1 00:00:00,050 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,820 under a Creative Commons license. 3 00:00:03,820 --> 00:00:06,540 Your support will help MIT OpenCourseWare continue 4 00:00:06,540 --> 00:00:10,120 to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,700 To make a donation, or to view additional materials 6 00:00:12,700 --> 00:00:16,585 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,585 --> 00:00:17,210 at ocw.mit.edu. 8 00:00:21,152 --> 00:00:22,652 PROFESSOR: I'm going to get started. 9 00:00:22,652 --> 00:00:27,520 And after a couple of minutes there will be an announcement. 10 00:00:27,520 --> 00:00:33,700 So today's lecture will begin by looking 11 00:00:33,700 --> 00:00:36,370 at the singlet state for a few minutes, 12 00:00:36,370 --> 00:00:37,990 a couple of its properties. 13 00:00:37,990 --> 00:00:43,370 And then we go into this Einstein-Podolsky-Rosen 14 00:00:43,370 --> 00:00:44,720 argument. 15 00:00:44,720 --> 00:00:47,810 We'll talk about what they said, what 16 00:00:47,810 --> 00:00:52,700 they wanted to criticize quantum mechanics for. 17 00:00:52,700 --> 00:00:56,940 And then we'll go through the answer 18 00:00:56,940 --> 00:01:00,190 given by Bell that actually demonstrated 19 00:01:00,190 --> 00:01:04,510 that Einstein, Podolsky, and Rosen were wrong. 20 00:01:04,510 --> 00:01:09,790 And after we'll do those Bell inequalities, 21 00:01:09,790 --> 00:01:14,990 we'll close this chapter on quantum states of spins 22 00:01:14,990 --> 00:01:19,600 and begin our treatment of angular momentum. 23 00:01:19,600 --> 00:01:22,780 So I want to remind you of a couple of things. 24 00:01:22,780 --> 00:01:26,740 We've been discussing this so-called singlet state. 25 00:01:32,030 --> 00:01:35,720 Plus, minus. 26 00:01:35,720 --> 00:01:42,060 For the second particle, minus minus, plus. 27 00:01:42,060 --> 00:01:45,340 First and second particle. 28 00:01:45,340 --> 00:01:45,955 This state. 29 00:01:51,710 --> 00:01:54,870 A few things we've discovered about this state--you've been 30 00:01:54,870 --> 00:01:55,650 working with it. 31 00:01:55,650 --> 00:02:01,160 We calculated its z-component of total angular momentum, 32 00:02:01,160 --> 00:02:03,200 the angular momentum of the first particle 33 00:02:03,200 --> 00:02:04,540 and the second particle. 34 00:02:04,540 --> 00:02:05,910 And it was 0. 35 00:02:05,910 --> 00:02:07,760 The x-component was 0. 36 00:02:07,760 --> 00:02:09,990 This y-component is 0. 37 00:02:09,990 --> 00:02:14,350 In fact it doesn't have any total angular momentum. 38 00:02:14,350 --> 00:02:18,250 So this state is rotationally invariant, we say, 39 00:02:18,250 --> 00:02:22,050 because it doesn't have angular momentum. 40 00:02:22,050 --> 00:02:24,750 You did very fine in the homework 41 00:02:24,750 --> 00:02:28,140 that the state is in fact rotationally invariant. 42 00:02:28,140 --> 00:02:31,660 Now this state is a very interesting state. 43 00:02:31,660 --> 00:02:34,320 It's one of those entangled states 44 00:02:34,320 --> 00:02:37,140 that we discussed last time when we 45 00:02:37,140 --> 00:02:42,000 were talking about teleportation and Bell states. 46 00:02:42,000 --> 00:02:47,870 And apart from that, it's a state 47 00:02:47,870 --> 00:02:51,380 that is not hard to realize physically. 48 00:02:51,380 --> 00:02:55,490 In fact, it typically takes place, for example, 49 00:02:55,490 --> 00:02:59,660 in reactions in decays of certain particles. 50 00:02:59,660 --> 00:03:04,110 For example, pi 0 can decay into two photons. 51 00:03:04,110 --> 00:03:07,690 Then the two photos can be in the state of total angular 52 00:03:07,690 --> 00:03:09,370 momentum 0. 