1 00:00:00,090 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,800 under a Creative Commons license. 3 00:00:03,800 --> 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 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:16,110 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,110 --> 00:00:17,070 at ocw.mit.edu. 8 00:00:21,590 --> 00:00:23,720 PROFESSOR: So, finally, before I get 9 00:00:23,720 --> 00:00:26,230 started on the new stuff, questions 10 00:00:26,230 --> 00:00:27,500 from the previous lectures? 11 00:00:34,820 --> 00:00:37,052 No questions? 12 00:00:37,052 --> 00:00:37,552 Yeah. 13 00:00:37,552 --> 00:00:38,025 AUDIENCE: I have a question. 14 00:00:38,025 --> 00:00:39,483 You might have said this last time, 15 00:00:39,483 --> 00:00:41,069 but when is the first exam? 16 00:00:41,069 --> 00:00:42,110 PROFESSOR: Ah, excellent. 17 00:00:42,110 --> 00:00:47,150 Those will be posted on the Stellar page later today. 18 00:00:47,150 --> 00:00:48,221 Yeah. 19 00:00:48,221 --> 00:00:50,145 AUDIENCE: OK, so we're associating operators 20 00:00:50,145 --> 00:00:51,444 with observables, right? 21 00:00:51,444 --> 00:00:52,069 PROFESSOR: Yes. 22 00:00:52,069 --> 00:00:53,568 AUDIENCE: And Professor [? Zugoff ?] 23 00:00:53,568 --> 00:00:55,676 mentioned that whenever we have done a wave 24 00:00:55,676 --> 00:01:00,082 function with an operator, it collapses. 25 00:01:00,082 --> 00:01:02,040 PROFESSOR: OK, so let me rephrase the question. 26 00:01:02,040 --> 00:01:04,640 This is a very valuable question to talk through. 27 00:01:04,640 --> 00:01:05,720 So, thanks for asking it. 28 00:01:05,720 --> 00:01:09,486 So, we've previously observed that observables are associated 29 00:01:09,486 --> 00:01:10,860 with operators-- and we'll review 30 00:01:10,860 --> 00:01:12,930 that in more detail in a second-- 31 00:01:12,930 --> 00:01:16,100 and the statement was then made, does that 32 00:01:16,100 --> 00:01:21,210 mean that acting on a wave function with an operator 33 00:01:21,210 --> 00:01:22,660 is like measuring the observable? 34 00:01:22,660 --> 00:01:25,540 And it's absolutely essential that you understand 35 00:01:25,540 --> 00:01:29,960 that acting on a wave function with an operator 36 00:01:29,960 --> 00:01:34,380 has nothing whatsoever to do with measuring that associated 37 00:01:34,380 --> 00:01:35,240 observable. 38 00:01:35,240 --> 00:01:36,360 Nothing. 39 00:01:36,360 --> 00:01:36,860 OK? 40 00:01:36,860 --> 00:01:37,895 And we'll talk about the relationship 41 00:01:37,895 --> 00:01:39,160 and what those things mean. 42 00:01:39,160 --> 00:01:41,400 But here's a very tempting thing to think. 43 00:01:41,400 --> 00:01:42,540 I have a wave function. 44 00:01:42,540 --> 00:01:43,890 I want to know the momentum. 45 00:01:43,890 --> 00:01:46,160 I will thus operate with the momentum operator. 46 00:01:46,160 --> 00:01:47,400 Completely wrong. 47 00:01:47,400 --> 00:01:50,430 So, before I even tell you what the right statement is, 48 00:01:50,430 --> 00:01:52,030 let me just get that out of your head, 49 00:01:52,030 --> 00:01:54,384 and then we'll talk through that in much more detail 50 00:01:54,384 --> 00:01:55,300 over the next lecture. 51 00:01:55,300 --> 00:01:55,970 Yeah. 52 00:01:55,970 --> 00:01:58,395 AUDIENCE: Why doesn't it collapse by special relativity? 53 00:01:58,395 --> 00:01:59,770 PROFESSOR: We're doing everything 54 00:01:59,770 --> 00:02:00,730 non-relativistically. 55 00:02:00,730 --> 00:02:02,620 Quantum Mechanics for 804 is going 56 00:02:02,620 --> 00:02:05,600 to be a universe in which there is no relativity. 57 00:02:05,600 --> 00:02:08,530 If you ask me that more precisely in my office hours, 58 00:02:08,530 --> 00:02:10,970 I will tell you a relativistic story. 59 00:02:10,970 --> 00:02:13,130 But it doesn't violate anything relativistic. 60 00:02:13,130 --> 00:02:13,700 At all. 61 00:02:13,700 --> 00:02:16,690 We'll talk about that-- just to be a little more detailed-- 62 00:02:16,690 --> 00:02:19,380 that will be a very important question that we'll 63 00:02:19,380 --> 00:02:22,140 deal with in the last two lectures of the course, 64 00:02:22,140 --> 00:02:24,894 when we come back to Bell's inequality and locality. 65 00:02:24,894 --> 00:02:25,560 Other questions? 66 00:02:29,580 --> 00:02:31,844 OK. 67 00:02:31,844 --> 00:02:32,760 So, let's get started. 68 00:02:32,760 --> 00:02:35,010 So, just to review where we are. 69 00:02:35,010 --> 00:02:38,292 In Quantum Mechanics according to 804, 70 00:02:38,292 --> 00:02:40,500 our first pass at the definition of quantum mechanics 71 00:02:40,500 --> 00:02:43,290 is that the configuration of any system-- and in particular, 72 00:02:43,290 --> 00:02:45,206 think about a single point particle-- 73 00:02:45,206 --> 00:02:46,580 the configuration of our particle 74 00:02:46,580 --> 00:02:49,696 is specified by giving a wave function, which 75 00:02:49,696 --> 00:02:51,320 is a function which may depend on time, 76 00:02:51,320 --> 00:02:52,445 but a function of position. 77 00:02:55,310 --> 00:02:57,956 Observables-- and this is a complete specification 78 00:02:57,956 --> 00:02:59,080 of the state of the system. 79 00:02:59,080 --> 00:03:01,540 If I know the wave function, I neither 80 00:03:01,540 --> 00:03:05,140 needed nor have access to any further information 81 00:03:05,140 --> 00:03:05,870 about the system. 82 00:03:05,870 --> 00:03:08,650 All the information specifying the configuration system 83 00:03:08,650 --> 00:03:13,420 is completely contained in the wave function. 84 00:03:13,420 --> 00:03:16,090 Secondly, observables in quantum mechanics 85 00:03:16,090 --> 00:03:17,840 are associated with operators. 86 00:03:17,840 --> 00:03:19,750 Something you can build an experiment 87 00:03:19,750 --> 00:03:24,920 to observe or to measure is associated with an operator. 88 00:03:24,920 --> 00:03:27,990 And by an operator, I mean a rule or a map, 89 00:03:27,990 --> 00:03:30,170 something that tells you if you give me a function, 90 00:03:30,170 --> 00:03:32,091 I will give you a different function back. 91 00:03:32,091 --> 00:03:32,590 OK? 92 00:03:32,590 --> 00:03:34,631 An operator is just a thing which eats a function 93 00:03:34,631 --> 00:03:37,280 and spits out another function. 94 00:03:37,280 --> 00:03:39,840 Now, operators-- which I will denote with a hat, 95 00:03:39,840 --> 00:03:43,730 as long as I can remember to do so-- operators 96 00:03:43,730 --> 00:03:46,230 come-- and in particular, the kinds of operators we're going 97 00:03:46,230 --> 00:03:48,521 to care about, linear operators, which you talked about 98 00:03:48,521 --> 00:03:53,300 in detail last lecture-- linear operators come endowed 99 00:03:53,300 --> 00:03:56,220 with a natural set of special functions 100 00:03:56,220 --> 00:03:59,930 called Eigenfunctions with the following property. 101 00:03:59,930 --> 00:04:03,930 Your operator, acting on its Eigenfunction, 102 00:04:03,930 --> 00:04:07,030 gives you that same function back times a constant. 103 00:04:09,920 --> 00:04:12,395 So, that's a very special generically. 104 00:04:12,395 --> 00:04:14,270 An operator will take a function and give you 105 00:04:14,270 --> 00:04:15,630 some other random function that doesn't 106 00:04:15,630 --> 00:04:17,200 look all like the original function. 107 00:04:17,200 --> 00:04:19,491 It's a very special thing to give you the same function 108 00:04:19,491 --> 00:04:21,180 back times a constant. 109 00:04:21,180 --> 00:04:23,070 So, a useful thing to think about here 110 00:04:23,070 --> 00:04:24,940 is just in the case of vector spaces. 111 00:04:24,940 --> 00:04:27,730 So, I'm going to consider the operation corresponding 112 00:04:27,730 --> 00:04:30,210 to rotation around the z-axis by a small angle. 113 00:04:30,210 --> 00:04:31,130 OK? 114 00:04:31,130 --> 00:04:34,810 So, under rotation around the z-axis by a small angle, 115 00:04:34,810 --> 00:04:37,287 I take an arbitrary vector to some other stupid vector. 116 00:04:37,287 --> 00:04:39,370 Which vector is completely determined by the rule? 117 00:04:39,370 --> 00:04:40,970 I rotate by a small amount, right? 118 00:04:40,970 --> 00:04:43,110 I take this vector and it gives me this one. 119 00:04:43,110 --> 00:04:45,750 I take that vector, it gives me this one. 120 00:04:45,750 --> 00:04:46,970 Everyone agree with that? 121 00:04:46,970 --> 00:04:50,350 What are the Eigenvectors of the rotation 122 00:04:50,350 --> 00:04:52,006 by a small angle around the z-axis? 123 00:04:52,006 --> 00:04:53,470 AUDIENCE: [INAUDIBLE] 124 00:04:53,470 --> 00:04:55,520 PROFESSOR: Yeah, it's got to be a vector that 125 00:04:55,520 --> 00:04:57,040 doesn't change its direction. 126 00:04:57,040 --> 00:04:58,720 It just changes by magnitude. 127 00:04:58,720 --> 00:05:00,660 So there's one, right? 128 00:05:00,660 --> 00:05:01,270 I rotate. 129 00:05:01,270 --> 00:05:02,897 And what's its Eigenvalue? 130 00:05:02,897 --> 00:05:03,480 AUDIENCE: One. 131 00:05:03,480 --> 00:05:05,490 PROFESSOR: One, because nothing changed, right? 132 00:05:05,490 --> 00:05:07,480 Now, let's consider the following operation. 133 00:05:07,480 --> 00:05:10,640 Rotate by small angle and double its length. 134 00:05:10,640 --> 00:05:12,020 OK, that's a different operator. 135 00:05:12,020 --> 00:05:13,430 I rotate and I double the length. 136 00:05:13,430 --> 00:05:15,550 I rotate and I double the length. 137 00:05:15,550 --> 00:05:17,930 I rotate and I double the length. 138 00:05:17,930 --> 00:05:20,360 Yeah, so what's the Eigenvalue under that operator? 139 00:05:20,360 --> 00:05:21,185 AUDIENCE: Two. 140 00:05:21,185 --> 00:05:21,810 PROFESSOR: Two. 141 00:05:21,810 --> 00:05:22,650 Right, exactly. 142 00:05:22,650 --> 00:05:25,482 So these are a very special set of functions. 143 00:05:25,482 --> 00:05:27,690 This is the same idea, but instead of having vectors, 144 00:05:27,690 --> 00:05:29,340 we have functions. 145 00:05:29,340 --> 00:05:30,722 Questions? 146 00:05:30,722 --> 00:05:32,580 I thought I saw a hand pop up. 147 00:05:32,580 --> 00:05:33,080 No? 148 00:05:33,080 --> 00:05:35,800 OK, cool. 149 00:05:35,800 --> 00:05:37,580 Third, superposition. 150 00:05:37,580 --> 00:05:41,620 Given any two viable wave functions 151 00:05:41,620 --> 00:05:43,120 that could describe our system, that 152 00:05:43,120 --> 00:05:46,620 could specify states or configurations of our system, 153 00:05:46,620 --> 00:05:50,130 an arbitrary superposition of them-- arbitrary linear sum-- 154 00:05:50,130 --> 00:05:52,900 could also be a valid physical configuration. 155 00:05:52,900 --> 00:05:55,172 There is also a state corresponding 156 00:05:55,172 --> 00:05:56,380 to being in an arbitrary sum. 157 00:05:56,380 --> 00:05:59,230 For example, if we know that the electron could be black 158 00:05:59,230 --> 00:06:00,980 and it could be white, it could also 159 00:06:00,980 --> 00:06:03,810 be in an arbitrary superposition of being black and white. 160 00:06:03,810 --> 00:06:06,975 And that is a statement in which the electron is not black. 161 00:06:06,975 --> 00:06:08,490 The electron is not white. 162 00:06:08,490 --> 00:06:10,530 It is in the superposition of the two. 163 00:06:10,530 --> 00:06:13,040 It does not have a definite color. 164 00:06:13,040 --> 00:06:15,680 And that is exactly the configuration 165 00:06:15,680 --> 00:06:19,740 we found inside our apparatus in the first lecture. 166 00:06:19,740 --> 00:06:20,800 Yeah. 167 00:06:20,800 --> 00:06:23,420 AUDIENCE: Are those Phi-A arbitrary functions, 168 00:06:23,420 --> 00:06:25,170 or are they supposed to be Eigenfunctions? 169 00:06:25,170 --> 00:06:25,670 PROFESSOR: Excellent. 170 00:06:25,670 --> 00:06:27,600 So, in general the superposition thank you. 171 00:06:27,600 --> 00:06:28,725 It's an excellent question. 172 00:06:28,725 --> 00:06:31,082 The question was are these Phi-As arbitrary functions, 173 00:06:31,082 --> 00:06:32,540 or are they specific Eigenfunctions 174 00:06:32,540 --> 00:06:33,704 of some operator? 175 00:06:33,704 --> 00:06:35,370 So, the superposition principle actually 176 00:06:35,370 --> 00:06:36,453 says a very general thing. 177 00:06:36,453 --> 00:06:39,860 It says, given any two viable wave functions, 178 00:06:39,860 --> 00:06:42,630 an arbitrary sum, an arbitrary linear combination, 179 00:06:42,630 --> 00:06:43,962 is also a viable wave function. 180 00:06:43,962 --> 00:06:46,170 But here I want to mark something slightly different. 181 00:06:46,170 --> 00:06:49,220 And this is why I chose the notation I did. 182 00:06:49,220 --> 00:06:51,310 Given an operator A, it comes endowed 183 00:06:51,310 --> 00:06:54,870 with a special set of functions, its Eigenfunctions, right? 184 00:06:54,870 --> 00:06:57,410 We saw the last time. 185 00:06:57,410 --> 00:07:00,480 And I claimed the following. 186 00:07:00,480 --> 00:07:03,050 Beyond just the usual superposition principle, 187 00:07:03,050 --> 00:07:05,670 the set of Eigenfunctions of operators 188 00:07:05,670 --> 00:07:07,810 corresponding to physical observables-- so, pick 189 00:07:07,810 --> 00:07:10,460 your observable, like momentum. 190 00:07:10,460 --> 00:07:12,050 That corresponds to an operator. 191 00:07:12,050 --> 00:07:14,632 Consider the Eigenfunctions of momentum. 192 00:07:14,632 --> 00:07:15,840 Those we know what those are. 193 00:07:15,840 --> 00:07:17,970 They're plane waves with definite wavelength, 194 00:07:17,970 --> 00:07:18,803 right? u to the ikx. 195 00:07:21,678 --> 00:07:25,240 Any function can be expressed as a superposition 196 00:07:25,240 --> 00:07:29,170 of those Eigenfunctions of your physical observable. 197 00:07:29,170 --> 00:07:32,280 We'll go over this in more detail in a minute. 198 00:07:32,280 --> 00:07:35,880 But here I want to emphasize that the Eigenfunctions have 199 00:07:35,880 --> 00:07:38,461 a special property that-- for observables, for operators 200 00:07:38,461 --> 00:07:40,710 corresponding to observables-- the Eigenfunctions form 201 00:07:40,710 --> 00:07:42,160 a basis. 202 00:07:42,160 --> 00:07:46,320 Any function can be expanded as some linear combination 203 00:07:46,320 --> 00:07:48,660 of these basis functions, the classic example 204 00:07:48,660 --> 00:07:51,070 being the Fourier expansion. 205 00:07:51,070 --> 00:07:53,370 Any function, any periodic function, 206 00:07:53,370 --> 00:07:55,890 can be expanded as a sum of sines and cosines, 207 00:07:55,890 --> 00:07:57,850 and any function on the real line 208 00:07:57,850 --> 00:08:02,010 can be expanded as a sum of exponentials, e to the ikx. 209 00:08:02,010 --> 00:08:03,210 This is the same statement. 210 00:08:03,210 --> 00:08:05,270 The Eigenfunctions of momentum are what? 211 00:08:05,270 --> 00:08:07,230 e to the ikx. 212 00:08:07,230 --> 00:08:09,320 So, this is the same that an arbitrary function-- 213 00:08:09,320 --> 00:08:11,304 when the observable is the momentum, 214 00:08:11,304 --> 00:08:13,470 this is the statement that an arbitrary function can 215 00:08:13,470 --> 00:08:17,550 be expanded as a superposition, or a sum of exponentials, 216 00:08:17,550 --> 00:08:19,480 and that's the Fourier theorem. 217 00:08:19,480 --> 00:08:20,360 Cool? 218 00:08:20,360 --> 00:08:21,806 Was there a question? 219 00:08:21,806 --> 00:08:22,770 AUDIENCE: [INAUDIBLE] 220 00:08:22,770 --> 00:08:24,920 PROFESSOR: OK, good. 221 00:08:24,920 --> 00:08:28,890 Other questions on these points? 222 00:08:28,890 --> 00:08:31,800 So, these should not yet be trivial and obvious to you. 223 00:08:31,800 --> 00:08:34,690 If they are, then that's great, but if they're not, 224 00:08:34,690 --> 00:08:36,440 we're going to be working through examples 225 00:08:36,440 --> 00:08:39,289 for the next several lectures and problem sets. 226 00:08:39,289 --> 00:08:43,010 The point now is to give you a grounding on which to stand. 227 00:08:43,010 --> 00:08:44,250 Fourth postulate. 228 00:08:44,250 --> 00:08:47,524 What these expansion coefficients mean. 229 00:08:47,524 --> 00:08:48,940 And this is also an interpretation 230 00:08:48,940 --> 00:08:50,439 of the meaning of the wave function. 231 00:08:50,439 --> 00:08:52,050 What these expansion coefficients mean 232 00:08:52,050 --> 00:08:55,550 is that the probability that I measure the observable 233 00:08:55,550 --> 00:08:58,810 to be a particular Eigenvalue is the norm squared 234 00:08:58,810 --> 00:09:00,635 of the expansion coefficient. 235 00:09:00,635 --> 00:09:01,910 OK? 236 00:09:01,910 --> 00:09:03,510 So, I tell you that any function can 237 00:09:03,510 --> 00:09:06,360 be expanded as a superposition of plane waves-- waves 238 00:09:06,360 --> 00:09:09,082 with definite momentum-- with some coefficients. 239 00:09:09,082 --> 00:09:11,040 And those coefficients depend on which function 240 00:09:11,040 --> 00:09:12,210 I'm talking about. 241 00:09:12,210 --> 00:09:14,442 What these coefficients tell me is the probability 242 00:09:14,442 --> 00:09:16,650 that I will measure the momentum to be the associated 243 00:09:16,650 --> 00:09:18,655 value, the Eigenvalue. 244 00:09:18,655 --> 00:09:19,290 OK? 245 00:09:19,290 --> 00:09:21,164 Take that coefficient, take its norm squared, 246 00:09:21,164 --> 00:09:23,790 that gives me the probability. 247 00:09:23,790 --> 00:09:26,902 How do we compute these expansion coefficients? 248 00:09:26,902 --> 00:09:29,110 I think Barton didn't introduce to you this notation, 249 00:09:29,110 --> 00:09:30,402 but he certainly told you this. 250 00:09:30,402 --> 00:09:32,943 So let me introduce to you this notation which I particularly 251 00:09:32,943 --> 00:09:33,780 like. 252 00:09:33,780 --> 00:09:36,460 We can extract the expansion coefficient 253 00:09:36,460 --> 00:09:42,030 if we know the wave function by taking this integral, 254 00:09:42,030 --> 00:09:43,690 taking the wave function, multiplying 255 00:09:43,690 --> 00:09:46,859 by the complex conjugate of the associated Eigenfunction, 256 00:09:46,859 --> 00:09:47,650 doing the integral. 257 00:09:47,650 --> 00:09:52,630 And that notation is this round brackets with Phi A 258 00:09:52,630 --> 00:09:56,880 and Psi is my notation for this integral. 