1 00:00:00,100 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,820 under a Creative Commons license. 3 00:00:03,820 --> 00:00:06,530 Your support will help MIT OpenCourseWare continue 4 00:00:06,530 --> 00:00:10,130 to offer high quality educational resources for free. 5 00:00:10,130 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:18,220 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:18,220 --> 00:00:18,720 ocw.mit.edu. 8 00:00:21,160 --> 00:00:23,660 PROFESSOR: Anything lingering and disturbing or bewildering? 9 00:00:28,750 --> 00:00:29,420 No? 10 00:00:29,420 --> 00:00:30,570 Nothing? 11 00:00:30,570 --> 00:00:32,430 All right. 12 00:00:32,430 --> 00:00:43,285 OK, so the story so far is basically three postulates. 13 00:00:46,140 --> 00:00:58,260 The first is that the configuration of a particle 14 00:00:58,260 --> 00:01:06,410 is given by, or described by, a wave function psi of x. 15 00:01:10,150 --> 00:01:11,310 Yeah? 16 00:01:11,310 --> 00:01:15,120 So in particular, just to flesh this out a little more, 17 00:01:15,120 --> 00:01:17,650 if we were in 3D, for example-- which we're not. 18 00:01:17,650 --> 00:01:19,490 We're currently in our one dimensional 19 00:01:19,490 --> 00:01:20,710 tripped out tricycles. 20 00:01:20,710 --> 00:01:23,150 In 3D, the wave function would be a function 21 00:01:23,150 --> 00:01:26,010 of all three positions x, y and z. 22 00:01:29,860 --> 00:01:36,180 If we had two particles, our wave function 23 00:01:36,180 --> 00:01:39,360 would be a function of the position of each particle. 24 00:01:39,360 --> 00:01:42,790 x1, x2, and so on. 25 00:01:42,790 --> 00:01:46,566 So we'll go through lots of details and examples later on. 26 00:01:46,566 --> 00:01:47,982 But for the most part, we're going 27 00:01:47,982 --> 00:01:49,731 to be sticking with single particle in one 28 00:01:49,731 --> 00:01:51,650 dimension for the next few weeks. 29 00:01:51,650 --> 00:01:54,290 Now again, I want to emphasize this is our first pass 30 00:01:54,290 --> 00:01:56,680 through our definition of quantum mechanics. 31 00:01:56,680 --> 00:01:59,720 Once we use the language and the machinery a little bit, 32 00:01:59,720 --> 00:02:04,460 we're going to develop a more general, more coherent set 33 00:02:04,460 --> 00:02:06,850 of rules or definition of quantum mechanics. 34 00:02:06,850 --> 00:02:09,210 But this is our first pass. 35 00:02:09,210 --> 00:02:12,950 Two, the meaning of the wave function 36 00:02:12,950 --> 00:02:18,440 is that the norm squared psi of x, norm squared, it's complex, 37 00:02:18,440 --> 00:02:29,450 dx is the probability of finding the particle- There's 38 00:02:29,450 --> 00:02:30,320 an n in their. 39 00:02:30,320 --> 00:02:38,900 Finding the particle-- in the region between x and x plus dx. 40 00:02:38,900 --> 00:02:40,690 So psi squared itself, norm squared, 41 00:02:40,690 --> 00:02:44,430 is the probability density. 42 00:02:44,430 --> 00:02:46,550 OK? 43 00:02:46,550 --> 00:02:52,980 And third, the superposition principle. 44 00:02:52,980 --> 00:02:55,130 If there are two possible configurations 45 00:02:55,130 --> 00:02:58,250 the system can be in, which in quantum mechanics 46 00:02:58,250 --> 00:03:00,570 means two different wave functions that 47 00:03:00,570 --> 00:03:05,394 could describe the system given psi 1 and psi 2, two 48 00:03:05,394 --> 00:03:06,810 wave functions that could describe 49 00:03:06,810 --> 00:03:09,190 two different configurations of the system. 50 00:03:09,190 --> 00:03:12,190 For example, the particles here or the particles over here. 51 00:03:12,190 --> 00:03:21,360 It's also possible to find the system in a superposition 52 00:03:21,360 --> 00:03:25,120 of those two psi is equal to some arbitrary 53 00:03:25,120 --> 00:03:34,780 linear combination alpha psi 1 plus beta psi 2 of x. 54 00:03:34,780 --> 00:03:35,280 OK? 55 00:03:41,490 --> 00:03:43,720 So some things to note-- so questions about those 56 00:03:43,720 --> 00:03:46,320 before we move on? 57 00:03:46,320 --> 00:03:49,030 No questions? 58 00:03:49,030 --> 00:03:50,330 Nothing? 59 00:03:50,330 --> 00:03:52,450 Really? 60 00:03:52,450 --> 00:03:55,105 You're going to make he threaten you with something. 61 00:03:55,105 --> 00:03:56,230 I know there are questions. 62 00:03:56,230 --> 00:03:57,480 This is not trivial stuff. 63 00:04:00,560 --> 00:04:02,440 OK. 64 00:04:02,440 --> 00:04:06,409 So some things to note. 65 00:04:06,409 --> 00:04:07,825 The first is we want to normalize. 66 00:04:11,455 --> 00:04:13,080 We will generally normalize and require 67 00:04:13,080 --> 00:04:16,640 that the integral over all possible positions 68 00:04:16,640 --> 00:04:19,839 of the probability density psi of x norm squared 69 00:04:19,839 --> 00:04:20,839 is equal to 1. 70 00:04:20,839 --> 00:04:23,830 This is just saying that the total probability that we find 71 00:04:23,830 --> 00:04:27,179 the particle somewhere had better be one. 72 00:04:27,179 --> 00:04:28,970 This is like saying if I know a particle is 73 00:04:28,970 --> 00:04:30,470 in one of two boxes, because I've 74 00:04:30,470 --> 00:04:31,950 put a particle in one of the boxes. 75 00:04:31,950 --> 00:04:33,510 I just don't remember which one. 76 00:04:33,510 --> 00:04:35,690 Then the probability that it's in the first box 77 00:04:35,690 --> 00:04:37,523 plus probability that it's in the second box 78 00:04:37,523 --> 00:04:38,720 must be 100% or one. 79 00:04:38,720 --> 00:04:42,020 If it's less, then the particle has simply disappeared. 80 00:04:42,020 --> 00:04:45,140 And basic rule, things don't just disappear. 81 00:04:45,140 --> 00:04:46,665 So probability should be normalized. 82 00:04:49,430 --> 00:04:50,890 And this is our prescription. 83 00:04:50,890 --> 00:04:57,740 So a second thing to note is that all reasonable, or non 84 00:04:57,740 --> 00:05:05,210 stupid, functions psi of x are equally 85 00:05:05,210 --> 00:05:06,770 reasonable as wave functions. 86 00:05:16,510 --> 00:05:17,280 OK? 87 00:05:17,280 --> 00:05:21,300 So this is a very reasonable function. 88 00:05:21,300 --> 00:05:22,420 It's nice and smooth. 89 00:05:22,420 --> 00:05:24,310 It converges to 0 infinity. 90 00:05:24,310 --> 00:05:27,140 It's got all the nice properties you might want. 91 00:05:27,140 --> 00:05:30,005 This is also a reasonable function. 92 00:05:30,005 --> 00:05:32,100 It's a little annoying, but there it is. 93 00:05:32,100 --> 00:05:36,240 And they're both perfectly reasonable as wave functions. 94 00:05:36,240 --> 00:05:38,940 This on the other hand, not so much. 95 00:05:38,940 --> 00:05:39,740 So for two reasons. 96 00:05:39,740 --> 00:05:40,990 First off, it's discontinuous. 97 00:05:40,990 --> 00:05:43,390 And as you're going to show in your problem set, 98 00:05:43,390 --> 00:05:45,462 discontinuities are very bad for wave functions. 99 00:05:45,462 --> 00:05:47,420 So we need our wave functions to be continuous. 100 00:05:47,420 --> 00:05:49,740 The second is over some domain it's multi valued. 101 00:05:49,740 --> 00:05:51,230 There are two different values of the function. 102 00:05:51,230 --> 00:05:52,780 That's also bad, because what's the probability? 103 00:05:52,780 --> 00:05:54,500 It's the norm squared, but if it two values, 104 00:05:54,500 --> 00:05:56,530 two values for the probability, that doesn't make any sense. 105 00:05:56,530 --> 00:05:58,071 What's the probability that I'm going 106 00:05:58,071 --> 00:06:00,150 to fall over in 10 seconds? 107 00:06:00,150 --> 00:06:05,000 Well, it's small, but it's not actually equal to 1% or 3%. 108 00:06:05,000 --> 00:06:07,600 It's one of those. 109 00:06:07,600 --> 00:06:10,290 Hopefully is much lower than that. 110 00:06:10,290 --> 00:06:14,640 So all reasonable functions are equally 111 00:06:14,640 --> 00:06:17,270 reasonable as wave functions. 112 00:06:17,270 --> 00:06:19,710 And in particular, what that means is all states 113 00:06:19,710 --> 00:06:21,520 corresponding to reasonable wave functions 114 00:06:21,520 --> 00:06:25,890 are equally reasonable as physical states. 115 00:06:25,890 --> 00:06:30,200 There's no primacy in wave functions or in states. 116 00:06:30,200 --> 00:06:36,430 However, with that said, some wave functions 117 00:06:36,430 --> 00:06:38,520 are more equal than others. 118 00:06:38,520 --> 00:06:40,120 OK? 119 00:06:40,120 --> 00:06:42,966 And this is important, and coming up 120 00:06:42,966 --> 00:06:44,340 with a good definition of this is 121 00:06:44,340 --> 00:06:45,696 going to be an important challenge for us 122 00:06:45,696 --> 00:06:47,200 in the next couple of lectures. 123 00:06:47,200 --> 00:06:49,346 So in particular, this wave function 124 00:06:49,346 --> 00:06:50,720 has a nice simple interpretation. 125 00:06:50,720 --> 00:06:52,560 If I tell you this is psi of x, then 126 00:06:52,560 --> 00:06:55,750 what can you tell me about the particle whose wave function is 127 00:06:55,750 --> 00:06:56,560 the psi of x? 128 00:06:58,240 --> 00:06:59,490 What can you tell me about it? 129 00:06:59,490 --> 00:07:00,603 What do you know? 130 00:07:00,603 --> 00:07:02,700 AUDIENCE: [INAUDIBLE]. 131 00:07:02,700 --> 00:07:04,330 PROFESSOR: It's here, right? 132 00:07:04,330 --> 00:07:05,166 It's not over here. 133 00:07:05,166 --> 00:07:06,290 Probability is basically 0. 134 00:07:06,290 --> 00:07:07,770 Probability is large. 135 00:07:07,770 --> 00:07:10,480 It's pretty much here with this great confidence. 136 00:07:10,480 --> 00:07:12,940 What about this guy? 137 00:07:12,940 --> 00:07:15,190 Less informative, right? 138 00:07:15,190 --> 00:07:17,540 It's less obvious what this wave function is telling me. 139 00:07:17,540 --> 00:07:19,623 So some wave functions are more equal in the sense 140 00:07:19,623 --> 00:07:21,030 that they have-- i.e. 141 00:07:21,030 --> 00:07:22,405 they have simple interpretations. 142 00:07:29,220 --> 00:07:35,650 So for example, this wave function 143 00:07:35,650 --> 00:07:38,970 continuing on infinitely, this wave function doesn't tell me 144 00:07:38,970 --> 00:07:41,116 where the particle is, but what does it tell me? 145 00:07:41,116 --> 00:07:42,030 AUDIENCE: Momentum. 146 00:07:42,030 --> 00:07:43,180 PROFESSOR: The momentum, exactly. 147 00:07:43,180 --> 00:07:45,000 So this is giving me information about the momentum 148 00:07:45,000 --> 00:07:47,375 of the particle because it has a well defined wavelength. 149 00:07:47,375 --> 00:07:50,810 So this one, I would also say is more equal than this one. 150 00:07:50,810 --> 00:07:52,790 They're both perfectly physical, but this one 151 00:07:52,790 --> 00:07:54,480 has a simple interpretation. 152 00:07:54,480 --> 00:07:57,670 And that's going to be important for us. 153 00:07:57,670 --> 00:08:11,310 Related to that is that any reasonable function psi of x 154 00:08:11,310 --> 00:08:24,980 can be expressed as a superposition of more 155 00:08:24,980 --> 00:08:36,169 equal wave functions, or more precisely easily 156 00:08:36,169 --> 00:08:46,056 interpretable wave functions. 157 00:08:46,056 --> 00:08:47,930 We saw this last time in the Fourier theorem. 158 00:08:47,930 --> 00:08:50,764 The Fourier theorem said look, take any wave function-- take 159 00:08:50,764 --> 00:08:52,430 any function, but I'm going to interpret 160 00:08:52,430 --> 00:08:53,570 in the language of quantum mechanics. 161 00:08:53,570 --> 00:08:56,069 Take any wave function which is given by some complex valued 162 00:08:56,069 --> 00:08:57,460 function, and it can be expressed 163 00:08:57,460 --> 00:09:00,530 as a superposition of plane waves. 164 00:09:00,530 --> 00:09:07,810 1 over 2pi in our normalization integral dk psi tilde of k, 165 00:09:07,810 --> 00:09:09,736 but this is a set of coefficients. 166 00:09:09,736 --> 00:09:10,797 e to the ikx. 167 00:09:10,797 --> 00:09:11,880 So what are we doing here? 168 00:09:11,880 --> 00:09:14,251 We're saying pick a value of k. 169 00:09:14,251 --> 00:09:15,750 There's a number associated with it, 170 00:09:15,750 --> 00:09:18,814 which is going to be an a magnitude and a phase. 171 00:09:18,814 --> 00:09:20,230 And that's the magnitude and phase 172 00:09:20,230 --> 00:09:22,380 of a plane wave, e to the ikx. 173 00:09:22,380 --> 00:09:25,980 Now remember that e to the ikx is 174 00:09:25,980 --> 00:09:33,007 equal to cos kx plus i sin kx. 175 00:09:33,007 --> 00:09:35,090 Which you should all know, but just to remind you. 176 00:09:35,090 --> 00:09:36,850 This is a periodic function. 177 00:09:36,850 --> 00:09:38,080 These are periodic functions. 178 00:09:38,080 --> 00:09:42,820 So this is a plane wave with a definite wavelength, 179 00:09:42,820 --> 00:09:45,490 2pi upon k. 180 00:09:45,490 --> 00:09:47,990 So this is a more equal wave function in the sense 181 00:09:47,990 --> 00:09:49,480 that it has a definite wavelength. 182 00:09:49,480 --> 00:09:50,770 We know what its momentum is. 183 00:09:50,770 --> 00:09:52,840 Its momentum is h bar k. 184 00:09:52,840 --> 00:09:56,030 Any function, we're saying, can be expressed as a superposition 185 00:09:56,030 --> 00:09:57,920 by summing over all possible values of k, 186 00:09:57,920 --> 00:09:59,560 all possible different wavelengths. 187 00:09:59,560 --> 00:10:03,030 Any function can be expressed as a superposition of wave 188 00:10:03,030 --> 00:10:05,622 functions with a definite momentum. 189 00:10:05,622 --> 00:10:07,300 That make sense? 190 00:10:07,300 --> 00:10:09,002 Fourier didn't think about it that way, 191 00:10:09,002 --> 00:10:10,460 but from quantum mechanics, this is 192 00:10:10,460 --> 00:10:11,960 the way we want to think about it. 193 00:10:11,960 --> 00:10:12,950 It's just a true statement. 194 00:10:12,950 --> 00:10:13,991 It's a mathematical fact. 195 00:10:17,206 --> 00:10:18,080 Questions about that? 196 00:10:20,750 --> 00:10:24,820 Similarly, I claim that I can expand the very same function, 197 00:10:24,820 --> 00:10:28,107 psi of x, as a superposition of states, 198 00:10:28,107 --> 00:10:29,815 not with definite momentum, but of states 199 00:10:29,815 --> 00:10:30,773 with definite position. 200 00:10:34,029 --> 00:10:35,820 So what's a state with a definite position? 201 00:10:35,820 --> 00:10:36,929 AUDIENCE: Delta. 202 00:10:36,929 --> 00:10:38,470 PROFESSOR: A delta function, exactly. 203 00:10:38,470 --> 00:10:40,510 So I claim that any function psi of x 204 00:10:40,510 --> 00:10:45,569 can be expanded a sum over all states 205 00:10:45,569 --> 00:10:46,610 with a definite position. 206 00:10:46,610 --> 00:10:49,470 So delta of-- well, what's a state with a definite position? 207 00:10:49,470 --> 00:10:50,670 x0. 