53 00:03:09,370 --> 00:03:13,430 But more precisely, since we're talking spin-1/2 particles, 54 00:03:13,430 --> 00:03:17,090 if you have a meson called an eta 0, 55 00:03:17,090 --> 00:03:23,480 it's a interacting particle of strong interactions. 56 00:03:23,480 --> 00:03:28,890 A meson decays rather quickly into a mu-plus plus a mu-minus. 57 00:03:28,890 --> 00:03:32,080 Actually it decays into other things as well. 58 00:03:32,080 --> 00:03:36,910 So it decays into two spin-1/2 particles. 59 00:03:36,910 --> 00:03:41,210 And this particle has 0 angular momentum. 60 00:03:41,210 --> 00:03:42,190 It's a scalar. 61 00:03:42,190 --> 00:03:44,540 It's not spinning. 62 00:03:44,540 --> 00:03:47,780 And therefore, if these two particles 63 00:03:47,780 --> 00:03:51,670 go into a state of 0 orbital angular momentum, 64 00:03:51,670 --> 00:03:55,220 conservation of angular momentum implies 65 00:03:55,220 --> 00:03:59,290 that these two particles are in the state of this form that 66 00:03:59,290 --> 00:04:05,660 has 0 spin angular momentum, total spin angular momentum. 67 00:04:05,660 --> 00:04:09,220 So the realization of an entangled state like that 68 00:04:09,220 --> 00:04:12,380 is fairly common and fairly easy. 69 00:04:12,380 --> 00:04:15,830 So you have a decay of this form and you 70 00:04:15,830 --> 00:04:18,540 get particles that are entangled this way. 71 00:04:21,420 --> 00:04:25,070 You showed that this state actually 72 00:04:25,070 --> 00:04:31,920 could be a written as one over square root of 2, n-plus, 73 00:04:31,920 --> 00:04:44,130 n-minus, 1, 2, minus n-minus, n-plus, 1, 2. 74 00:04:44,130 --> 00:04:48,060 Precisely because this invariant and the rotations, 75 00:04:48,060 --> 00:04:56,620 you could use instead of the plus-minus basis, any basis n. 76 00:04:56,620 --> 00:05:00,730 For any direction n, you have this state. 77 00:05:00,730 --> 00:05:06,540 Now we'll talk about the probability 78 00:05:06,540 --> 00:05:09,230 that is of interest to us. 79 00:05:09,230 --> 00:05:16,410 We'll write to this symbol, probability to get a plus, 80 00:05:16,410 --> 00:05:20,410 say, b plus. 81 00:05:20,410 --> 00:05:31,480 And this means that probability to get particle-- 82 00:05:31,480 --> 00:05:40,860 let me see-- to find the first particle-- particle 83 00:05:40,860 --> 00:05:54,280 with spin along a so that this first particle to be 84 00:05:54,280 --> 00:05:56,830 in the state a plus. 85 00:06:00,020 --> 00:06:09,890 And the second particle-- particle 86 00:06:09,890 --> 00:06:17,430 will spin along-- along b. 87 00:06:17,430 --> 00:06:23,910 So in this state, b plus. 88 00:06:23,910 --> 00:06:26,780 Now that looks-- this calculation 89 00:06:26,780 --> 00:06:30,730 of this probability-- So you're going to do some measurement. 90 00:06:30,730 --> 00:06:32,300 And you ask, what is the probability 91 00:06:32,300 --> 00:06:35,620 I find the first particle in this direction, second particle 92 00:06:35,620 --> 00:06:38,900 pointing in this direction? 93 00:06:38,900 --> 00:06:43,120 It may look like a somewhat non-trivial calculation. 94 00:06:43,120 --> 00:06:44,180 And it is. 95 00:06:44,180 --> 00:06:47,730 But if you use the fact this the state over 96 00:06:47,730 --> 00:06:53,020 there-- because we're asking for this probability on this state. 97 00:06:53,020 --> 00:06:55,360 We're going to be talking about this state. 98 00:06:55,360 --> 00:07:03,040 So if you put the state and you write the state in the form, 99 00:07:03,040 --> 00:07:06,070 you pick one of the two vectors, say, a. 100 00:07:06,070 --> 00:07:15,470 Well, the state is a-plus, a-minus, 1, 2, minus a-minus, 101 00:07:15,470 --> 00:07:16,750 a-plus. 102 00:07:16,750 --> 00:07:20,600 Because I could choose n to be anything. 