259 00:09:59,390 --> 00:10:02,200 And again, we'll still see this in more detail later on. 260 00:10:02,200 --> 00:10:04,650 And finally we have collapse, the statement that, 261 00:10:04,650 --> 00:10:08,630 if we go about measuring some observable A, 262 00:10:08,630 --> 00:10:13,100 then we will always, always observe precisely one 263 00:10:13,100 --> 00:10:15,140 of the Eigenvalues of that operator. 264 00:10:15,140 --> 00:10:17,360 We will never measure anything else. 265 00:10:17,360 --> 00:10:20,110 If the Eigenvalues are one, two, three, four, and five, 266 00:10:20,110 --> 00:10:24,380 you will never measure half, 13 halves. 267 00:10:24,380 --> 00:10:27,280 You will always measure an Eigenvalue. 268 00:10:27,280 --> 00:10:30,060 And upon measuring that Eigenvalue, 269 00:10:30,060 --> 00:10:33,720 you can be confident that that's the actual value of the system. 270 00:10:33,720 --> 00:10:36,180 I observe that it's a white electron, 271 00:10:36,180 --> 00:10:39,950 then it will remain white if I subsequently measure its color. 272 00:10:39,950 --> 00:10:43,230 What that's telling you is it's no longer a superposition 273 00:10:43,230 --> 00:10:45,570 of white and black, but it's wave function 274 00:10:45,570 --> 00:10:48,800 is that corresponding to a definite value 275 00:10:48,800 --> 00:10:50,790 of the observable. 276 00:10:50,790 --> 00:10:53,090 So, somehow the process of measurement-- and this 277 00:10:53,090 --> 00:10:56,740 is a disturbing statement, to which we'll return-- somehow 278 00:10:56,740 --> 00:10:59,780 the process of measuring the observable changes 279 00:10:59,780 --> 00:11:02,980 the wave function from our arbitrary superposition 280 00:11:02,980 --> 00:11:06,937 to a specific Eigenfunction, one particular Eigenfunction 281 00:11:06,937 --> 00:11:08,270 of the operator we're measuring. 282 00:11:10,972 --> 00:11:13,180 And this is called the collapse of the wave function. 283 00:11:13,180 --> 00:11:15,060 It collapses from being a superposition 284 00:11:15,060 --> 00:11:18,050 over possible states to being in a definite state 285 00:11:18,050 --> 00:11:18,900 upon measurement. 286 00:11:18,900 --> 00:11:20,680 And the definite state is that state 287 00:11:20,680 --> 00:11:25,020 corresponding to the value we observed or measured. 288 00:11:25,020 --> 00:11:25,727 Yeah. 289 00:11:25,727 --> 00:11:27,685 AUDIENCE: So, when the wave function collapses, 290 00:11:27,685 --> 00:11:29,790 does it instantaneously not become a function of time 291 00:11:29,790 --> 00:11:30,290 anymore? 292 00:11:30,290 --> 00:11:32,510 Because originally we had Psi of (x,t). 293 00:11:32,510 --> 00:11:34,468 PROFESSOR: Yeah, that's a really good question. 294 00:11:34,468 --> 00:11:38,234 So I wrote this only in terms of position, 295 00:11:38,234 --> 00:11:39,650 but I should more precisely write. 296 00:11:39,650 --> 00:11:42,390 So, the question was, does this happen instantaneously, or more 297 00:11:42,390 --> 00:11:45,250 precisely, does it cease to be a function of time? 298 00:11:45,250 --> 00:11:45,750 Thank you. 299 00:11:45,750 --> 00:11:46,750 It's very good question. 300 00:11:46,750 --> 00:11:48,833 So, no, it doesn't cease to be a function of time. 301 00:11:48,833 --> 00:11:50,460 It just says that Psi at x-- what 302 00:11:50,460 --> 00:11:51,960 you know upon doing this measurement 303 00:11:51,960 --> 00:11:55,360 is that Psi, as a function of x, at the time which I'll 304 00:11:55,360 --> 00:11:58,460 call T star, at what you've done the measurement 305 00:11:58,460 --> 00:12:00,067 is equal to this wave function. 306 00:12:00,067 --> 00:12:02,400 And so that leaves us with the following question, which 307 00:12:02,400 --> 00:12:04,608 is another way of asking the question you just asked. 308 00:12:04,608 --> 00:12:05,810 What happens next? 309 00:12:05,810 --> 00:12:08,550 How does the system evolve subsequently? 310 00:12:08,550 --> 00:12:10,450 And at the very end of the last lecture, 311 00:12:10,450 --> 00:12:13,720 we answered that-- or rather, Barton 312 00:12:13,720 --> 00:12:16,850 answered that-- by introducing the Schrodinger equation. 313 00:12:16,850 --> 00:12:19,760 And the Schrodinger equation, we don't derive, we just posit. 314 00:12:19,760 --> 00:12:21,650 Much like Newton posits f equals ma. 315 00:12:21,650 --> 00:12:25,110 You can motivate it, but you can't derive it. 316 00:12:25,110 --> 00:12:28,900 It's just what we mean by the quantum mechanical model. 317 00:12:28,900 --> 00:12:32,550 And Schrodinger's equation says, given a wave function, 318 00:12:32,550 --> 00:12:34,510 I can determine the time derivative, the time 319 00:12:34,510 --> 00:12:36,530 rate of changes of that wave function, 320 00:12:36,530 --> 00:12:38,950 and determine its time evolution, 321 00:12:38,950 --> 00:12:42,070 and its time derivative, its slope-- its velocity, 322 00:12:42,070 --> 00:12:48,090 if you will-- is one upon I h bar, the energy operator acting 323 00:12:48,090 --> 00:12:49,460 on that wave function. 324 00:12:49,460 --> 00:12:52,090 So, suppose we measure that our observable capital 325 00:12:52,090 --> 00:12:53,580 A takes the value of little a, one 326 00:12:53,580 --> 00:12:55,870 of the Eigenvalues of the associated operators. 327 00:12:55,870 --> 00:12:58,550 Suppose we measure that A equals little a 328 00:12:58,550 --> 00:13:00,602 at some particular moment T start. 329 00:13:00,602 --> 00:13:02,060 Then we know that the wave function 330 00:13:02,060 --> 00:13:04,735 is Psi of x at that moment in time. 331 00:13:04,735 --> 00:13:06,360 We can then compute the time derivative 332 00:13:06,360 --> 00:13:08,151 of the wave function at that moment in time 333 00:13:08,151 --> 00:13:11,150 by acting on this wave function with the operator e 334 00:13:11,150 --> 00:13:12,690 hat, the energy operator. 335 00:13:12,690 --> 00:13:15,190 And we can then integrate that differential equation forward 336 00:13:15,190 --> 00:13:19,100 in time and determine how the wave function evolves. 337 00:13:19,100 --> 00:13:21,370 The point of today's lecture is going 338 00:13:21,370 --> 00:13:24,980 to be to study how time evolution works in quantum 339 00:13:24,980 --> 00:13:27,950 mechanics, and to look at some basic examples 340 00:13:27,950 --> 00:13:31,060 and basic strategies for solving the time evolution 341 00:13:31,060 --> 00:13:33,540 problem in quantum mechanics. 342 00:13:33,540 --> 00:13:35,590 One of the great surprises in quantum mechanics-- 343 00:13:35,590 --> 00:13:38,048 hold on just one sec-- one of the real surprises in quantum 344 00:13:38,048 --> 00:13:39,770 mechanics is that time evolution is 345 00:13:39,770 --> 00:13:43,070 in a very specific sense trivial in quantum mechanics. 346 00:13:43,070 --> 00:13:44,680 It's preposterously simple. 347 00:13:44,680 --> 00:13:49,010 In particular, time evolution is governed by a linear equation. 348 00:13:49,010 --> 00:13:52,760 How many of you have studied a classical mechanical system 349 00:13:52,760 --> 00:13:57,181 where the time evolution is governed by a linear equation? 350 00:13:57,181 --> 00:13:57,680 Right. 351 00:13:57,680 --> 00:13:58,305 OK, all of you. 352 00:13:58,305 --> 00:13:59,740 The harmonic oscillator. 353 00:13:59,740 --> 00:14:02,600 But otherwise, not at all. 354 00:14:02,600 --> 00:14:04,890 Otherwise, the equations in classical mechanics 355 00:14:04,890 --> 00:14:06,860 are generically highly nonlinear. 356 00:14:06,860 --> 00:14:09,762 The time rate of change of position of a particle 357 00:14:09,762 --> 00:14:12,095 is the gradient of the force, and the force is generally 358 00:14:12,095 --> 00:14:13,940 some complicated function of position. 359 00:14:13,940 --> 00:14:16,050 You've got some capacitors over here, and maybe 360 00:14:16,050 --> 00:14:16,980 some magnetic field. 361 00:14:16,980 --> 00:14:18,062 It's very nonlinear. 362 00:14:18,062 --> 00:14:19,770 Evolution in quantum mechanics is linear, 363 00:14:19,770 --> 00:14:21,921 and this is going to be surprising. 364 00:14:21,921 --> 00:14:24,170 It's going to lead to some surprising simplifications. 365 00:14:24,170 --> 00:14:25,350 And we'll turn back to that, but I 366 00:14:25,350 --> 00:14:27,319 want to put that your mind like a little hook, 367 00:14:27,319 --> 00:14:28,860 that that's something you should mark 368 00:14:28,860 --> 00:14:30,724 on to as different from classical mechanics. 369 00:14:30,724 --> 00:14:31,890 And we'll come back to that. 370 00:14:31,890 --> 00:14:32,380 Yeah. 371 00:14:32,380 --> 00:14:34,005 AUDIENCE: If a particle is continuously 372 00:14:34,005 --> 00:14:35,600 observed as a not evolving particle? 373 00:14:35,600 --> 00:14:37,183 PROFESSOR: That's an awesome question. 374 00:14:37,183 --> 00:14:39,780 The question is, look, imagine I observe-- 375 00:14:39,780 --> 00:14:42,030 I'm going to paraphrase-- imagine I observe a particle 376 00:14:42,030 --> 00:14:43,320 and I observe that it's here. 377 00:14:43,320 --> 00:14:43,820 OK? 378 00:14:43,820 --> 00:14:45,710 Subsequently, its wave function will evolve in some way-- 379 00:14:45,710 --> 00:14:47,995 and we'll actually study that later today-- its wave function 380 00:14:47,995 --> 00:14:49,780 will evolve in some way, and it'll change. 381 00:14:49,780 --> 00:14:51,780 It won't necessarily be definitely here anymore. 382 00:14:51,780 --> 00:14:54,640 But if I just keep measuring it over and over and over again, 383 00:14:54,640 --> 00:14:56,348 I just keep measure it to be right there. 384 00:14:56,348 --> 00:14:58,290 It can't possibly evolve. 385 00:14:58,290 --> 00:15:00,180 And that's actually true, and it's 386 00:15:00,180 --> 00:15:02,210 called the Quantum Zeno problem. 387 00:15:02,210 --> 00:15:04,660 So, it's the observation that if you continuously 388 00:15:04,660 --> 00:15:06,480 measure a thing, you can't possibly 389 00:15:06,480 --> 00:15:09,200 have its wave function evolve significantly. 390 00:15:09,200 --> 00:15:11,780 And not only is it a cute idea, but it's something people 391 00:15:11,780 --> 00:15:13,340 do in the laboratory. 392 00:15:13,340 --> 00:15:15,850 So, Martin-- well, OK. 393 00:15:15,850 --> 00:15:18,040 People do it in a laboratory and it's cool. 394 00:15:18,040 --> 00:15:19,240 Come ask me and I'll tell you about the experiments. 395 00:15:19,240 --> 00:15:19,580 Other questions? 396 00:15:19,580 --> 00:15:20,040 There were a bunch. 397 00:15:20,040 --> 00:15:20,450 Yeah. 398 00:15:20,450 --> 00:15:22,334 AUDIENCE: So after you measure, the Schrodinger equation 399 00:15:22,334 --> 00:15:24,540 also gives you the evolution backwards in time? 400 00:15:24,540 --> 00:15:25,810 PROFESSOR: Oh, crap! 401 00:15:25,810 --> 00:15:26,390 Yes. 402 00:15:26,390 --> 00:15:27,590 That's such a good question. 403 00:15:27,590 --> 00:15:28,160 OK. 404 00:15:28,160 --> 00:15:29,670 I hate it when people ask that at this point, 405 00:15:29,670 --> 00:15:31,211 because I had to then say more words. 406 00:15:31,211 --> 00:15:32,920 That's a very good question. 407 00:15:32,920 --> 00:15:34,390 So the question goes like this. 408 00:15:34,390 --> 00:15:37,300 So this was going to be a punchline later on in the 409 00:15:37,300 --> 00:15:41,910 in the lecture but you stole my thunder, so that's awesome. 410 00:15:41,910 --> 00:15:43,520 So, here's the deal. 411 00:15:43,520 --> 00:15:46,960 We have a rule for time evolution of a wave function, 412 00:15:46,960 --> 00:15:48,840 and it has some lovely properties. 413 00:15:48,840 --> 00:15:55,550 In particular-- let me talk through this-- in particular, 414 00:15:55,550 --> 00:15:59,167 this equation is linear. 415 00:15:59,167 --> 00:16:00,500 So what properties does it have? 416 00:16:00,500 --> 00:16:01,680 Let me just-- I'm going to come back 417 00:16:01,680 --> 00:16:02,800 to your question in just a second, 418 00:16:02,800 --> 00:16:05,280 but first I want to set it up so we have a little more meat 419 00:16:05,280 --> 00:16:07,460 to answer your question precisely. 420 00:16:07,460 --> 00:16:09,730 So we note some properties of this equation, this time 421 00:16:09,730 --> 00:16:11,860 evolution equation. 422 00:16:11,860 --> 00:16:15,330 The first is that it's a linear equation. 423 00:16:15,330 --> 00:16:17,660 The derivative of a sum of function 424 00:16:17,660 --> 00:16:19,224 is a sum of the derivatives. 425 00:16:19,224 --> 00:16:20,890 The energy operator's a linear operator, 426 00:16:20,890 --> 00:16:23,370 meaning the energy operator acting on a sum of functions 427 00:16:23,370 --> 00:16:26,370 is a sum of the energy operator acting on each function. 428 00:16:26,370 --> 00:16:28,620 You guys studied linear operators in your problem set, 429 00:16:28,620 --> 00:16:29,786 right? 430 00:16:29,786 --> 00:16:30,660 So, these are linear. 431 00:16:30,660 --> 00:16:34,780 What that tells you is if Psi 1 of x and t 432 00:16:34,780 --> 00:16:38,580 solves the Schrodinger equation, and Psi 2 of x and t-- two 433 00:16:38,580 --> 00:16:40,870 different functions of position in time-- both 434 00:16:40,870 --> 00:16:46,000 solve the Schrodinger equation, then any combination of them-- 435 00:16:46,000 --> 00:16:51,630 alpha Psi 1 plus Beta Psi 2-- also solves-- which I will call 436 00:16:51,630 --> 00:16:56,130 Psi, and I'll make it a capital Psi for fun-- 437 00:16:56,130 --> 00:17:01,040 solves the Schrodinger equation automatically. 438 00:17:01,040 --> 00:17:03,329 Given two solutions of the Schrodinger equation, 439 00:17:03,329 --> 00:17:05,579 a superposition of them-- an arbitrary superposition-- 440 00:17:05,579 --> 00:17:07,660 also solves the Schrodinger equation. 441 00:17:07,660 --> 00:17:10,390 This is linearity. 442 00:17:10,390 --> 00:17:12,500 Cool? 443 00:17:12,500 --> 00:17:14,450 Next property. 444 00:17:14,450 --> 00:17:15,869 It's unitary. 445 00:17:15,869 --> 00:17:18,800 What I mean by unitary is this. 446 00:17:18,800 --> 00:17:21,089 It concerns probability. 447 00:17:21,089 --> 00:17:23,579 And you'll give a precise derivation 448 00:17:23,579 --> 00:17:25,859 of what I mean by unitary and you'll 449 00:17:25,859 --> 00:17:27,859 demonstrate that, in fact, Schrodinger evolution 450 00:17:27,859 --> 00:17:30,075 is unitary on your next problem set. 451 00:17:30,075 --> 00:17:31,570 It's not on the current one. 452 00:17:31,570 --> 00:17:33,995 But what I mean by unitary is that conserves probability. 453 00:17:33,995 --> 00:17:35,160 Whoops, that's an o. 454 00:17:35,160 --> 00:17:39,120 Conserves probability. 455 00:17:39,120 --> 00:17:42,180 IE, if there's an electron here, or if we 456 00:17:42,180 --> 00:17:43,910 have an object, a piece of chalk-- which 457 00:17:43,910 --> 00:17:45,365 I'm treating as a quantum mechanical point particle-- 458 00:17:45,365 --> 00:17:47,430 it's described by the wave function. 459 00:17:47,430 --> 00:17:50,530 The integral, the probability distribution over all the 460 00:17:50,530 --> 00:17:52,550 places it could possibly be had better be one, 461 00:17:52,550 --> 00:17:57,410 because it had better be somewhere with probability one. 462 00:17:57,410 --> 00:18:00,230 That had better not change in time. 463 00:18:00,230 --> 00:18:02,705 If I solve the Schrodinger equation evolve the system 464 00:18:02,705 --> 00:18:05,007 forward for half an hour, it had better not 465 00:18:05,007 --> 00:18:06,590 be the case that the total probability 466 00:18:06,590 --> 00:18:08,610 of finding the particle is one half. 467 00:18:08,610 --> 00:18:10,570 That means things disappear in the universe. 468 00:18:10,570 --> 00:18:12,240 And much as my socks would seem to be 469 00:18:12,240 --> 00:18:15,690 a counter example of that, things don't disappear, right? 470 00:18:15,690 --> 00:18:17,020 It just doesn't happen. 471 00:18:17,020 --> 00:18:20,180 So, quantum mechanics is demonstrably-- well, 472 00:18:20,180 --> 00:18:22,330 quantum mechanics is unitary, and this 473 00:18:22,330 --> 00:18:26,050 is a demonstrably good description of the real world. 474 00:18:26,050 --> 00:18:28,750 It fits all the observations we've ever made. 475 00:18:28,750 --> 00:18:32,050 No one's ever discovered an experimental violation 476 00:18:32,050 --> 00:18:33,850 of unitarity of quantum mechanics. 477 00:18:33,850 --> 00:18:37,490 I will note that there is a theoretical violation 478 00:18:37,490 --> 00:18:39,710 of unitarity in quantum mechanics, which 479 00:18:39,710 --> 00:18:40,750 is dear to my heart. 480 00:18:40,750 --> 00:18:43,670 It's called the Hawking Effect, and it's an observation that, 481 00:18:43,670 --> 00:18:47,520 due quantum mechanics, black holes in general relativity-- 482 00:18:47,520 --> 00:18:52,520 places from which light cannot escape-- evaporate. 483 00:18:52,520 --> 00:18:54,500 So you throw stuff and you form a black hole. 484 00:18:54,500 --> 00:18:55,310 It's got a horizon. 485 00:18:55,310 --> 00:18:56,684 If you fall through that horizon, 486 00:18:56,684 --> 00:18:57,730 we never see you again. 487 00:18:57,730 --> 00:19:00,290 Surprisingly, a black hole's a hot object like an iron, 488 00:19:00,290 --> 00:19:02,667 and it sends off radiation. 489 00:19:02,667 --> 00:19:04,750 As it sends off radiation, it's losing its energy. 490 00:19:04,750 --> 00:19:05,760 It's shrinking. 491 00:19:05,760 --> 00:19:07,760 And eventually it will, like the classical atom, 492 00:19:07,760 --> 00:19:09,160 collapse to nothing. 493 00:19:09,160 --> 00:19:10,660 There's a quibble going on right now 494 00:19:10,660 --> 00:19:12,040 over whether it really collapses to nothing, 495 00:19:12,040 --> 00:19:13,655 or whether there's a little granule 496 00:19:13,655 --> 00:19:15,577 nugget of quantum goodness. 497 00:19:15,577 --> 00:19:16,710 [LAUGHTER] 498 00:19:16,710 --> 00:19:18,220 We argue about this. 499 00:19:18,220 --> 00:19:19,930 We get paid to argue about this. 500 00:19:19,930 --> 00:19:20,430 [LAUGHTER] 501 00:19:20,430 --> 00:19:22,440 So, but here's the funny thing. 