208 00:10:50,670 --> 00:10:53,810 Delta of x minus x0. 209 00:10:53,810 --> 00:10:54,420 OK? 210 00:10:54,420 --> 00:10:56,795 This goes bing when x0 is equal to x. 211 00:10:56,795 --> 00:10:58,920 But I want a sum over all possible delta functions. 212 00:10:58,920 --> 00:11:00,920 That means all possible positions. 213 00:11:00,920 --> 00:11:04,930 That means all possible values of x0, dx0. 214 00:11:04,930 --> 00:11:06,900 And I need some coefficient function here. 215 00:11:06,900 --> 00:11:09,490 Well, the coefficient function I'm going to call psi of x0. 216 00:11:13,210 --> 00:11:16,015 So is this true? 217 00:11:16,015 --> 00:11:17,640 Is it true that I can take any function 218 00:11:17,640 --> 00:11:21,730 and expand it in a superposition of delta functions? 219 00:11:21,730 --> 00:11:22,410 Absolutely. 220 00:11:22,410 --> 00:11:24,364 Because look at what this equation does. 221 00:11:24,364 --> 00:11:26,030 Remember, delta function is your friend. 222 00:11:26,030 --> 00:11:29,210 It's a map from integrals to numbers or functions. 223 00:11:29,210 --> 00:11:31,280 So this integral, is an integral over x0. 224 00:11:31,280 --> 00:11:33,000 Here we have a delta of x minus x0. 225 00:11:33,000 --> 00:11:35,946 So this basically says the value of this integral 226 00:11:35,946 --> 00:11:37,570 is what you get by taking the integrand 227 00:11:37,570 --> 00:11:39,040 and replacing x by x0. 228 00:11:39,040 --> 00:11:41,250 Set x equals x0, that's when delta equals 0. 229 00:11:41,250 --> 00:11:45,020 So this is equal to the argument evaluated at x0 is equal to x. 230 00:11:45,020 --> 00:11:47,620 That's your psi of x. 231 00:11:47,620 --> 00:11:48,670 OK? 232 00:11:48,670 --> 00:11:52,310 Any arbitrarily ugly function can be expressed either 233 00:11:52,310 --> 00:11:55,650 as a superposition of states with definite momentum 234 00:11:55,650 --> 00:11:59,280 or a superposition of states with definite position. 235 00:11:59,280 --> 00:12:00,022 OK? 236 00:12:00,022 --> 00:12:01,230 And this is going to be true. 237 00:12:01,230 --> 00:12:03,188 We're going to find this is a general statement 238 00:12:03,188 --> 00:12:06,780 that any state can be expressed as a superposition of states 239 00:12:06,780 --> 00:12:10,780 with a well defined observable quantity 240 00:12:10,780 --> 00:12:12,457 for any observable quantity you want. 241 00:12:12,457 --> 00:12:14,790 So let me give you just a quick little bit of intuition. 242 00:12:14,790 --> 00:12:19,644 In 2D, this is a perfectly good vector, right? 243 00:12:19,644 --> 00:12:21,310 Now here's a question I want to ask you. 244 00:12:21,310 --> 00:12:22,310 Is that a superposition? 245 00:12:25,730 --> 00:12:26,230 Yeah. 246 00:12:26,230 --> 00:12:28,070 I mean every vector can be written 247 00:12:28,070 --> 00:12:30,900 as the sum of other vectors, right? 248 00:12:30,900 --> 00:12:34,052 And it can be done in an infinite number of ways, right? 249 00:12:34,052 --> 00:12:35,510 So there's no such thing as a state 250 00:12:35,510 --> 00:12:38,410 which is not a superposition. 251 00:12:38,410 --> 00:12:40,740 Every vector is a superposition of other vectors. 252 00:12:40,740 --> 00:12:43,980 It's a sum of other vector. 253 00:12:43,980 --> 00:12:48,380 So in particular we often find it useful to pick a basis 254 00:12:48,380 --> 00:12:50,992 and say look, I know what I mean by the vector y, 255 00:12:50,992 --> 00:12:52,700 y hat is a unit vector in this direction. 256 00:12:52,700 --> 00:12:54,325 I know what I mean by the vector x hat. 257 00:12:54,325 --> 00:12:55,990 It's a unit vector in this direction. 258 00:12:55,990 --> 00:12:59,630 And now I can ask, given that these are my natural guys, 259 00:12:59,630 --> 00:13:03,660 the guys I want to attend to, is this a superposition 260 00:13:03,660 --> 00:13:04,510 of x and y? 261 00:13:04,510 --> 00:13:06,585 Or is it just x or y? 262 00:13:06,585 --> 00:13:09,990 Well, that's a superposition. 263 00:13:09,990 --> 00:13:11,950 Whereas x hat itself is not. 264 00:13:11,950 --> 00:13:16,760 So this somehow is about finding convenient choice of basis. 265 00:13:16,760 --> 00:13:18,920 But any given vector can be expressed 266 00:13:18,920 --> 00:13:21,730 as a superposition of some pair of basis vectors 267 00:13:21,730 --> 00:13:24,900 or a different pair of basis vectors. 268 00:13:24,900 --> 00:13:28,060 There's nothing hallowed about your choice of basis. 269 00:13:28,060 --> 00:13:30,392 There's no God given basis for the universe. 270 00:13:30,392 --> 00:13:32,600 We look out in the universe in the Hubble deep field, 271 00:13:32,600 --> 00:13:34,766 and you don't see somewhere in the Hubble deep field 272 00:13:34,766 --> 00:13:37,470 an arrow going x, right? 273 00:13:37,470 --> 00:13:39,060 So there's no natural choice of basis, 274 00:13:39,060 --> 00:13:41,090 but it's sometimes convenient to pick a basis. 275 00:13:41,090 --> 00:13:42,590 This is the direction of the surface of the earth. 276 00:13:42,590 --> 00:13:44,640 This is the direction perpendicular to it. 277 00:13:44,640 --> 00:13:46,900 So sometimes particular basis sets 278 00:13:46,900 --> 00:13:49,730 have particular meanings to us. 279 00:13:49,730 --> 00:13:50,821 That's true in vectors. 280 00:13:50,821 --> 00:13:51,820 This is along the earth. 281 00:13:51,820 --> 00:13:52,890 This is perpendicular to it. 282 00:13:52,890 --> 00:13:54,260 This would be slightly strange. 283 00:13:54,260 --> 00:13:56,450 Maybe if you're leaning. 284 00:13:56,450 --> 00:13:58,910 And similarly, this is an expansion 285 00:13:58,910 --> 00:14:01,790 of a function as a sum, as a superposition 286 00:14:01,790 --> 00:14:03,140 of other functions. 287 00:14:03,140 --> 00:14:06,030 And you could have done this in any good space of functions. 288 00:14:06,030 --> 00:14:07,291 We'll talk about that more. 289 00:14:07,291 --> 00:14:08,790 These are particularly natural ones. 290 00:14:08,790 --> 00:14:09,581 They're more equal. 291 00:14:09,581 --> 00:14:12,590 These are ones with different definite values of position, 292 00:14:12,590 --> 00:14:14,270 different definite values of momentum. 293 00:14:14,270 --> 00:14:15,520 Everyone cool? 294 00:14:15,520 --> 00:14:17,936 Quickly what's the momentum associated to the plane wave e 295 00:14:17,936 --> 00:14:18,724 to the ikx? 296 00:14:18,724 --> 00:14:19,640 AUDIENCE: [INAUDIBLE]. 297 00:14:22,220 --> 00:14:23,100 PROFESSOR: h bar k. 298 00:14:23,100 --> 00:14:23,600 Good. 299 00:14:33,180 --> 00:14:35,140 So now I want to just quickly run over 300 00:14:35,140 --> 00:14:37,850 some concept questions for you. 301 00:14:37,850 --> 00:14:39,390 So whip out your clickers. 302 00:14:39,390 --> 00:14:41,030 OK, we'll do this verbally. 303 00:14:49,052 --> 00:14:50,385 All right, let's try this again. 304 00:14:50,385 --> 00:14:53,200 So how would you interpret this wave function? 305 00:14:53,200 --> 00:14:54,452 AUDIENCE: e. 306 00:14:54,452 --> 00:14:55,160 PROFESSOR: Solid. 307 00:14:58,580 --> 00:15:00,080 How do you know whether the particle 308 00:15:00,080 --> 00:15:02,332 is big or small by looking at the wave function? 309 00:15:02,332 --> 00:15:03,248 AUDIENCE: [INAUDIBLE]. 310 00:15:05,640 --> 00:15:06,780 PROFESSOR: All right. 311 00:15:06,780 --> 00:15:09,306 Two particles described by a plane wave of the form e 312 00:15:09,306 --> 00:15:10,820 to the ikx. 313 00:15:10,820 --> 00:15:13,979 Particle one is a smaller wavelength than particle two. 314 00:15:13,979 --> 00:15:15,520 Which particle has a larger momentum? 315 00:15:15,520 --> 00:15:17,803 Think about it, but don't say it out loud. 316 00:15:22,230 --> 00:15:24,647 And this sort of defeats the purpose of the clicker thing, 317 00:15:24,647 --> 00:15:27,146 because now I'm supposed to be able to know without you guys 318 00:15:27,146 --> 00:15:28,040 saying anything. 319 00:15:28,040 --> 00:15:29,549 So instead of saying it out loud, 320 00:15:29,549 --> 00:15:30,840 here's what I'd like you to do. 321 00:15:30,840 --> 00:15:33,350 Talk to the person next to you and discuss 322 00:15:33,350 --> 00:15:37,271 which one has the larger 323 00:15:37,271 --> 00:15:38,104 AUDIENCE: [CHATTER]. 324 00:16:00,550 --> 00:16:02,000 All right. 325 00:16:02,000 --> 00:16:07,350 Cool, so which one has the larger momentum? 326 00:16:07,350 --> 00:16:09,460 AUDIENCE: A. 327 00:16:09,460 --> 00:16:10,485 PROFESSOR: How come? 328 00:16:10,485 --> 00:16:12,760 [INTERPOSING VOICES] 329 00:16:12,760 --> 00:16:15,540 PROFESSOR: RIght, smaller wavelength. 330 00:16:15,540 --> 00:16:18,262 P equals h bar k. 331 00:16:18,262 --> 00:16:22,330 k equals 2pi over lambda. 332 00:16:22,330 --> 00:16:23,880 Solid? 333 00:16:23,880 --> 00:16:25,869 Smaller wavelength, higher momentum. 334 00:16:25,869 --> 00:16:27,660 If it has higher momentum, what do you just 335 00:16:27,660 --> 00:16:30,642 intuitively expect to know about its energy? 336 00:16:30,642 --> 00:16:31,940 It's probably higher. 337 00:16:31,940 --> 00:16:33,310 Are you positive about that? 338 00:16:33,310 --> 00:16:35,810 No, you need to know how the energy depends on the momentum, 339 00:16:35,810 --> 00:16:38,130 but it's probably higher. 340 00:16:38,130 --> 00:16:39,990 So this is an important little lesson 341 00:16:39,990 --> 00:16:41,990 that you probably all know from optics and maybe 342 00:16:41,990 --> 00:16:43,030 from core mechanics. 343 00:16:43,030 --> 00:16:45,670 Shorter wavelength thing, higher energy. 344 00:16:45,670 --> 00:16:47,010 Higher momentum for sure. 345 00:16:47,010 --> 00:16:50,550 Usually higher energy as well. 346 00:16:50,550 --> 00:16:53,190 Very useful rule of thumb to keep in mind. 347 00:16:53,190 --> 00:16:54,490 Indeed, it's particle one. 348 00:16:54,490 --> 00:16:55,730 OK next one. 349 00:16:55,730 --> 00:16:59,490 Compared to the wave function psi of x, it's Fourier 350 00:16:59,490 --> 00:17:02,660 transform, psi tilde of x contains more information, 351 00:17:02,660 --> 00:17:06,740 or less, or the same, or something. 352 00:17:06,740 --> 00:17:08,569 Don't say it out loud. 353 00:17:08,569 --> 00:17:12,579 OK, so how many people know the answer? 354 00:17:12,579 --> 00:17:13,079 Awesome. 355 00:17:13,079 --> 00:17:16,380 And how many people are not sure. 356 00:17:16,380 --> 00:17:17,869 OK, good. 357 00:17:17,869 --> 00:17:22,053 So talk to the person next to you and convince them briefly. 358 00:17:43,100 --> 00:17:43,640 All right. 359 00:17:48,800 --> 00:17:49,590 So let's vote. 360 00:17:49,590 --> 00:17:52,980 A, more information. 361 00:17:52,980 --> 00:17:55,350 B, less information. 362 00:17:55,350 --> 00:17:57,690 C, same. 363 00:17:57,690 --> 00:17:59,586 OK, good you got it. 364 00:17:59,586 --> 00:18:00,710 So these are not hard ones. 365 00:18:04,600 --> 00:18:08,530 This function, which is a sine wave of length l, 366 00:18:08,530 --> 00:18:10,080 0 outside of that region. 367 00:18:10,080 --> 00:18:12,340 Which is closer to true? 368 00:18:12,340 --> 00:18:15,370 f has a single well defined wavelength for the most part? 369 00:18:15,370 --> 00:18:16,280 It's closer to true. 370 00:18:16,280 --> 00:18:17,570 This doesn't have to be exact. 371 00:18:17,570 --> 00:18:19,440 f has a single well defined wavelengths. 372 00:18:19,440 --> 00:18:21,996 Or f is made up of a wide range of wavelengths? 373 00:18:26,350 --> 00:18:28,382 Think it to yourself. 374 00:18:28,382 --> 00:18:29,590 Ponder that one for a minute. 375 00:18:40,380 --> 00:18:42,430 OK, now before we get talking about it. 376 00:18:42,430 --> 00:18:44,152 Hold on, hold on, hold on. 377 00:18:44,152 --> 00:18:45,610 Since we don't have clickers, but I 378 00:18:45,610 --> 00:18:47,410 want to pull off the same effect, 379 00:18:47,410 --> 00:18:50,170 and we can do this, because it's binary here. 380 00:18:50,170 --> 00:18:53,780 I want everyone close your eyes. 381 00:18:53,780 --> 00:18:56,020 Just close your eyes, just for a moment. 382 00:18:56,020 --> 00:18:56,777 Yeah. 383 00:18:56,777 --> 00:18:58,610 Or close the eyes of the person next to you. 384 00:18:58,610 --> 00:18:59,720 That's fine. 385 00:18:59,720 --> 00:19:02,610 And now and I want you to vote. 386 00:19:02,610 --> 00:19:05,200 A is f has a single well defined wavelength. 387 00:19:05,200 --> 00:19:07,830 B is f has a wide range of wavelengths. 388 00:19:07,830 --> 00:19:10,840 So how many people think A, a single wavelength? 389 00:19:10,840 --> 00:19:12,540 OK. 390 00:19:12,540 --> 00:19:14,556 Lower your hands, good. 391 00:19:14,556 --> 00:19:18,000 And how many people think B, a wide range of wavelengths? 392 00:19:18,000 --> 00:19:18,550 Awesome. 393 00:19:18,550 --> 00:19:20,340 So this is exactly what happens when we actually use clickers. 394 00:19:20,340 --> 00:19:21,470 It's 50/50. 395 00:19:21,470 --> 00:19:24,040 So now you guys need to talk to the person next to you 396 00:19:24,040 --> 00:19:26,225 and convince each other of the truth. 397 00:19:26,225 --> 00:19:27,058 AUDIENCE: [CHATTER]. 398 00:20:23,840 --> 00:20:30,719 All right, so the volume sort of tones down as people, I think, 399 00:20:30,719 --> 00:20:31,510 come to resolution. 400 00:20:31,510 --> 00:20:33,020 Close your eyes again. 401 00:20:33,020 --> 00:20:34,700 Once more into the breach, my friends. 402 00:20:34,700 --> 00:20:37,980 So close your eyes, and now let's vote again. 403 00:20:37,980 --> 00:20:41,530 f of x has a single, well defined wavelength. 404 00:20:41,530 --> 00:20:45,370 And now f of x is made up of a range of wavelengths? 405 00:20:45,370 --> 00:20:46,090 OK. 406 00:20:46,090 --> 00:20:48,940 There's a dramatic shift in the field to B, 407 00:20:48,940 --> 00:20:51,600 it has a wide range of wavelengths, not 408 00:20:51,600 --> 00:20:52,920 a single wavelength. 409 00:20:52,920 --> 00:20:55,350 And that is, in fact, the correct answer. 410 00:20:55,350 --> 00:20:58,010 OK, so learning happens. 411 00:20:58,010 --> 00:21:00,810 That was an empirical test. 412 00:21:00,810 --> 00:21:02,700 So does anyone want to defend this view 413 00:21:02,700 --> 00:21:04,900 that f is made of a wide range of wavelengths? 414 00:21:04,900 --> 00:21:06,190 Sure, bring it. 415 00:21:06,190 --> 00:21:08,650 AUDIENCE: So, the sine wave is an infinite, 416 00:21:08,650 --> 00:21:11,217 and it cancels out past minus l over 2 417 00:21:11,217 --> 00:21:12,800 and positive l over 2, which means you 418 00:21:12,800 --> 00:21:13,830 need to add a bunch of wavelengths 419 00:21:13,830 --> 00:21:15,464 to actually cancel it out there. 420 00:21:15,464 --> 00:21:16,630 PROFESSOR: Awesome, exactly. 421 00:21:16,630 --> 00:21:17,240 Exactly. 