103 00:07:20,600 --> 00:07:26,400 So might this well choose one of the two vectors to be a. 104 00:07:26,400 --> 00:07:29,470 And then you ask what is the probability 105 00:07:29,470 --> 00:07:34,743 to find first particle in this state 106 00:07:34,743 --> 00:07:37,320 and second particle in this state? 107 00:07:37,320 --> 00:07:40,210 Well, when somebody tells you, what is the probability 108 00:07:40,210 --> 00:07:42,520 to find a particle in a state, you 109 00:07:42,520 --> 00:07:46,810 put that state into-- you sandwich your psi 110 00:07:46,810 --> 00:07:48,500 with that state. 111 00:07:48,500 --> 00:07:53,080 And this overlap, which is a number, you square it. 112 00:07:53,080 --> 00:07:56,070 So we're going to do the same thing here. 113 00:07:56,070 --> 00:07:59,590 So what is this probability? 114 00:07:59,590 --> 00:08:04,740 Probability to find a plus, b plus, 115 00:08:04,740 --> 00:08:06,820 would be the absolute value. 116 00:08:06,820 --> 00:08:11,850 And then we put here a plus in the first state. 117 00:08:14,470 --> 00:08:16,550 Tensor product, we could say. 118 00:08:16,550 --> 00:08:21,830 Well, b plus in the second state. 119 00:08:21,830 --> 00:08:24,960 And we should put-- because this is what we want to find, 120 00:08:24,960 --> 00:08:29,840 the state on the tensor product that we're looking for. 121 00:08:29,840 --> 00:08:31,890 And we put the psi here. 122 00:08:31,890 --> 00:08:36,340 So we must put the 1 over square root of 2 times 123 00:08:36,340 --> 00:08:45,990 the a-plus 1, a-minus 2, minus a-minus 1, a-plus 2. 124 00:08:49,440 --> 00:08:56,210 So [? overlap ?] time, well, we go 1 with 1, 2 with 2. 125 00:08:56,210 --> 00:09:01,150 So well, a-plus with a-plus will give me 1. 126 00:09:01,150 --> 00:09:04,110 b-plus with a-minus, I don't know. 127 00:09:04,110 --> 00:09:09,630 Second term, a-plus with a-minus gives me 0. 128 00:09:09,630 --> 00:09:12,170 So this term is irrelevant. 129 00:09:12,170 --> 00:09:14,280 I just need this. 130 00:09:14,280 --> 00:09:19,310 So the a-plus with the a-plus gives me 1. 131 00:09:19,310 --> 00:09:23,120 I have the 1 over square root of 2 here. 132 00:09:23,120 --> 00:09:27,010 And I must close this and square. 133 00:09:27,010 --> 00:09:30,140 I forgot to write that. 134 00:09:30,140 --> 00:09:32,410 So what is this probability? 135 00:09:32,410 --> 00:09:42,300 A-plus, b-plus is equal to-- the 1 over square root becomes 1/2. 136 00:09:42,300 --> 00:09:49,660 And then all we have left is b-plus with a-minus. 137 00:09:49,660 --> 00:09:55,040 In the second state space, in the particle 2 state space, 138 00:09:55,040 --> 00:09:57,410 the label doesn't matter at the end of the day 139 00:09:57,410 --> 00:10:00,570 now that we've disentangled the 1 and 2. 140 00:10:00,570 --> 00:10:02,810 So it's b-plus a-minus. 141 00:10:02,810 --> 00:10:04,970 I don't have to write that it's 2, 2. 142 00:10:04,970 --> 00:10:06,760 Or you can write it. 143 00:10:06,760 --> 00:10:08,220 And it's this squared. 144 00:10:11,080 --> 00:10:16,380 So it's simplified a lot but not quite yet the answer. 145 00:10:16,380 --> 00:10:20,370 What we need here is the overlap between these two spin states. 146 00:10:20,370 --> 00:10:25,970 And I remind you that when you had any arbitrary spin 147 00:10:25,970 --> 00:10:31,960 states with n and n prime, long, long ago, homework three 148 00:10:31,960 --> 00:10:34,520 or four, something like that, you calculated 149 00:10:34,520 --> 00:10:39,060 the overlap between these two spin states. 