502 00:19:22,440 --> 00:19:26,107 If you threw in a dictionary and then the black hole evaporates, 503 00:19:26,107 --> 00:19:28,440 where did the information about what made the black hole 504 00:19:28,440 --> 00:19:30,660 go if it's just thermal radiation coming out? 505 00:19:30,660 --> 00:19:32,480 So, this is a classic calculation, 506 00:19:32,480 --> 00:19:34,350 which to a theorist says, ah ha! 507 00:19:34,350 --> 00:19:35,810 Maybe unitarity isn't conserved. 508 00:19:35,810 --> 00:19:37,960 But, look. 509 00:19:37,960 --> 00:19:39,610 Black holes, theorists. 510 00:19:39,610 --> 00:19:41,680 There's no experimental violation 511 00:19:41,680 --> 00:19:43,420 of unitarity anywhere. 512 00:19:43,420 --> 00:19:45,390 And if anyone ever did find such a violation, 513 00:19:45,390 --> 00:19:48,210 it would shatter the basic tenets of quantum mechanics, 514 00:19:48,210 --> 00:19:50,450 in particular the Schrodinger equation. 515 00:19:50,450 --> 00:19:52,925 So that's something we would love to see but never have. 516 00:19:52,925 --> 00:19:54,300 It depends on your point of view. 517 00:19:54,300 --> 00:19:55,469 You might hate to see it. 518 00:19:55,469 --> 00:19:57,010 And the third-- and this is, I think, 519 00:19:57,010 --> 00:20:00,904 the most important-- is that the Schrodinger evolution, this 520 00:20:00,904 --> 00:20:02,730 is a time derivative. 521 00:20:02,730 --> 00:20:04,150 It's a differential equation. 522 00:20:04,150 --> 00:20:05,567 If you know the initial condition, 523 00:20:05,567 --> 00:20:07,941 and you know the derivative, you can integrate it forward 524 00:20:07,941 --> 00:20:08,470 in time. 525 00:20:08,470 --> 00:20:11,836 And they're existence and uniqueness theorems for this. 526 00:20:11,836 --> 00:20:14,740 The system is deterministic. 527 00:20:14,740 --> 00:20:19,600 What that means is that if I have 528 00:20:19,600 --> 00:20:23,795 complete knowledge of the system at some moment in time, 529 00:20:23,795 --> 00:20:25,920 if I know the wave function at some moment in time, 530 00:20:25,920 --> 00:20:27,720 I can determine unambiguously the wave 531 00:20:27,720 --> 00:20:29,840 function in all subsequent moments of time. 532 00:20:29,840 --> 00:20:30,720 Unambiguously. 533 00:20:30,720 --> 00:20:34,870 There's no probability, there's no likelihood, it's determined. 534 00:20:34,870 --> 00:20:36,150 Completely determined. 535 00:20:36,150 --> 00:20:39,680 Given full knowledge now, I will have full knowledge later. 536 00:20:39,680 --> 00:20:41,450 Does everyone agree that this equation 537 00:20:41,450 --> 00:20:45,380 is a deterministic equation in that sense? 538 00:20:45,380 --> 00:20:46,110 Question. 539 00:20:46,110 --> 00:20:47,193 AUDIENCE: It's also local? 540 00:20:47,193 --> 00:20:48,510 PROFESSOR: It's all-- well, OK. 541 00:20:48,510 --> 00:20:51,350 This one happens to be-- you need 542 00:20:51,350 --> 00:20:54,250 to give me a better definition of local. 543 00:20:54,250 --> 00:20:56,926 So give me a definition of local that you want. 544 00:20:56,926 --> 00:20:59,854 AUDIENCE: The time evolution of the wave function 545 00:20:59,854 --> 00:21:03,270 happens only at a point that depends only 546 00:21:03,270 --> 00:21:06,055 on the value of the derivatives of the wave function 547 00:21:06,055 --> 00:21:07,680 and its potential energy at that point. 548 00:21:07,680 --> 00:21:08,240 PROFESSOR: No. 549 00:21:08,240 --> 00:21:09,420 Unfortunately, that's not the case. 550 00:21:09,420 --> 00:21:11,050 We'll see counter examples of that. 551 00:21:11,050 --> 00:21:13,165 The wave function-- the energy operator. 552 00:21:13,165 --> 00:21:15,040 So let's think about what this equation says. 553 00:21:15,040 --> 00:21:16,748 What this says is the time rate of change 554 00:21:16,748 --> 00:21:19,110 of the value of the wave function at some position 555 00:21:19,110 --> 00:21:23,875 and some moment in time is the energy operator acting on Psi 556 00:21:23,875 --> 00:21:24,927 at x of t. 557 00:21:24,927 --> 00:21:27,010 But I didn't tell you what the energy operator is. 558 00:21:27,010 --> 00:21:29,057 The energy operator just has to be linear. 559 00:21:29,057 --> 00:21:31,390 But it doesn't have to be-- it could know about the wave 560 00:21:31,390 --> 00:21:32,223 function everywhere. 561 00:21:32,223 --> 00:21:35,850 The energy operator's a map that takes the wave function 562 00:21:35,850 --> 00:21:38,410 and tells you what it should be later. 563 00:21:38,410 --> 00:21:41,550 And so, at this level there's nothing about locality built 564 00:21:41,550 --> 00:21:43,110 in to the energy operator, and we'll 565 00:21:43,110 --> 00:21:45,530 see just how bad that can be. 566 00:21:45,530 --> 00:21:47,100 So, this is related to your question 567 00:21:47,100 --> 00:21:50,760 about special relativity, and so those are deeply intertwined. 568 00:21:50,760 --> 00:21:52,900 We don't have that property here yet. 569 00:21:52,900 --> 00:21:55,259 But keep that in your mind, and ask questions 570 00:21:55,259 --> 00:21:56,300 when it seems to come up. 571 00:21:56,300 --> 00:21:57,870 Because it's a very, very, very important 572 00:21:57,870 --> 00:21:59,610 question when we talk about relativity. 573 00:21:59,610 --> 00:22:00,385 Yeah. 574 00:22:00,385 --> 00:22:02,385 AUDIENCE: Are postulates six and three redundant 575 00:22:02,385 --> 00:22:05,077 if the Schrodinger equation has superposition in it? 576 00:22:05,077 --> 00:22:05,660 PROFESSOR: No. 577 00:22:05,660 --> 00:22:06,624 Excellent question. 578 00:22:06,624 --> 00:22:07,790 That's a very good question. 579 00:22:07,790 --> 00:22:10,849 The question is, look, there's postulate three, which says, 580 00:22:10,849 --> 00:22:12,890 given any two wave functions that are viable wave 581 00:22:12,890 --> 00:22:15,340 functions of the system, then there's 582 00:22:15,340 --> 00:22:16,930 another state which is a viable wave 583 00:22:16,930 --> 00:22:19,570 function at some moment in time, which is also a viable wave 584 00:22:19,570 --> 00:22:21,120 function. 585 00:22:21,120 --> 00:22:25,711 But number six, the Schrodinger equation-- or sorry, 586 00:22:25,711 --> 00:22:27,710 really the linearity property of the Schrodinger 587 00:22:27,710 --> 00:22:31,236 equation-- so it needs to be the case for the Schrodinger 588 00:22:31,236 --> 00:22:33,360 question, but it says something slightly different. 589 00:22:33,360 --> 00:22:42,070 It doesn't just say that any any plausible or viable wave 590 00:22:42,070 --> 00:22:45,360 function and another can be superposed. 591 00:22:45,360 --> 00:22:48,210 It says that, specifically, any solution of the Schrodinger 592 00:22:48,210 --> 00:22:51,150 equation plus any other solution of the Schrodinger equation 593 00:22:51,150 --> 00:22:52,760 is again the Schrodinger operation. 594 00:22:52,760 --> 00:22:54,690 So, it's a slightly more specific thing 595 00:22:54,690 --> 00:22:55,970 than postulate three. 596 00:22:55,970 --> 00:23:00,180 However, your question is excellent because could it 597 00:23:00,180 --> 00:23:02,230 have been that the Schrodinger evolution didn't 598 00:23:02,230 --> 00:23:05,680 respect superposition? 599 00:23:05,680 --> 00:23:07,410 Well, you could imagine something, sure. 600 00:23:07,410 --> 00:23:09,160 We could've done a differ equation, right? 601 00:23:09,160 --> 00:23:10,110 It might not have been linear. 602 00:23:10,110 --> 00:23:12,068 We could have had that Schrodinger equation was 603 00:23:12,068 --> 00:23:12,970 equal to dt Psi. 604 00:23:12,970 --> 00:23:14,240 So imagine this equation. 605 00:23:14,240 --> 00:23:17,230 How do we have blown linearity while preserving determinism? 606 00:23:17,230 --> 00:23:25,500 So we could have added plus, I don't know, PSI squared of x. 607 00:23:25,500 --> 00:23:27,250 So that would now be a nonlinear equation. 608 00:23:27,250 --> 00:23:29,820 It's actually refer to as the nonlinear Schrodinger equation. 609 00:23:32,425 --> 00:23:34,050 Well, people mean many different things 610 00:23:34,050 --> 00:23:35,410 by the nonlinear Schrodinger equation, 611 00:23:35,410 --> 00:23:37,390 but that's a nonlinear Schrodinger equation. 612 00:23:37,390 --> 00:23:40,380 So you could certainly write this down. 613 00:23:40,380 --> 00:23:43,120 It's not linear. 614 00:23:43,120 --> 00:23:46,340 Does it violate the statement three 615 00:23:46,340 --> 00:23:49,540 that any two states of the system 616 00:23:49,540 --> 00:23:52,400 could be superposed to give another viable state 617 00:23:52,400 --> 00:23:54,080 at a moment in time? 618 00:23:54,080 --> 00:23:54,914 No, right? 619 00:23:54,914 --> 00:23:56,080 It doesn't directly violate. 620 00:23:56,080 --> 00:23:58,272 It violates the spirit of it. 621 00:23:58,272 --> 00:23:59,730 And as we'll see later, it actually 622 00:23:59,730 --> 00:24:01,090 would cause dramatic problems. 623 00:24:01,090 --> 00:24:03,896 It's something we don't usually emphasize-- something 624 00:24:03,896 --> 00:24:05,770 I don't usually emphasize in lectures of 804, 625 00:24:05,770 --> 00:24:07,410 but I will make a specific effort 626 00:24:07,410 --> 00:24:11,740 to mark the places where this would cause disasters. 627 00:24:11,740 --> 00:24:14,090 But, so this is actually a logically independent, 628 00:24:14,090 --> 00:24:16,400 although morally-- and in some sense 629 00:24:16,400 --> 00:24:19,410 is a technically related point to the superposition principle 630 00:24:19,410 --> 00:24:20,710 number three. 631 00:24:20,710 --> 00:24:21,896 Yeah. 632 00:24:21,896 --> 00:24:28,175 AUDIENCE: For postulate three, can that sum be infinite sum? 633 00:24:28,175 --> 00:24:29,624 PROFESSOR: Absolutely. 634 00:24:29,624 --> 00:24:31,556 AUDIENCE: Can you do bad things, then, 635 00:24:31,556 --> 00:24:33,488 like creating discontinuous wave functions? 636 00:24:33,488 --> 00:24:34,454 PROFESSOR: Oh yes. 637 00:24:34,454 --> 00:24:36,150 Oh, yes you can. 638 00:24:36,150 --> 00:24:37,090 So here's the thing. 639 00:24:37,090 --> 00:24:40,310 Look, if you have two functions and you add them together-- 640 00:24:40,310 --> 00:24:41,950 like two smooth continuous functions, 641 00:24:41,950 --> 00:24:43,616 you add them together-- what do you get? 642 00:24:43,616 --> 00:24:45,970 You get another smooth continuous function, right? 643 00:24:45,970 --> 00:24:47,390 Take seven. 644 00:24:47,390 --> 00:24:49,270 You get another. 645 00:24:49,270 --> 00:24:51,960 But if you take an infinite number-- look, 646 00:24:51,960 --> 00:24:53,230 mathematicians are sneaky. 647 00:24:53,230 --> 00:24:55,170 There's a reason we keep them down that hall, 648 00:24:55,170 --> 00:24:56,043 far away from us. 649 00:24:56,043 --> 00:24:56,850 [LAUGHTER] 650 00:24:56,850 --> 00:24:57,690 They're very sneaky. 651 00:24:57,690 --> 00:25:02,680 And if you give them an infinite number of continuous functions, 652 00:25:02,680 --> 00:25:06,035 they'll build for you a discontinuous function, right? 653 00:25:06,035 --> 00:25:06,535 Sneaky. 654 00:25:09,810 --> 00:25:13,170 Does that seem terribly physical? 655 00:25:13,170 --> 00:25:13,670 No. 656 00:25:13,670 --> 00:25:16,003 It's what happens when you give a mathematician too much 657 00:25:16,003 --> 00:25:19,010 paper and time, right? 658 00:25:19,010 --> 00:25:21,760 So, I mean this less flippantly than I'm saying it, 659 00:25:21,760 --> 00:25:24,910 but it's worth being a little flippant here. 660 00:25:24,910 --> 00:25:27,290 In a physical setting, we will often 661 00:25:27,290 --> 00:25:30,050 find that there are effectively an infinite number 662 00:25:30,050 --> 00:25:32,010 of possible things that could happen. 663 00:25:32,010 --> 00:25:33,970 So, for example in this room, where is this piece of chalk? 664 00:25:33,970 --> 00:25:35,430 It's described by a continuous variable. 665 00:25:35,430 --> 00:25:37,700 That's an uncountable infinite number of positions. 666 00:25:37,700 --> 00:25:38,800 Now, in practice, you can't really 667 00:25:38,800 --> 00:25:40,220 build an experiment that does that, 668 00:25:40,220 --> 00:25:42,136 but it is in principle an uncountable infinity 669 00:25:42,136 --> 00:25:44,100 of possible positions, right? 670 00:25:44,100 --> 00:25:46,560 You will never get a discontinuous wave function 671 00:25:46,560 --> 00:25:47,590 for this guy, because it would correspond 672 00:25:47,590 --> 00:25:49,000 to divergent amounts of momentum, 673 00:25:49,000 --> 00:25:51,790 as you showed on the previous problem set. 674 00:25:51,790 --> 00:25:56,000 So, in general, we will often be in a situation as physicists 675 00:25:56,000 --> 00:25:58,690 where there's the possibility of using 676 00:25:58,690 --> 00:26:00,640 the machinery-- the mathematical machinery-- 677 00:26:00,640 --> 00:26:02,790 to create pathological examples. 678 00:26:02,790 --> 00:26:05,380 And yes, that is a risk. 679 00:26:05,380 --> 00:26:07,260 But physically it never happens. 680 00:26:07,260 --> 00:26:09,120 Physically it's extraordinarily rare 681 00:26:09,120 --> 00:26:12,910 that such infinite divergences could matter. 682 00:26:12,910 --> 00:26:15,080 Now, I'm not saying that they never do. 683 00:26:15,080 --> 00:26:18,060 But we're going to be very carefree and casual in 804 684 00:26:18,060 --> 00:26:20,910 and just assume that when problems can arise from, 685 00:26:20,910 --> 00:26:23,244 say, superposing an infinite number of smooth functions, 686 00:26:23,244 --> 00:26:24,826 leading potentially to discontinuities 687 00:26:24,826 --> 00:26:27,790 or singularities, that they will either not happen for us-- not 688 00:26:27,790 --> 00:26:30,480 be relevant-- or they will happen because they're 689 00:26:30,480 --> 00:26:32,310 forced too, so for physical reasons 690 00:26:32,310 --> 00:26:34,402 we'll be able to identify. 691 00:26:34,402 --> 00:26:35,860 So, this is a very important point. 692 00:26:35,860 --> 00:26:37,526 We're not proving mathematical theorems. 693 00:26:37,526 --> 00:26:39,060 We're not trying to be rigorous. 694 00:26:39,060 --> 00:26:40,590 To prove a mathematical theorem you 695 00:26:40,590 --> 00:26:42,530 have to look at all the exceptional cases 696 00:26:42,530 --> 00:26:44,700 and say, those exceptional cases, 697 00:26:44,700 --> 00:26:46,600 we can deal with them mathematically. 698 00:26:46,600 --> 00:26:48,939 To a physicist, exceptional cases are exceptional. 699 00:26:48,939 --> 00:26:49,730 They're irrelevant. 700 00:26:49,730 --> 00:26:50,480 They don't happen. 701 00:26:50,480 --> 00:26:51,600 It doesn't matter. 702 00:26:51,600 --> 00:26:52,126 OK? 703 00:26:52,126 --> 00:26:53,500 And it doesn't mean that we don't 704 00:26:53,500 --> 00:26:55,375 care about the mathematical precision, right? 705 00:26:55,375 --> 00:26:57,290 I mean, I publish papers in math journals, 706 00:26:57,290 --> 00:26:59,800 so I have a deep love for these questions. 707 00:26:59,800 --> 00:27:02,780 But they're not salient for most of the physical questions 708 00:27:02,780 --> 00:27:03,580 we care about. 709 00:27:03,580 --> 00:27:08,440 So, do your best to try not to let those special cases get 710 00:27:08,440 --> 00:27:11,022 in the way of your understanding of the general case. 711 00:27:11,022 --> 00:27:12,730 I don't want you to not think about them, 712 00:27:12,730 --> 00:27:16,700 I just want you not let them stop you, OK? 713 00:27:16,700 --> 00:27:17,561 Yeah. 714 00:27:17,561 --> 00:27:19,590 AUDIENCE: So, in postulate five, you 715 00:27:19,590 --> 00:27:21,256 mentioned that [? functions ?] in effect 716 00:27:21,256 --> 00:27:24,669 was a experiment that more or less proves 717 00:27:24,669 --> 00:27:26,100 this collapse [INAUDIBLE] 718 00:27:26,100 --> 00:27:30,410 But, so I read that it is not [? complicit. ?] 719 00:27:30,410 --> 00:27:33,050 PROFESSOR: Yeah, so as with many things in quantum mechanics-- 720 00:27:33,050 --> 00:27:34,008 that's a fair question. 721 00:27:34,008 --> 00:27:37,170 So, let me make a slightly more general statement 722 00:27:37,170 --> 00:27:40,540 than answering that question directly. 723 00:27:40,540 --> 00:27:50,460 Many things will-- how to say-- so, we will not prove-- 724 00:27:50,460 --> 00:27:53,150 and experimentally you almost never prove a positive thing. 725 00:27:53,150 --> 00:27:56,504 You can show that a prediction is violated by experiment. 726 00:27:56,504 --> 00:27:58,420 So there's always going to be some uncertainty 727 00:27:58,420 --> 00:27:59,166 in your measurements, there's always 728 00:27:59,166 --> 00:28:01,124 going to be some uncertainty in your arguments. 729 00:28:01,124 --> 00:28:04,070 However, in the absence of a compelling alternate 730 00:28:04,070 --> 00:28:05,680 theoretical description, you cling 731 00:28:05,680 --> 00:28:07,930 on to what you've got it as long as it fits your data, 732 00:28:07,930 --> 00:28:09,790 and this fits the data like a champ. 733 00:28:09,790 --> 00:28:10,290 Right? 734 00:28:10,290 --> 00:28:11,630 So, does it prove? 735 00:28:11,630 --> 00:28:12,939 No. 736 00:28:12,939 --> 00:28:14,480 It fits pretty well, and nothing else 737 00:28:14,480 --> 00:28:15,870 comes even within the ballpark. 738 00:28:15,870 --> 00:28:17,260 And there's no explicit violation 739 00:28:17,260 --> 00:28:19,510 that's better than our experimental uncertainties. 740 00:28:19,510 --> 00:28:21,590 So, I don't know if I'd say, well, we 741 00:28:21,590 --> 00:28:25,550 could prove such a thing, but it fits. 742 00:28:25,550 --> 00:28:26,610 And I'm a physicist. 743 00:28:26,610 --> 00:28:28,400 I'm looking for things that fit. 744 00:28:28,400 --> 00:28:29,750 I'm not a metaphysicist. 745 00:28:29,750 --> 00:28:33,320 I'm not trying to give you some ontological commitment 746 00:28:33,320 --> 00:28:35,800 about what things are true and exist in the world, right? 747 00:28:35,800 --> 00:28:36,610 That's not my job. 748 00:28:39,910 --> 00:28:42,462 OK. 749 00:28:42,462 --> 00:28:43,420 So much for our review. 750 00:28:43,420 --> 00:28:45,010 But let me finally come back to-- now 751 00:28:45,010 --> 00:28:46,280 that we've observed that it's determinist, 752 00:28:46,280 --> 00:28:47,821 let me come back to the question that 753 00:28:47,821 --> 00:28:50,960 was asked a few minutes ago, which is, look, 754 00:28:50,960 --> 00:28:54,637 suppose we take our superposition. 