422 00:21:17,240 --> 00:21:19,470 If you only had the thing of a single wavelength, 423 00:21:19,470 --> 00:21:21,720 it would continue with a single wavelength all the way out. 424 00:21:21,720 --> 00:21:23,386 In fact, there's a nice way to say this. 425 00:21:23,386 --> 00:21:28,441 When you have a sine wave, what can you say about it's-- we 426 00:21:28,441 --> 00:21:29,940 know that a sine wave is continuous, 427 00:21:29,940 --> 00:21:32,580 and it's continuous everywhere, right? 428 00:21:32,580 --> 00:21:34,220 It's also differentiable everywhere. 429 00:21:34,220 --> 00:21:37,210 Its derivative is continuous and differentiable everywhere, 430 00:21:37,210 --> 00:21:39,510 because it's a cosine, right? 431 00:21:39,510 --> 00:21:42,050 So if yo you take a superposition of sines 432 00:21:42,050 --> 00:21:46,180 and cosines, do you ever get a discontinuity? 433 00:21:46,180 --> 00:21:47,090 No. 434 00:21:47,090 --> 00:21:49,990 Do you ever get something whose derivative is discontinuous? 435 00:21:49,990 --> 00:21:50,690 No. 436 00:21:50,690 --> 00:21:53,630 So how would you ever reproduce a thing 437 00:21:53,630 --> 00:21:57,320 with a discontinuity using sines and cosines? 438 00:21:57,320 --> 00:22:00,390 Well, you'd need some infinite sum of sines and cosines 439 00:22:00,390 --> 00:22:03,400 where there's some technicality about the infinite limit being 440 00:22:03,400 --> 00:22:04,900 singular, because you can't do it 441 00:22:04,900 --> 00:22:07,200 a finite number of sines and cosines. 442 00:22:07,200 --> 00:22:10,390 That function is continuous, but its derivative 443 00:22:10,390 --> 00:22:12,240 is discontinuous. 444 00:22:12,240 --> 00:22:13,409 Yeah? 445 00:22:13,409 --> 00:22:15,450 So it's going to take an infinite number of sines 446 00:22:15,450 --> 00:22:18,940 and cosines to reproduce that little kink at the edge. 447 00:22:18,940 --> 00:22:19,658 Yeah? 448 00:22:19,658 --> 00:22:21,898 AUDIENCE: So a finite number of sines and cosines 449 00:22:21,898 --> 00:22:26,176 doesn't mean finding-- or an infinite number of sines 450 00:22:26,176 --> 00:22:28,384 and cosines doesn't mean infinite [? regular ?] sines 451 00:22:28,384 --> 00:22:29,204 and cosines, right? 452 00:22:29,204 --> 00:22:33,430 Because over a finite region [INAUDIBLE]. 453 00:22:33,430 --> 00:22:35,864 PROFESSOR: That's true, but you need arbitrarily-- so 454 00:22:35,864 --> 00:22:36,780 let's talk about that. 455 00:22:36,780 --> 00:22:38,804 That's an excellent question. 456 00:22:38,804 --> 00:22:39,970 That's a very good question. 457 00:22:39,970 --> 00:22:41,780 The question here is look, there's 458 00:22:41,780 --> 00:22:43,696 two different things you can be talking about. 459 00:22:43,696 --> 00:22:46,280 One is arbitrarily large and arbitrarily short wavelengths, 460 00:22:46,280 --> 00:22:47,640 so an arbitrary range of wavelengths. 461 00:22:47,640 --> 00:22:49,170 And the other is an infinite number. 462 00:22:49,170 --> 00:22:50,520 But an infinite number is silly, because there's 463 00:22:50,520 --> 00:22:51,932 a continuous variable here k. 464 00:22:51,932 --> 00:22:53,640 You got an infinite number of wavelengths 465 00:22:53,640 --> 00:22:56,170 between one and 1.2, right? 466 00:22:56,170 --> 00:22:56,970 It's continuous. 467 00:22:56,970 --> 00:22:58,190 So which one do you mean? 468 00:22:58,190 --> 00:23:00,470 So let's go back to this connection 469 00:23:00,470 --> 00:23:02,950 that we got a minute ago from short distance 470 00:23:02,950 --> 00:23:04,080 and high momentum. 471 00:23:07,499 --> 00:23:09,790 That thing looks like it has one particular wavelength. 472 00:23:09,790 --> 00:23:11,540 But I claim, in order to reproduce 473 00:23:11,540 --> 00:23:14,980 that as a superposition of states with definite momentum, 474 00:23:14,980 --> 00:23:17,714 I need arbitrarily high wavelength. 475 00:23:17,714 --> 00:23:19,880 And why do I need arbitrarily high wavelength modes? 476 00:23:19,880 --> 00:23:22,290 Why do we need to arbitrarily high momentum modes? 477 00:23:22,290 --> 00:23:24,170 Well, it's because of this. 478 00:23:24,170 --> 00:23:24,950 We have a kink. 479 00:23:27,610 --> 00:23:32,330 And this feature, what's the length scale of that feature? 480 00:23:32,330 --> 00:23:34,170 It's infinitesimally small, which 481 00:23:34,170 --> 00:23:35,840 means I'm going to have to-- in order to reproduce that, 482 00:23:35,840 --> 00:23:37,465 in order to probe it, I'm going to need 483 00:23:37,465 --> 00:23:40,130 a momentum that's arbitrarily large. 484 00:23:40,130 --> 00:23:42,940 So it's really about the range, not just the number. 485 00:23:42,940 --> 00:23:45,930 But you need arbitrarily large momentum. 486 00:23:45,930 --> 00:23:50,890 To construct or detect an arbitrarily small feature 487 00:23:50,890 --> 00:23:52,640 you need arbitrarily large momentum modes. 488 00:23:52,640 --> 00:23:53,140 Yeah? 489 00:23:53,140 --> 00:23:56,390 AUDIENCE: Why do you [INAUDIBLE]? 490 00:23:56,390 --> 00:23:58,330 Why don't you just say, oh you need 491 00:23:58,330 --> 00:24:00,020 an arbitrary small wavelength? 492 00:24:00,020 --> 00:24:02,754 Why wouldn't you just phrase that [INAUDIBLE]? 493 00:24:02,754 --> 00:24:04,420 PROFESSOR: I chose to phrase it that way 494 00:24:04,420 --> 00:24:06,711 because I want an emphasize and encourage-- I emphasize 495 00:24:06,711 --> 00:24:10,740 you to think and encourage you to conflate 496 00:24:10,740 --> 00:24:13,140 short distance and large momentum. 497 00:24:13,140 --> 00:24:16,764 I want the connection between momentum and the length scale 498 00:24:16,764 --> 00:24:18,680 to be something that becomes intuitive to you. 499 00:24:18,680 --> 00:24:20,440 So when I talk about something with short features, 500 00:24:20,440 --> 00:24:22,940 I'm going to talk about it as something with large momentum. 501 00:24:22,940 --> 00:24:25,960 And that's because in a quantum mechanical system, 502 00:24:25,960 --> 00:24:27,990 something with short wavelength is 503 00:24:27,990 --> 00:24:31,236 something that carries large momentum. 504 00:24:31,236 --> 00:24:32,351 That cool? 505 00:24:32,351 --> 00:24:32,850 Great. 506 00:24:32,850 --> 00:24:33,930 Good question. 507 00:24:33,930 --> 00:24:36,270 AUDIENCE: So earlier you said that any reasonable wave 508 00:24:36,270 --> 00:24:40,122 function, a possible wave function, 509 00:24:40,122 --> 00:24:41,580 does that mean they're not supposed 510 00:24:41,580 --> 00:24:43,100 to be Fourier transformable? 511 00:24:43,100 --> 00:24:45,640 PROFESSOR: That's usually a condition. 512 00:24:45,640 --> 00:24:46,360 Yeah, exactly. 513 00:24:46,360 --> 00:24:47,600 We don't quite phrase it that way. 514 00:24:47,600 --> 00:24:49,683 And in fact, there's a problem on your problem set 515 00:24:49,683 --> 00:24:52,060 that will walk you through what we will mean. 516 00:24:52,060 --> 00:24:53,700 What should be true of the Fourier 517 00:24:53,700 --> 00:24:56,410 transform in order for this to reasonably function. 518 00:24:56,410 --> 00:24:58,450 And among other things-- and your intuition 519 00:24:58,450 --> 00:25:00,710 here is exactly right-- among other things, 520 00:25:00,710 --> 00:25:03,190 being able to have a Fourier transform where you don't have 521 00:25:03,190 --> 00:25:04,822 arbitrarily high momentum modes is 522 00:25:04,822 --> 00:25:06,280 going to be an important condition. 523 00:25:06,280 --> 00:25:09,460 That's going to turn to be related to the derivative 524 00:25:09,460 --> 00:25:11,014 being continuous. 525 00:25:11,014 --> 00:25:12,180 That's a very good question. 526 00:25:12,180 --> 00:25:17,144 So that's the optional problem 8 on problem set 2. 527 00:25:17,144 --> 00:25:17,810 Other questions? 528 00:25:22,340 --> 00:25:25,130 PROFESSOR: Cool, so that's it for the clicker questions. 529 00:25:25,130 --> 00:25:26,490 Sorry for the technology fail. 530 00:25:29,400 --> 00:25:32,505 So I'm just going to turn this off in disgust. 531 00:25:37,384 --> 00:25:38,425 That's really irritating. 532 00:25:41,580 --> 00:25:44,880 So today what I want to start on is pick up 533 00:25:44,880 --> 00:25:47,070 on the discussion of the uncertainty principle 534 00:25:47,070 --> 00:25:49,440 that we sort of outlined previously. 535 00:25:49,440 --> 00:25:51,950 The fact that when we have a wave function with reasonably 536 00:25:51,950 --> 00:25:53,460 well defined position corresponding 537 00:25:53,460 --> 00:25:55,744 to a particle with reasonably well defined position, 538 00:25:55,744 --> 00:25:58,160 it didn't have a reasonably well defined momentum and vice 539 00:25:58,160 --> 00:25:58,850 versa. 540 00:25:58,850 --> 00:26:00,630 The certainty of the momentum seems 541 00:26:00,630 --> 00:26:04,570 to imply lack of knowledge about the position and vice versa. 542 00:26:04,570 --> 00:26:09,520 So in order to do that, we need to define uncertainty. 543 00:26:09,520 --> 00:26:11,930 So I need to define for you delta x and delta p. 544 00:26:14,470 --> 00:26:16,740 So first I just want to run through what 545 00:26:16,740 --> 00:26:19,290 should be totally remedial probability, 546 00:26:19,290 --> 00:26:23,570 but it's always useful to just remember 547 00:26:23,570 --> 00:26:25,310 how these basic things work. 548 00:26:25,310 --> 00:26:28,620 So consider a set of people in a room, 549 00:26:28,620 --> 00:26:32,370 and I want to plot the number of people with a particular age 550 00:26:32,370 --> 00:26:36,240 as a function of the age of possible ages. 551 00:26:36,240 --> 00:26:41,540 So let's say we have 16 people, and at 14 we have one, 552 00:26:41,540 --> 00:26:46,720 and at 15 we have 1, and at 16 we have 3. 553 00:26:46,720 --> 00:26:50,170 And that's 16. 554 00:26:50,170 --> 00:26:53,705 And at 20 we have 2. 555 00:26:56,680 --> 00:26:58,715 And at 21 we have 4. 556 00:27:02,270 --> 00:27:03,935 And at 22 we have 5. 557 00:27:09,040 --> 00:27:10,070 And that's it. 558 00:27:10,070 --> 00:27:12,180 OK. 559 00:27:12,180 --> 00:27:23,020 So 1, 1, 3, 2, 4, 5. 560 00:27:23,020 --> 00:27:26,950 OK, so what's the probability that any given 561 00:27:26,950 --> 00:27:29,750 person in this group of 16 has a particular age? 562 00:27:29,750 --> 00:27:30,647 I'll call it a. 563 00:27:33,340 --> 00:27:34,840 So how do we compute the probability 564 00:27:34,840 --> 00:27:36,630 that they have age a? 565 00:27:36,630 --> 00:27:37,380 Well this is easy. 566 00:27:37,380 --> 00:27:40,460 It's the number that have age a over the total number. 567 00:27:44,660 --> 00:27:47,140 So note an important thing, an important side note, 568 00:27:47,140 --> 00:27:49,860 which is that the sum over all possible ages 569 00:27:49,860 --> 00:27:52,955 of the probability that you have age a is equal to 1, 570 00:27:52,955 --> 00:27:55,080 because it's just going to be the sum of the number 571 00:27:55,080 --> 00:27:57,340 with a particular age over the total number, which is just 572 00:27:57,340 --> 00:27:59,048 the sum of the number with any given age. 573 00:28:04,540 --> 00:28:05,750 So here's some questions. 574 00:28:05,750 --> 00:28:09,264 So what's the most likely age? 575 00:28:09,264 --> 00:28:10,680 If you grabbed one of these people 576 00:28:10,680 --> 00:28:12,495 from the room with a giant Erector set, 577 00:28:12,495 --> 00:28:14,030 and pull out a person, and let them dangle, 578 00:28:14,030 --> 00:28:15,613 and ask them what their age is, what's 579 00:28:15,613 --> 00:28:17,288 the most likely they'll have? 580 00:28:17,288 --> 00:28:18,382 AUDIENCE: 22. 581 00:28:18,382 --> 00:28:18,965 PROFESSOR: 22. 582 00:28:21,211 --> 00:28:22,960 On the other hand, what's the average age? 583 00:28:31,700 --> 00:28:35,420 Well, just by eyeball roughly what do you think it is? 584 00:28:35,420 --> 00:28:37,030 So around 19 or 20. 585 00:28:37,030 --> 00:28:40,220 It turns out to be 19.2 for this. 586 00:28:40,220 --> 00:28:40,840 OK. 587 00:28:40,840 --> 00:28:43,490 But if everyone had a little sticker on their lapel 588 00:28:43,490 --> 00:28:47,770 that says I'm 14, 15, 16, 20, 21 or 22, how many people have 589 00:28:47,770 --> 00:28:49,910 the age 19.2? 590 00:28:49,910 --> 00:28:51,010 None, right? 591 00:28:51,010 --> 00:28:54,020 So a useful thing is that the average need not 592 00:28:54,020 --> 00:28:55,940 be an observable value. 593 00:28:57,985 --> 00:28:59,610 This is going to come back to haunt us. 594 00:28:59,610 --> 00:29:02,210 Oops, 19.4. 595 00:29:02,210 --> 00:29:03,550 That's what I got. 596 00:29:03,550 --> 00:29:11,300 So in particular how did I get the average? 597 00:29:11,300 --> 00:29:13,050 I'm going to define some notation. 598 00:29:13,050 --> 00:29:14,320 This notation is going to stick with us 599 00:29:14,320 --> 00:29:15,736 for the rest of quantum mechanics. 600 00:29:15,736 --> 00:29:18,800 The average age, how do I compute it? 601 00:29:18,800 --> 00:29:21,330 So we all know this, but let me just be explicit about it. 602 00:29:21,330 --> 00:29:25,240 It's the sum over all possible ages 603 00:29:25,240 --> 00:29:29,610 of the number of the number of people 604 00:29:29,610 --> 00:29:32,950 with that age times the age divided 605 00:29:32,950 --> 00:29:34,800 by the total number of people. 606 00:29:37,100 --> 00:29:37,600 OK? 607 00:29:41,000 --> 00:29:46,892 So in this case, I'd go 14,14, 16, 16, 16, 20, 20, 21, 21, 608 00:29:46,892 --> 00:29:49,620 21 21, 22, 22, 22, 22, 22. 609 00:29:49,620 --> 00:29:51,662 And so that's all I've written here. 610 00:29:51,662 --> 00:29:53,620 But notice that I can write this in a nice way. 611 00:29:53,620 --> 00:29:56,462 This is equal to the sum over all possible ages 612 00:29:56,462 --> 00:30:01,759 of a times the ratio of Na to N with a ratio of Na to n total. 613 00:30:01,759 --> 00:30:03,800 That's just the probability that any given person 614 00:30:03,800 --> 00:30:05,490 has a probability a. 615 00:30:05,490 --> 00:30:07,610 a times probability of a. 616 00:30:07,610 --> 00:30:11,680 So the expected value is the sum over all possible values 617 00:30:11,680 --> 00:30:14,460 of the value times the probability to get that value. 618 00:30:14,460 --> 00:30:15,840 Yeah? 619 00:30:15,840 --> 00:30:18,520 This is the same equation, but I'm going to box it. 620 00:30:18,520 --> 00:30:21,820 It's a very useful relation. 621 00:30:21,820 --> 00:30:24,290 And so, again, does the average have to be measurable? 622 00:30:24,290 --> 00:30:26,225 No, it certainly doesn't. 623 00:30:26,225 --> 00:30:29,310 And it usually isn't. 624 00:30:29,310 --> 00:30:33,590 So let's ask the same thing for the square of ages. 625 00:30:33,590 --> 00:30:39,190 What is the average of a squared? 