150 00:10:39,060 --> 00:10:40,990 And the answer was that you would 151 00:10:40,990 --> 00:10:44,192 take cosine of 1/2 of the angle-- 152 00:10:44,192 --> 00:10:46,580 if there is an angle gamma. 153 00:10:46,580 --> 00:10:53,010 The overlap between the spin states 154 00:10:53,010 --> 00:10:57,280 squared was cosine squared of 1/2 the angle. 155 00:11:02,770 --> 00:11:12,590 So here we have the vector a, see, the vector b. 156 00:11:12,590 --> 00:11:20,195 Here is the vector a-minus, a-minus direction. 157 00:11:23,240 --> 00:11:30,680 And if we call this theta ab, this is pi minus theta ab. 158 00:11:30,680 --> 00:11:34,490 So this should be 1/2 cos squared 159 00:11:34,490 --> 00:11:40,230 1/2 of pi minus theta ab. 160 00:11:40,230 --> 00:11:44,020 1/2 cosine square of half of the angle 161 00:11:44,020 --> 00:11:46,610 between the two relevant vectors. 162 00:11:46,610 --> 00:11:50,670 So this is cosine of pi over 2 minus theta over 2. 163 00:11:50,670 --> 00:11:57,750 That's sine squared of theta ab over 2. 164 00:11:57,750 --> 00:11:58,840 So here it is. 165 00:11:58,840 --> 00:12:06,120 It's our calculation of this thing. 166 00:12:06,120 --> 00:12:11,920 It's a neat formula that we're going to need later today. 167 00:12:11,920 --> 00:12:17,050 So one more comment before a little stop, 168 00:12:17,050 --> 00:12:20,650 if b is equal to minus a. 169 00:12:27,710 --> 00:12:32,140 Now in that case you should be in luck because precisely 170 00:12:32,140 --> 00:12:38,930 what's happening here is that if one spin is 171 00:12:38,930 --> 00:12:40,890 along the plus direction the other 172 00:12:40,890 --> 00:12:43,430 has to be along the minus direction, whichever 173 00:12:43,430 --> 00:12:44,220 you choose. 174 00:12:44,220 --> 00:12:50,640 So in order to have that a be in plus and b be in plus, 175 00:12:50,640 --> 00:12:54,200 well, this first term would do it. 176 00:12:54,200 --> 00:12:57,350 a is in plus, the first particle. 177 00:12:57,350 --> 00:12:59,550 Here, no. 178 00:12:59,550 --> 00:13:03,100 And b, which is minus a, would be in plus. 179 00:13:03,100 --> 00:13:05,120 So the probably should be 1/2. 180 00:13:05,120 --> 00:13:16,590 So if b is minus a, the angle theta ab is equal to pi. 181 00:13:16,590 --> 00:13:24,130 And the probability of ab is 1/2, correctly. 182 00:13:27,440 --> 00:13:33,510 And it's 1/2 because half of the cases a is in plus. 183 00:13:33,510 --> 00:13:36,780 The other cases, a is in minus. 184 00:13:36,780 --> 00:13:40,730 So the other case, for example, that could be interesting is, 185 00:13:40,730 --> 00:13:44,590 what is the probability that the first particle is 186 00:13:44,590 --> 00:13:51,115 in z-plus and the second particle is in x-plus? 187 00:13:55,510 --> 00:13:58,770 Well, these two vectors form 90 degrees. 188 00:13:58,770 --> 00:14:03,570 So you should have 1/2 of the sine of half that. 189 00:14:03,570 --> 00:14:14,740 So that should give you 1/2 of the sine squared of pi over 4. 190 00:14:17,310 --> 00:14:22,050 And it's 1/2, then it's 1 over square root of 2, that's 191 00:14:22,050 --> 00:14:23,130 another 1/2. 192 00:14:23,130 --> 00:14:27,570 So it's one quarter, for example. 193 00:14:27,570 --> 00:14:30,440 OK, I went long enough for a moment. 194 00:14:30,440 --> 00:14:32,620 There's an announcement that [? Preshanth ?] wants 195 00:14:32,620 --> 00:14:36,630 to make so please listen to him. 196 00:14:36,630 --> 00:14:41,820 [? PRESHANTH: Hi ?] I'm back for another announcement. 197 00:14:41,820 --> 00:14:45,930 So tonight, the MIT SPS is going to be holding 198 00:14:45,930 --> 00:14:49,450 its fall UROP lightning lectures. 