755 00:28:54,637 --> 00:28:56,970 We evolve it forward for some time using the Schrodinger 756 00:28:56,970 --> 00:28:58,180 evolution. 757 00:28:58,180 --> 00:28:59,805 Notice that it's time reversal. 758 00:28:59,805 --> 00:29:01,180 If we know it's time reverted, we 759 00:29:01,180 --> 00:29:02,430 could run it backwards just as well 760 00:29:02,430 --> 00:29:03,620 as we could run it forwards, right? 761 00:29:03,620 --> 00:29:04,780 We could integrate that in time back, 762 00:29:04,780 --> 00:29:06,280 or we could integrate that in time forward. 763 00:29:06,280 --> 00:29:07,730 So, if we know the wave function at some moment in time, 764 00:29:07,730 --> 00:29:09,670 we can integrate it forward, and we can integrate it back 765 00:29:09,670 --> 00:29:10,490 in time. 766 00:29:10,490 --> 00:29:13,360 But, If at some point we measure, 767 00:29:13,360 --> 00:29:16,720 then the wave function collapses. 768 00:29:16,720 --> 00:29:18,900 And subsequently, the system evolves according 769 00:29:18,900 --> 00:29:20,441 to the Schrodinger equation, but with 770 00:29:20,441 --> 00:29:22,100 this new initial condition. 771 00:29:22,100 --> 00:29:24,400 So now we seem to have a problem. 772 00:29:24,400 --> 00:29:25,901 We seem to have-- and I believe this 773 00:29:25,901 --> 00:29:27,233 was the question that was asked. 774 00:29:27,233 --> 00:29:28,570 I don't remember who asked it. 775 00:29:28,570 --> 00:29:29,111 Who asked it? 776 00:29:31,847 --> 00:29:32,680 So someone asked it. 777 00:29:32,680 --> 00:29:33,990 It was a good question. 778 00:29:33,990 --> 00:29:35,580 We have this problem that there seem 779 00:29:35,580 --> 00:29:39,300 to be two definitions of time evolution in quantum mechanics. 780 00:29:39,300 --> 00:29:41,064 One is the Schrodinger equation, which 781 00:29:41,064 --> 00:29:42,480 says that things deterministically 782 00:29:42,480 --> 00:29:44,970 evolve forward in time. 783 00:29:44,970 --> 00:29:48,530 And the second is collapse, that if you do a measurement, 784 00:29:48,530 --> 00:29:52,610 things non-deterministically by probabilities collapse 785 00:29:52,610 --> 00:29:56,020 to some possible state. 786 00:29:56,020 --> 00:29:56,520 Yeah? 787 00:29:56,520 --> 00:29:57,895 And the probability is determined 788 00:29:57,895 --> 00:30:01,190 by which wave function you have. 789 00:30:01,190 --> 00:30:04,010 How can these things both be true? 790 00:30:04,010 --> 00:30:07,770 How can you have two different definitions of time evolution? 791 00:30:07,770 --> 00:30:09,700 So, this sort of frustration lies 792 00:30:09,700 --> 00:30:12,180 at the heart of much of the sort of spiel 793 00:30:12,180 --> 00:30:14,247 about the interpretation of quantum mechanics. 794 00:30:14,247 --> 00:30:15,830 On the one hand, we want to say, well, 795 00:30:15,830 --> 00:30:18,520 the world is inescapably probabilistic. 796 00:30:18,520 --> 00:30:21,909 Measurement comes with probabilistic outcomes 797 00:30:21,909 --> 00:30:23,700 and leads to collapse of the wave function. 798 00:30:23,700 --> 00:30:25,491 On the other hand, when you're not looking, 799 00:30:25,491 --> 00:30:27,340 the system evolves deterministically. 800 00:30:27,340 --> 00:30:29,167 And this sounds horrible. 801 00:30:29,167 --> 00:30:31,000 It sounds horrible to a classical physicist. 802 00:30:31,000 --> 00:30:32,830 It sounds horrible to me. 803 00:30:32,830 --> 00:30:34,790 It just sounds awful. 804 00:30:34,790 --> 00:30:35,920 It sounds arbitrary. 805 00:30:35,920 --> 00:30:38,030 Meanwhile, it makes it sound like the world cares. 806 00:30:38,030 --> 00:30:39,930 It evolves differently depending on whether you're looking 807 00:30:39,930 --> 00:30:40,700 or not. 808 00:30:40,700 --> 00:30:42,000 And that-- come on. 809 00:30:42,000 --> 00:30:45,560 I mean, I think we can all agree that that's just crazy. 810 00:30:45,560 --> 00:30:46,920 So what's going on? 811 00:30:46,920 --> 00:30:51,940 So for a long time, physicists in practice-- and still 812 00:30:51,940 --> 00:30:54,550 in practice-- for a long time physicists 813 00:30:54,550 --> 00:30:56,490 almost exclusively looked at this problem 814 00:30:56,490 --> 00:30:59,710 and said, look, don't worry about. 815 00:30:59,710 --> 00:31:00,970 It fits the data. 816 00:31:00,970 --> 00:31:03,820 It makes good predictions. 817 00:31:03,820 --> 00:31:04,950 Work with me here. 818 00:31:04,950 --> 00:31:05,590 Right? 819 00:31:05,590 --> 00:31:09,210 And it's really hard to argue against that attitude. 820 00:31:09,210 --> 00:31:10,290 You have a set of rules. 821 00:31:10,290 --> 00:31:11,350 It allows you to compute things. 822 00:31:11,350 --> 00:31:12,058 You compute them. 823 00:31:12,058 --> 00:31:13,300 They fit the data. 824 00:31:13,300 --> 00:31:13,900 Done. 825 00:31:13,900 --> 00:31:15,140 That is triumph. 826 00:31:15,140 --> 00:31:17,870 But it's deeply disconcerting. 827 00:31:17,870 --> 00:31:21,900 So, over the last, I don't know, in the second 828 00:31:21,900 --> 00:31:24,670 or the last quarter, roughly, the last third 829 00:31:24,670 --> 00:31:27,080 of the 20th century, various people 830 00:31:27,080 --> 00:31:29,530 started getting more upset about this. 831 00:31:29,530 --> 00:31:33,530 So, this notion of just shut up and calculate, 832 00:31:33,530 --> 00:31:36,870 which has been enshrined in the physics literature, 833 00:31:36,870 --> 00:31:39,880 goes under the name of the Copenhagen interpretation, 834 00:31:39,880 --> 00:31:42,540 which roughly says, look, just do this. 835 00:31:42,540 --> 00:31:43,270 Don't ask. 836 00:31:43,270 --> 00:31:47,100 Compute the numbers, and get what you will. 837 00:31:47,100 --> 00:31:55,200 And people have questioned the sanity or wisdom of doing that. 838 00:31:55,200 --> 00:31:57,061 And in particular, there's an idea-- 839 00:31:57,061 --> 00:31:59,560 so I refer to the Copenhagen interpretation with my students 840 00:31:59,560 --> 00:32:03,000 as the cop out, because it's basically 841 00:32:03,000 --> 00:32:04,240 disavowal of responsibility. 842 00:32:04,240 --> 00:32:05,620 Look, it doesn't make sense, but I'm not 843 00:32:05,620 --> 00:32:06,545 responsible for making sense. 844 00:32:06,545 --> 00:32:08,378 I'm just responsible for making predictions. 845 00:32:08,378 --> 00:32:09,050 Come on. 846 00:32:09,050 --> 00:32:14,680 So, more recently has come the theory of decoherence. 847 00:32:14,680 --> 00:32:17,650 And we're not going to talk about it in any detail 848 00:32:17,650 --> 00:32:19,530 until the last couple lectures of 804. 849 00:32:19,530 --> 00:32:22,060 Decoherence. 850 00:32:22,060 --> 00:32:23,310 I can't spell to save my life. 851 00:32:23,310 --> 00:32:24,740 So, the theory of decoherence. 852 00:32:24,740 --> 00:32:26,580 And here's roughly what the theory says. 853 00:32:26,580 --> 00:32:28,670 The theory says, look, the reason 854 00:32:28,670 --> 00:32:33,530 you have this problem between on the one hand, Schrodinger 855 00:32:33,530 --> 00:32:35,850 evolution of a quantum system, and on the other hand, 856 00:32:35,850 --> 00:32:37,420 measurement leading to collapse, is 857 00:32:37,420 --> 00:32:39,760 that in the case of measurement meaning to collapse, 858 00:32:39,760 --> 00:32:42,480 you're not really studying the evolution of a quantum system. 859 00:32:42,480 --> 00:32:45,360 You're studying the evolution of a quantum system-- ie 860 00:32:45,360 --> 00:32:47,977 a little thing that you're measuring-- interacting 861 00:32:47,977 --> 00:32:50,060 with your experimental apparatus, which is made up 862 00:32:50,060 --> 00:32:53,600 of 10 to the 27th particles, and you made up of 10 863 00:32:53,600 --> 00:32:55,320 to the 28 particles. 864 00:32:55,320 --> 00:32:55,820 Whatever. 865 00:32:55,820 --> 00:32:56,865 It's a large number. 866 00:32:56,865 --> 00:32:58,750 OK, a lot more than that. 867 00:32:58,750 --> 00:33:04,272 You, a macroscopic object, where classical dynamics 868 00:33:04,272 --> 00:33:05,230 are a good description. 869 00:33:05,230 --> 00:33:06,605 In particular, what that means is 870 00:33:06,605 --> 00:33:08,600 that the quantum effects are being washed out. 871 00:33:08,600 --> 00:33:10,110 You're washing out the interference 872 00:33:10,110 --> 00:33:12,400 of fringes, which is why I can catch this thing 873 00:33:12,400 --> 00:33:14,910 and not have it split into many different possible wave 874 00:33:14,910 --> 00:33:16,940 functions and where it went. 875 00:33:16,940 --> 00:33:20,694 So, dealing with that is hard, because now if you really 876 00:33:20,694 --> 00:33:22,860 want to treat the system with Schrodinger evolution, 877 00:33:22,860 --> 00:33:24,859 you have to study the trajectory and the motion, 878 00:33:24,859 --> 00:33:27,640 the dynamics, of every particle in the system, every degree 879 00:33:27,640 --> 00:33:28,930 of freedom in the system. 880 00:33:28,930 --> 00:33:31,831 So here's the question that decoherence is trying to ask. 881 00:33:31,831 --> 00:33:34,080 If you take a system where you have one little quantum 882 00:33:34,080 --> 00:33:35,760 subsystem that you're trying to measure, 883 00:33:35,760 --> 00:33:37,990 and then again a gagillion other degrees of freedom, 884 00:33:37,990 --> 00:33:39,850 some of which you care about-- they're made of you-- 885 00:33:39,850 --> 00:33:40,880 some of which you don't, like the particles 886 00:33:40,880 --> 00:33:42,760 of gas in the room, the environment. 887 00:33:42,760 --> 00:33:46,700 If you take that whole system, does Schrodinger 888 00:33:46,700 --> 00:33:51,180 evolution in the end boil down to collapse 889 00:33:51,180 --> 00:33:53,650 for that single quantum microsystem? 890 00:33:53,650 --> 00:33:55,460 And the answer is yes. 891 00:33:55,460 --> 00:33:57,720 Showing that take some work, and we'll 892 00:33:57,720 --> 00:33:59,300 touch on it at the end of 804. 893 00:33:59,300 --> 00:34:01,010 But I want to mark right here that this 894 00:34:01,010 --> 00:34:03,690 is one of the most deeply unsatisfying points 895 00:34:03,690 --> 00:34:06,070 in the basic story of quantum mechanics, 896 00:34:06,070 --> 00:34:08,239 and that it's deeply unsatisfying because of the way 897 00:34:08,239 --> 00:34:09,750 that we're presenting it. 898 00:34:09,750 --> 00:34:11,550 And there's a much more satisfying-- 899 00:34:11,550 --> 00:34:14,540 although still you never escape the fact 900 00:34:14,540 --> 00:34:16,909 that quantum mechanics violates your intuition. 901 00:34:16,909 --> 00:34:17,889 That's inescapable. 902 00:34:17,889 --> 00:34:19,750 But at least it's not illogical. 903 00:34:19,750 --> 00:34:21,659 it doesn't directly contradict itself. 904 00:34:21,659 --> 00:34:25,500 So that story is the story of decoherence. 905 00:34:25,500 --> 00:34:27,139 And if we're very lucky, I think we'll 906 00:34:27,139 --> 00:34:31,679 try to get one of my friends who's a quantum computing 907 00:34:31,679 --> 00:34:33,610 guy to talk about it. 908 00:34:33,610 --> 00:34:34,540 Yeah. 909 00:34:34,540 --> 00:34:37,040 AUDIENCE: [INAUDIBLE] Is it possible 910 00:34:37,040 --> 00:34:39,639 that we get two different results? 911 00:34:39,639 --> 00:34:40,560 PROFESSOR: No. 912 00:34:40,560 --> 00:34:41,060 No. 913 00:34:41,060 --> 00:34:41,560 No. 914 00:34:41,560 --> 00:34:43,889 There's never any ambiguity about what result you got. 915 00:34:43,889 --> 00:34:46,139 You never end up in a state of-- and this is also something 916 00:34:46,139 --> 00:34:47,805 that decoherence is supposed to explain. 917 00:34:47,805 --> 00:34:50,850 You never end up in a situation where you go like, wait, wait. 918 00:34:50,850 --> 00:34:51,710 I don't know. 919 00:34:51,710 --> 00:34:53,293 Maybe it was here, maybe it was there. 920 00:34:53,293 --> 00:34:54,320 I'm really confused. 921 00:34:54,320 --> 00:34:55,986 I mean, you can get up in that situation 922 00:34:55,986 --> 00:34:57,550 because you did a bad job, but you 923 00:34:57,550 --> 00:34:58,900 don't end up in that situation because you're 924 00:34:58,900 --> 00:35:00,220 in a superposition state. 925 00:35:00,220 --> 00:35:02,442 You always end up when you're a classical beast doing 926 00:35:02,442 --> 00:35:03,900 a classical measurement, you always 927 00:35:03,900 --> 00:35:06,070 end up in some definite state. 928 00:35:06,070 --> 00:35:08,060 Now, what wave function describes 929 00:35:08,060 --> 00:35:09,520 you doesn't necessarily correspond 930 00:35:09,520 --> 00:35:10,400 to you being in a simple state. 931 00:35:10,400 --> 00:35:12,400 You might be in a superposition of thinking this and thinking 932 00:35:12,400 --> 00:35:13,080 that. 933 00:35:13,080 --> 00:35:15,690 But, when you think this, that's in fact what happened. 934 00:35:15,690 --> 00:35:18,651 And when you think that, that's in fact what happened. 935 00:35:18,651 --> 00:35:19,150 OK. 936 00:35:19,150 --> 00:35:21,190 So I'm going to leave this alone for the moment, 937 00:35:21,190 --> 00:35:23,350 but I just wanted to mark that as an important part 938 00:35:23,350 --> 00:35:26,120 of the quantum mechanical story. 939 00:35:26,120 --> 00:35:28,430 OK. 940 00:35:28,430 --> 00:35:32,771 So let's go on to solving the Schrodinger equation. 941 00:35:32,771 --> 00:35:34,520 So what I want to do for the rest of today 942 00:35:34,520 --> 00:35:36,478 is talk about solving the Schrodinger equation. 943 00:35:43,740 --> 00:35:46,115 So when we set about solving the Schrodinger equation, 944 00:35:46,115 --> 00:35:47,490 the first thing we should realize 945 00:35:47,490 --> 00:35:49,490 is that at the end of the day, the Schrodinger equation 946 00:35:49,490 --> 00:35:50,890 is just some differential equation. 947 00:35:50,890 --> 00:35:53,390 And in fact, it's a particularly easy differential equation. 948 00:35:53,390 --> 00:35:56,460 It's a first order linear differential equation. 949 00:35:56,460 --> 00:35:56,960 Right? 950 00:35:56,960 --> 00:35:59,750 We know how to solve those. 951 00:35:59,750 --> 00:36:02,104 But, while it's first order in time, 952 00:36:02,104 --> 00:36:04,270 we have to think about what this energy operator is. 953 00:36:04,270 --> 00:36:06,370 So, just like the Newton equation f equals ma, 954 00:36:06,370 --> 00:36:07,995 we have to specify the energy operative 955 00:36:07,995 --> 00:36:10,330 before we can actually solve the dynamics of the system. 956 00:36:10,330 --> 00:36:11,788 In f equals ma, we have to tell you 957 00:36:11,788 --> 00:36:14,280 what the force is before we can solve for p, 958 00:36:14,280 --> 00:36:17,554 from p is equal to f. 959 00:36:17,554 --> 00:36:18,220 So, for example. 960 00:36:20,880 --> 00:36:23,930 So one strategy to solve the Schrodinger equation 961 00:36:23,930 --> 00:36:26,605 is to say, look, it's just a differential equation, 962 00:36:26,605 --> 00:36:28,480 and I'll solve it using differential equation 963 00:36:28,480 --> 00:36:29,510 techniques. 964 00:36:29,510 --> 00:36:34,690 So let me specify, for example, the energy operator. 965 00:36:34,690 --> 00:36:38,190 What's an easy energy operator? 966 00:36:38,190 --> 00:36:41,230 Well, imagine you had a harmonic oscillator, which, you know, 967 00:36:41,230 --> 00:36:42,930 physicists, that's your go-to. 968 00:36:42,930 --> 00:36:45,380 So, harmonic oscillator has energy p 969 00:36:45,380 --> 00:36:49,840 squared over 2m plus M Omega squared upon 2x squared. 970 00:36:49,840 --> 00:36:51,480 But we're going quantum mechanics, 971 00:36:51,480 --> 00:36:53,882 so we replace these guys by operators. 972 00:36:53,882 --> 00:36:55,090 So that's an energy operator. 973 00:36:55,090 --> 00:36:58,520 It's a perfectly viable operator. 974 00:36:58,520 --> 00:37:01,240 And what is the differential equation that this leads to? 975 00:37:01,240 --> 00:37:03,050 What's the Schrodinger equation leads to? 976 00:37:03,050 --> 00:37:05,216 Well, I'm going to put the ih bar on the other side. 977 00:37:05,216 --> 00:37:10,300 ih bar derivative with respect to time of Psi of x and t 978 00:37:10,300 --> 00:37:12,210 is equal to p squared. 979 00:37:12,210 --> 00:37:14,200 Well, we remember that p is equal to h bar 980 00:37:14,200 --> 00:37:17,160 upon i, derivative with respect to x. 981 00:37:17,160 --> 00:37:20,120 So p squared is minus h bar squared derivative with respect 982 00:37:20,120 --> 00:37:24,270 to x squared upon 2m, or minus h bar squared upon 2m. 983 00:37:27,042 --> 00:37:28,425 Psi prime prime. 984 00:37:28,425 --> 00:37:31,200 Let me write this as dx squared. 985 00:37:31,200 --> 00:37:37,570 Two spatial derivatives acting on Psi of x and t plus m 986 00:37:37,570 --> 00:37:44,720 omega squared upon 2x squared Psi of x and t. 987 00:37:44,720 --> 00:37:47,160 So here's a differential equation. 988 00:37:47,160 --> 00:37:51,241 And if we want to know how does a system evolve in time, 989 00:37:51,241 --> 00:37:53,240 ie given some initial wave function, how does it 990 00:37:53,240 --> 00:37:55,531 evolve in time, we just take this differential equation 991 00:37:55,531 --> 00:37:56,280 and we solve it. 992 00:37:56,280 --> 00:37:58,540 And there are many tools to solve this partial differential 993 00:37:58,540 --> 00:37:59,040 equation. 994 00:37:59,040 --> 00:38:01,300 For example, you could put it on Mathematica 995 00:38:01,300 --> 00:38:03,175 and just use NDSolve, right? 996 00:38:03,175 --> 00:38:04,550 This wasn't available, of course, 997 00:38:04,550 --> 00:38:06,480 to the physicists at the turn of the century, 998 00:38:06,480 --> 00:38:09,467 but they were less timid about differential equations 999 00:38:09,467 --> 00:38:11,550 than we are, because they didn't have Mathematica. 1000 00:38:11,550 --> 00:38:13,130 So, this is a very straightforward differential 1001 00:38:13,130 --> 00:38:13,950 equation to solve, and we're going 1002 00:38:13,950 --> 00:38:15,510 to solve it in a couple of lectures. 1003 00:38:15,510 --> 00:38:17,801 We're going to study the harmonic oscillator in detail. 1004 00:38:17,801 --> 00:38:20,430 What I want to emphasize for you is that any system has have 1005 00:38:20,430 --> 00:38:23,540 some specified energy operator, just like any classical system, 1006 00:38:23,540 --> 00:38:26,980 has some definite force function. 