626 00:30:39,190 --> 00:30:39,920 Square the ages. 627 00:30:39,920 --> 00:30:42,500 You might say, well, why would I ever care about that? 628 00:30:42,500 --> 00:30:43,999 But let's just be explicit about it. 629 00:30:43,999 --> 00:30:45,630 So following the same logic here, 630 00:30:45,630 --> 00:30:48,460 the average of a squared, the average value 631 00:30:48,460 --> 00:30:50,092 of the square of the ages is, well, I'm 632 00:30:50,092 --> 00:30:51,550 going to do exactly the same thing. 633 00:30:51,550 --> 00:30:53,060 It's just a squared, right? 634 00:30:53,060 --> 00:30:56,317 14 squared, 15 squared, 16 square, 16 squared, 16 squared. 635 00:30:56,317 --> 00:30:58,650 So this is going to give me exactly the same expression. 636 00:30:58,650 --> 00:31:03,330 So over a of a squared probability of measuring a. 637 00:31:07,140 --> 00:31:12,230 And more generally, the expected value, or the average value 638 00:31:12,230 --> 00:31:14,802 of some function of a is equal-- and this 639 00:31:14,802 --> 00:31:16,260 is something you don't usually do-- 640 00:31:16,260 --> 00:31:21,610 is equal to the sum over a of f of a, the value of f given 641 00:31:21,610 --> 00:31:24,309 a particular value of a, times the probability that you 642 00:31:24,309 --> 00:31:26,100 measure that value of a in the first place. 643 00:31:29,320 --> 00:31:33,855 It's exactly the same logic as averages. 644 00:31:37,340 --> 00:31:39,330 Right, cool. 645 00:31:39,330 --> 00:31:41,940 So here's a quick question. 646 00:31:41,940 --> 00:31:47,700 Is a squared equal to the expected value of a squared? 647 00:31:47,700 --> 00:31:48,844 AUDIENCE: No. 648 00:31:48,844 --> 00:31:50,885 PROFESSOR: Right, in general no, not necessarily. 649 00:31:56,810 --> 00:32:00,480 So for example, the average value-- 650 00:32:00,480 --> 00:32:02,780 suppose we have a Gaussian centered at the origin. 651 00:32:02,780 --> 00:32:04,760 So here's a. 652 00:32:04,760 --> 00:32:07,076 Now a isn't age, but it's something-- I don't know. 653 00:32:07,076 --> 00:32:09,570 You include infants or whatever. 654 00:32:09,570 --> 00:32:10,320 It's not age. 655 00:32:10,320 --> 00:32:13,340 Its happiness on a given day. 656 00:32:13,340 --> 00:32:17,920 So what's the average value? 657 00:32:17,920 --> 00:32:19,000 Meh. 658 00:32:19,000 --> 00:32:19,580 Right? 659 00:32:19,580 --> 00:32:21,910 Sort of vaguely neutral, right? 660 00:32:21,910 --> 00:32:24,080 But on the other hand, if you take a squared, 661 00:32:24,080 --> 00:32:26,080 very few people have a squared as zero. 662 00:32:26,080 --> 00:32:28,340 Most people have a squared as not a 0 value. 663 00:32:28,340 --> 00:32:30,120 And most people are sort of in the middle. 664 00:32:30,120 --> 00:32:34,130 Most people are sort of hazy on what the day is. 665 00:32:34,130 --> 00:32:37,430 So in this case, the expected value 666 00:32:37,430 --> 00:32:39,170 of a, or the average value of a is 0. 667 00:32:39,170 --> 00:32:43,620 The average value of a squared is not equal to 0. 668 00:32:43,620 --> 00:32:44,500 Yeah? 669 00:32:44,500 --> 00:32:46,791 And that's because the squared has everything positive. 670 00:32:48,810 --> 00:32:51,560 So how do we characterize-- this gives us 671 00:32:51,560 --> 00:32:54,390 a useful tool for characterizing the width of a distribution. 672 00:32:54,390 --> 00:32:56,890 So here we have a distribution where its average value is 0, 673 00:32:56,890 --> 00:32:58,000 but its width is non-zero. 674 00:32:58,000 --> 00:33:00,810 And then the expectation value of a squared, 675 00:33:00,810 --> 00:33:03,310 the expected value of a squared, is non-zero. 676 00:33:03,310 --> 00:33:07,700 So how do we define the width of a distribution? 677 00:33:07,700 --> 00:33:10,020 This is going to be like our uncertainty. 678 00:33:10,020 --> 00:33:11,100 How happy are you today? 679 00:33:11,100 --> 00:33:11,910 Well, I'm not sure. 680 00:33:11,910 --> 00:33:12,880 How unsure are you? 681 00:33:12,880 --> 00:33:14,980 Well, that should give us a precise measure. 682 00:33:14,980 --> 00:33:17,170 So let me define three things. 683 00:33:17,170 --> 00:33:19,060 First the deviation. 684 00:33:19,060 --> 00:33:22,770 So the deviation is going to be a minus the average value of a. 685 00:33:22,770 --> 00:33:24,520 So this is just take the actual value of a 686 00:33:24,520 --> 00:33:26,341 and subtract off the average value of a. 687 00:33:26,341 --> 00:33:28,340 So we always get something that's centered at 0. 688 00:33:32,087 --> 00:33:33,420 I'm going to write it like this. 689 00:33:36,240 --> 00:33:38,730 Note, by the way, just a convenient thing to note. 690 00:33:38,730 --> 00:33:43,920 The average value of a minus it's average value. 691 00:33:43,920 --> 00:33:46,564 Well, what's the average value of 7? 692 00:33:46,564 --> 00:33:47,447 AUDIENCE: 7. 693 00:33:47,447 --> 00:33:48,280 PROFESSOR: OK, good. 694 00:33:48,280 --> 00:33:52,832 So that first term is the average value of a. 695 00:33:52,832 --> 00:33:54,540 And that second term is the average value 696 00:33:54,540 --> 00:33:58,090 of this number, which is just this number minus a. 697 00:33:58,090 --> 00:33:59,931 So this is 0. 698 00:33:59,931 --> 00:34:00,430 Yeah? 699 00:34:04,092 --> 00:34:05,550 The average value of a number is 0. 700 00:34:05,550 --> 00:34:06,966 The average value of this variable 701 00:34:06,966 --> 00:34:10,469 is the average value of that variable, but that's 0. 702 00:34:10,469 --> 00:34:12,750 So deviation is not a terribly good thing on average, 703 00:34:12,750 --> 00:34:14,625 because on average the deviation is always 0. 704 00:34:14,625 --> 00:34:17,602 That's what it means to say this is the average. 705 00:34:17,602 --> 00:34:19,060 So the derivation is saying how far 706 00:34:19,060 --> 00:34:21,560 is any particular instance from the average. 707 00:34:21,560 --> 00:34:23,060 And if you average those deviations, 708 00:34:23,060 --> 00:34:24,260 they always give you 0. 709 00:34:24,260 --> 00:34:25,909 So this is not a very good measure 710 00:34:25,909 --> 00:34:28,630 of the actual width of the system. 711 00:34:28,630 --> 00:34:31,559 But we can get a nice measure by getting the deviation squared. 712 00:34:34,600 --> 00:34:36,880 And let's take the mean of the derivation squared. 713 00:34:36,880 --> 00:34:40,829 So the mean of the derivation squared, mean of a minus 714 00:34:40,829 --> 00:34:42,120 the average value of a squared. 715 00:34:45,179 --> 00:34:48,669 This is what I'm going to call the standard deviation. 716 00:34:48,669 --> 00:34:50,460 Which is a little odd, because really you'd 717 00:34:50,460 --> 00:34:52,418 want to call it the standard deviation squared. 718 00:34:52,418 --> 00:34:53,886 But whatever. 719 00:34:53,886 --> 00:34:55,219 We use funny words. 720 00:34:59,450 --> 00:35:03,225 So now what does it mean if the average value of a is 0? 721 00:35:03,225 --> 00:35:05,100 It means it's centered at 0, but what does it 722 00:35:05,100 --> 00:35:07,830 mean if the standard deviation of a is 0? 723 00:35:10,880 --> 00:35:15,610 So if the standard deviation is 0, 724 00:35:15,610 --> 00:35:21,280 one then the distribution has no width, right? 725 00:35:21,280 --> 00:35:23,450 Because if there was any amplitude away 726 00:35:23,450 --> 00:35:25,600 from the average value, then that 727 00:35:25,600 --> 00:35:27,960 would give a non-zero strictly positive contribution 728 00:35:27,960 --> 00:35:31,280 to this average expectation, and this wouldn't be 0 anymore. 729 00:35:31,280 --> 00:35:33,120 So standard deviation is 0, as long 730 00:35:33,120 --> 00:35:35,578 as there's no width, which is why the standard deviation is 731 00:35:35,578 --> 00:35:40,970 a good useful measure of width or uncertainty. 732 00:35:40,970 --> 00:35:45,660 And just as a note, taking this seriously and taking 733 00:35:45,660 --> 00:35:49,200 the square, so standard deviation squared, 734 00:35:49,200 --> 00:35:52,320 this is equal to the average value of a squared 735 00:35:52,320 --> 00:35:57,150 minus twice a times the average value of a plus average value 736 00:35:57,150 --> 00:35:59,500 of a quantity squared. 737 00:35:59,500 --> 00:36:01,110 But if you do this out, this is going 738 00:36:01,110 --> 00:36:05,770 to be equal to a squared minus 2 average value 739 00:36:05,770 --> 00:36:07,260 of a average value of a. 740 00:36:07,260 --> 00:36:11,060 That's just minus twice the average value 741 00:36:11,060 --> 00:36:14,180 of a quantity squared. 742 00:36:14,180 --> 00:36:16,360 And then plus average value of a squared. 743 00:36:16,360 --> 00:36:19,230 So this is an alternate way of writing the standard deviation. 744 00:36:19,230 --> 00:36:19,730 OK? 745 00:36:19,730 --> 00:36:23,170 So we can either write it in this fashion or this fashion. 746 00:36:23,170 --> 00:36:29,075 And the notation for this is delta a squared. 747 00:36:32,750 --> 00:36:33,580 OK? 748 00:36:33,580 --> 00:36:35,840 So when I talk about an uncertainty, what 749 00:36:35,840 --> 00:36:38,520 I mean is, given my distribution, 750 00:36:38,520 --> 00:36:40,094 I compute the standard deviation. 751 00:36:40,094 --> 00:36:41,510 And the uncertainty is going to be 752 00:36:41,510 --> 00:36:44,302 the square root of the standard deviations squared. 753 00:36:44,302 --> 00:36:45,990 OK? 754 00:36:45,990 --> 00:36:48,880 So delta a, the words I'm going to use for this 755 00:36:48,880 --> 00:36:59,430 is the uncertainty in a given some probability distribution. 756 00:37:01,612 --> 00:37:03,820 Different probability distributions are going to give 757 00:37:03,820 --> 00:37:05,020 me different delta a's. 758 00:37:09,240 --> 00:37:10,880 So one thing that's sort of annoying 759 00:37:10,880 --> 00:37:12,706 is that when you write delta a, there's 760 00:37:12,706 --> 00:37:14,330 nothing in the notation that says which 761 00:37:14,330 --> 00:37:16,360 distribution you were talking about. 762 00:37:16,360 --> 00:37:18,030 When you have multiple distributions, 763 00:37:18,030 --> 00:37:21,582 or multiple possible probability distributions, sometimes it's 764 00:37:21,582 --> 00:37:23,790 useful to just put given the probability distribution 765 00:37:23,790 --> 00:37:25,670 p of a. 766 00:37:25,670 --> 00:37:28,170 This is not very often used, but sometimes it's very helpful 767 00:37:28,170 --> 00:37:30,984 when you're doing calculations just to keep track. 768 00:37:30,984 --> 00:37:34,010 Everyone cool with that? 769 00:37:34,010 --> 00:37:35,702 Yeah, questions? 770 00:37:35,702 --> 00:37:37,590 AUDIENCE: [INAUDIBLE] delta a squared, right? 771 00:37:37,590 --> 00:37:38,980 PROFESSOR: Yeah, exactly. 772 00:37:38,980 --> 00:37:40,145 Of delta a squared. 773 00:37:40,145 --> 00:37:41,450 Yeah. 774 00:37:41,450 --> 00:37:43,550 Other questions? 775 00:37:43,550 --> 00:37:44,100 Yeah? 776 00:37:44,100 --> 00:37:45,890 AUDIENCE: So really it should be parentheses [INAUDIBLE]. 777 00:37:45,890 --> 00:37:47,620 PROFESSOR: Yeah, it's just this is notation that's 778 00:37:47,620 --> 00:37:50,010 used typically, so I didn't put the parentheses around 779 00:37:50,010 --> 00:37:53,140 precisely to alert you to the stupidities of this notation. 780 00:37:59,580 --> 00:38:02,810 So any other questions? 781 00:38:02,810 --> 00:38:03,310 Good. 782 00:38:03,310 --> 00:38:06,970 OK, so let's just do the same thing for continuous variables. 783 00:38:06,970 --> 00:38:08,610 Now for continuous variables. 784 00:38:14,516 --> 00:38:16,140 I'm just going to write the expressions 785 00:38:16,140 --> 00:38:17,790 and just get them out of the way. 786 00:38:17,790 --> 00:38:20,900 So the average value of some x, given a probability 787 00:38:20,900 --> 00:38:23,220 distribution on x where x is a continuous variable, 788 00:38:23,220 --> 00:38:24,952 is going to be equal to the integral. 789 00:38:24,952 --> 00:38:26,910 Let's just say x is defined from minus infinity 790 00:38:26,910 --> 00:38:31,500 to infinity, which is pretty useful, or pretty typical. 791 00:38:31,500 --> 00:38:37,644 dx probability distribution of x times x. 792 00:38:37,644 --> 00:38:38,560 I shouldn't use curvy. 793 00:38:38,560 --> 00:38:39,620 I should just use x. 794 00:38:42,150 --> 00:38:44,960 And similarly for x squared, or more 795 00:38:44,960 --> 00:38:47,920 generally, for f of x, the average value of f 796 00:38:47,920 --> 00:38:52,050 of x, or the expected value of f of x given this probability 797 00:38:52,050 --> 00:38:54,900 distribution, is going to be equal to the integral dx 798 00:38:54,900 --> 00:38:56,960 minus infinity to infinity. 799 00:38:56,960 --> 00:39:01,124 The probability distribution of x times f of x. 800 00:39:01,124 --> 00:39:02,790 In direct analogy to what we had before. 801 00:39:05,890 --> 00:39:08,850 So this is all just mathematics. 802 00:39:08,850 --> 00:39:11,620 And we define the uncertainty in x 803 00:39:11,620 --> 00:39:15,940 is equal to the expectation value of x squared 804 00:39:15,940 --> 00:39:20,155 minus the expected value of x quantity squared. 805 00:39:23,560 --> 00:39:24,760 And this is delta x squared. 806 00:39:27,430 --> 00:39:30,210 If you see me dropping an exponent or a factor of 2, 807 00:39:30,210 --> 00:39:33,110 please, please, please tell me. 808 00:39:33,110 --> 00:39:36,060 So thank you for that. 809 00:39:36,060 --> 00:39:39,720 All of that is just straight up classical probability theory. 810 00:39:39,720 --> 00:39:42,750 And I just want to write this in the notation of quantum 811 00:39:42,750 --> 00:39:43,660 mechanics. 812 00:39:43,660 --> 00:39:45,850 Given that the system is in a state 813 00:39:45,850 --> 00:39:50,210 described by the wave function psi of x, the average value, 814 00:39:50,210 --> 00:39:53,560 the expected value of x, the typical value if you just 815 00:39:53,560 --> 00:39:56,160 observe the particle at some moment, 816 00:39:56,160 --> 00:40:01,200 is equal to the integral over all possible values of x. 817 00:40:01,200 --> 00:40:05,790 The probability distribution, psi of x norm squared x. 818 00:40:08,480 --> 00:40:12,480 And similarly, for any function of x, 819 00:40:12,480 --> 00:40:15,250 the expected value is going to be equal to the integral dx. 820 00:40:15,250 --> 00:40:17,583 The probability distribution, which is given by the norm 821 00:40:17,583 --> 00:40:23,710 squared of the wave function times f of x minus infinity 822 00:40:23,710 --> 00:40:26,140 to infinity. 823 00:40:26,140 --> 00:40:30,010 And same definition for uncertainty. 824 00:40:30,010 --> 00:40:33,070 And again, this notation is really dangerous, 825 00:40:33,070 --> 00:40:37,010 because the expected value of x depends on the probability 826 00:40:37,010 --> 00:40:37,790 distribution. 