199 00:14:49,450 --> 00:14:54,210 So if you don't have a UROP it's a great opportunity for you 200 00:14:54,210 --> 00:14:59,440 to come and see what other research your classmates are 201 00:14:59,440 --> 00:15:01,320 doing in physics. 202 00:15:01,320 --> 00:15:03,645 If you do have a UROP, I would encourage 203 00:15:03,645 --> 00:15:06,820 you to come as well because you can actually come and share 204 00:15:06,820 --> 00:15:10,380 your stories about the technical content of what 205 00:15:10,380 --> 00:15:12,936 you're doing in your UROP. 206 00:15:12,936 --> 00:15:17,480 It's 7:30 this evening in the PCR 8329. 207 00:15:17,480 --> 00:15:21,110 There will be free food, so please join us then. 208 00:15:21,110 --> 00:15:23,970 And see you all then. 209 00:15:23,970 --> 00:15:24,470 Thanks. 210 00:15:24,470 --> 00:15:25,345 PROFESSOR: Thank you. 211 00:15:32,030 --> 00:15:34,820 All right. 212 00:15:34,820 --> 00:15:41,460 So, this was the introduction to what we really 213 00:15:41,460 --> 00:15:44,850 need to do today. 214 00:15:44,850 --> 00:15:49,640 So before we get started, are there any questions 215 00:15:49,640 --> 00:15:52,910 on what we've done so far? 216 00:15:52,910 --> 00:15:58,540 On these properties of this entangled state? 217 00:15:58,540 --> 00:16:03,870 So this is a measurement of an entangled state. 218 00:16:03,870 --> 00:16:07,320 These two particles could have flown away a big distance. 219 00:16:07,320 --> 00:16:11,660 Two observers, one tries to see what is the probability. 220 00:16:11,660 --> 00:16:15,080 The first observer sees the spine 221 00:16:15,080 --> 00:16:16,470 pointing in some direction. 222 00:16:16,470 --> 00:16:20,110 The other observer sees the spin pointing in another direction. 223 00:16:20,110 --> 00:16:21,640 It's a the natural question which 224 00:16:21,640 --> 00:16:25,440 can be done experimentally. 225 00:16:25,440 --> 00:16:28,325 And we've calculated that answer. 226 00:16:36,430 --> 00:16:41,156 Now so let's begin with this EPR story. 227 00:16:44,700 --> 00:16:51,210 Now, you've seen some of EPR last semester in 804. 228 00:16:51,210 --> 00:16:54,330 The only complication with that is that you really 229 00:16:54,330 --> 00:16:59,290 needed to have these mathematics to appreciate it completely. 230 00:16:59,290 --> 00:17:03,860 So this second look at EPR should 231 00:17:03,860 --> 00:17:08,470 be fairly complete in that we won't leave almost anything out 232 00:17:08,470 --> 00:17:09,910 of the story. 233 00:17:09,910 --> 00:17:15,520 There are many ways of doing EPR and essentially these Bell 234 00:17:15,520 --> 00:17:19,540 inequalities, which is the really non-trivial thing that 235 00:17:19,540 --> 00:17:20,980 comes after that. 236 00:17:20,980 --> 00:17:25,230 So some are in the homework, some elaborations. 237 00:17:25,230 --> 00:17:28,960 And probably in recitation later in the course 238 00:17:28,960 --> 00:17:31,700 we'll see a little more. 239 00:17:31,700 --> 00:17:38,330 But it all begins with a strange thing, the kind of thing 240 00:17:38,330 --> 00:17:42,420 that you wouldn't expect people in physics to discuss. 241 00:17:42,420 --> 00:17:46,490 And it's the point of this so-called-- 242 00:17:46,490 --> 00:17:49,480 Einstein, Polosky, and Rosen wrote a paper. 243 00:17:49,480 --> 00:17:52,993 And they talked about local realism. 244 00:17:58,240 --> 00:18:01,270 Now, that sounds like philosophy. 245 00:18:01,270 --> 00:18:04,400 And for awhile people thought, well, this 246 00:18:04,400 --> 00:18:09,200 is interesting, but undecidable. 247 00:18:09,200 --> 00:18:12,290 Can't really do anything with it. 248 00:18:12,290 --> 00:18:13,973 So what is local realism? 