1007 00:38:26,980 --> 00:38:28,780 And given that energy operator, that's 1008 00:38:28,780 --> 00:38:30,100 going to lead to a differential equation. 1009 00:38:30,100 --> 00:38:31,975 So one way to solve the differential equation 1010 00:38:31,975 --> 00:38:35,040 is just to go ahead and brute force solve it. 1011 00:38:35,040 --> 00:38:36,770 But, at the end of the day, solving 1012 00:38:36,770 --> 00:38:38,520 the Schrodinger equation is always, always 1013 00:38:38,520 --> 00:38:40,180 going to boil down to some version 1014 00:38:40,180 --> 00:38:43,900 morally of solve this differential equation. 1015 00:38:43,900 --> 00:38:46,119 Questions about that? 1016 00:38:46,119 --> 00:38:48,900 OK. 1017 00:38:48,900 --> 00:38:51,129 But when we actually look at a differential equation 1018 00:38:51,129 --> 00:38:53,420 like this-- so, say we have this differential equation. 1019 00:38:53,420 --> 00:38:55,211 It's got a derivative with respect to time, 1020 00:38:55,211 --> 00:38:57,210 so we have to specify some initial condition. 1021 00:38:57,210 --> 00:38:59,010 There are many ways to solve it. 1022 00:38:59,010 --> 00:39:06,250 So given E, given some specific E, 1023 00:39:06,250 --> 00:39:08,190 given some specific energy operator, 1024 00:39:08,190 --> 00:39:09,420 there are many ways to solve. 1025 00:39:13,570 --> 00:39:16,444 The resulting differential equation. 1026 00:39:16,444 --> 00:39:18,485 And I'm just going to mark that, in general, it's 1027 00:39:18,485 --> 00:39:20,100 a PDE, because it's got derivatives with respect 1028 00:39:20,100 --> 00:39:22,016 to time and derivatives with respect to space. 1029 00:39:25,810 --> 00:39:27,950 And roughly speaking, all these techniques 1030 00:39:27,950 --> 00:39:29,300 fall into three camps. 1031 00:39:29,300 --> 00:39:30,700 The first is just brute force. 1032 00:39:33,440 --> 00:39:37,230 That means some analog of throw it on Mathematica, 1033 00:39:37,230 --> 00:39:39,262 go to the closet and pull out your mathematician 1034 00:39:39,262 --> 00:39:41,560 and tie them to the chalkboard until they're done, 1035 00:39:41,560 --> 00:39:42,520 and then put them back. 1036 00:39:42,520 --> 00:39:46,600 But some version of a brute force, which is just 1037 00:39:46,600 --> 00:39:48,700 use, by hook or by crook, some technique 1038 00:39:48,700 --> 00:39:51,710 that allows you to solve the differential equation. 1039 00:39:51,710 --> 00:39:52,500 OK. 1040 00:39:52,500 --> 00:39:55,810 The second is extreme cleverness. 1041 00:39:55,810 --> 00:39:57,990 And you'd be amazed how often this comes in handy. 1042 00:39:57,990 --> 00:39:59,490 So, extreme cleverness-- which we'll 1043 00:39:59,490 --> 00:40:02,610 see both of these techniques used 1044 00:40:02,610 --> 00:40:05,860 for the harmonic oscillator. 1045 00:40:05,860 --> 00:40:07,200 That's what we'll do next week. 1046 00:40:07,200 --> 00:40:08,790 First, the brute force, and secondly, 1047 00:40:08,790 --> 00:40:10,909 the clever way of solving the harmonic oscillator. 1048 00:40:10,909 --> 00:40:12,950 When I say extreme cleverness, what I really mean 1049 00:40:12,950 --> 00:40:15,310 is a more elegant use of your mathematician. 1050 00:40:15,310 --> 00:40:17,890 You know something about the structure, 1051 00:40:17,890 --> 00:40:21,430 the mathematical structure of your differential equation. 1052 00:40:21,430 --> 00:40:23,690 And you're going to use that structure 1053 00:40:23,690 --> 00:40:27,240 to figure out a good way to organize the differential 1054 00:40:27,240 --> 00:40:29,500 equation, the good way to organize the problem. 1055 00:40:29,500 --> 00:40:30,977 And that will teach you physics. 1056 00:40:30,977 --> 00:40:33,310 And the reason I distinguish brute force from cleverness 1057 00:40:33,310 --> 00:40:34,850 in this sense is that brute force, 1058 00:40:34,850 --> 00:40:35,890 you just get a list of numbers. 1059 00:40:35,890 --> 00:40:37,764 Cleverness, you learn something about the way 1060 00:40:37,764 --> 00:40:39,750 the physics of the system operates. 1061 00:40:39,750 --> 00:40:43,264 We'll see this at work in the next two lectures. 1062 00:40:43,264 --> 00:40:45,555 And see, I really should separate this out numerically. 1063 00:40:48,760 --> 00:40:52,100 And here I don't just mean sticking it into MATLAB. 1064 00:40:52,100 --> 00:40:53,850 Numerically, it can be enormously valuable 1065 00:40:53,850 --> 00:40:54,690 for a bunch of reasons. 1066 00:40:54,690 --> 00:40:56,130 First off, there are often situations 1067 00:40:56,130 --> 00:40:58,330 where no classic technique in differential equations 1068 00:40:58,330 --> 00:41:00,760 or no simple mathematical structure that would just 1069 00:41:00,760 --> 00:41:02,780 leap to the imagination comes to use. 1070 00:41:02,780 --> 00:41:04,780 And you have some horrible differential you just 1071 00:41:04,780 --> 00:41:07,100 have to solve, and you can solve it numerically. 1072 00:41:07,100 --> 00:41:09,220 Very useful lesson, and a reason to not even-- 1073 00:41:09,220 --> 00:41:11,720 how many of y'all are thinking about being theorists of some 1074 00:41:11,720 --> 00:41:13,540 stripe or other? 1075 00:41:13,540 --> 00:41:14,040 OK. 1076 00:41:14,040 --> 00:41:15,080 And how many of y'all are thinking about being 1077 00:41:15,080 --> 00:41:17,260 experimentalists of some stripe or another? 1078 00:41:17,260 --> 00:41:17,990 OK, cool. 1079 00:41:17,990 --> 00:41:21,380 So, look, there's this deep, deep prejudice 1080 00:41:21,380 --> 00:41:27,030 in theory against numerical solutions of problems. 1081 00:41:27,030 --> 00:41:28,640 It's myopia. 1082 00:41:28,640 --> 00:41:31,140 It's a terrible attitude, and here's the reason. 1083 00:41:31,140 --> 00:41:34,220 Computers are stupid. 1084 00:41:34,220 --> 00:41:36,050 Computers are breathtakingly dumb. 1085 00:41:36,050 --> 00:41:37,800 They will do whatever you tell them to do, 1086 00:41:37,800 --> 00:41:40,091 but they will not tell you that was a dumb thing to do. 1087 00:41:40,091 --> 00:41:40,940 They have no idea. 1088 00:41:40,940 --> 00:41:43,830 So, in order to solve an interesting physical problem, 1089 00:41:43,830 --> 00:41:46,480 you have to first extract all the physics 1090 00:41:46,480 --> 00:41:48,770 and organize the problem in such a way 1091 00:41:48,770 --> 00:41:52,510 that a stupid computer can do the solution. 1092 00:41:52,510 --> 00:41:55,430 As a consequence, you learn the physics about the problem. 1093 00:41:55,430 --> 00:41:58,019 It's extremely valuable to learn how to solve problems 1094 00:41:58,019 --> 00:42:00,310 numerically, and we're going to have problem sets later 1095 00:42:00,310 --> 00:42:01,100 in the course in which you're going 1096 00:42:01,100 --> 00:42:03,010 to be required to numerically solve 1097 00:42:03,010 --> 00:42:05,020 some of these differential equations. 1098 00:42:05,020 --> 00:42:07,340 But it's useful because you get numbers, 1099 00:42:07,340 --> 00:42:09,810 and you can check against data, but also it 1100 00:42:09,810 --> 00:42:11,640 lets you in the process of understanding 1101 00:42:11,640 --> 00:42:12,681 how to solve the problem. 1102 00:42:12,681 --> 00:42:17,360 You learn things about the problem. 1103 00:42:17,360 --> 00:42:20,640 So I want to mark that as a separate logical way to do it. 1104 00:42:20,640 --> 00:42:24,690 So today, I want to start our analysis 1105 00:42:24,690 --> 00:42:27,070 by looking at a couple of examples 1106 00:42:27,070 --> 00:42:33,430 of solving the Schrodinger equation. 1107 00:42:33,430 --> 00:42:38,740 And I want to start by looking at energy Eigenfunctions. 1108 00:42:44,540 --> 00:42:54,780 And then once we understand how a single energy Eigenfunction 1109 00:42:54,780 --> 00:42:56,975 evolves in time, once we understand that solution 1110 00:42:56,975 --> 00:42:58,570 to the Schrodinger equation, we're 1111 00:42:58,570 --> 00:43:01,660 going to use the linearity of the Schrodinger equation 1112 00:43:01,660 --> 00:43:06,380 to write down a general solution of the Schrodinger equation. 1113 00:43:06,380 --> 00:43:07,940 OK. 1114 00:43:07,940 --> 00:43:10,840 So, first. 1115 00:43:10,840 --> 00:43:13,090 What happens if we have a single energy Eigenfunction? 1116 00:43:13,090 --> 00:43:16,820 So, suppose our wave function as a function of x at time t 1117 00:43:16,820 --> 00:43:20,570 equals zero is in a known configuration, which 1118 00:43:20,570 --> 00:43:24,480 is an energy Eigenfunction Phi sub E of x. 1119 00:43:24,480 --> 00:43:28,270 What I mean by Phi sub E of x is if I take the energy operator, 1120 00:43:28,270 --> 00:43:32,050 and I act on Phi sub E of x, this gives me back 1121 00:43:32,050 --> 00:43:34,620 the number E Phi sub E of x. 1122 00:43:37,994 --> 00:43:38,920 OK? 1123 00:43:38,920 --> 00:43:41,830 So it's an Eigenfunction of the energy operator, the Eigenvalue 1124 00:43:41,830 --> 00:43:42,460 E. 1125 00:43:42,460 --> 00:43:44,560 So, suppose our initial condition is 1126 00:43:44,560 --> 00:43:48,090 that our system began life at time t 1127 00:43:48,090 --> 00:43:51,306 equals zero in this state with definite energy E. Everyone 1128 00:43:51,306 --> 00:43:52,360 cool with that? 1129 00:43:52,360 --> 00:43:53,810 First off, question. 1130 00:43:53,810 --> 00:43:56,400 Suppose I immediately at time zero 1131 00:43:56,400 --> 00:43:59,210 measure the energy of this system. 1132 00:43:59,210 --> 00:44:00,310 What will I get? 1133 00:44:00,310 --> 00:44:01,726 AUDIENCE: E. 1134 00:44:01,726 --> 00:44:03,100 PROFESSOR: With what probability? 1135 00:44:03,100 --> 00:44:04,000 AUDIENCE: 100% 1136 00:44:04,000 --> 00:44:06,920 PROFESSOR: 100%, because this is, in fact, of this form, 1137 00:44:06,920 --> 00:44:09,890 it's a superposition of energy Eigenstates, 1138 00:44:09,890 --> 00:44:11,350 except there's only one term. 1139 00:44:11,350 --> 00:44:12,850 And the coefficient of that one term 1140 00:44:12,850 --> 00:44:16,170 is one, and the probability that I measure the energy 1141 00:44:16,170 --> 00:44:19,170 to be equal to that value is the coefficient norm squared, 1142 00:44:19,170 --> 00:44:21,040 and that's one norm squared. 1143 00:44:21,040 --> 00:44:22,499 Everyone cool with that? 1144 00:44:22,499 --> 00:44:25,040 Consider on the other hand, if I had taken this wave function 1145 00:44:25,040 --> 00:44:28,390 and I had multiplied it by phase E to the i Alpha. 1146 00:44:28,390 --> 00:44:31,370 What now is the probability where alpha is just a number? 1147 00:44:31,370 --> 00:44:34,380 What now is the probability that I measured the state 1148 00:44:34,380 --> 00:44:35,710 to have energy E? 1149 00:44:35,710 --> 00:44:36,455 AUDIENCE: One. 1150 00:44:36,455 --> 00:44:38,580 PROFESSOR: It's still one, because the norm squared 1151 00:44:38,580 --> 00:44:40,430 of a phase is one. 1152 00:44:40,430 --> 00:44:41,266 Right? 1153 00:44:41,266 --> 00:44:42,180 OK. 1154 00:44:42,180 --> 00:44:46,530 The overall phase does not matter. 1155 00:44:46,530 --> 00:44:48,540 So, suppose I have this as my initial condition. 1156 00:44:48,540 --> 00:44:49,630 Let's take away the overall phase 1157 00:44:49,630 --> 00:44:50,921 because my life will be easier. 1158 00:44:53,094 --> 00:44:54,260 So here's the wave function. 1159 00:44:54,260 --> 00:44:55,635 What is the Schrodinger equation? 1160 00:44:55,635 --> 00:44:57,330 Well, the Schrodinger equation says 1161 00:44:57,330 --> 00:45:01,550 that ih bar time derivative of Psi 1162 00:45:01,550 --> 00:45:04,409 is equal to the energy operator acting on Psi. 1163 00:45:04,409 --> 00:45:05,450 And I should be specific. 1164 00:45:05,450 --> 00:45:11,740 This is Psi at x at time t, Eigenvalued at this time zero 1165 00:45:11,740 --> 00:45:15,610 is equal to the energy operator acting on this wave function. 1166 00:45:15,610 --> 00:45:19,880 But what's the energy operator acting on this wave function? 1167 00:45:19,880 --> 00:45:20,380 AUDIENCE: E. 1168 00:45:20,380 --> 00:45:22,420 PROFESSOR: E. E on Psi is equal to E 1169 00:45:22,420 --> 00:45:25,750 on Phi sub E, which is just E the number. 1170 00:45:25,750 --> 00:45:28,720 This is the number E, the Eigenvalue E times 1171 00:45:28,720 --> 00:45:31,867 Psi at x zero. 1172 00:45:31,867 --> 00:45:34,200 And now, instead of having an operator on the right hand 1173 00:45:34,200 --> 00:45:35,814 side, we just have a number. 1174 00:45:35,814 --> 00:45:38,230 So, I'm going to write this differential equation slightly 1175 00:45:38,230 --> 00:45:44,470 differently, ie time derivative of Psi 1176 00:45:44,470 --> 00:45:54,450 is equal to E upon ih bar, or minus ie over h bar Psi. 1177 00:45:58,941 --> 00:45:59,440 Yeah? 1178 00:45:59,440 --> 00:46:00,439 Everyone cool with that? 1179 00:46:00,439 --> 00:46:04,050 This is the easiest differential equation in the world to solve. 1180 00:46:04,050 --> 00:46:06,700 So, the time derivative is a constant. 1181 00:46:06,700 --> 00:46:08,090 Well, times itself. 1182 00:46:08,090 --> 00:46:13,660 That means that therefore Psi at x and t 1183 00:46:13,660 --> 00:46:23,329 is equal to E to the minus i ET over h bar Psi at x zero. 1184 00:46:23,329 --> 00:46:25,620 Where I've imposed the initial condition that at time t 1185 00:46:25,620 --> 00:46:27,310 equals zero, the wave function is 1186 00:46:27,310 --> 00:46:28,980 just equal to Psi of x at zero. 1187 00:46:32,340 --> 00:46:35,250 And in particular, I know what Psi of x and zero is. 1188 00:46:35,250 --> 00:46:37,360 It's Phi E of x. 1189 00:46:37,360 --> 00:46:39,836 So I can simply write this as Phi E of x. 1190 00:46:44,970 --> 00:46:47,850 Are we cool with that? 1191 00:46:47,850 --> 00:46:51,420 So, what this tells me is that under time evolution, 1192 00:46:51,420 --> 00:46:55,284 a state which is initially in an energy Eigenstate 1193 00:46:55,284 --> 00:46:57,450 remains in an energy Eigenstate with the same energy 1194 00:46:57,450 --> 00:46:57,975 Eigenvalue. 1195 00:46:57,975 --> 00:47:00,100 The only thing that changes about the wave function 1196 00:47:00,100 --> 00:47:02,320 is that its phase changes, and its phase 1197 00:47:02,320 --> 00:47:05,300 changes by rotating with a constant velocity. 1198 00:47:05,300 --> 00:47:08,270 E to the minus i, the energy Eigenvalue, 1199 00:47:08,270 --> 00:47:10,674 times time upon h bar. 1200 00:47:10,674 --> 00:47:12,840 Now, first off, before we do anything else as usual, 1201 00:47:12,840 --> 00:47:14,923 we should first check the dimensions of our result 1202 00:47:14,923 --> 00:47:16,680 to make sure we didn't make a goof. 1203 00:47:16,680 --> 00:47:20,860 So, does this make sense dimensionally? 1204 00:47:20,860 --> 00:47:22,806 Let's quickly check. 1205 00:47:22,806 --> 00:47:23,449 Yeah, it does. 1206 00:47:23,449 --> 00:47:24,490 Let's just quickly check. 1207 00:47:24,490 --> 00:47:30,791 So we have that the exponent there is Et over h bar. 1208 00:47:30,791 --> 00:47:31,290 OK? 1209 00:47:31,290 --> 00:47:33,650 And this should have dimensions of what in order to make sense? 1210 00:47:33,650 --> 00:47:34,630 AUDIENCE: Nothing. 1211 00:47:34,630 --> 00:47:35,220 PROFESSOR: Nothing, exactly. 1212 00:47:35,220 --> 00:47:36,410 It should be dimensionless. 1213 00:47:36,410 --> 00:47:38,554 So what are the dimensions of h bar? 1214 00:47:38,554 --> 00:47:39,490 AUDIENCE: [INAUDIBLE] 1215 00:47:39,490 --> 00:47:40,800 PROFESSOR: Oh, no, the dimensions, guys, not 1216 00:47:40,800 --> 00:47:41,240 the units. 1217 00:47:41,240 --> 00:47:42,239 What are the dimensions? 1218 00:47:42,239 --> 00:47:44,205 AUDIENCE: [INAUDIBLE] 1219 00:47:44,205 --> 00:47:46,330 PROFESSOR: It's an action, which is energy of time. 1220 00:47:46,330 --> 00:47:49,146 So the units of the dimensions of h 1221 00:47:49,146 --> 00:47:51,480 are an energy times a time, also known 1222 00:47:51,480 --> 00:47:54,651 as a momentum times a position. 1223 00:47:54,651 --> 00:47:55,150 OK? 1224 00:48:03,350 --> 00:48:06,280 So, this has dimensions of action or energy times time, 1225 00:48:06,280 --> 00:48:07,780 and then upstairs we have dimensions 1226 00:48:07,780 --> 00:48:09,590 of energy times time. 1227 00:48:09,590 --> 00:48:11,510 So that's consistent. 1228 00:48:11,510 --> 00:48:14,510 So this in fact is dimensionally sensible, which is good. 1229 00:48:14,510 --> 00:48:16,310 Now, this tells you a very important thing. 1230 00:48:16,310 --> 00:48:17,590 In fact, we just answered this equation. 1231 00:48:17,590 --> 00:48:19,100 At time t equals zero, what will we 1232 00:48:19,100 --> 00:48:22,290 get if we measure the energy? 1233 00:48:22,290 --> 00:48:26,480 E. At time t prime-- some subsequent time-- what energy 1234 00:48:26,480 --> 00:48:27,180 will we measure? 1235 00:48:27,180 --> 00:48:27,874 AUDIENCE: E. 1236 00:48:27,874 --> 00:48:28,540 PROFESSOR: Yeah. 1237 00:48:28,540 --> 00:48:31,810 Does the energy change over time? 1238 00:48:31,810 --> 00:48:32,440 No. 1239 00:48:32,440 --> 00:48:34,530 When I say that, what I mean is, does the energy 1240 00:48:34,530 --> 00:48:36,410 that you expect to measure change over time? 1241 00:48:36,410 --> 00:48:36,730 No. 1242 00:48:36,730 --> 00:48:38,645 Does the probability that you measure energy E change? 1243 00:48:38,645 --> 00:48:41,228 No, because it's just a phase, and the norm squared of a phase 1244 00:48:41,228 --> 00:48:42,400 is one. 1245 00:48:42,400 --> 00:48:43,440 Yeah? 1246 00:48:43,440 --> 00:48:45,160 Everyone cool with that? 1247 00:48:45,160 --> 00:48:48,109 Questions at this point. 1248 00:48:48,109 --> 00:48:49,650 This is very simple example, but it's 1249 00:48:49,650 --> 00:48:50,270 going to have a lot of power. 1250 00:48:50,270 --> 00:48:51,061 Oh, yeah, question. 1251 00:48:51,061 --> 00:48:51,850 Thank you. 1252 00:48:51,850 --> 00:48:54,224 AUDIENCE: Are we going to deal with energy operators that 1253 00:48:54,224 --> 00:48:56,100 change over time? 1254 00:48:56,100 --> 00:48:57,350 PROFESSOR: Excellent question. 1255 00:48:57,350 --> 00:48:59,470 We will later, but not in 804. 1256 00:48:59,470 --> 00:49:02,020 In 805, you'll discuss it in more detail. 