827 00:40:37,790 --> 00:40:39,985 In a physical system, the expected value of x 828 00:40:39,985 --> 00:40:41,610 depends on what the state of the system 829 00:40:41,610 --> 00:40:43,610 is, what the wave function is, and this notation 830 00:40:43,610 --> 00:40:44,970 doesn't indicate that. 831 00:40:44,970 --> 00:40:47,450 So there are a couple of ways to improve this notation. 832 00:40:47,450 --> 00:40:52,540 One of which is-- so this is, again, a sort of side note. 833 00:40:52,540 --> 00:40:54,810 One way to improve this notation x 834 00:40:54,810 --> 00:40:59,450 is to write the expected value of x in the state psi, 835 00:40:59,450 --> 00:41:01,036 so you write psi as a subscript. 836 00:41:01,036 --> 00:41:02,910 Another notation that will come back-- you'll 837 00:41:02,910 --> 00:41:05,040 see why this is a useful notation later 838 00:41:05,040 --> 00:41:10,511 in the semester-- is this notation, psi. 839 00:41:10,511 --> 00:41:12,510 And we will give meaning to this notation later, 840 00:41:12,510 --> 00:41:13,610 but I just want to alert you that it's 841 00:41:13,610 --> 00:41:15,690 used throughout books, and it means the same thing 842 00:41:15,690 --> 00:41:17,814 as what we're talking about the expected value of x 843 00:41:17,814 --> 00:41:19,880 given a particular state psi. 844 00:41:19,880 --> 00:41:20,640 OK? 845 00:41:20,640 --> 00:41:21,140 Yeah? 846 00:41:21,140 --> 00:41:23,480 AUDIENCE: To calculate the expected value of momentum 847 00:41:23,480 --> 00:41:25,630 do you need to transform the-- 848 00:41:25,630 --> 00:41:26,880 PROFESSOR: Excellent question. 849 00:41:26,880 --> 00:41:27,840 Excellent, excellent question. 850 00:41:27,840 --> 00:41:29,290 OK, so the question is, how do we 851 00:41:29,290 --> 00:41:30,961 do the same thing for momentum? 852 00:41:30,961 --> 00:41:33,210 If you want to compute the expected value of momentum, 853 00:41:33,210 --> 00:41:34,170 what do you have to do? 854 00:41:34,170 --> 00:41:36,790 Do you have to do some Fourier transform to the wave function? 855 00:41:36,790 --> 00:41:39,830 So this is a question that you're 856 00:41:39,830 --> 00:41:41,360 going to answer on the problem set 857 00:41:41,360 --> 00:41:43,334 and that we made a guess for last time. 858 00:41:43,334 --> 00:41:45,000 But quickly, let's just think about what 859 00:41:45,000 --> 00:41:46,820 it's going to be purely formally. 860 00:41:46,820 --> 00:41:50,540 Formally, if we want to know the likely value of the momentum, 861 00:41:50,540 --> 00:41:53,210 the likely value the momentum, it's a continuous variable. 862 00:41:53,210 --> 00:41:55,400 Just like any other observable variable, 863 00:41:55,400 --> 00:41:58,700 we can write as the integral over all possible values 864 00:41:58,700 --> 00:42:01,270 of momentum from, let's say, it could 865 00:42:01,270 --> 00:42:03,980 be minus infinity to infinity. 866 00:42:03,980 --> 00:42:09,141 The probability of having that momentum times momentum, right? 867 00:42:09,141 --> 00:42:10,140 Everyone cool with that? 868 00:42:10,140 --> 00:42:11,590 This is a tautology, right? 869 00:42:11,590 --> 00:42:14,930 This is what you mean by probability. 870 00:42:14,930 --> 00:42:17,600 But we need to know if we have a quantum mechanical system 871 00:42:17,600 --> 00:42:19,830 described by state psi of x, how do 872 00:42:19,830 --> 00:42:23,066 we can get the probability that you measure p? 873 00:42:23,066 --> 00:42:25,096 Do I want to do this now? 874 00:42:25,096 --> 00:42:27,430 Yeah, OK I do. 875 00:42:27,430 --> 00:42:30,780 And we need a guess. 876 00:42:30,780 --> 00:42:31,391 Question mark. 877 00:42:31,391 --> 00:42:33,140 We made a guess at the end of last lecture 878 00:42:33,140 --> 00:42:34,765 that, in quantum mechanics, this should 879 00:42:34,765 --> 00:42:41,820 be dp minus infinity to infinity of the Fourier transform. 880 00:42:41,820 --> 00:42:46,700 Psi tilde of p up to an h bar factor. 881 00:42:46,700 --> 00:42:53,160 Psi tilde of p, the Fourier transform p norm squared. 882 00:42:53,160 --> 00:42:56,730 OK, so we're guessing that the Fourier transform norm squared 883 00:42:56,730 --> 00:42:59,190 is equal to the probability of measuring the associated 884 00:42:59,190 --> 00:43:00,850 momentum. 885 00:43:00,850 --> 00:43:03,370 So that's a guess. 886 00:43:03,370 --> 00:43:04,090 That's a guess. 887 00:43:04,090 --> 00:43:06,400 And so on your problem set you're going to prove it. 888 00:43:06,400 --> 00:43:07,220 OK? 889 00:43:07,220 --> 00:43:08,940 So exactly the same logic goes through. 890 00:43:08,940 --> 00:43:10,050 It's a very good question, thanks. 891 00:43:10,050 --> 00:43:10,850 Other questions? 892 00:43:10,850 --> 00:43:12,698 Yeah? 893 00:43:12,698 --> 00:43:14,674 AUDIENCE: Is that p the momentum itself? 894 00:43:14,674 --> 00:43:17,150 Or is that the probability? 895 00:43:17,150 --> 00:43:19,100 PROFESSOR: So this is the probability 896 00:43:19,100 --> 00:43:20,990 of measuring momentum p. 897 00:43:20,990 --> 00:43:22,770 And that's the value p. 898 00:43:22,770 --> 00:43:23,980 We're summing over all p's. 899 00:43:23,980 --> 00:43:27,660 This is the probability, and that's actually p. 900 00:43:27,660 --> 00:43:29,351 So the Fourier transform is a function 901 00:43:29,351 --> 00:43:31,600 of the momentum in the same way that the wave function 902 00:43:31,600 --> 00:43:34,380 is a function of the position, right? 903 00:43:34,380 --> 00:43:36,080 So this is a function of the momentum. 904 00:43:36,080 --> 00:43:39,917 It's norm squared defines the probability. 905 00:43:39,917 --> 00:43:41,500 And then the p on the right is this p, 906 00:43:41,500 --> 00:43:43,500 because we're computing the expected value of p, 907 00:43:43,500 --> 00:43:44,640 or the average value of p. 908 00:43:44,640 --> 00:43:46,100 That make sense? 909 00:43:46,100 --> 00:43:47,530 Cool. 910 00:43:47,530 --> 00:43:48,426 Yeah? 911 00:43:48,426 --> 00:43:50,590 AUDIENCE: Are we then multiplying by p squared 912 00:43:50,590 --> 00:43:52,027 if we're doing all p's? 913 00:43:52,027 --> 00:43:56,340 Because we have the dp times p for each [INAUDIBLE]. 914 00:43:56,340 --> 00:43:56,960 PROFESSOR: No. 915 00:43:56,960 --> 00:43:58,100 So that's a very good question. 916 00:43:58,100 --> 00:43:58,920 So let's go back. 917 00:43:58,920 --> 00:44:00,490 Very good question. 918 00:44:00,490 --> 00:44:02,470 Let me phrase it in terms of position, 919 00:44:02,470 --> 00:44:03,430 because the same question comes up. 920 00:44:03,430 --> 00:44:04,580 Thank you for asking that. 921 00:44:04,580 --> 00:44:05,250 Look at this. 922 00:44:05,250 --> 00:44:06,070 This is weird. 923 00:44:06,070 --> 00:44:08,570 I'm going to phrase this as a dimensional analysis question. 924 00:44:08,570 --> 00:44:10,520 Tell me if this is the same question as you're asking. 925 00:44:10,520 --> 00:44:12,310 This is a thing with dimensions of what? 926 00:44:12,310 --> 00:44:13,542 Length, right? 927 00:44:13,542 --> 00:44:15,000 But over on the right hand side, we 928 00:44:15,000 --> 00:44:19,305 have a length and a probability, which is a number, and then 929 00:44:19,305 --> 00:44:19,930 another length. 930 00:44:19,930 --> 00:44:21,940 That looks like x squared, right? 931 00:44:21,940 --> 00:44:23,570 So why are we getting something with dimensions of length, 932 00:44:23,570 --> 00:44:25,610 not something with dimensions of length squared? 933 00:44:25,610 --> 00:44:27,443 And the answer is this is not a probability. 934 00:44:27,443 --> 00:44:29,970 It is a probability density. 935 00:44:29,970 --> 00:44:34,180 So it's got units of probability per unit length. 936 00:44:34,180 --> 00:44:36,240 So this has dimensions of one over length. 937 00:44:36,240 --> 00:44:39,380 So this quantity, p of x dx, tells me 938 00:44:39,380 --> 00:44:41,930 the probability, which is a pure number, no dimensions. 939 00:44:41,930 --> 00:44:44,690 The probability to find the particle between x and x 940 00:44:44,690 --> 00:44:46,550 plus dx. 941 00:44:46,550 --> 00:44:47,110 Cool? 942 00:44:47,110 --> 00:44:50,100 So that was our second postulate. 943 00:44:50,100 --> 00:44:52,455 Psi of x dx squared is the probability 944 00:44:52,455 --> 00:44:54,640 of finding it in this domain. 945 00:44:54,640 --> 00:44:59,040 And so what we're doing is we're summing over all such domains 946 00:44:59,040 --> 00:45:02,610 the probability times the value. 947 00:45:02,610 --> 00:45:03,490 Cool? 948 00:45:03,490 --> 00:45:05,730 So this is the difference between discrete, 949 00:45:05,730 --> 00:45:09,115 where we didn't have these probability densities, 950 00:45:09,115 --> 00:45:11,490 we just had numbers, pure numbers and pure probabilities. 951 00:45:11,490 --> 00:45:14,776 Now we have probability densities per unit whatever. 952 00:45:14,776 --> 00:45:15,405 Yeah? 953 00:45:15,405 --> 00:45:17,405 AUDIENCE: How do you pronounce the last notation 954 00:45:17,405 --> 00:45:18,707 that you wrote? 955 00:45:18,707 --> 00:45:20,040 PROFESSOR: How do you pronounce? 956 00:45:20,040 --> 00:45:21,248 Good, that's a good question. 957 00:45:21,248 --> 00:45:23,500 The question is, how do we pronounce these things. 958 00:45:23,500 --> 00:45:25,160 So this is called the expected value 959 00:45:25,160 --> 00:45:28,190 of x, or the average value of x, or most typically in quantum 960 00:45:28,190 --> 00:45:31,130 mechanics, the expectation value of x. 961 00:45:31,130 --> 00:45:33,160 So you can call it anything you want. 962 00:45:33,160 --> 00:45:34,830 This is the same thing. 963 00:45:34,830 --> 00:45:38,029 The psi is just to denote that this is in the state psi. 964 00:45:38,029 --> 00:45:39,570 And it can be pronounced in two ways. 965 00:45:39,570 --> 00:45:41,236 You can either say the expectation value 966 00:45:41,236 --> 00:45:45,600 of x, or the expectation of x in the state psi. 967 00:45:45,600 --> 00:45:48,610 And this would be pronounced one of two ways. 968 00:45:48,610 --> 00:45:55,451 The expectation value of x in the state psi, or psi x psi. 969 00:45:55,451 --> 00:45:55,950 Yeah. 970 00:45:55,950 --> 00:45:58,030 That's a very good question. 971 00:45:58,030 --> 00:45:59,240 But they mean the same thing. 972 00:45:59,240 --> 00:46:02,812 Now, I should emphasize that you can have two ways of describing 973 00:46:02,812 --> 00:46:04,270 something that mean the same thing, 974 00:46:04,270 --> 00:46:06,144 but they carry different connotations, right? 975 00:46:08,500 --> 00:46:12,727 Like have a friend who's a really nice guy. 976 00:46:12,727 --> 00:46:13,310 He's a mensch. 977 00:46:13,310 --> 00:46:14,310 He's a good guy. 978 00:46:14,310 --> 00:46:16,020 And so I could see he's a nice guy, 979 00:46:16,020 --> 00:46:18,200 I could say he's [? carinoso ?], and they 980 00:46:18,200 --> 00:46:21,490 mean different things in different languages. 981 00:46:21,490 --> 00:46:24,500 It's the same idea, but they have different flavors, right? 982 00:46:24,500 --> 00:46:27,310 So whatever your native language is, 983 00:46:27,310 --> 00:46:29,082 you've got some analog of this. 984 00:46:29,082 --> 00:46:32,045 This means something in a particular mathematical 985 00:46:32,045 --> 00:46:33,920 language for talking about quantum mechanics. 986 00:46:33,920 --> 00:46:35,450 And this has a different flavor. 987 00:46:35,450 --> 00:46:36,896 It carries different implications, 988 00:46:36,896 --> 00:46:38,270 and we'll see what that is later. 989 00:46:38,270 --> 00:46:39,120 We haven't got there yet. 990 00:46:39,120 --> 00:46:39,620 Yeah? 991 00:46:39,620 --> 00:46:41,890 AUDIENCE: Why is there a double notation of psi? 992 00:46:41,890 --> 00:46:44,460 PROFESSOR: Why is there a double notation of psi? 993 00:46:44,460 --> 00:46:47,830 Yeah, we'll see later. 994 00:46:47,830 --> 00:46:50,530 Roughly speaking, it's because in computing this expectation 995 00:46:50,530 --> 00:46:52,480 value, there's a psi squared. 996 00:46:52,480 --> 00:46:55,940 And so this is to remind you of that. 997 00:46:55,940 --> 00:46:58,010 Other questions? 998 00:46:58,010 --> 00:47:00,060 Terminology is one of the most annoying features 999 00:47:00,060 --> 00:47:00,640 of quantum mechanics. 1000 00:47:00,640 --> 00:47:01,139 Yeah? 1001 00:47:01,139 --> 00:47:04,435 AUDIENCE: So it seems like this [INAUDIBLE] variance 1002 00:47:04,435 --> 00:47:07,301 is a really convenient way of doing it. 1003 00:47:07,301 --> 00:47:08,800 How is it the Heisenberg uncertainty 1004 00:47:08,800 --> 00:47:13,480 works exactly as it does for this definition of variance. 1005 00:47:13,480 --> 00:47:15,105 PROFESSOR: That's a very good question. 1006 00:47:17,285 --> 00:47:18,660 In order to answer that question, 1007 00:47:18,660 --> 00:47:20,440 we need to actually work out the Heisenberg uncertainty 1008 00:47:20,440 --> 00:47:20,940 relation. 1009 00:47:20,940 --> 00:47:25,182 So the question is, look, this is some choice of uncertainty. 1010 00:47:25,182 --> 00:47:27,640 You could have chosen some other definition of uncertainly. 1011 00:47:27,640 --> 00:47:29,590 We could have considered the expectation 1012 00:47:29,590 --> 00:47:32,230 value of x to the fourth minus x to the fourth 1013 00:47:32,230 --> 00:47:33,920 and taken the fourth root of that. 1014 00:47:33,920 --> 00:47:36,100 So why this one? 1015 00:47:36,100 --> 00:47:38,460 And one answer is, indeed, the uncertainty relation 1016 00:47:38,460 --> 00:47:40,050 works out quite nicely. 1017 00:47:40,050 --> 00:47:43,170 But then I think important to say 1018 00:47:43,170 --> 00:47:45,837 here is that there are many ways you could construct quantities. 1019 00:47:45,837 --> 00:47:47,378 This is a convenient one, and we will 1020 00:47:47,378 --> 00:47:49,530 discover that it has nice properties that we like. 1021 00:47:49,530 --> 00:47:53,662 There is no God given reason why this had to be the right thing. 1022 00:47:53,662 --> 00:47:56,120 I can say more, but I don't want to take the time to do it, 1023 00:47:56,120 --> 00:47:58,560 so ask in office hours. 1024 00:47:58,560 --> 00:48:01,180 OK, good. 1025 00:48:01,180 --> 00:48:02,680 The second part of your question was 1026 00:48:02,680 --> 00:48:04,690 why does the Heisenberg relation work out 1027 00:48:04,690 --> 00:48:05,750 nicely in terms of these guys, and we 1028 00:48:05,750 --> 00:48:07,400 will study that in extraordinary detail. 1029 00:48:07,400 --> 00:48:08,025 We'll see that. 1030 00:48:08,025 --> 00:48:11,240 So we're going to derive it twice soon and then later. 1031 00:48:11,240 --> 00:48:12,690 The later version is better. 