249 00:18:17,140 --> 00:18:20,290 Now, again, not being exactly physics, 250 00:18:20,290 --> 00:18:23,650 it's not all that easy to say what it is. 251 00:18:23,650 --> 00:18:27,340 And people discussed that. 252 00:18:27,340 --> 00:18:31,570 But some notion of it is fairly clear. 253 00:18:31,570 --> 00:18:37,480 The notion is that this reflects something-- it's basically 254 00:18:37,480 --> 00:18:50,165 two assumptions about measurement results. 255 00:18:56,560 --> 00:19:04,130 So you measure something and obtain a number. 256 00:19:04,130 --> 00:19:12,780 And the first assumption, one, is that these measurement 257 00:19:12,780 --> 00:19:19,090 results correspond to some aspects of reality. 258 00:19:19,090 --> 00:19:23,165 Just said like that it seems a little funny. 259 00:19:23,165 --> 00:19:25,890 That you measure something, if get some numbers, 260 00:19:25,890 --> 00:19:29,110 because that was something real about this object, 261 00:19:29,110 --> 00:19:30,710 it had this property. 262 00:19:30,710 --> 00:19:37,100 And so measurement corresponds to some aspect of reality. 263 00:19:37,100 --> 00:19:42,720 So measurements-- assumptions about measurement results. 264 00:19:42,720 --> 00:19:46,540 So measurements. 265 00:19:46,540 --> 00:20:01,783 m, correspond to some aspect of reality. 266 00:20:05,980 --> 00:20:13,860 Two, the measurements that you do in you lab are not 267 00:20:13,860 --> 00:20:18,370 affected whatsoever by the measurements 268 00:20:18,370 --> 00:20:23,970 that somebody else is doing at the same time at the moon. 269 00:20:23,970 --> 00:20:26,340 There's no time for the information 270 00:20:26,340 --> 00:20:31,960 of what that result in the moon has given to reach you. 271 00:20:31,960 --> 00:20:36,750 So at that instant of time, what they are doing at the moon 272 00:20:36,750 --> 00:20:39,080 doesn't affect the result of your experiment. 273 00:20:39,080 --> 00:20:51,380 So it's measurement is independent of actions 274 00:20:51,380 --> 00:21:09,500 performed at a distant location at the same time. 275 00:21:13,790 --> 00:21:18,770 Now to Einstein and Polosky and Rosen-- but Einstein 276 00:21:18,770 --> 00:21:26,500 was very vocal-- Physics must satisfy that. 277 00:21:26,500 --> 00:21:29,730 It's kind of sad I think, actually. 278 00:21:29,730 --> 00:21:36,640 The person that managed to see through and discover how nature 279 00:21:36,640 --> 00:21:42,540 works at so many deep levels-- the photoelectric effect, 280 00:21:42,540 --> 00:21:45,920 special relatively, general relativity-- 281 00:21:45,920 --> 00:21:50,530 somehow became convinced that this had to be true. 282 00:21:50,530 --> 00:21:55,110 And unfortunately, he was wrong. 283 00:21:55,110 --> 00:21:56,900 Or fortunately, I guess. 284 00:21:56,900 --> 00:21:59,990 It's not worth trying to qualify that. 285 00:21:59,990 --> 00:22:05,490 But these two things that seem just so reasonable 286 00:22:05,490 --> 00:22:07,770 are just not true. 287 00:22:07,770 --> 00:22:10,610 This one, measurements correspond 288 00:22:10,610 --> 00:22:14,640 to some aspect of reality-- you see, 289 00:22:14,640 --> 00:22:18,650 you have a Stern-Gerlach apparatus, you throw a spin, 290 00:22:18,650 --> 00:22:20,330 it goes up. 291 00:22:20,330 --> 00:22:23,670 You say it ended up with spin up. 292 00:22:23,670 --> 00:22:27,320 Well, Einstein would say it always had spin up. 293 00:22:27,320 --> 00:22:32,090 It was a reality about that object, at it had spin up. 294 00:22:32,090 --> 00:22:33,300 You just didn't know. 295 00:22:33,300 --> 00:22:37,850 You did the experiment to you discovered it. 296 00:22:37,850 --> 00:22:43,600 So the thing that people try to do in order 297 00:22:43,600 --> 00:22:46,920 to understand