1257 00:49:02,020 --> 00:49:04,650 Nothing dramatic happens, but you just 1258 00:49:04,650 --> 00:49:06,405 have to add more symbols. 1259 00:49:06,405 --> 00:49:07,700 There's nothing deep about it. 1260 00:49:07,700 --> 00:49:08,370 It's a very good question. 1261 00:49:08,370 --> 00:49:09,953 The question was, are we going to deal 1262 00:49:09,953 --> 00:49:12,877 with energy operators that change in time? 1263 00:49:12,877 --> 00:49:14,960 My answer was no, not in 804, but you will in 805. 1264 00:49:14,960 --> 00:49:17,550 And what you'll find is that it's not a big deal. 1265 00:49:17,550 --> 00:49:19,200 Nothing particularly dramatic happens. 1266 00:49:19,200 --> 00:49:23,530 We will deal with systems where the energy operator changes 1267 00:49:23,530 --> 00:49:24,455 instantaneously. 1268 00:49:24,455 --> 00:49:26,080 So not a continuous function, but we're 1269 00:49:26,080 --> 00:49:27,830 at some of them you turn on the electric field, 1270 00:49:27,830 --> 00:49:28,890 or something like that. 1271 00:49:28,890 --> 00:49:30,719 So we'll deal with that later on. 1272 00:49:30,719 --> 00:49:32,760 But we won't develop a theory of energy operators 1273 00:49:32,760 --> 00:49:35,490 that depend on time. 1274 00:49:35,490 --> 00:49:38,110 But you could do it, and you will do in 805. 1275 00:49:38,110 --> 00:49:39,610 There's nothing mysterious about it. 1276 00:49:39,610 --> 00:49:42,400 Other questions? 1277 00:49:42,400 --> 00:49:43,760 OK. 1278 00:49:43,760 --> 00:49:48,770 So, these states-- a state Psi of x and t, 1279 00:49:48,770 --> 00:49:52,690 which is of the form e to the minus i Omega t, 1280 00:49:52,690 --> 00:49:56,117 where Omega is equal to E over h bar. 1281 00:49:56,117 --> 00:49:57,200 This should look familiar. 1282 00:49:57,200 --> 00:50:00,770 It's the de Broglie relation, [INAUDIBLE] relation, whatever. 1283 00:50:00,770 --> 00:50:04,370 Times some Phi E of x, where this 1284 00:50:04,370 --> 00:50:07,290 is an energy Eigenfunction. 1285 00:50:07,290 --> 00:50:10,390 These states are called stationary states. 1286 00:50:19,459 --> 00:50:20,750 And what's the reason for that? 1287 00:50:20,750 --> 00:50:22,333 Why are they called stationary states? 1288 00:50:22,333 --> 00:50:24,160 I'm going to erase this. 1289 00:50:24,160 --> 00:50:27,990 Well, suppose this is my wave function as a function of time. 1290 00:50:27,990 --> 00:50:31,030 What is the probability that at time t 1291 00:50:31,030 --> 00:50:33,080 I will measure the particle to be at position x, 1292 00:50:33,080 --> 00:50:34,205 or the probability density? 1293 00:50:36,980 --> 00:50:40,580 Well, the probability density we know from our postulates, 1294 00:50:40,580 --> 00:50:42,580 it's just the norm squared of the wave function. 1295 00:50:42,580 --> 00:50:47,090 This is Psi at x t norm squared. 1296 00:50:47,090 --> 00:50:49,500 But this is equal to the norm squared of e 1297 00:50:49,500 --> 00:50:53,024 to the minus Psi Omega t Phi E by the Schrodinger equation. 1298 00:50:53,024 --> 00:50:54,440 But when we take the norm squared, 1299 00:50:54,440 --> 00:50:56,720 this phase cancels out, as we already saw. 1300 00:50:56,720 --> 00:51:01,600 So this is just equal to Phi E of x norm squared, 1301 00:51:01,600 --> 00:51:04,820 the energy Eigenfunction norm squared independent of time. 1302 00:51:11,700 --> 00:51:14,100 So, if we happen to know that our state is in an energy 1303 00:51:14,100 --> 00:51:17,100 Eigenfunction, then the probability density 1304 00:51:17,100 --> 00:51:19,180 for finding the particle at any given position 1305 00:51:19,180 --> 00:51:20,550 does not change in time. 1306 00:51:20,550 --> 00:51:22,520 It remains invariant. 1307 00:51:22,520 --> 00:51:24,620 The wave function rotates by an overall phase, 1308 00:51:24,620 --> 00:51:26,620 but the probability density is the norm squared. 1309 00:51:26,620 --> 00:51:28,070 It's insensitive to that overall phase, 1310 00:51:28,070 --> 00:51:29,650 and so the probability density just 1311 00:51:29,650 --> 00:51:31,358 remains constant in whatever shape it is. 1312 00:51:33,439 --> 00:51:34,980 Hence it's called a stationary state. 1313 00:51:34,980 --> 00:51:37,470 Notice its consequence. 1314 00:51:37,470 --> 00:51:39,440 What can you say about the expectation value 1315 00:51:39,440 --> 00:51:41,030 of the position as a function of time? 1316 00:51:44,090 --> 00:51:47,360 Well, this is equal to the integral dx 1317 00:51:47,360 --> 00:51:50,702 in the state Psi of x and t. 1318 00:51:50,702 --> 00:51:52,740 And I'll call this Psi sub E just to emphasize. 1319 00:51:52,740 --> 00:51:54,990 It's the integral of the x, integral over all possible 1320 00:51:54,990 --> 00:51:57,520 positions of the probability distribution, 1321 00:51:57,520 --> 00:52:00,570 probability of x at time t times x. 1322 00:52:00,570 --> 00:52:03,610 But this is equal to the integral dx 1323 00:52:03,610 --> 00:52:08,760 of Phi E of x squared x. 1324 00:52:08,760 --> 00:52:13,680 But that's equal to expectation value of x at any time, 1325 00:52:13,680 --> 00:52:15,640 or time zero. t equals zero. 1326 00:52:15,640 --> 00:52:20,030 And maybe the best way to write this is as a function of time. 1327 00:52:20,030 --> 00:52:22,940 So, the expectation value of x doesn't change. 1328 00:52:22,940 --> 00:52:26,010 In a stationary state, expected positions, 1329 00:52:26,010 --> 00:52:28,840 energy-- these things don't change. 1330 00:52:28,840 --> 00:52:30,430 Everyone cool with that? 1331 00:52:30,430 --> 00:52:32,240 And it's because of this basic fact 1332 00:52:32,240 --> 00:52:34,260 that the wave function only rotates 1333 00:52:34,260 --> 00:52:36,512 by a phase under time evolution when 1334 00:52:36,512 --> 00:52:37,970 the system is an energy Eigenstate. 1335 00:52:42,010 --> 00:52:44,320 Questions? 1336 00:52:44,320 --> 00:52:46,200 OK. 1337 00:52:46,200 --> 00:52:49,960 So, here's a couple of questions for you guys. 1338 00:52:54,320 --> 00:52:57,740 Are all systems always in energy Eigenstates? 1339 00:53:00,950 --> 00:53:02,320 Am I in an energy Eigenstate? 1340 00:53:05,530 --> 00:53:06,490 AUDIENCE: No. 1341 00:53:06,490 --> 00:53:07,690 PROFESSOR: No, right? 1342 00:53:07,690 --> 00:53:09,950 OK, expected position of my hand is changing in time. 1343 00:53:09,950 --> 00:53:14,240 I am not in-- so obviously, things change in time. 1344 00:53:14,240 --> 00:53:16,267 Energies change in time. 1345 00:53:16,267 --> 00:53:18,600 Positions-- expected typical positions-- change in time. 1346 00:53:18,600 --> 00:53:21,740 We are not in energy Eigenstates. 1347 00:53:21,740 --> 00:53:23,590 That's a highly non-generic state. 1348 00:53:23,590 --> 00:53:25,160 So here's another question. 1349 00:53:25,160 --> 00:53:28,790 Are any states ever truly in energy Eigenstates? 1350 00:53:32,670 --> 00:53:34,790 Can you imagine an object in the world 1351 00:53:34,790 --> 00:53:36,680 that is truly described precisely 1352 00:53:36,680 --> 00:53:41,717 by an energy Eigenstate in the real world? 1353 00:53:41,717 --> 00:53:42,258 AUDIENCE: No. 1354 00:53:44,962 --> 00:53:46,670 PROFESSOR: Ok, there have been a few nos. 1355 00:53:46,670 --> 00:53:47,410 Why? 1356 00:53:47,410 --> 00:53:49,050 Why not? 1357 00:53:49,050 --> 00:53:53,360 Does anything really remain invariant in time? 1358 00:53:53,360 --> 00:53:54,370 No, right? 1359 00:53:54,370 --> 00:53:56,060 Everything is getting buffeted around 1360 00:53:56,060 --> 00:53:57,480 by the rest of the universe. 1361 00:53:57,480 --> 00:54:01,030 So, not only are these not typical states, 1362 00:54:01,030 --> 00:54:03,050 not only are stationary states not typical, 1363 00:54:03,050 --> 00:54:05,970 but they actually never exist in the real world. 1364 00:54:05,970 --> 00:54:07,930 So why am I talking about them at all? 1365 00:54:11,030 --> 00:54:13,600 So here's why. 1366 00:54:13,600 --> 00:54:16,350 And actually I'm going to do this here. 1367 00:54:16,350 --> 00:54:16,950 So here's why. 1368 00:54:16,950 --> 00:54:20,810 The reason is this guy, the superposition principle, 1369 00:54:20,810 --> 00:54:25,940 which tells me that if I have possible states, 1370 00:54:25,940 --> 00:54:27,557 I can build superpositions of them. 1371 00:54:27,557 --> 00:54:29,182 And this statement-- and in particular, 1372 00:54:29,182 --> 00:54:31,180 linearity-- which says that given any two 1373 00:54:31,180 --> 00:54:34,250 solutions of the Schrodinger equation, 1374 00:54:34,250 --> 00:54:36,210 I can take a superposition and build 1375 00:54:36,210 --> 00:54:38,170 a new solution of the Schrodinger equation. 1376 00:54:38,170 --> 00:54:39,220 So, let me build it. 1377 00:54:39,220 --> 00:54:40,720 So, in particular, I want to exploit 1378 00:54:40,720 --> 00:54:46,770 the linearity of the Schrodinger equation to do the following. 1379 00:54:49,470 --> 00:54:52,850 Suppose Psi. 1380 00:54:52,850 --> 00:54:54,400 And I'm going to label these by n. 1381 00:54:54,400 --> 00:55:02,590 Psi n of x and t is equal to e to the minus i Omega nt 1382 00:55:02,590 --> 00:55:09,632 Phi sub En of x, where En is equal to h bar Omega n. 1383 00:55:09,632 --> 00:55:11,840 n labels the various different energy Eigenfunctions. 1384 00:55:11,840 --> 00:55:15,290 So, consider all the energy Eigenfunctions Phi sub En. 1385 00:55:15,290 --> 00:55:18,690 n is a number which labels them. 1386 00:55:18,690 --> 00:55:21,540 And this is the solution to the Schrodinger equation, which 1387 00:55:21,540 --> 00:55:24,310 at time zero is just equal to the energy 1388 00:55:24,310 --> 00:55:25,440 Eigenfunction of interest. 1389 00:55:25,440 --> 00:55:26,345 Cool? 1390 00:55:26,345 --> 00:55:30,060 So, consider these guys. 1391 00:55:30,060 --> 00:55:34,280 So, suppose we have these guys such that they 1392 00:55:34,280 --> 00:55:40,160 solve the Schrodinger equation. 1393 00:55:40,160 --> 00:55:42,127 Solve the Schrodinger equation. 1394 00:55:42,127 --> 00:55:44,210 Suppose these guys solve the Schrodinger equation. 1395 00:55:44,210 --> 00:55:48,790 Then, by linearity, we can take Psi of x and t 1396 00:55:48,790 --> 00:55:54,950 to be an arbitrary superposition sum over n, c sub n, Psi sub 1397 00:55:54,950 --> 00:55:57,480 n of x and t. 1398 00:55:57,480 --> 00:56:01,460 And this will automatically solve the Schrodinger equation 1399 00:56:01,460 --> 00:56:03,660 by linearity of the Schrodinger equation. 1400 00:56:03,660 --> 00:56:04,194 Yeah. 1401 00:56:04,194 --> 00:56:05,569 AUDIENCE: But can't we just get n 1402 00:56:05,569 --> 00:56:07,902 as the sum of the energy Eigenstate 1403 00:56:07,902 --> 00:56:10,840 by just applying that and by just measuring that? 1404 00:56:10,840 --> 00:56:13,080 PROFESSOR: Excellent. 1405 00:56:13,080 --> 00:56:14,080 So, here's the question. 1406 00:56:14,080 --> 00:56:15,660 The question is, look, a minute ago 1407 00:56:15,660 --> 00:56:21,010 you said no system is truly in an energy Eigenstate, right? 1408 00:56:21,010 --> 00:56:23,435 But can't we put a system in an energy Eigenstate 1409 00:56:23,435 --> 00:56:26,630 by just measuring the energy? 1410 00:56:26,630 --> 00:56:27,130 Right? 1411 00:56:27,130 --> 00:56:30,720 Isn't that exactly what the collapse postulate says? 1412 00:56:30,720 --> 00:56:31,806 So here's my question. 1413 00:56:31,806 --> 00:56:33,430 How confident are you that you actually 1414 00:56:33,430 --> 00:56:36,370 measure the energy precisely? 1415 00:56:36,370 --> 00:56:38,930 With what accuracy can we measure the energy? 1416 00:56:38,930 --> 00:56:41,430 So here's the unfortunate truth, the unfortunate practical 1417 00:56:41,430 --> 00:56:41,585 truth. 1418 00:56:41,585 --> 00:56:42,610 And I'm not talking about in principle things. 1419 00:56:42,610 --> 00:56:45,151 I'm talking about it in practice things in the real universe. 1420 00:56:45,151 --> 00:56:47,840 When you measure the energy of something, you've got a box, 1421 00:56:47,840 --> 00:56:50,430 and the box has a dial, and the dial has a needle, 1422 00:56:50,430 --> 00:56:52,570 it has a finite width, and your current meter 1423 00:56:52,570 --> 00:56:54,800 has a finite sensitivity to the current. 1424 00:56:54,800 --> 00:56:57,040 So you never truly measure the energy exactly. 1425 00:56:57,040 --> 00:56:59,440 You measure it to within some tolerance. 1426 00:56:59,440 --> 00:56:59,940 And 1427 00:56:59,940 --> 00:57:02,180 In fact, there's a fundamental bound-- 1428 00:57:02,180 --> 00:57:06,140 there's a fundamental bound on the accuracy with which you can 1429 00:57:06,140 --> 00:57:08,425 make a measurement, which is just the following. 1430 00:57:08,425 --> 00:57:10,550 And this is the analog of the uncertainty equation. 1431 00:57:10,550 --> 00:57:11,760 We'll talk about this more later, 1432 00:57:11,760 --> 00:57:13,470 but let me just jump ahead a little bit. 1433 00:57:13,470 --> 00:57:17,600 Suppose I want to measure frequency. 1434 00:57:17,600 --> 00:57:20,180 So I have some signal, and I look 1435 00:57:20,180 --> 00:57:22,610 at that signal for 10 minutes. 1436 00:57:22,610 --> 00:57:23,200 OK? 1437 00:57:23,200 --> 00:57:26,480 Can I be absolutely confident that this signal is in fact 1438 00:57:26,480 --> 00:57:29,210 a plane wave with the given frequency that I just did? 1439 00:57:29,210 --> 00:57:30,390 No, because it could change outside that. 1440 00:57:30,390 --> 00:57:31,806 But more to the point, there might 1441 00:57:31,806 --> 00:57:33,300 have been small variations inside. 1442 00:57:33,300 --> 00:57:35,510 There could've been a wavelength that 1443 00:57:35,510 --> 00:57:38,140 could change on a time scale longer than the time 1444 00:57:38,140 --> 00:57:38,900 that I measured. 1445 00:57:38,900 --> 00:57:41,180 So, to know that the system doesn't change 1446 00:57:41,180 --> 00:57:43,930 on a arbitrary-- that it's strictly fixed Omega, 1447 00:57:43,930 --> 00:57:46,640 I have to wait a very long time. 1448 00:57:46,640 --> 00:57:49,470 And in particular, how confident you can be of the frequency 1449 00:57:49,470 --> 00:57:55,430 is bounded by the time over which-- so your confidence, 1450 00:57:55,430 --> 00:57:57,290 your uncertainty in the frequency, 1451 00:57:57,290 --> 00:58:01,460 is bounded in the following fashion. 1452 00:58:01,460 --> 00:58:04,180 Delta Omega, Delta t is always greater than or equal to one, 1453 00:58:04,180 --> 00:58:05,904 approximately. 1454 00:58:05,904 --> 00:58:07,320 What this says is that if you want 1455 00:58:07,320 --> 00:58:09,440 to be absolute confident of the frequency, 1456 00:58:09,440 --> 00:58:11,817 you have to wait an arbitrarily long time. 1457 00:58:11,817 --> 00:58:13,650 Now if I multiply this whole thing by h bar, 1458 00:58:13,650 --> 00:58:14,640 I get the following. 1459 00:58:14,640 --> 00:58:17,010 Delta E-- so this is a classic equation 1460 00:58:17,010 --> 00:58:20,480 that signals analysis-- Delta E, Delta t 1461 00:58:20,480 --> 00:58:22,817 is greater than or approximately equal to h bar. 1462 00:58:22,817 --> 00:58:25,025 This is a hallowed time- energy uncertainty relation, 1463 00:58:25,025 --> 00:58:27,580 which we haven't talked about. 1464 00:58:27,580 --> 00:58:30,400 So, in fact, it is possible to make 1465 00:58:30,400 --> 00:58:32,510 an arbitrarily precise measurement of the energy. 1466 00:58:32,510 --> 00:58:34,240 What do I have to do? 1467 00:58:34,240 --> 00:58:36,070 I have to wait forever. 1468 00:58:36,070 --> 00:58:38,450 How patient are you, right? 1469 00:58:38,450 --> 00:58:39,400 So, that's the issue. 1470 00:58:39,400 --> 00:58:41,630 In the real world, we can't make arbitrarily long measurements, 1471 00:58:41,630 --> 00:58:43,800 and we can't isolate systems for an arbitrarily long amount 1472 00:58:43,800 --> 00:58:44,379 of time. 1473 00:58:44,379 --> 00:58:46,670 So, we can't put things in a definite energy Eigenstate 1474 00:58:46,670 --> 00:58:48,017 by measurement. 1475 00:58:48,017 --> 00:58:49,100 That answer your question? 1476 00:58:49,100 --> 00:58:49,683 AUDIENCE: Yes. 1477 00:58:49,683 --> 00:58:50,815 PROFESSOR: Great. 1478 00:58:50,815 --> 00:58:52,190 How many people have seen signals 1479 00:58:52,190 --> 00:58:55,520 in this expression, the bound on the frequency? 1480 00:58:55,520 --> 00:58:56,150 Oh, good. 1481 00:58:56,150 --> 00:58:59,240 So we'll talk about that later in the course. 1482 00:58:59,240 --> 00:59:02,030 OK, so coming back to this. 1483 00:59:02,030 --> 00:59:06,120 So, we have our solutions of the Schrodinger equation 1484 00:59:06,120 --> 00:59:07,786 that are initially energy Eigenstates. 1485 00:59:07,786 --> 00:59:10,035 I claim I can take an arbitrary superposition of them, 1486 00:59:10,035 --> 00:59:14,705 and by linearity derive that this is also 1487 00:59:14,705 --> 00:59:16,330 a solution to the Schrodinger equation. 1488 00:59:18,840 --> 00:59:29,300 And in particular, what that tells me is-- well, 1489 00:59:29,300 --> 00:59:31,000 another way to say this is that if I 1490 00:59:31,000 --> 00:59:37,660 know that Psi of x times zero is equal to sum over n-- so 1491 00:59:37,660 --> 00:59:40,706 if sum Psi of x-- if the wave function 1492 00:59:40,706 --> 00:59:42,080 at some particular moment in time 1493 00:59:42,080 --> 00:59:50,064 can be expanded as sum over n Cn Phi E of x, if this is 1494 00:59:50,064 --> 00:59:51,980 my initial condition, my initial wave function 1495 00:59:51,980 --> 00:59:56,007 is some superposition, then I know what the wave function is 1496 00:59:56,007 --> 00:59:56,840 at subsequent times. 1497 00:59:56,840 --> 01:00:00,530 The wave function by superposition Psi of x and t 1498 01:00:00,530 --> 01:00:05,400 is equal to sum over n Cn e to the minus i 1499 01:00:05,400 --> 01:00:12,830 Omega nt Phi n-- sorry, this should've been Phi sub n-- Phi 1500 01:00:12,830 --> 01:00:13,750 n of x. 1501 01:00:21,109 --> 01:00:22,900 And I know this has to be true because this 1502 01:00:22,900 --> 01:00:25,710 is a solution to the Schrodinger equation by construction, 1503 01:00:25,710 --> 01:00:28,475 and at time t equals zero, it reduces to this. 1504 01:00:28,475 --> 01:00:30,600 So, this is a solution to the Schrodinger equation, 1505 01:00:30,600 --> 01:00:34,950 satisfying this condition at the initial time t equals zero. 1506 01:00:34,950 --> 01:00:36,520 Don't even have to do a calculation. 