,279 x as a function of the coordinate, the turning points. 667 00:49:03,279 --> 00:49:04,570 So here are the turning points. 668 00:49:04,570 --> 00:49:07,330 And we can calculate what is going 669 00:49:07,330 --> 00:49:13,131 to be the probability of finding the particle at one turning 670 00:49:13,131 --> 00:49:13,630 point. 671 00:49:13,630 --> 00:49:15,640 Well, it comes down from infinity, 672 00:49:15,640 --> 00:49:18,620 goes back up to infinity. 673 00:49:18,620 --> 00:49:28,660 So here at the turning points, the particle is stopped. 674 00:49:28,660 --> 00:49:32,410 And so the probability of finding it at a turning point 675 00:49:32,410 --> 00:49:35,530 is infinite, times dx. 676 00:49:35,530 --> 00:49:39,636 So we don't have an infinite number. 677 00:49:39,636 --> 00:49:41,010 In quantum mechanics, we're going 678 00:49:41,010 --> 00:49:44,204 to discover that quantum mechanics is smarter than that. 679 00:49:44,204 --> 00:49:45,870 And what quantum mechanics does, suppose 680 00:49:45,870 --> 00:49:47,325 we have this is the energy. 681 00:49:51,040 --> 00:49:54,100 What quantum mechanics does is that the probability 682 00:49:54,100 --> 00:49:57,940 at a turning point is not 0. 683 00:49:57,940 --> 00:50:02,600 And there's something-- what happens 684 00:50:02,600 --> 00:50:08,490 is there's tunneling tails that instead of going to infinity, 685 00:50:08,490 --> 00:50:09,950 it has a finite value. 686 00:50:09,950 --> 00:50:14,270 And it reaches out into the forbidden region 687 00:50:14,270 --> 00:50:17,300 in an exponentially decreasing way. 688 00:50:17,300 --> 00:50:19,610 And that's basically the difference 689 00:50:19,610 --> 00:50:21,980 between classical mechanics and quantum mechanics. 690 00:50:21,980 --> 00:50:24,470 And there's an awful lot of important stuff 691 00:50:24,470 --> 00:50:25,780 that happens there. 692 00:50:25,780 --> 00:50:27,920 Now, I've been very fast through all of this 693 00:50:27,920 --> 00:50:32,084 because it's built on stuff that you're supposed to know. 694 00:50:32,084 --> 00:50:33,500 And it's built on stuff that we're 695 00:50:33,500 --> 00:50:36,840 going to work hard to understand from a quantum mechanical point 696 00:50:36,840 --> 00:50:37,340 of view. 697 00:50:37,340 --> 00:50:39,740 But this sets the stage for, what 698 00:50:39,740 --> 00:50:43,700 are the things we have to look for in quantum mechanics? 699 00:50:43,700 --> 00:50:47,880 So I do recommend that you look at the notes 700 00:50:47,880 --> 00:50:51,330 and make sure you can follow all the steps, which I went 701 00:50:51,330 --> 00:50:52,855 through incredibly rapidly. 702 00:50:56,260 --> 00:50:57,730 And it'll be really helpful. 703 00:50:57,730 --> 00:51:02,680 Because your job will be to draw cartoons. 704 00:51:02,680 --> 00:51:04,770 And these will guide you through it. 705 00:51:04,770 --> 00:51:09,989 OK, so I'll see you on Friday.