1032 00:48:12,690 --> 00:48:14,115 So let me work out some examples. 1033 00:48:17,330 --> 00:48:21,380 Or actually, I'm going to skip the examples 1034 00:48:21,380 --> 00:48:22,740 in the interest of time. 1035 00:48:22,740 --> 00:48:23,850 They're in the notes, and so they'll 1036 00:48:23,850 --> 00:48:24,933 be posted on the web page. 1037 00:48:24,933 --> 00:48:28,341 By the way, the first 18 lectures of notes are posted. 1038 00:48:28,341 --> 00:48:29,590 I had a busy night last night. 1039 00:48:32,600 --> 00:48:34,547 So let's come back to computing expectation 1040 00:48:34,547 --> 00:48:35,380 values for momentum. 1041 00:48:46,020 --> 00:48:48,000 So I want to go back to this and ask 1042 00:48:48,000 --> 00:48:51,230 a silly-- I want to make some progress towards deriving 1043 00:48:51,230 --> 00:48:51,910 this relation. 1044 00:48:51,910 --> 00:48:55,070 So I want to start over on the definition of the expected 1045 00:48:55,070 --> 00:48:55,820 value of momentum. 1046 00:48:58,119 --> 00:49:00,660 And I'd like to do it directly in terms of the wave function. 1047 00:49:00,660 --> 00:49:01,990 So how would we do this? 1048 00:49:01,990 --> 00:49:04,900 So one way of saying this is what's the average value of p. 1049 00:49:04,900 --> 00:49:06,910 Well, I can phrase this in terms of the wave 1050 00:49:06,910 --> 00:49:08,035 function the following way. 1051 00:49:08,035 --> 00:49:10,630 I'm going to sum over all positions dx. 1052 00:49:10,630 --> 00:49:15,140 Expectation value of x squared from minus infinity 1053 00:49:15,140 --> 00:49:16,620 to infinity. 1054 00:49:16,620 --> 00:49:19,065 And then the momentum associated to the value x. 1055 00:49:23,287 --> 00:49:25,370 So it's tempting to write something like this down 1056 00:49:25,370 --> 00:49:26,950 to think maybe there's some p of x. 1057 00:49:29,476 --> 00:49:31,100 This is a tempting thing to write down. 1058 00:49:33,700 --> 00:49:34,200 Can we? 1059 00:49:40,490 --> 00:49:46,490 Are we ever in a position to say intelligently 1060 00:49:46,490 --> 00:49:50,010 that a particle-- that an electron 1061 00:49:50,010 --> 00:49:55,094 is both hard and white? 1062 00:49:55,094 --> 00:49:56,010 AUDIENCE: No. 1063 00:49:56,010 --> 00:49:58,060 PROFESSOR: No, because being hard 1064 00:49:58,060 --> 00:50:01,676 is a superposition of being black and white, right? 1065 00:50:01,676 --> 00:50:06,210 Are we ever in a position to say that our particle has 1066 00:50:06,210 --> 00:50:08,850 a definite position x and correspondingly 1067 00:50:08,850 --> 00:50:11,450 a definite momentum p. 1068 00:50:11,450 --> 00:50:12,930 It's not that we don't get too. 1069 00:50:12,930 --> 00:50:15,664 It's that it doesn't make sense to do so. 1070 00:50:15,664 --> 00:50:17,330 In general, being in a definite position 1071 00:50:17,330 --> 00:50:21,230 means being in a superposition of having 1072 00:50:21,230 --> 00:50:24,520 different values for momentum. 1073 00:50:24,520 --> 00:50:27,100 And if you want a sharp way of saying this, 1074 00:50:27,100 --> 00:50:29,065 look at these relations. 1075 00:50:29,065 --> 00:50:32,730 They claim that any function can be expressed 1076 00:50:32,730 --> 00:50:37,040 as a superposition of states with definite momentum, right? 1077 00:50:37,040 --> 00:50:41,880 Well, among other things a state with definite position, 1078 00:50:41,880 --> 00:50:46,990 x0, can be written as a superposition, 1 1079 00:50:46,990 --> 00:50:52,100 over 2pi integral dk. 1080 00:50:52,100 --> 00:50:56,659 I'll call this delta tilde of k. 1081 00:50:56,659 --> 00:50:57,200 e to the ikx. 1082 00:51:05,009 --> 00:51:07,050 If you haven't played with delta functions before 1083 00:51:07,050 --> 00:51:09,655 and you haven't seen this, then you will on the problem 1084 00:51:09,655 --> 00:51:11,230 set, because we have a problem that 1085 00:51:11,230 --> 00:51:13,020 works through a great many details. 1086 00:51:13,020 --> 00:51:17,840 But in particular, it's clear that this is not-- 1087 00:51:17,840 --> 00:51:20,672 this quantity can't be a delta function of k, 1088 00:51:20,672 --> 00:51:22,880 because, if it were, this would be just e to the ikx. 1089 00:51:22,880 --> 00:51:25,060 And that's definitely not a delta function. 1090 00:51:25,060 --> 00:51:29,910 Meanwhile, what can you say about the continuity structure 1091 00:51:29,910 --> 00:51:31,060 of a delta function. 1092 00:51:31,060 --> 00:51:33,300 Is it continuous? 1093 00:51:33,300 --> 00:51:33,960 No. 1094 00:51:33,960 --> 00:51:35,364 Its derivative isn't continuous. 1095 00:51:35,364 --> 00:51:36,280 Its second derivative. 1096 00:51:36,280 --> 00:51:38,363 None of its derivatives are in any way continuous. 1097 00:51:38,363 --> 00:51:40,879 They're all absolutely horrible, OK? 1098 00:51:40,879 --> 00:51:42,420 So how many momentum modes am I going 1099 00:51:42,420 --> 00:51:44,919 to need to superimpose in order to reproduce a function that 1100 00:51:44,919 --> 00:51:47,900 has this sort of structure? 1101 00:51:47,900 --> 00:51:48,790 An infinite number. 1102 00:51:48,790 --> 00:51:49,650 And it turns out it's going to be 1103 00:51:49,650 --> 00:51:51,970 an infinite number with the same amplitude, slightly 1104 00:51:51,970 --> 00:51:54,420 different phase, OK? 1105 00:51:54,420 --> 00:51:57,885 So you can never say that you're in a state 1106 00:51:57,885 --> 00:51:59,760 with definite position and definite momentum. 1107 00:51:59,760 --> 00:52:01,400 Being in a state with definite position 1108 00:52:01,400 --> 00:52:07,210 means being in a superposition of being in a superposition. 1109 00:52:07,210 --> 00:52:11,570 In fact, I'm just going right down the answer here. 1110 00:52:11,570 --> 00:52:12,702 e to the ikx0. 1111 00:52:17,030 --> 00:52:21,060 Being in a state with definite position 1112 00:52:21,060 --> 00:52:22,850 means being in a superposition of states 1113 00:52:22,850 --> 00:52:26,420 with arbitrary momentum and vice versa. 1114 00:52:26,420 --> 00:52:28,651 You cannot be in a state with definite position, 1115 00:52:28,651 --> 00:52:29,400 definite momentum. 1116 00:52:29,400 --> 00:52:30,274 So this doesn't work. 1117 00:52:30,274 --> 00:52:33,280 So what we want is we want some good definition. 1118 00:52:33,280 --> 00:52:34,780 So this does not work. 1119 00:52:34,780 --> 00:52:37,500 We want some good definition of p 1120 00:52:37,500 --> 00:52:40,950 given that we're working with a wave function which 1121 00:52:40,950 --> 00:52:43,500 is a function of x. 1122 00:52:43,500 --> 00:52:45,415 What is that good definition of the momentum? 1123 00:52:51,310 --> 00:52:54,560 We have a couple of hints. 1124 00:52:54,560 --> 00:52:55,930 So hint the first. 1125 00:53:02,860 --> 00:53:04,810 So this is what we're after. 1126 00:53:04,810 --> 00:53:06,520 Hint the first is that a wave-- we 1127 00:53:06,520 --> 00:53:09,950 know that given a wave with wave number k, which is equal 2pi 1128 00:53:09,950 --> 00:53:15,470 over lambda, is associated, according to de Broglie 1129 00:53:15,470 --> 00:53:19,460 and according to Davisson-Germer experiments, 1130 00:53:19,460 --> 00:53:22,070 to a particle-- so having a particle-- 1131 00:53:22,070 --> 00:53:25,280 a wave, with wave number k or wavelength lambda associated 1132 00:53:25,280 --> 00:53:29,590 particle with momentum p is equal to h bar k. 1133 00:53:29,590 --> 00:53:31,670 Yeah? 1134 00:53:31,670 --> 00:53:34,540 But in particular, what is a plane with wavelength lambda 1135 00:53:34,540 --> 00:53:36,570 or wave number k look like? 1136 00:53:36,570 --> 00:53:39,500 That's e to the iks. 1137 00:53:39,500 --> 00:53:41,890 And if I have a wave, a plane wave 1138 00:53:41,890 --> 00:53:45,190 e to the iks, how do I get h bar k out of it? 1139 00:53:52,650 --> 00:53:58,504 Note the following, the derivative with respect to x. 1140 00:53:58,504 --> 00:53:59,920 Actually let me do this down here. 1141 00:54:08,060 --> 00:54:12,910 Note that the derivative with respect to x of e to the ikx 1142 00:54:12,910 --> 00:54:17,230 is equal to ik e to the ikx. 1143 00:54:24,030 --> 00:54:27,050 There's nothing up my sleeves. 1144 00:54:27,050 --> 00:54:30,140 So in particular, if I want to get h bar k, 1145 00:54:30,140 --> 00:54:32,820 I can multiply by h bar and divide by i. 1146 00:54:32,820 --> 00:54:35,510 Multiply by h bar, divide by i, derivative with respect 1147 00:54:35,510 --> 00:54:39,140 to x e to the ikx. 1148 00:54:39,140 --> 00:54:42,490 And this is equal to h bar k e to the ikx. 1149 00:54:45,160 --> 00:54:45,960 That's suggestive. 1150 00:54:49,200 --> 00:54:53,330 And I can write this as p e to the ikx. 1151 00:55:00,620 --> 00:55:02,196 So let's quickly check the units. 1152 00:55:05,170 --> 00:55:08,740 So first off, what are the units of h bar? 1153 00:55:08,740 --> 00:55:10,130 Here's the super easy to remember 1154 00:55:10,130 --> 00:55:13,450 the units of-- or dimensions of h bar are. 1155 00:55:13,450 --> 00:55:18,550 Delta x delta p is h bar. 1156 00:55:18,550 --> 00:55:19,050 OK? 1157 00:55:19,050 --> 00:55:21,300 If you're ever in doubt, if you just remember, 1158 00:55:21,300 --> 00:55:24,852 h bar has units of momentum times length. 1159 00:55:24,852 --> 00:55:26,560 It's just the easiest way to remember it. 1160 00:55:26,560 --> 00:55:27,893 You'll never forget it that way. 1161 00:55:27,893 --> 00:55:31,070 So if h bar has units of momentum times length, 1162 00:55:31,070 --> 00:55:33,130 what are the units of k? 1163 00:55:33,130 --> 00:55:33,850 1 over length. 1164 00:55:33,850 --> 00:55:35,433 So does this dimensionally make sense? 1165 00:55:35,433 --> 00:55:36,170 Yeah. 1166 00:55:36,170 --> 00:55:40,630 Momentum times length divided by length number momentum. 1167 00:55:40,630 --> 00:55:41,130 Good. 1168 00:55:41,130 --> 00:55:42,781 So dimensionally we haven't lied yet. 1169 00:55:46,000 --> 00:55:48,250 So this makes it tempting to say something like, well, 1170 00:55:48,250 --> 00:55:51,530 hell h bar upon i derivative with respect 1171 00:55:51,530 --> 00:55:57,425 to x is equal in some-- question mark, quotation mark-- p. 1172 00:56:01,110 --> 00:56:03,510 Right? 1173 00:56:03,510 --> 00:56:06,250 So at this point it's just tempting to say, look, trust 1174 00:56:06,250 --> 00:56:07,915 me, p is h bar upon idx. 1175 00:56:07,915 --> 00:56:09,290 But I don't know about you, but I 1176 00:56:09,290 --> 00:56:11,910 find that deeply, deeply unsatisfying. 1177 00:56:11,910 --> 00:56:14,540 So let me ask the question slightly differently. 1178 00:56:14,540 --> 00:56:16,580 We've followed the de Broglie relations, 1179 00:56:16,580 --> 00:56:18,586 and we've been led to the idea that 1180 00:56:18,586 --> 00:56:19,960 using wave functions that there's 1181 00:56:19,960 --> 00:56:22,400 some relationship between the momentum, 1182 00:56:22,400 --> 00:56:24,850 the observable quantity that you measure with sticks, 1183 00:56:24,850 --> 00:56:28,450 and meters, and stuff, and this operator, this differential 1184 00:56:28,450 --> 00:56:32,170 operator, h bar upon on i derivative with respect to x. 1185 00:56:32,170 --> 00:56:36,560 By the way, my notation for dx is the partial derivative 1186 00:56:36,560 --> 00:56:38,652 with respect to x. 1187 00:56:38,652 --> 00:56:39,235 Just notation. 1188 00:56:42,350 --> 00:56:45,670 So if this is supposed to be true in some sense, 1189 00:56:45,670 --> 00:56:49,590 what is momentum have to do with a derivative? 1190 00:56:49,590 --> 00:56:51,090 Momentum is about velocities, which 1191 00:56:51,090 --> 00:56:53,090 is like derivatives with respect to time, right? 1192 00:56:53,090 --> 00:56:53,860 Times mass. 1193 00:56:53,860 --> 00:56:56,180 Mass times derivative with respect to time, velocity. 1194 00:56:56,180 --> 00:56:57,960 So what does it have to do with the derivative with respect 1195 00:56:57,960 --> 00:56:58,735 to position? 1196 00:57:02,480 --> 00:57:07,550 And this ties into the most beautiful theorem 1197 00:57:07,550 --> 00:57:13,627 in classical mechanics, which is the Noether's theorem, named 1198 00:57:13,627 --> 00:57:15,960 after the mathematician who discovered it, Emmy Noether. 1199 00:57:19,070 --> 00:57:22,870 And just out of curiosity, how many people 1200 00:57:22,870 --> 00:57:25,626 have seen Noether's theorem in class. 1201 00:57:25,626 --> 00:57:26,970 Oh that's so sad. 1202 00:57:26,970 --> 00:57:27,640 That's a sin. 1203 00:57:27,640 --> 00:57:31,212 OK, so here's a statement of Noether's theorem, 1204 00:57:31,212 --> 00:57:33,670 and it underlies an enormous amount of classical mechanics, 1205 00:57:33,670 --> 00:57:34,920 but also of quantum mechanics. 1206 00:57:37,839 --> 00:57:39,630 Noether, incidentally, was a mathematician. 1207 00:57:39,630 --> 00:57:41,899 There's a whole wonderful story about Emmy Noether. 1208 00:57:41,899 --> 00:57:43,440 Ville went to her and was like, look, 1209 00:57:43,440 --> 00:57:45,690 I'm trying to understand the notion of energy. 1210 00:57:45,690 --> 00:57:48,130 And this guy down the hall, Einstein, he 1211 00:57:48,130 --> 00:57:50,660 has a theory called general relativity about curved space 1212 00:57:50,660 --> 00:57:52,650 times and how that has something to do with gravity. 1213 00:57:52,650 --> 00:57:53,246 But it doesn't make a lot of sense 1214 00:57:53,246 --> 00:57:54,870 to me, because I don't even know how to define the energy. 1215 00:57:54,870 --> 00:57:56,620 So how do you define momentum and energy 1216 00:57:56,620 --> 00:57:59,860 in this guy's crazy theory? 1217 00:57:59,860 --> 00:58:02,390 And so Noether, who was a mathematician, 1218 00:58:02,390 --> 00:58:05,050 did all sorts of beautiful stuff in algebra, 1219 00:58:05,050 --> 00:58:06,080 looked at the problem and was like I don't even 1220 00:58:06,080 --> 00:58:07,450 know what it means in classical mechanics. 1221 00:58:07,450 --> 00:58:08,835 So what is a mean in classical mechanics? 1222 00:58:08,835 --> 00:58:09,970 So she went back to classical mechanics 1223 00:58:09,970 --> 00:58:11,520 and, from first principles, came up 1224 00:58:11,520 --> 00:58:13,680 with a good definition of momentum, which turns out 1225 00:58:13,680 --> 00:58:16,190 to underlie the modern idea of conserved quantities 1226 00:58:16,190 --> 00:58:17,250 and symmetries. 1227 00:58:17,250 --> 00:58:21,210 And it's had enormous far reaching impact, 1228 00:58:21,210 --> 00:58:24,430 and say her name would praise. 1229 00:58:24,430 --> 00:58:28,780 So Noether tells us the following statement, 1230 00:58:28,780 --> 00:58:39,230 to every symmetry-- and I should say continuous symmetry-- 1231 00:58:39,230 --> 00:58:46,060 to every symmetry is associated a conserved quantity. 