1507 01:00:36,520 --> 01:00:38,646 So, having solved the Schrodinger equation once 1508 01:00:38,646 --> 01:00:40,020 for energy, Eigenstates allows me 1509 01:00:40,020 --> 01:00:41,860 to solve it for general superposition. 1510 01:00:41,860 --> 01:00:43,950 However, what I just said isn't quite enough. 1511 01:00:43,950 --> 01:00:46,120 I need one more argument. 1512 01:00:46,120 --> 01:00:52,260 And that one more argument is really the stronger version 1513 01:00:52,260 --> 01:00:54,600 of three that we talked about before, which 1514 01:00:54,600 --> 01:01:05,090 is that, given an energy operator E, 1515 01:01:05,090 --> 01:01:07,720 we find the set of wave functions Phi sub 1516 01:01:07,720 --> 01:01:10,610 E, the Eigenfunctions of the energy operator, 1517 01:01:10,610 --> 01:01:14,680 with Eigenvalue E. 1518 01:01:14,680 --> 01:01:17,620 So, given the energy operator, we find its Eigenfunctions. 1519 01:01:17,620 --> 01:01:22,300 Any wave function Psi at x-- we'll say at time zero-- 1520 01:01:22,300 --> 01:01:25,930 any function of x can be expanded as a sum. 1521 01:01:25,930 --> 01:01:34,530 Specific superposition sum over n Cn Phi E sub n of x. 1522 01:01:38,010 --> 01:01:39,860 And if any function can be expanded 1523 01:01:39,860 --> 01:01:42,140 as a superposition of energy Eigenfunctions, 1524 01:01:42,140 --> 01:01:43,985 and we know how to take a superposition, 1525 01:01:43,985 --> 01:01:45,620 an arbitrary superposition of energy 1526 01:01:45,620 --> 01:01:47,974 Eigenfunctions, and find the corresponding solution 1527 01:01:47,974 --> 01:01:49,140 to the Schrodinger equation. 1528 01:01:49,140 --> 01:01:52,540 What this means is, we can take an arbitrary initial condition 1529 01:01:52,540 --> 01:01:55,990 and compute the full solution of the Schrodinger equation. 1530 01:01:55,990 --> 01:02:00,770 All we have to do is figure out what these coefficients Cn are. 1531 01:02:00,770 --> 01:02:02,300 Everyone cool with that? 1532 01:02:02,300 --> 01:02:04,950 So, we have thus, using superposition and energy 1533 01:02:04,950 --> 01:02:08,280 Eigenvalues, totally solved the Schrodinger equation, 1534 01:02:08,280 --> 01:02:10,860 and reduced it to the problem of finding these expansion 1535 01:02:10,860 --> 01:02:12,530 coefficients. 1536 01:02:12,530 --> 01:02:14,850 Meanwhile, these expansion coefficients have a meaning. 1537 01:02:14,850 --> 01:02:16,510 They correspond to the probability 1538 01:02:16,510 --> 01:02:18,240 that we measure the energy to be equal 1539 01:02:18,240 --> 01:02:21,740 to the corresponding energy E sub n. 1540 01:02:21,740 --> 01:02:23,960 And it's just the norm squared of that coefficient. 1541 01:02:23,960 --> 01:02:28,380 So those coefficients mean something. 1542 01:02:28,380 --> 01:02:31,790 And they allow us to solve the problem. 1543 01:02:31,790 --> 01:02:32,850 Cool? 1544 01:02:32,850 --> 01:02:34,940 So this is fairly abstract. 1545 01:02:34,940 --> 01:02:37,540 So let's make it concrete by looking at some examples. 1546 01:02:37,540 --> 01:02:39,660 So, just as a quick aside. 1547 01:02:39,660 --> 01:02:42,400 This should sound an awful lot like the Fourier theorem. 1548 01:02:42,400 --> 01:02:44,750 And let me comment on that. 1549 01:02:44,750 --> 01:02:47,245 This statement originally was about a general observable 1550 01:02:47,245 --> 01:02:48,120 and general operator. 1551 01:02:48,120 --> 01:02:49,700 Here I'm talking about the energy. 1552 01:02:49,700 --> 01:02:52,559 But let's think about a slightly more special example, 1553 01:02:52,559 --> 01:02:53,600 or more familiar example. 1554 01:02:53,600 --> 01:02:54,766 Let's consider the momentum. 1555 01:02:54,766 --> 01:02:57,390 Given the momentum, we can find a set of Eigenstates. 1556 01:02:57,390 --> 01:02:59,510 What are the set of good, properly normalized 1557 01:02:59,510 --> 01:03:00,750 Eigenfunctions of momentum? 1558 01:03:03,382 --> 01:03:05,590 What are the Eigenfunctions of the momentum operator? 1559 01:03:05,590 --> 01:03:06,841 AUDIENCE: E to the ikx. 1560 01:03:06,841 --> 01:03:07,930 PROFESSOR: E to the ikx. 1561 01:03:07,930 --> 01:03:08,430 Exactly. 1562 01:03:08,430 --> 01:03:11,940 In particular, one over 2 pi e to the ikx. 1563 01:03:16,160 --> 01:03:18,160 So I claim that, for every different value of k, 1564 01:03:18,160 --> 01:03:20,650 I get a different value of p, and the Eigenvalue associated 1565 01:03:20,650 --> 01:03:24,450 to this guy is p is equal to h bar k. 1566 01:03:24,450 --> 01:03:25,477 That's the Eigenvalue. 1567 01:03:25,477 --> 01:03:27,560 And we get that by acting with the momentum, which 1568 01:03:27,560 --> 01:03:31,600 is h bar upon i, h bar times derivative with respect to x. 1569 01:03:31,600 --> 01:03:33,225 Derivative with respect to x pulls down 1570 01:03:33,225 --> 01:03:35,230 an ik times the same thing. 1571 01:03:35,230 --> 01:03:38,200 H bar multiplies the k over i, kills the i, 1572 01:03:38,200 --> 01:03:40,470 and leaves us with an overall coefficient of h bar k. 1573 01:03:40,470 --> 01:03:42,330 This is an Eigenfunction of the momentum 1574 01:03:42,330 --> 01:03:45,740 operator with Eigenvalue h bar k. 1575 01:03:45,740 --> 01:03:48,600 And that statement three is the statement 1576 01:03:48,600 --> 01:03:51,000 that an arbitrary function f of x 1577 01:03:51,000 --> 01:03:52,940 can be expanded as a superposition 1578 01:03:52,940 --> 01:03:54,790 of all possible energy Eigenvalues. 1579 01:03:54,790 --> 01:03:58,200 But k is continuously valued and the momentum, 1580 01:03:58,200 --> 01:04:03,850 so that's an integral dk one over 2 pi, 1581 01:04:03,850 --> 01:04:06,150 e to the ikx times some coefficients. 1582 01:04:06,150 --> 01:04:08,042 And those coefficients are labeled by k, 1583 01:04:08,042 --> 01:04:10,500 but since k is continuous, I'm going to call it a function. 1584 01:04:10,500 --> 01:04:12,791 And just to give it a name, instead of calling C sub k, 1585 01:04:12,791 --> 01:04:15,530 I'll call it f tilde of k. 1586 01:04:15,530 --> 01:04:17,180 This is of exactly the same form. 1587 01:04:17,180 --> 01:04:22,110 Here is the expansion-- there's the Eigenfunction, here 1588 01:04:22,110 --> 01:04:24,940 is the Eigenfunction, here is the expansion coefficient, 1589 01:04:24,940 --> 01:04:26,250 here is expansion coefficient. 1590 01:04:26,250 --> 01:04:27,490 And this has a familiar name. 1591 01:04:27,490 --> 01:04:30,280 It's the Fourier theorem. 1592 01:04:30,280 --> 01:04:31,740 So, we see that the Fourier theorem 1593 01:04:31,740 --> 01:04:33,948 is this statement, statement three, the superposition 1594 01:04:33,948 --> 01:04:37,700 principal, for the momentum operator. 1595 01:04:37,700 --> 01:04:40,502 We also see that it's true for the energy operator. 1596 01:04:40,502 --> 01:04:42,210 And what we're claiming here is that it's 1597 01:04:42,210 --> 01:04:43,320 true for any observable. 1598 01:04:43,320 --> 01:04:47,260 Given any observable, you can find its Eigenfunctions, 1599 01:04:47,260 --> 01:04:50,410 and they form a basis on the space of all good functions, 1600 01:04:50,410 --> 01:04:54,310 and an arbitrary function can be expanded in that basis. 1601 01:04:54,310 --> 01:04:58,785 So, as a last example, consider the following. 1602 01:04:58,785 --> 01:04:59,535 We've done energy. 1603 01:04:59,535 --> 01:05:00,368 We've done momentum. 1604 01:05:00,368 --> 01:05:02,240 What's another operator we care about? 1605 01:05:02,240 --> 01:05:04,620 What about position? 1606 01:05:04,620 --> 01:05:07,661 What are the Eigenfunctions of position? 1607 01:05:07,661 --> 01:05:12,640 Well, x hat on Delta of x minus y 1608 01:05:12,640 --> 01:05:18,547 is equal to y Delta x minus y. 1609 01:05:18,547 --> 01:05:20,880 So, these are the states with definite value of position 1610 01:05:20,880 --> 01:05:22,600 x is equal to y. 1611 01:05:22,600 --> 01:05:26,020 And the reason this is true is that when x is equal to y, 1612 01:05:26,020 --> 01:05:28,320 x is the operator that multiplies by the variable x. 1613 01:05:28,320 --> 01:05:31,610 But it's zero, except at x is equal to y, 1614 01:05:31,610 --> 01:05:35,100 so we might as well replace x by y. 1615 01:05:35,100 --> 01:05:36,500 So, there are the Eigenfunctions. 1616 01:05:36,500 --> 01:05:38,000 And this statement is a statement 1617 01:05:38,000 --> 01:05:39,875 that we can represent an arbitrary function f 1618 01:05:39,875 --> 01:05:43,515 of x in a superposition of these states of definite x. 1619 01:05:43,515 --> 01:05:47,170 f of x is equal to the integral over all possible expansion 1620 01:05:47,170 --> 01:05:51,920 coefficients dy delta x minus y times some expansion 1621 01:05:51,920 --> 01:05:52,420 coefficient. 1622 01:05:52,420 --> 01:05:55,069 And what's the expansion coefficient? 1623 01:05:55,069 --> 01:05:56,360 It's got to be a function of y. 1624 01:05:56,360 --> 01:05:57,870 And what function of y must it be? 1625 01:06:01,430 --> 01:06:03,000 Just f of y. 1626 01:06:03,000 --> 01:06:05,240 Because this integral against this delta function 1627 01:06:05,240 --> 01:06:06,537 had better give me f of x. 1628 01:06:06,537 --> 01:06:08,411 And that will only be true if this is f of x. 1629 01:06:11,060 --> 01:06:12,994 So here we see, in some sense, the definition 1630 01:06:12,994 --> 01:06:13,910 of the delta function. 1631 01:06:13,910 --> 01:06:16,702 But really, this is a statement of the superposition principle, 1632 01:06:16,702 --> 01:06:18,660 the statement that any function can be expanded 1633 01:06:18,660 --> 01:06:21,210 as a superposition of Eigenfunctions of the position 1634 01:06:21,210 --> 01:06:22,300 operator. 1635 01:06:22,300 --> 01:06:24,620 Any function can be expanded as a superposition 1636 01:06:24,620 --> 01:06:26,830 of Eigenfunctions of momentum. 1637 01:06:26,830 --> 01:06:29,630 Any function can be expanded as a superposition 1638 01:06:29,630 --> 01:06:31,780 of Eigenfunctions of energy. 1639 01:06:31,780 --> 01:06:34,910 Any function can be expanded as a superposition 1640 01:06:34,910 --> 01:06:40,110 of Eigenfunctions of any operator of your choice. 1641 01:06:40,110 --> 01:06:40,950 OK? 1642 01:06:40,950 --> 01:06:44,650 The special cases-- the Fourier theorem, the general cases, 1643 01:06:44,650 --> 01:06:47,170 the superposition postulate. 1644 01:06:47,170 --> 01:06:48,660 Cool? 1645 01:06:48,660 --> 01:06:49,930 Powerful tool. 1646 01:06:49,930 --> 01:06:51,640 And we've used this powerful tool 1647 01:06:51,640 --> 01:06:54,550 to write down a general expression for a solution 1648 01:06:54,550 --> 01:06:56,050 to the Schrodinger equation. 1649 01:06:56,050 --> 01:06:57,060 That's good. 1650 01:06:57,060 --> 01:06:58,156 That's progress. 1651 01:06:58,156 --> 01:06:59,780 So let's look at some examples of this. 1652 01:07:02,957 --> 01:07:03,790 I can leave this up. 1653 01:07:03,790 --> 01:07:07,900 So, our first example is going to be for the free particle. 1654 01:07:07,900 --> 01:07:12,720 So, a particle whose energy operator 1655 01:07:12,720 --> 01:07:14,870 has no potential whatsoever. 1656 01:07:14,870 --> 01:07:16,960 So the energy operator is going to be 1657 01:07:16,960 --> 01:07:19,340 just equal to p squared upon 2m. 1658 01:07:22,060 --> 01:07:22,880 Kinetic energy. 1659 01:07:22,880 --> 01:07:25,100 Yeah. 1660 01:07:25,100 --> 01:07:28,030 AUDIENCE: When you say any wave function can 1661 01:07:28,030 --> 01:07:29,925 be expanded in terms of-- 1662 01:07:29,925 --> 01:07:31,300 PROFESSOR: Energy Eigenfunctions, 1663 01:07:31,300 --> 01:07:33,430 position Eigenfunctions, momentum Eigenfunctions-- 1664 01:07:33,430 --> 01:07:35,350 AUDIENCE: Eigenbasis, does the Eigenbasis 1665 01:07:35,350 --> 01:07:38,565 have to come from an operator corresponding to an observable? 1666 01:07:38,565 --> 01:07:39,190 PROFESSOR: Yes. 1667 01:07:39,190 --> 01:07:39,690 Absolutely. 1668 01:07:39,690 --> 01:07:41,106 I'm starting with that assumption. 1669 01:07:41,106 --> 01:07:42,420 AUDIENCE: OK. 1670 01:07:42,420 --> 01:07:44,590 PROFESSOR: So, again, this is a first pass 1671 01:07:44,590 --> 01:07:46,720 of the axioms of quantum mechanics. 1672 01:07:46,720 --> 01:07:50,492 We'll make this more precise, and we'll make it more general, 1673 01:07:50,492 --> 01:07:52,950 later on in the course, as we go through a second iteration 1674 01:07:52,950 --> 01:07:53,660 of this. 1675 01:07:53,660 --> 01:07:55,280 And there we'll talk about exactly what we need, 1676 01:07:55,280 --> 01:07:57,160 and what operators are appropriate operators. 1677 01:07:57,160 --> 01:07:59,201 But for the moment, the sufficient and physically 1678 01:07:59,201 --> 01:08:00,850 correct answer is, operators correspond 1679 01:08:00,850 --> 01:08:02,070 to each observable values. 1680 01:08:02,070 --> 01:08:02,569 Yeah. 1681 01:08:02,569 --> 01:08:05,150 AUDIENCE: So are the set of all reasonable wave functions 1682 01:08:05,150 --> 01:08:07,320 in the vector space that is the same 1683 01:08:07,320 --> 01:08:10,674 as the one with the Eigenfunctions? 1684 01:08:10,674 --> 01:08:12,340 PROFESSOR: That's an excellent question. 1685 01:08:12,340 --> 01:08:14,322 In general, no. 1686 01:08:14,322 --> 01:08:15,280 So here's the question. 1687 01:08:15,280 --> 01:08:17,470 The question is, look, if this is true, 1688 01:08:17,470 --> 01:08:20,100 shouldn't it be that the Eigenfunctions, since they're 1689 01:08:20,100 --> 01:08:21,660 our basis for the good functions, 1690 01:08:21,660 --> 01:08:24,430 are inside the space of reasonable functions, 1691 01:08:24,430 --> 01:08:26,430 they should also be reasonable functions, right? 1692 01:08:26,430 --> 01:08:28,710 Because if you're going to expand-- for example, 1693 01:08:28,710 --> 01:08:29,819 consider two dimensional vector space. 1694 01:08:29,819 --> 01:08:31,235 And you want to say any vector can 1695 01:08:31,235 --> 01:08:34,990 be expanded in a basis of pairs of vectors in two dimensions, 1696 01:08:34,990 --> 01:08:36,176 like x and y. 1697 01:08:36,176 --> 01:08:38,300 You really want to make sure that those vectors are 1698 01:08:38,300 --> 01:08:39,341 inside your vector space. 1699 01:08:39,341 --> 01:08:41,250 But if you say this vector in this space 1700 01:08:41,250 --> 01:08:43,580 can be expanded in terms of two vectors, this vector 1701 01:08:43,580 --> 01:08:46,400 and that vector, you're in trouble, right? 1702 01:08:46,400 --> 01:08:47,774 That's not going to work so well. 1703 01:08:47,774 --> 01:08:50,024 So you want to make sure that your vectors, your basis 1704 01:08:50,024 --> 01:08:51,819 vectors, are in the space. 1705 01:08:51,819 --> 01:08:55,069 For position, the basis vector's a delta function. 1706 01:08:55,069 --> 01:08:58,130 Is that a smooth, continuous normalizable function? 1707 01:08:58,130 --> 01:08:58,930 No. 1708 01:08:58,930 --> 01:09:01,029 For momentum, the basis functions 1709 01:09:01,029 --> 01:09:03,200 are plane waves that extend off to infinity 1710 01:09:03,200 --> 01:09:05,399 and have support everywhere. 1711 01:09:05,399 --> 01:09:08,020 Is that a normalizable reasonable function? 1712 01:09:08,020 --> 01:09:08,520 No. 1713 01:09:08,520 --> 01:09:10,634 So, both of these sets are really bad. 1714 01:09:10,634 --> 01:09:13,300 So, at that point you might say, look, this is clearly nonsense. 1715 01:09:13,300 --> 01:09:14,550 But here's an important thing. 1716 01:09:17,029 --> 01:09:18,695 So this is a totally mathematical aside, 1717 01:09:18,695 --> 01:09:19,840 and for those of you who don't care about the math, 1718 01:09:19,840 --> 01:09:20,689 don't worry about it. 1719 01:09:20,689 --> 01:09:22,050 Well, these guys don't technically 1720 01:09:22,050 --> 01:09:23,841 live in the space of non-stupid functions-- 1721 01:09:23,841 --> 01:09:27,800 reasonable, smooth, normalizable functions. 1722 01:09:27,800 --> 01:09:31,050 What you can show is that they exist in the closure, 1723 01:09:31,050 --> 01:09:33,700 in the completion of that space. 1724 01:09:33,700 --> 01:09:34,550 OK? 1725 01:09:34,550 --> 01:09:38,439 So, you can find a sequence of wave functions 1726 01:09:38,439 --> 01:09:40,970 that are good wave functions, an infinite sequence, 1727 01:09:40,970 --> 01:09:43,238 that eventually that infinite sequence converges 1728 01:09:43,238 --> 01:09:45,029 to these guys, even though these are silly. 1729 01:09:45,029 --> 01:09:47,090 So, for example, for the position Eigenstates, 1730 01:09:47,090 --> 01:09:50,398 the delta function is not a continuous smooth function. 1731 01:09:50,398 --> 01:09:51,439 It's not even a function. 1732 01:09:51,439 --> 01:09:53,540 Really, it's some god- awful thing called a distribution. 1733 01:09:53,540 --> 01:09:54,360 It's some horrible thing. 1734 01:09:54,360 --> 01:09:56,485 It's the thing that tells you, give it an integral, 1735 01:09:56,485 --> 01:09:57,800 it'll give you a number. 1736 01:09:57,800 --> 01:09:59,722 Or a function. 1737 01:09:59,722 --> 01:10:01,180 But how do we build this as a limit 1738 01:10:01,180 --> 01:10:02,513 of totally reasonable functions? 1739 01:10:02,513 --> 01:10:03,960 We've already done that. 1740 01:10:03,960 --> 01:10:07,670 Take this function with area one, and if you want, 1741 01:10:07,670 --> 01:10:10,230 you can round this out by making it hyperbolic tangents. 1742 01:10:10,230 --> 01:10:11,190 OK? 1743 01:10:11,190 --> 01:10:13,360 We did it on one of the problem sets. 1744 01:10:13,360 --> 01:10:15,360 And then just make it more narrow and more tall. 1745 01:10:15,360 --> 01:10:17,901 And keep making it more narrow and more tall, and more narrow 1746 01:10:17,901 --> 01:10:19,930 and more tall, keeping its area to be one. 1747 01:10:19,930 --> 01:10:24,190 And I claim that eventually that series, that sequence 1748 01:10:24,190 --> 01:10:26,340 of functions, converges to the delta function. 