this concept, that it corresponds to something 298 00:22:46,920 --> 00:22:49,540 having to do with the reality, is 299 00:22:49,540 --> 00:22:55,220 that you admit that half of the particles 300 00:22:55,220 --> 00:22:58,350 go up and half go down. 301 00:22:58,350 --> 00:23:01,110 But you say, actually, there's something 302 00:23:01,110 --> 00:23:03,080 about these particles you don't know. 303 00:23:03,080 --> 00:23:08,450 And if you knew that, you would just be able to tell. 304 00:23:08,450 --> 00:23:11,320 This is a particle that has spin up. 305 00:23:11,320 --> 00:23:13,260 And it will go up. 306 00:23:13,260 --> 00:23:15,930 But in quantum mechanics, we have abandoned that. 307 00:23:15,930 --> 00:23:19,650 We've said these particles here are a superposition of a state 308 00:23:19,650 --> 00:23:21,270 up and a state down. 309 00:23:21,270 --> 00:23:25,070 And there's nothing definitely up about this particle 310 00:23:25,070 --> 00:23:27,670 or definitely down. 311 00:23:27,670 --> 00:23:32,800 So the way people do this, to correspond 312 00:23:32,800 --> 00:23:35,530 to this aspect of reality that you don't know, 313 00:23:35,530 --> 00:23:40,640 is by introducing what is called hidden variables. 314 00:23:40,640 --> 00:23:44,360 Some things that allow you-- there's 315 00:23:44,360 --> 00:23:50,710 something hidden about this spin particle that you don't know. 316 00:23:50,710 --> 00:23:54,670 But if you knew it you would know exactly how 317 00:23:54,670 --> 00:23:59,970 it's going to come out through this Stern-Gerlach experiment. 318 00:23:59,970 --> 00:24:02,560 And you say, well, that sounds fairly untestable. 319 00:24:02,560 --> 00:24:04,310 But the fact is that it's not. 320 00:24:04,310 --> 00:24:09,100 Now, so this is implemented, this assumption 321 00:24:09,100 --> 00:24:12,080 is-- when people try to modify quantum mechanics, 322 00:24:12,080 --> 00:24:14,450 they use what is called hidden variables. 323 00:24:14,450 --> 00:24:17,630 Some things that you don't know about the particle, 324 00:24:17,630 --> 00:24:21,700 but if you knew, you would see that in fact this particle 325 00:24:21,700 --> 00:24:23,510 has spin up. 326 00:24:26,040 --> 00:24:31,190 This second is in some ways even more disturbing because we got 327 00:24:31,190 --> 00:24:34,430 accustomed to the idea that, locally, 328 00:24:34,430 --> 00:24:38,580 simultaneous things that cannot be reached-- 329 00:24:38,580 --> 00:24:42,500 events that cannot talk to each other via the exchange of light 330 00:24:42,500 --> 00:24:43,800 cannot effect each other. 331 00:24:43,800 --> 00:24:46,810 So simultaneous things that's happened far away 332 00:24:46,810 --> 00:24:48,150 can't effect each other. 333 00:24:48,150 --> 00:24:51,830 So this also sounds very reasonable. 334 00:24:51,830 --> 00:24:53,235 But that's also wrong. 335 00:24:56,250 --> 00:24:59,470 And there's the obvious question, so if this is wrong, 336 00:24:59,470 --> 00:25:04,230 can you send information faster than the speed of light? 337 00:25:04,230 --> 00:25:08,200 And people looked at it in many, many ways. 338 00:25:08,200 --> 00:25:09,840 And it's very interesting. 339 00:25:09,840 --> 00:25:11,420 And we could discuss that. 340 00:25:11,420 --> 00:25:14,330 But it would take us long, so I will leave it 341 00:25:14,330 --> 00:25:18,250 to, maybe, the recitations, maybe other forums. 342 00:25:18,250 --> 00:25:22,810 But here, actually, there's no contradiction, 343 00:25:22,810 --> 00:25:28,960 no way of finding real information going faster 344 00:25:28,960 --> 00:25:31,720 than the speed of light, even though things 345 00:25:31,720 --> 00:25:35,310 far away at the same time can affect you. 346 00:25:35,310 --> 00:25:38,820 So two very interesting things that 347 00:25:38,820 --> 00:25:44,570 seemed very dangerous to discard but turned out to be wrong. 