1232 00:58:52,590 --> 00:58:53,310 OK? 1233 00:58:53,310 --> 00:58:54,920 So in particular, what do I mean by symmetry? 1234 00:58:54,920 --> 00:58:56,253 Well, for example, translations. 1235 00:58:56,253 --> 00:58:59,530 x goes to x plus some length l. 1236 00:58:59,530 --> 00:59:01,280 This could be done for arbitrary length l. 1237 00:59:01,280 --> 00:59:03,571 So for example, translation by this much or translation 1238 00:59:03,571 --> 00:59:04,150 by that much. 1239 00:59:04,150 --> 00:59:05,440 These are translations. 1240 00:59:05,440 --> 00:59:08,870 To every symmetry is associated a conserved quantity. 1241 00:59:08,870 --> 00:59:11,165 What symmetry is associated to translations? 1242 00:59:16,270 --> 00:59:18,142 Conservation of momentum, p dot. 1243 00:59:20,980 --> 00:59:27,910 Time translations, t goes to t plus capital 1244 00:59:27,910 --> 00:59:30,790 T. What's a conserved quantity associated 1245 00:59:30,790 --> 00:59:32,660 with time translational symmetry? 1246 00:59:32,660 --> 00:59:35,230 Energy, which is time independent. 1247 00:59:38,270 --> 00:59:40,290 And rotations. 1248 00:59:40,290 --> 00:59:41,280 Rotational symmetries. 1249 00:59:45,350 --> 00:59:51,680 x, as a vector, goes to some rotation times x. 1250 00:59:54,540 --> 00:59:56,875 What's conserved by virtue of rotational symmetry? 1251 00:59:56,875 --> 00:59:58,000 AUDIENCE: Angular momentum. 1252 00:59:58,000 --> 00:59:58,970 PROFESSOR: Angular momentum. 1253 00:59:58,970 --> 00:59:59,500 Rock on. 1254 01:00:03,760 --> 01:00:08,910 OK So quickly, I'm not going to prove to you Noether's theorem. 1255 01:00:08,910 --> 01:00:12,800 It's one of the most beautiful and important theorems 1256 01:00:12,800 --> 01:00:14,700 in physics, and you should all study it. 1257 01:00:14,700 --> 01:00:16,200 But let me just convince you quickly 1258 01:00:16,200 --> 01:00:18,002 that it's true in classical mechanics. 1259 01:00:18,002 --> 01:00:20,210 And this was observed long before Noether pointed out 1260 01:00:20,210 --> 01:00:23,080 why it was true in general. 1261 01:00:23,080 --> 01:00:25,930 What does it mean to have transitional symmetry? 1262 01:00:25,930 --> 01:00:28,570 It means that, if I do an experiment here 1263 01:00:28,570 --> 01:00:31,650 and I do it here, I get exactly the same results. 1264 01:00:31,650 --> 01:00:33,820 I translate the system and nothing changes. 1265 01:00:33,820 --> 01:00:34,320 Cool? 1266 01:00:34,320 --> 01:00:36,278 That's what I mean by saying I have a symmetry. 1267 01:00:36,278 --> 01:00:38,540 You do this thing, and nothing changes. 1268 01:00:38,540 --> 01:00:42,324 OK, so imagine I have a particle, a classical particle, 1269 01:00:42,324 --> 01:00:43,740 and it's moving in some potential. 1270 01:00:46,890 --> 01:00:49,564 This is u of x, right? 1271 01:00:49,564 --> 01:00:51,230 And we know what the equations of motion 1272 01:00:51,230 --> 01:00:53,550 are in classical mechanics from f 1273 01:00:53,550 --> 01:00:58,560 equals ma p dot is equal to the force, which is minus 1274 01:00:58,560 --> 01:00:59,440 the gradient of u. 1275 01:00:59,440 --> 01:01:01,510 Minus the gradient of u. 1276 01:01:01,510 --> 01:01:02,010 Right? 1277 01:01:02,010 --> 01:01:05,680 That's f equals ma in terms of the potential. 1278 01:01:05,680 --> 01:01:08,710 Now is the gradient of u 0? 1279 01:01:08,710 --> 01:01:10,175 No. 1280 01:01:10,175 --> 01:01:11,425 In this case, there's a force. 1281 01:01:13,671 --> 01:01:15,920 So if I do an experiment here, do I get the same thing 1282 01:01:15,920 --> 01:01:17,583 as doing my experiment here? 1283 01:01:17,583 --> 01:01:18,430 AUDIENCE: No. 1284 01:01:18,430 --> 01:01:19,080 PROFESSOR: Certainly not. 1285 01:01:19,080 --> 01:01:21,163 The [? system ?] is not translationally invariant. 1286 01:01:21,163 --> 01:01:23,210 The potential breaks that translational symmetry. 1287 01:01:23,210 --> 01:01:26,490 What potential has translational symmetry? 1288 01:01:26,490 --> 01:01:27,490 AUDIENCE: [INAUDIBLE]. 1289 01:01:27,490 --> 01:01:28,870 PROFESSOR: Yeah, constant. 1290 01:01:28,870 --> 01:01:32,100 The only potential that has full translational symmetry in one 1291 01:01:32,100 --> 01:01:35,970 dimension is translation invariant, i.e. 1292 01:01:35,970 --> 01:01:37,500 constant. 1293 01:01:37,500 --> 01:01:38,650 OK? 1294 01:01:38,650 --> 01:01:40,180 What's the force? 1295 01:01:40,180 --> 01:01:40,680 AUDIENCE: 0. 1296 01:01:40,680 --> 01:01:41,221 PROFESSOR: 0. 1297 01:01:41,221 --> 01:01:41,890 0 gradient. 1298 01:01:41,890 --> 01:01:43,810 So what's p dot? 1299 01:01:43,810 --> 01:01:44,910 Yep. 1300 01:01:44,910 --> 01:01:45,720 Noether's theorem. 1301 01:01:45,720 --> 01:01:48,040 Solid. 1302 01:01:48,040 --> 01:01:49,270 OK. 1303 01:01:49,270 --> 01:01:51,360 Less trivial is conservation of energy. 1304 01:01:51,360 --> 01:01:53,670 I claim and she claims-- and she's right-- 1305 01:01:53,670 --> 01:01:56,770 that if the system has the same dynamics at one 1306 01:01:56,770 --> 01:01:58,562 moment and a few moments later and, indeed, 1307 01:01:58,562 --> 01:02:00,561 any amount of time later, if the laws of physics 1308 01:02:00,561 --> 01:02:02,550 don't change in time, then there must 1309 01:02:02,550 --> 01:02:05,260 be a conserved quantity called energy. 1310 01:02:05,260 --> 01:02:08,740 There must be a conserved quantity. 1311 01:02:08,740 --> 01:02:09,960 And that's Noether's theorem. 1312 01:02:09,960 --> 01:02:11,870 So this is the first step, but this still 1313 01:02:11,870 --> 01:02:14,190 doesn't tell us what momentum exactly 1314 01:02:14,190 --> 01:02:16,874 has to do with a derivative with respect to space. 1315 01:02:16,874 --> 01:02:18,290 We see that there's a relationship 1316 01:02:18,290 --> 01:02:22,380 between translations and momentum conservation, 1317 01:02:22,380 --> 01:02:24,890 but what's the relationship? 1318 01:02:24,890 --> 01:02:26,270 So let's do this. 1319 01:02:26,270 --> 01:02:29,060 I'm going to define an operation called translate by L. 1320 01:02:29,060 --> 01:02:30,520 And what translate by L does is it 1321 01:02:30,520 --> 01:02:39,211 takes f of x and it maps it to f of x minus L. 1322 01:02:39,211 --> 01:02:41,210 So this is a thing that affects the translation. 1323 01:02:41,210 --> 01:02:43,880 And why do I say that's a translation by L rather 1324 01:02:43,880 --> 01:02:47,490 than minus L. Well, the point-- if you have 1325 01:02:47,490 --> 01:02:50,420 some function like this, and it has a peak at 0, 1326 01:02:50,420 --> 01:02:57,630 then after the translation, the peak is when x is equal to L. 1327 01:02:57,630 --> 01:02:58,130 OK? 1328 01:02:58,130 --> 01:02:59,940 So just to get the signs straight. 1329 01:02:59,940 --> 01:03:02,660 So define this operation, which takes a function of x 1330 01:03:02,660 --> 01:03:05,235 and translates it by L, but leaves it otherwise identical. 1331 01:03:09,084 --> 01:03:10,500 So let's consider how translations 1332 01:03:10,500 --> 01:03:13,160 behave on functions. 1333 01:03:13,160 --> 01:03:15,160 And this is really cute. 1334 01:03:15,160 --> 01:03:20,180 f of x minus L can be written as a Taylor expansion 1335 01:03:20,180 --> 01:03:23,820 around the point x-- around the point L equals 0. 1336 01:03:23,820 --> 01:03:26,250 So let's do Taylor expansion for small L. 1337 01:03:26,250 --> 01:03:32,210 So this is equal to f of x minus L derivative with respect 1338 01:03:32,210 --> 01:03:39,050 to x of f of x plus L squared over 2 derivative squared, 1339 01:03:39,050 --> 01:03:42,780 two derivatives of x, f of x plus dot, dot, dot. 1340 01:03:42,780 --> 01:03:43,280 Right? 1341 01:03:43,280 --> 01:03:46,306 I'm just Taylor expanding. 1342 01:03:46,306 --> 01:03:46,930 Nothing sneaky. 1343 01:03:49,510 --> 01:03:52,550 Let's add the next term, actually. 1344 01:03:52,550 --> 01:03:54,820 Let me do this on a whole new board. 1345 01:04:06,210 --> 01:04:08,865 All right, so we have translate by L 1346 01:04:08,865 --> 01:04:15,130 on f of x is equal to f of x minus L is equal to f of x. 1347 01:04:15,130 --> 01:04:17,530 Now Taylor expanding minus L derivative 1348 01:04:17,530 --> 01:04:26,376 with respect to x of f plus L squared over 2-- 1349 01:04:26,376 --> 01:04:27,900 I'm not giving myself enough space. 1350 01:04:27,900 --> 01:04:28,400 I'm sorry. 1351 01:04:30,990 --> 01:04:37,600 f of x minus L is equal to f of x minus L with respect 1352 01:04:37,600 --> 01:04:46,150 to x of f of x plus L squared over 2 to derivatives 1353 01:04:46,150 --> 01:04:51,430 of x f of x minus L cubed over 6-- 1354 01:04:51,430 --> 01:04:55,030 we're just Taylor expanding-- cubed with respect 1355 01:04:55,030 --> 01:05:00,800 to x of f of x and so on. 1356 01:05:00,800 --> 01:05:01,871 Yeah? 1357 01:05:01,871 --> 01:05:04,370 But I'm going to write this in the following suggestive way. 1358 01:05:04,370 --> 01:05:13,610 This is equal to 1 times f of x minus L derivative 1359 01:05:13,610 --> 01:05:17,730 with respect to x f of x plus L squared 1360 01:05:17,730 --> 01:05:22,740 over 2 derivative with respect to x squared times f 1361 01:05:22,740 --> 01:05:28,000 of x minus L cubed over 6 derivative cubed with respect 1362 01:05:28,000 --> 01:05:31,470 to x plus dot, dot, dot. 1363 01:05:31,470 --> 01:05:32,580 Everybody good with that? 1364 01:05:35,750 --> 01:05:38,740 But this is a series that you should recognize, 1365 01:05:38,740 --> 01:05:41,700 a particular Taylor series for a particular function. 1366 01:05:41,700 --> 01:05:44,754 It's a Taylor expansion for the 1367 01:05:44,754 --> 01:05:45,670 AUDIENCE: Exponential. 1368 01:05:45,670 --> 01:05:46,880 PROFESSOR: Exponential. 1369 01:05:46,880 --> 01:05:51,230 e to the minus L derivative with respect to x f of x. 1370 01:05:59,689 --> 01:06:00,730 Which is kind of awesome. 1371 01:06:00,730 --> 01:06:02,220 So let's just check to make sure that this makes sense 1372 01:06:02,220 --> 01:06:03,460 from dimensional grounds. 1373 01:06:03,460 --> 01:06:05,710 So that's a derivative with respect to x as units of 1 1374 01:06:05,710 --> 01:06:06,120 over length. 1375 01:06:06,120 --> 01:06:07,870 That's a length, so this is dimensionless, 1376 01:06:07,870 --> 01:06:09,370 so we can exponentiate it. 1377 01:06:09,370 --> 01:06:12,450 Now you might look at me and say, look, this is silly. 1378 01:06:12,450 --> 01:06:14,680 You've taken an operation like derivative 1379 01:06:14,680 --> 01:06:15,760 and exponentiated it. 1380 01:06:15,760 --> 01:06:17,631 What does that mean? 1381 01:06:17,631 --> 01:06:19,290 And that is what it means? 1382 01:06:19,290 --> 01:06:21,000 [LAUGHTER] 1383 01:06:21,000 --> 01:06:21,610 OK? 1384 01:06:21,610 --> 01:06:24,180 So we're going to do this all the time in quantum mechanics. 1385 01:06:24,180 --> 01:06:26,452 We're going to do things like exponentiate operations. 1386 01:06:26,452 --> 01:06:27,910 We'll talk about it in more detail, 1387 01:06:27,910 --> 01:06:29,368 but we're always going to define it 1388 01:06:29,368 --> 01:06:31,925 in this fashion as a formal power series. 1389 01:06:31,925 --> 01:06:32,425 Questions? 1390 01:06:35,019 --> 01:06:36,560 AUDIENCE: Can you transform operators 1391 01:06:36,560 --> 01:06:38,170 from one space to another? 1392 01:06:38,170 --> 01:06:39,680 PROFESSOR: Oh, you totally can. 1393 01:06:39,680 --> 01:06:40,930 But we'll come back to that. 1394 01:06:40,930 --> 01:06:43,400 We're going to talk about operators next time. 1395 01:06:43,400 --> 01:06:48,750 OK, so here's where we are. 1396 01:06:48,750 --> 01:06:55,880 So from this what is a derivative 1397 01:06:55,880 --> 01:06:56,950 with respect to x mean? 1398 01:06:56,950 --> 01:06:58,820 What does a derivative with respect to x do? 1399 01:06:58,820 --> 01:07:00,280 Well a derivative with respect to x 1400 01:07:00,280 --> 01:07:03,850 is something that generates translations with respect 1401 01:07:03,850 --> 01:07:06,285 to x through a Taylor expansion. 1402 01:07:09,550 --> 01:07:12,240 If we have L be arbitrarily small, right? 1403 01:07:12,240 --> 01:07:14,951 L is arbitrarily small. 1404 01:07:14,951 --> 01:07:17,200 What is the translation by an arbitrarily small amount 1405 01:07:17,200 --> 01:07:18,140 of f of x? 1406 01:07:18,140 --> 01:07:19,610 Well, if L is arbitrarily small, we 1407 01:07:19,610 --> 01:07:21,110 can drop all the higher order terms, 1408 01:07:21,110 --> 01:07:24,542 and the change is just Ldx. 1409 01:07:24,542 --> 01:07:26,000 So the derivative with respect to x 1410 01:07:26,000 --> 01:07:30,330 is telling us about infinitesimal translations. 1411 01:07:30,330 --> 01:07:31,360 Cool? 1412 01:07:31,360 --> 01:07:33,490 The derivative with respect to a position 1413 01:07:33,490 --> 01:07:35,520 is something that tells you, or controls, 1414 01:07:35,520 --> 01:07:39,429 or generates infinitesimal translations. 1415 01:07:39,429 --> 01:07:41,220 And if you exponentiate it, you do it many, 1416 01:07:41,220 --> 01:07:43,350 many, many times in a particular way, 1417 01:07:43,350 --> 01:07:46,410 you get a macroscopic finite translation. 1418 01:07:46,410 --> 01:07:48,410 Cool? 1419 01:07:48,410 --> 01:07:50,820 So this gives us three things. 1420 01:07:50,820 --> 01:08:02,730 Translations in x are generated by derivative with respect 1421 01:08:02,730 --> 01:08:03,230 to x. 1422 01:08:05,920 --> 01:08:14,410 But through Noether's theorem translations, 1423 01:08:14,410 --> 01:08:20,390 in x are associated to conservation of momentum. 1424 01:08:27,270 --> 01:08:30,720 So you shouldn't be so shocked-- it's really not 1425 01:08:30,720 --> 01:08:44,050 totally shocking-- that in quantum mechanics, where we're 1426 01:08:44,050 --> 01:08:48,500 very interested in the action of things on functions, 1427 01:08:48,500 --> 01:08:51,319 not just in positions, but on functions of position, 1428 01:08:51,319 --> 01:08:56,415 it shouldn't be totally shocking that in quantum mechanics, 1429 01:08:56,415 --> 01:08:58,970 the derivative with respect to x is related 1430 01:08:58,970 --> 01:09:00,810 to the momentum in some particular way. 1431 01:09:03,950 --> 01:09:07,968 Similarly, translations in t are going 1432 01:09:07,968 --> 01:09:09,384 to be generated by what operation? 1433 01:09:12,380 --> 01:09:15,140 Derivative with respect to time. 1434 01:09:15,140 --> 01:09:18,520 So derivative with respect to time from Noether's theorem 1435 01:09:18,520 --> 01:09:20,779 is associated with conservation of energy. 1436 01:09:23,640 --> 01:09:25,810 That seems plausible. 1437 01:09:25,810 --> 01:09:27,689 Derivative with respect to, I don't know, 1438 01:09:27,689 --> 01:09:31,779 an angle, a rotation. 