1749 01:10:26,340 --> 01:10:29,430 So, while this function is not technically in our space, 1750 01:10:29,430 --> 01:10:31,600 it's in the completion of our space, 1751 01:10:31,600 --> 01:10:34,290 in the sense that we take a series and they converge to it. 1752 01:10:34,290 --> 01:10:36,715 And that's what you need for this theorem to work out. 1753 01:10:36,715 --> 01:10:38,590 That's what you need for the Fourier theorem. 1754 01:10:38,590 --> 01:10:40,470 And in some sense, that observation 1755 01:10:40,470 --> 01:10:42,070 was really the genius of Fourier, 1756 01:10:42,070 --> 01:10:44,571 understanding that that could be done. 1757 01:10:44,571 --> 01:10:46,070 That was totally mathematical aside. 1758 01:10:46,070 --> 01:10:47,180 But that answer your question? 1759 01:10:47,180 --> 01:10:47,480 AUDIENCE: Yes. 1760 01:10:47,480 --> 01:10:48,330 PROFESSOR: OK. 1761 01:10:48,330 --> 01:10:49,891 Every once in a while I can't resist 1762 01:10:49,891 --> 01:10:51,390 talking about these sort of details, 1763 01:10:51,390 --> 01:10:52,515 because I really like them. 1764 01:10:52,515 --> 01:10:56,350 But it's good to know that stupid things like this 1765 01:10:56,350 --> 01:10:58,930 can't matter for us, and they don't. 1766 01:10:58,930 --> 01:11:01,820 But it's a very good question. 1767 01:11:01,820 --> 01:11:03,920 If you're confused about some mathematical detail, 1768 01:11:03,920 --> 01:11:05,970 no matter how elementary, ask. 1769 01:11:05,970 --> 01:11:09,410 If you're confused, someone else the room is also confused. 1770 01:11:09,410 --> 01:11:11,350 So please don't hesitate. 1771 01:11:11,350 --> 01:11:14,990 OK, so our first example's going to be the free particle. 1772 01:11:14,990 --> 01:11:17,820 And this operator can be written in a nice way. 1773 01:11:17,820 --> 01:11:21,980 We can write it as minus-- so p is h bar upon iddx, 1774 01:11:21,980 --> 01:11:23,550 so this line is minus h bar squared 1775 01:11:23,550 --> 01:11:27,434 upon 2m to the derivative with respect to x. 1776 01:11:27,434 --> 01:11:28,600 There's the energy operator. 1777 01:11:31,550 --> 01:11:35,750 So, we want to solve for the wave functions. 1778 01:11:35,750 --> 01:11:37,670 So let's solve it using an expansion 1779 01:11:37,670 --> 01:11:38,660 in terms of energy Eigenfunctions. 1780 01:11:38,660 --> 01:11:40,243 So what are the energy Eigenfunctions? 1781 01:11:40,243 --> 01:11:45,280 We want to find the functions E on Phi sub E 1782 01:11:45,280 --> 01:11:47,007 such that this is equal to-- whoops. 1783 01:11:47,007 --> 01:11:47,840 That's not a vector. 1784 01:11:47,840 --> 01:11:54,510 That's a hat-- such as this is equal to a number E Phi sub E. 1785 01:11:54,510 --> 01:11:56,290 But given this energy operator, this 1786 01:11:56,290 --> 01:12:00,950 says that minus h bar squared over 2m-- whoops, that's a 2. 1787 01:12:00,950 --> 01:12:10,060 2m-- Phi prime prime of x is equal to E Phi of x. 1788 01:12:10,060 --> 01:12:18,240 Or equivalently, Phi prime prime of x plus 2me over h bar 1789 01:12:18,240 --> 01:12:21,720 squared Phi of x is equal to zero. 1790 01:12:27,140 --> 01:12:30,280 So I'm just going to call 2me-- because it's 1791 01:12:30,280 --> 01:12:32,030 annoying to write it over and over again-- 1792 01:12:32,030 --> 01:12:32,460 over h bar squared. 1793 01:12:32,460 --> 01:12:34,170 Well, first off, what are its units? 1794 01:12:34,170 --> 01:12:35,810 What are the units of this coefficient? 1795 01:12:35,810 --> 01:12:36,770 Well, you could do it two ways. 1796 01:12:36,770 --> 01:12:38,770 You could either do dimensional analysis of each thing here, 1797 01:12:38,770 --> 01:12:41,228 or you could just know that we started with a dimensionally 1798 01:12:41,228 --> 01:12:42,910 sensible equation, and this has units 1799 01:12:42,910 --> 01:12:46,160 of this divided by length twice. 1800 01:12:46,160 --> 01:12:48,182 So this must have to whatever length squared. 1801 01:12:48,182 --> 01:12:49,640 So I'm going to call this something 1802 01:12:49,640 --> 01:12:51,130 like k squared, something that has 1803 01:12:51,130 --> 01:12:52,504 units of one over length squared. 1804 01:12:54,940 --> 01:12:56,410 And the general solution of this is 1805 01:12:56,410 --> 01:13:02,750 that phi E E of x-- well, this is a second order differential 1806 01:13:02,750 --> 01:13:05,240 equation that will have two solutions with two expansion 1807 01:13:05,240 --> 01:13:15,960 coefficients-- A e to the ikx plus B e to the minus iks. 1808 01:13:15,960 --> 01:13:19,070 A state with definite momentum and definite negative momentum 1809 01:13:19,070 --> 01:13:26,736 where such that E is equal to h bar squared k squared upon 2m. 1810 01:13:26,736 --> 01:13:30,720 And we get that just from this. 1811 01:13:30,720 --> 01:13:36,386 So, this is the solution of the energy Eigenfunction equation. 1812 01:13:36,386 --> 01:13:37,510 Just a note of terminology. 1813 01:13:37,510 --> 01:13:38,950 People sometimes call the equation 1814 01:13:38,950 --> 01:13:40,984 determining an energy Eigenfunction-- the energy 1815 01:13:40,984 --> 01:13:42,400 Eigenfunction equation-- sometimes 1816 01:13:42,400 --> 01:13:45,110 that's refer to as the Schrodinger equation. 1817 01:13:45,110 --> 01:13:48,060 That's a sort of cruel thing to do to language, 1818 01:13:48,060 --> 01:13:50,380 because the Schrodinger's equation is about time 1819 01:13:50,380 --> 01:13:54,500 evolution, and this equation is about energy Eigenfunctions. 1820 01:13:54,500 --> 01:13:57,670 Now, it's true that energy Eigenfunctions evolve 1821 01:13:57,670 --> 01:14:00,151 in a particularly simple way under time evolution, 1822 01:14:00,151 --> 01:14:01,400 but it's a different equation. 1823 01:14:01,400 --> 01:14:03,890 This is telling you about energy Eigenstates, OK? 1824 01:14:07,370 --> 01:14:09,170 And then more discussion of this is 1825 01:14:09,170 --> 01:14:12,100 done in the notes, which I will leave aside for the moment. 1826 01:14:12,100 --> 01:14:15,250 But I want to do one more example before we take off. 1827 01:14:15,250 --> 01:14:16,384 Wow. 1828 01:14:16,384 --> 01:14:18,300 We got through a lot less than expected today. 1829 01:14:21,410 --> 01:14:23,190 And the one last example is the following. 1830 01:14:23,190 --> 01:14:25,085 It's a particle in a box And this 1831 01:14:25,085 --> 01:14:26,500 is going to be important for your problem sets, 1832 01:14:26,500 --> 01:14:28,833 so I'm going to go ahead and get this one out of the way 1833 01:14:28,833 --> 01:14:30,620 as quickly as possible. 1834 01:14:30,620 --> 01:14:31,680 So, example two. 1835 01:14:35,130 --> 01:14:36,290 Particle in a box. 1836 01:14:40,979 --> 01:14:42,520 So, what I mean by particle in a box. 1837 01:14:42,520 --> 01:14:45,551 I'm going to take a system that has a deep well. 1838 01:14:45,551 --> 01:14:47,550 So what I'm drawing here is the potential energy 1839 01:14:47,550 --> 01:14:55,130 U of x, where this is some energy E naught, 1840 01:14:55,130 --> 01:14:57,270 and this is the energy zero, and this 1841 01:14:57,270 --> 01:14:59,330 is the position x equals zero, and this 1842 01:14:59,330 --> 01:15:00,724 is position x equals l. 1843 01:15:00,724 --> 01:15:02,640 And I'm going to idealize this by saying look, 1844 01:15:02,640 --> 01:15:05,300 I'm going to be interested in low energy physics, 1845 01:15:05,300 --> 01:15:08,030 so I'm going to just treat this as infinitely deep. 1846 01:15:08,030 --> 01:15:09,770 And meanwhile, my life is easier if I 1847 01:15:09,770 --> 01:15:11,180 don't think about curvy bottoms but I just 1848 01:15:11,180 --> 01:15:12,730 think about things as being constant. 1849 01:15:12,730 --> 01:15:16,460 So, my idealization is going to be 1850 01:15:16,460 --> 01:15:22,030 that the well is infinitely high and square. 1851 01:15:22,030 --> 01:15:27,630 So out here the potential is infinite, 1852 01:15:27,630 --> 01:15:29,205 and in here the potential is zero. 1853 01:15:29,205 --> 01:15:32,630 U equals inside, between zero and l for x. 1854 01:15:36,270 --> 01:15:38,310 So that's my system particle in a box. 1855 01:15:38,310 --> 01:15:40,987 So, let's find the energy Eigenfunctions. 1856 01:15:40,987 --> 01:15:42,945 And again, it's the same differential equations 1857 01:15:42,945 --> 01:15:43,444 as before. 1858 01:15:47,264 --> 01:15:49,430 So, first off, before we even solve anything, what's 1859 01:15:49,430 --> 01:15:51,820 the probability that I find x less than zero, 1860 01:15:51,820 --> 01:15:58,822 or find the particle at x greater than l? 1861 01:15:58,822 --> 01:15:59,735 AUDIENCE: Zero. 1862 01:15:59,735 --> 01:16:01,360 PROFESSOR: Right, because the potential 1863 01:16:01,360 --> 01:16:02,770 is infinitely large out there. 1864 01:16:02,770 --> 01:16:03,650 It's just not going to happen. 1865 01:16:03,650 --> 01:16:04,810 If you found it there, that would correspond 1866 01:16:04,810 --> 01:16:06,590 to a particle of infinite energy, 1867 01:16:06,590 --> 01:16:08,180 and that's not going to happen. 1868 01:16:08,180 --> 01:16:09,770 So, our this tells us effectively 1869 01:16:09,770 --> 01:16:12,530 the boundary condition Psi of x is 1870 01:16:12,530 --> 01:16:14,200 equal to zero outside the box. 1871 01:16:16,984 --> 01:16:18,400 So all we have to do is figure out 1872 01:16:18,400 --> 01:16:22,077 what the wave function is inside the box between zero and l. 1873 01:16:22,077 --> 01:16:23,910 And meanwhile, what must be true of the wave 1874 01:16:23,910 --> 01:16:25,934 function at zero and at l? 1875 01:16:25,934 --> 01:16:27,850 It's got to actually vanish at the boundaries. 1876 01:16:27,850 --> 01:16:30,830 So this gives us boundary conditions outside the box 1877 01:16:30,830 --> 01:16:35,914 and at the boundaries x equals zero, x equals l. 1878 01:16:40,570 --> 01:16:43,310 But, what's our differential equation inside the box? 1879 01:16:43,310 --> 01:16:46,470 Inside the box, well, the potential is zero. 1880 01:16:46,470 --> 01:16:48,410 So the equation is the same as the equation 1881 01:16:48,410 --> 01:16:49,950 for a free particle. 1882 01:16:49,950 --> 01:16:51,540 It's just this guy. 1883 01:16:51,540 --> 01:16:53,900 And we know what the solutions are. 1884 01:16:53,900 --> 01:16:56,150 So the solutions can be written in the following form. 1885 01:16:56,150 --> 01:17:01,780 Therefore inside the wave function-- whoops. 1886 01:17:01,780 --> 01:17:03,490 Let me write this as Phi sub E-- Phi sub 1887 01:17:03,490 --> 01:17:05,390 E is a superposition of two. 1888 01:17:05,390 --> 01:17:07,220 And instead of writing it as exponentials, 1889 01:17:07,220 --> 01:17:10,800 I'm going to write it as sines and cosines, 1890 01:17:10,800 --> 01:17:13,190 because you can express them in terms of each other. 1891 01:17:13,190 --> 01:17:23,030 Alpha cosine of kx plus Beta sine of kx, 1892 01:17:23,030 --> 01:17:27,950 where again Alpha and Beta are general complex numbers. 1893 01:17:27,950 --> 01:17:33,210 But, we must satisfy the boundary conditions imposed 1894 01:17:33,210 --> 01:17:35,830 by our potential at x equals zero and x equals l. 1895 01:17:35,830 --> 01:17:38,989 So from x equals zero, we find that Phi 1896 01:17:38,989 --> 01:17:40,280 must vanish when x equals zero. 1897 01:17:40,280 --> 01:17:42,238 When x equals zero, this is automatically zero. 1898 01:17:42,238 --> 01:17:43,520 Sine of zero is zero. 1899 01:17:43,520 --> 01:17:45,820 Cosine of zero is one. 1900 01:17:45,820 --> 01:17:49,630 So that tells us that Alpha is equal to zero. 1901 01:17:49,630 --> 01:17:53,490 Meanwhile, the condition that at x equals l-- the wave function 1902 01:17:53,490 --> 01:17:57,355 must also vanish-- tells us that-- so this term is gone, 1903 01:17:57,355 --> 01:17:59,700 set off with zero-- this term, when x is equal to l, 1904 01:17:59,700 --> 01:18:01,580 had better also be zero. 1905 01:18:01,580 --> 01:18:03,630 We can solve that by setting Beta equal to zero, 1906 01:18:03,630 --> 01:18:05,690 but then our wave function is just zero. 1907 01:18:05,690 --> 01:18:07,790 And that's a really stupid wave function. 1908 01:18:07,790 --> 01:18:08,360 So we don't want to do that. 1909 01:18:08,360 --> 01:18:09,776 We don't want to set Beta to zero. 1910 01:18:09,776 --> 01:18:12,840 Instead, what must we do? 1911 01:18:12,840 --> 01:18:16,566 Well, we've got a sine, and depending on what k is, 1912 01:18:16,566 --> 01:18:18,440 it starts at zero and it ends somewhere else. 1913 01:18:18,440 --> 01:18:19,810 But we need it hit zero. 1914 01:18:19,810 --> 01:18:21,960 So only for a very special value of k will 1915 01:18:21,960 --> 01:18:25,400 it actually hit zero at the end of l. 1916 01:18:25,400 --> 01:18:26,600 We need kl equals zero. 1917 01:18:29,170 --> 01:18:31,560 Or really, kl is a multiple of pi. 1918 01:18:31,560 --> 01:18:33,980 Kl is equal-- and we want it to not be zero, 1919 01:18:33,980 --> 01:18:37,880 so I'll call it n plus 1, an integer, times pi. 1920 01:18:37,880 --> 01:18:42,940 Or equivalently, k is equal to sub n is equal to n plus 1, 1921 01:18:42,940 --> 01:18:46,540 where n goes from zero to any large positive integer, pi 1922 01:18:46,540 --> 01:18:49,500 over l. 1923 01:18:49,500 --> 01:18:51,210 So the energy Eigenfunction's here. 1924 01:18:54,711 --> 01:18:56,710 The energy Eigenfunction is some normalization-- 1925 01:18:56,710 --> 01:19:05,940 whoops-- a sub n sine of k and x. 1926 01:19:14,470 --> 01:19:17,720 And where kn is equal to this-- and as a consequence, 1927 01:19:17,720 --> 01:19:21,820 E is equal to h bar squared kn can squared E sub 1928 01:19:21,820 --> 01:19:27,160 n is h bar squared kn squared over 2m, which is equal to h 1929 01:19:27,160 --> 01:19:32,860 bar squared-- just plugging in-- pi squared n plus 1 squared 1930 01:19:32,860 --> 01:19:34,260 over 2ml squared. 1931 01:19:37,180 --> 01:19:39,860 And what we found is something really interesting. 1932 01:19:39,860 --> 01:19:42,810 What we found is, first off, that the wave functions 1933 01:19:42,810 --> 01:19:45,170 look like-- well, the ground state, 1934 01:19:45,170 --> 01:19:48,350 the lowest possible energy there is n equals zero. 1935 01:19:48,350 --> 01:19:51,510 For n equals zero, this is just a single half a sine wave. 1936 01:19:51,510 --> 01:19:52,260 It does this. 1937 01:19:52,260 --> 01:19:54,200 This is the n equals zero state. 1938 01:19:54,200 --> 01:19:56,330 And it has some energy, which is E zero. 1939 01:19:56,330 --> 01:19:58,820 And in particular, E zero is not equal to zero. 1940 01:19:58,820 --> 01:20:03,905 E zero is equal to h bar squared pi squared over 2ml squared. 1941 01:20:08,784 --> 01:20:10,450 It is impossible for a particle in a box 1942 01:20:10,450 --> 01:20:14,800 to have an energy lower than some minimal value E 1943 01:20:14,800 --> 01:20:17,260 naught, which is not zero. 1944 01:20:17,260 --> 01:20:21,060 You cannot have less energy than this. 1945 01:20:21,060 --> 01:20:22,620 Everyone agree with that? 1946 01:20:22,620 --> 01:20:24,990 There is no such Eigenstate with energy less than this. 1947 01:20:24,990 --> 01:20:26,510 Meanwhile, it's worse. 1948 01:20:26,510 --> 01:20:28,540 The next energy is when n is equal to 1, 1949 01:20:28,540 --> 01:20:32,730 because if we decrease the wavelength or increase 1950 01:20:32,730 --> 01:20:35,010 k a little bit, we get something that looks like this, 1951 01:20:35,010 --> 01:20:36,810 and that doesn't satisfy our boundary condition. 1952 01:20:36,810 --> 01:20:38,630 In order to satisfy our boundary condition, 1953 01:20:38,630 --> 01:20:40,055 we're going to have to eventually have it cross over 1954 01:20:40,055 --> 01:20:41,640 and get to zero again. 1955 01:20:41,640 --> 01:20:47,390 And if I could only draw-- I'll draw it up here-- 1956 01:20:47,390 --> 01:20:48,480 it looks like this. 1957 01:20:48,480 --> 01:20:50,080 And this has an energy E one, which 1958 01:20:50,080 --> 01:20:51,750 you can get by plugging one in here. 1959 01:20:51,750 --> 01:20:55,160 And that differs by one, two, four, a factor of four 1960 01:20:55,160 --> 01:20:56,230 from this guy. 1961 01:20:56,230 --> 01:20:59,110 E one is four E zero. 1962 01:20:59,110 --> 01:21:00,330 And so on and so forth. 1963 01:21:00,330 --> 01:21:03,180 The energies are gaped. 1964 01:21:03,180 --> 01:21:04,770 They're spread away from each other. 1965 01:21:04,770 --> 01:21:06,872 The energies are discrete. 1966 01:21:06,872 --> 01:21:09,080 And they get further and further away from each other 1967 01:21:09,080 --> 01:21:12,640 as we go to higher and higher energies. 1968 01:21:12,640 --> 01:21:15,180 So this is already a peculiar fact, 1969 01:21:15,180 --> 01:21:18,087 and we'll explore some of its consequences later on. 1970 01:21:18,087 --> 01:21:19,920 But here's that I want to emphasize for you. 1971 01:21:19,920 --> 01:21:23,630 Already in the first most trivial example 1972 01:21:23,630 --> 01:21:25,460 of solving a Schrodinger equation, 1973 01:21:25,460 --> 01:21:27,850 or actually even before that, just finding the energy 1974 01:21:27,850 --> 01:21:31,090 Eigenvalues and the energy of Eigenfunctions of the simplest 1975 01:21:31,090 --> 01:21:33,575 system you possibly could, either a free particular, 1976 01:21:33,575 --> 01:21:35,950 or a particle in a box, a particle 1977 01:21:35,950 --> 01:21:38,800 trapped inside a potential well, what we discovered 1978 01:21:38,800 --> 01:21:41,220 is that the energy Eigenvalues, the allowed values 1979 01:21:41,220 --> 01:21:46,520 of the energy, are discrete, and that they're greater than zero. 1980 01:21:46,520 --> 01:21:48,522 You can never have zero energy. 1981 01:21:48,522 --> 01:21:49,980 And if that doesn't sound familiar, 1982 01:21:49,980 --> 01:21:51,340 let me remind you of something. 1983 01:21:51,340 --> 01:21:55,380 The spectrum of light coming off of a gas of hot hydrogen 1984 01:21:55,380 --> 01:21:57,420 is discrete. 1985 01:21:57,420 --> 01:21:59,870 And no one's ever found a zero energy 1986 01:21:59,870 --> 01:22:02,440 beam of light coming out of it. 1987 01:22:02,440 --> 01:22:05,550 And we're going to make contact with that experimental data. 1988 01:22:05,550 --> 01:22:07,550 That's going to be part of the job for the rest. 1989 01:22:07,550 --> 01:22:08,300 See you next time. 1990 01:22:11,270 --> 01:22:15,220 [APPLAUSE]