1439 01:09:31,779 --> 01:09:34,260 That's going to be associated with what? 1440 01:09:34,260 --> 01:09:36,390 Angular momentum? 1441 01:09:36,390 --> 01:09:38,080 But angular momentum around the axis 1442 01:09:38,080 --> 01:09:39,240 for whom this is the angle, so I'll 1443 01:09:39,240 --> 01:09:40,364 call that z for the moment. 1444 01:09:43,859 --> 01:09:48,140 And we're going to see these pop up over and over again. 1445 01:09:48,140 --> 01:09:49,520 But here's the thing. 1446 01:09:52,819 --> 01:09:58,880 We started out with these three principles today, 1447 01:09:58,880 --> 01:10:02,140 and we've let ourselves to some sort of association 1448 01:10:02,140 --> 01:10:08,350 between the momentum and the derivative like this. 1449 01:10:08,350 --> 01:10:08,850 OK? 1450 01:10:08,850 --> 01:10:10,558 And I've given you some reason to believe 1451 01:10:10,558 --> 01:10:11,950 that this isn't totally insane. 1452 01:10:11,950 --> 01:10:13,440 Translations are deeply connected 1453 01:10:13,440 --> 01:10:14,500 with conservation of momentum. 1454 01:10:14,500 --> 01:10:15,950 Transitional symmetry is deeply connected 1455 01:10:15,950 --> 01:10:17,310 with conservation momentum. 1456 01:10:17,310 --> 01:10:18,770 And an infinitesimal translation is 1457 01:10:18,770 --> 01:10:22,070 nothing but a derivative with respect to position. 1458 01:10:22,070 --> 01:10:24,330 Those are deeply linked concepts. 1459 01:10:24,330 --> 01:10:27,900 But I didn't derive anything. 1460 01:10:27,900 --> 01:10:29,570 I gave you no derivation whatsoever 1461 01:10:29,570 --> 01:10:33,000 of the relationship between d dx and the momentum. 1462 01:10:33,000 --> 01:10:36,170 Instead, I'm simply going to declare it. 1463 01:10:36,170 --> 01:10:39,620 I'm going to declare that, in quantum mechanics-- 1464 01:10:39,620 --> 01:10:42,150 you cannot stop me-- in quantum mechanics, 1465 01:10:42,150 --> 01:10:48,300 p is represented by an operator, it's represented 1466 01:10:48,300 --> 01:10:51,720 by the specific operator h bar upon I derivative with respect 1467 01:10:51,720 --> 01:10:53,840 to x. 1468 01:10:53,840 --> 01:10:54,990 And this is a declaration. 1469 01:10:58,680 --> 01:11:00,890 OK? 1470 01:11:00,890 --> 01:11:02,700 It is simply a fact. 1471 01:11:02,700 --> 01:11:05,730 And when they say it's a fact, I mean two things by that. 1472 01:11:05,730 --> 01:11:08,037 The first is it is a fact that, in quantum mechanics, 1473 01:11:08,037 --> 01:11:10,120 momentum is represented by derivative with respect 1474 01:11:10,120 --> 01:11:12,220 to x times h bar upon i. 1475 01:11:12,220 --> 01:11:15,639 Secondly, it is a fact that, if you take this expression 1476 01:11:15,639 --> 01:11:17,930 and you work with the rest of the postulates of quantum 1477 01:11:17,930 --> 01:11:19,346 mechanics, including what's coming 1478 01:11:19,346 --> 01:11:22,730 next lecture about operators and time evolution, 1479 01:11:22,730 --> 01:11:24,580 you reproduce the physics of the real world. 1480 01:11:24,580 --> 01:11:26,030 You reproduce it beautifully. 1481 01:11:26,030 --> 01:11:28,540 You reproduce it so well that no other models have even 1482 01:11:28,540 --> 01:11:32,000 ever vaguely come close to the explanatory power of quantum 1483 01:11:32,000 --> 01:11:32,971 mechanics. 1484 01:11:32,971 --> 01:11:33,470 OK? 1485 01:11:33,470 --> 01:11:34,210 It is a fact. 1486 01:11:34,210 --> 01:11:36,710 It is not true in some epistemic sense. 1487 01:11:36,710 --> 01:11:38,910 You can't sit back and say, ah a priori 1488 01:11:38,910 --> 01:11:41,990 starting with the integers we derive that p is equal to-- no, 1489 01:11:41,990 --> 01:11:43,050 it's a model. 1490 01:11:43,050 --> 01:11:44,509 But that's what physics does. 1491 01:11:44,509 --> 01:11:46,050 Physics doesn't tell you what's true. 1492 01:11:46,050 --> 01:11:48,390 Physics doesn't tell you what a priori 1493 01:11:48,390 --> 01:11:49,820 did the world have to look like. 1494 01:11:49,820 --> 01:11:52,412 Physics tells you this is a good model, 1495 01:11:52,412 --> 01:11:54,370 and it works really well, and it fits the data. 1496 01:11:54,370 --> 01:11:56,328 And to the degree that it doesn't fit the data, 1497 01:11:56,328 --> 01:11:57,850 it's wrong. 1498 01:11:57,850 --> 01:11:58,414 OK? 1499 01:11:58,414 --> 01:11:59,705 This isn't something we derive. 1500 01:11:59,705 --> 01:12:01,250 This is something we declare. 1501 01:12:01,250 --> 01:12:03,940 We call it our model, and then we use it to calculate stuff, 1502 01:12:03,940 --> 01:12:06,100 and we see if it fits the real world. 1503 01:12:09,600 --> 01:12:12,041 Out, please, please leave. 1504 01:12:12,041 --> 01:12:12,540 Thank you. 1505 01:12:15,473 --> 01:12:24,730 [LAUGHTER] 1506 01:12:24,730 --> 01:12:25,750 I love MIT. 1507 01:12:25,750 --> 01:12:26,390 I really do. 1508 01:12:43,340 --> 01:12:45,280 So let me close off at this point 1509 01:12:45,280 --> 01:12:48,450 with the following observation. 1510 01:12:48,450 --> 01:12:51,180 [LAUGHTER] 1511 01:12:51,180 --> 01:12:55,540 We live in a world governed by probabilities. 1512 01:12:55,540 --> 01:12:57,910 There's a finite probability that, at any given moment, 1513 01:12:57,910 --> 01:13:00,976 that two pirates might walk into a room, OK? 1514 01:13:00,976 --> 01:13:01,475 [LAUGHTER] 1515 01:13:01,475 --> 01:13:03,340 You just never know. 1516 01:13:03,340 --> 01:13:08,350 [APPLAUSE] 1517 01:13:08,350 --> 01:13:13,240 But those probabilities can be computed in quantum mechanics. 1518 01:13:13,240 --> 01:13:15,302 And they're computed in the following ways. 1519 01:13:15,302 --> 01:13:16,760 They're computed the following ways 1520 01:13:16,760 --> 01:13:18,420 as we'll study in great detail. 1521 01:13:18,420 --> 01:13:24,840 If I take a state, psi of x, which is equal to e to the ikx, 1522 01:13:24,840 --> 01:13:28,000 this is a state that has definite momentum h bar k. 1523 01:13:28,000 --> 01:13:28,990 Right? 1524 01:13:28,990 --> 01:13:29,720 We claimed this. 1525 01:13:29,720 --> 01:13:33,090 This was de Broglie and Davisson-Germer. 1526 01:13:33,090 --> 01:13:35,450 Note the following, take this operator 1527 01:13:35,450 --> 01:13:37,770 and act on this wave function with this operator. 1528 01:13:37,770 --> 01:13:38,925 What do you get? 1529 01:13:38,925 --> 01:13:40,300 Well, we already know, because we 1530 01:13:40,300 --> 01:13:42,760 constructed it to have this property. 1531 01:13:42,760 --> 01:13:44,605 P hat on psi of x-- and I'm going 1532 01:13:44,605 --> 01:13:46,230 to call this psi sub k of x, because it 1533 01:13:46,230 --> 01:13:51,070 has a definite k-- is equal to h bar k psi k of x. 1534 01:13:55,110 --> 01:13:58,455 A state with a definite momentum has the property 1535 01:13:58,455 --> 01:14:00,580 that, when you hit it with the operation associated 1536 01:14:00,580 --> 01:14:03,204 with momentum, you get back the same function times a constant, 1537 01:14:03,204 --> 01:14:06,120 and that constant is exactly the momentum we 1538 01:14:06,120 --> 01:14:09,486 ascribe to that plane wave. 1539 01:14:09,486 --> 01:14:10,730 Is that cool? 1540 01:14:10,730 --> 01:14:11,280 Yeah? 1541 01:14:11,280 --> 01:14:12,071 AUDIENCE: Question. 1542 01:14:12,071 --> 01:14:13,735 Just with notation, what does the hat 1543 01:14:13,735 --> 01:14:14,744 above the p [INAUDIBLE]? 1544 01:14:14,744 --> 01:14:15,410 PROFESSOR: Good. 1545 01:14:15,410 --> 01:14:15,610 Excellent. 1546 01:14:15,610 --> 01:14:17,360 So the hat above the P is to remind you 1547 01:14:17,360 --> 01:14:18,740 that P is on a number. 1548 01:14:18,740 --> 01:14:20,250 It's an operation. 1549 01:14:20,250 --> 01:14:23,370 It's a rule for acting on functions. 1550 01:14:23,370 --> 01:14:25,670 We'll talk about that in great detail next time. 1551 01:14:25,670 --> 01:14:27,560 But here's what I want to emphasize. 1552 01:14:27,560 --> 01:14:30,792 This is a state which is equal to all others in the sense 1553 01:14:30,792 --> 01:14:32,750 that it's a perfectly reasonable wave function, 1554 01:14:32,750 --> 01:14:35,980 but it's more equal because it has a simple interpretation. 1555 01:14:35,980 --> 01:14:37,350 Right? 1556 01:14:37,350 --> 01:14:39,904 The probability that I measure the momentum to be h bar k 1557 01:14:39,904 --> 01:14:41,320 is one, and the probability that I 1558 01:14:41,320 --> 01:14:45,010 measure it to be anything else is 0, correct? 1559 01:14:45,010 --> 01:14:47,870 But I can always consider a state which is a superposition. 1560 01:14:47,870 --> 01:14:54,426 Psi is equal to alpha, let's just do 1 over 2 e to the ikx. 1561 01:14:54,426 --> 01:14:59,180 k1 x plus 1 over root 2 e to the minus ikx. 1562 01:15:09,070 --> 01:15:12,930 Is this state a state with definite momentum? 1563 01:15:12,930 --> 01:15:15,170 If I act on this state-- I'll call this i 1564 01:15:15,170 --> 01:15:18,590 sub s-- if I act on this state with the momentum operator, 1565 01:15:18,590 --> 01:15:21,771 do I get back this state times a constant? 1566 01:15:21,771 --> 01:15:22,270 No. 1567 01:15:22,270 --> 01:15:23,080 That's interesting. 1568 01:15:23,080 --> 01:15:24,659 And so it seems to be that if we have 1569 01:15:24,659 --> 01:15:27,200 a state with definite momentum and we act on it with momentum 1570 01:15:27,200 --> 01:15:28,810 operator, we get back its momentum. 1571 01:15:28,810 --> 01:15:30,340 If we have a state that's a superposition 1572 01:15:30,340 --> 01:15:33,006 of different momentum and we act on it with a momentum operator, 1573 01:15:33,006 --> 01:15:35,330 this gives us h bar k 1, this gives us h bar k2. 1574 01:15:35,330 --> 01:15:36,957 So it changes which superposition 1575 01:15:36,957 --> 01:15:37,790 we're talking about. 1576 01:15:37,790 --> 01:15:41,330 We don't get back our same state. 1577 01:15:41,330 --> 01:15:43,065 So the action of this operator on a state 1578 01:15:43,065 --> 01:15:45,440 is going to tell us something about whether the state has 1579 01:15:45,440 --> 01:15:48,570 definite value of the momentum. 1580 01:15:48,570 --> 01:15:50,490 And these coefficients are going to turn out 1581 01:15:50,490 --> 01:15:52,910 to contain all the information about the probability 1582 01:15:52,910 --> 01:15:53,660 of the system. 1583 01:15:53,660 --> 01:15:55,560 This is the probability when norm 1584 01:15:55,560 --> 01:15:59,000 squared that will measure the system to have momentum k1. 1585 01:15:59,000 --> 01:16:00,530 And this coefficient norm squared 1586 01:16:00,530 --> 01:16:02,450 is going to tell us the probability that we 1587 01:16:02,450 --> 01:16:05,750 have momentum k2. 1588 01:16:05,750 --> 01:16:10,730 So I think the current wave function 1589 01:16:10,730 --> 01:16:21,020 is something like a superposition of 1/10 psi 1590 01:16:21,020 --> 01:16:32,890 pirates plus 1 minus is 1/100 square root. 1591 01:16:32,890 --> 01:16:35,760 To normalize it properly psi no pirates. 1592 01:16:41,340 --> 01:16:43,670 And I'll leave you with pondering this probability. 1593 01:16:43,670 --> 01:16:47,330 See you guys next time. 1594 01:16:47,330 --> 01:17:11,525 [APPLAUSE] 1595 01:17:11,525 --> 01:17:13,150 CHRISTOPHER SMITH: We've come for Prof. 1596 01:17:13,150 --> 01:17:15,140 Allan Adams. 1597 01:17:15,140 --> 01:17:16,880 PROFESSOR: It is I. 1598 01:17:16,880 --> 01:17:21,110 CHRISTOPHER SMITH: When in the chronicles of wasted time, 1599 01:17:21,110 --> 01:17:24,900 I see descriptions of fairest rights, 1600 01:17:24,900 --> 01:17:29,650 and I see lovely shows of lovely dames. 1601 01:17:29,650 --> 01:17:34,440 And descriptions of ladies dead and lovely nights. 1602 01:17:34,440 --> 01:17:37,860 Then in the bosom of fair loves depths. 1603 01:17:37,860 --> 01:17:43,730 Of eyes, of foot, of eye, of brow. 1604 01:17:43,730 --> 01:17:48,090 I see the antique pens do but express the beauty 1605 01:17:48,090 --> 01:17:50,770 that you master now. 1606 01:17:50,770 --> 01:17:55,550 So are all their praises but prophecies of this, our time. 1607 01:17:55,550 --> 01:17:59,120 All you prefiguring. 1608 01:17:59,120 --> 01:18:02,942 But though they had but diving eyes-- 1609 01:18:02,942 --> 01:18:04,900 PROFESSOR: I was wrong about the probabilities. 1610 01:18:04,900 --> 01:18:05,376 [LAUGHTER] 1611 01:18:05,376 --> 01:18:06,792 CHRISTOPHER SMITH: But though they 1612 01:18:06,792 --> 01:18:09,120 had but diving eyes, they had not skill 1613 01:18:09,120 --> 01:18:11,710 enough you're worth to sing. 1614 01:18:11,710 --> 01:18:15,020 For we which now behold these present days 1615 01:18:15,020 --> 01:18:17,600 have eyes to behold. 1616 01:18:17,600 --> 01:18:21,440 [LAUGHTER] 1617 01:18:21,440 --> 01:18:23,180 But not tongues to praise. 1618 01:18:25,840 --> 01:18:28,200 [APPLAUSE] 1619 01:18:28,200 --> 01:18:29,144 It's not over. 1620 01:18:29,144 --> 01:18:31,990 You wait. 1621 01:18:31,990 --> 01:18:35,020 ARSHIA SURTI: Not marbled with gilded monuments of princes 1622 01:18:35,020 --> 01:18:37,160 shall outlive this powerful rhyme. 1623 01:18:37,160 --> 01:18:40,690 But you shall shine more bright in these contents 1624 01:18:40,690 --> 01:18:43,370 that unswept stone besmear its sluttish tide. 1625 01:18:43,370 --> 01:18:48,970 When wasteful war shall statues overturn and broils 1626 01:18:48,970 --> 01:18:50,230 root out the work of masonry. 1627 01:18:50,230 --> 01:18:53,060 Nor Mars his sword. 1628 01:18:53,060 --> 01:18:57,390 Nor war's quick fire shall burn the living record 1629 01:18:57,390 --> 01:18:59,640 of your memory. 1630 01:18:59,640 --> 01:19:04,810 Gainst death and all oblivious enmity shall you pace forth. 1631 01:19:04,810 --> 01:19:06,570 Your praise shall still find room, 1632 01:19:06,570 --> 01:19:08,233 even in the eyes of all posterity. 1633 01:19:11,220 --> 01:19:15,910 So no judgment arise till you yourself judgment arise. 1634 01:19:15,910 --> 01:19:18,060 You live in this and dwell in lover's eyes. 1635 01:19:20,862 --> 01:19:23,200 [APPLAUSE] 1636 01:19:23,200 --> 01:19:26,220 CHRISTOPHER SMITH: Verily happy Valentine's day upon you. 1637 01:19:26,220 --> 01:19:28,170 May your day be filled with love and poetry. 1638 01:19:28,170 --> 01:19:31,448 Whatever state you're in, we will always love you. 1639 01:19:31,448 --> 01:19:33,918 [LAUGHTER] 1640 01:19:33,918 --> 01:19:37,880 [APPLAUSE] 1641 01:19:37,880 --> 01:19:41,690 Signed, Jack Florian, James [INAUDIBLE]. 1642 01:19:41,690 --> 01:19:42,590 [LAUGHTER] 1643 01:19:42,590 --> 01:19:44,390 PROFESSOR: Thank you, sir. 1644 01:19:44,390 --> 01:19:45,582 Thank you. 1645 01:19:45,582 --> 01:19:46,790 CHRISTOPHER SMITH: Now we go. 1646 01:19:50,390 --> 01:19:51,940 [APPLAUSE]