348 00:25:44,570 --> 00:25:48,920 So this is what EPR did. 349 00:25:48,920 --> 00:25:52,740 And they made some thought experiments 350 00:25:52,740 --> 00:25:55,250 that we're going to review to some degree 351 00:25:55,250 --> 00:26:00,150 and see if we can discard these assumptions. 352 00:26:00,150 --> 00:26:03,520 So that's what we're going to do now. 353 00:26:03,520 --> 00:26:05,120 We're going to try to understand that. 354 00:26:05,120 --> 00:26:09,200 Now, if you're interested in what hidden variables are, 355 00:26:09,200 --> 00:26:13,080 my discussion will not use hidden variables. 356 00:26:13,080 --> 00:26:15,160 Although they are kind of implicit, 357 00:26:15,160 --> 00:26:18,390 you will see, as I state some things. 358 00:26:18,390 --> 00:26:21,530 I'm basically going to be explicit on the fact that 359 00:26:21,530 --> 00:26:25,450 things, [? the ?] [? real ?] [? facts, ?] as Einstein would 360 00:26:25,450 --> 00:26:27,970 like you to think. 361 00:26:27,970 --> 00:26:34,580 And we'll try to see if those real qualities about particles 362 00:26:34,580 --> 00:26:37,070 get us in trouble. 363 00:26:37,070 --> 00:26:45,520 So let's begin and try to discuss this first experiment. 364 00:26:45,520 --> 00:26:52,870 You see observer one, Alice and Bob again, if you want. 365 00:26:52,870 --> 00:26:56,210 Alice is going to measure about the z-axis. 366 00:26:56,210 --> 00:26:59,380 Bob is going to measure about the z-axis. 367 00:26:59,380 --> 00:27:08,440 And therefore, if Alice finds spin up, Bob finds spin down. 368 00:27:08,440 --> 00:27:13,610 If Alice finds spin down, Bob finds spin up. 369 00:27:13,610 --> 00:27:15,900 And that's a correlation. 370 00:27:15,900 --> 00:27:18,980 And it's very interesting. 371 00:27:18,980 --> 00:27:23,110 But Einstein, Bell, and Polosky would say, look, 372 00:27:23,110 --> 00:27:25,740 it's not all that interesting. 373 00:27:25,740 --> 00:27:31,170 There's nothing all that mysterious happening here. 374 00:27:31,170 --> 00:27:41,300 EPR would says, the pairs that you've built, 375 00:27:41,300 --> 00:27:44,410 the so-called entangled pairs, are 376 00:27:44,410 --> 00:28:00,680 pairs of particles with definite spins-- spin 377 00:28:00,680 --> 00:28:02,330 directions, spin vectors. 378 00:28:02,330 --> 00:28:06,170 So what you have built, EPR would say, 379 00:28:06,170 --> 00:28:14,800 if you have here particle 1 and particle 2-- 380 00:28:14,800 --> 00:28:16,480 I'm going to list the properties. 381 00:28:16,480 --> 00:28:21,176 Suppose you have a particle 1, you've created. 382 00:28:21,176 --> 00:28:24,750 Einstein would say some particle 1s 383 00:28:24,750 --> 00:28:30,360 with spin up in z and particle 2 with spin down in z. 384 00:28:30,360 --> 00:28:35,970 Or you've created particles 1 with spin down in z 385 00:28:35,970 --> 00:28:39,400 and particle 2 with spin up in z. 386 00:28:39,400 --> 00:28:43,810 And you've created [INAUDIBLE] 50% of the particles 387 00:28:43,810 --> 00:28:46,150 are of this type, of the pairs. 388 00:28:46,150 --> 00:28:50,340 And 50% of the pairs are of this kind. 389 00:28:50,340 --> 00:28:56,706 So don't tell me all this superposition hocus pocus. 390 00:28:56,706 --> 00:29:00,220 Half of your pairs, one particle has spin up, 391 00:29:00,220 --> 00:29:02,420 one particle has spin down. 392 00:29:02,420 --> 00:29:09,410 The other 50% of your pairs, particle one is down, 393 00:29:09,410 --> 00:29:12,300 the other particle is up. 394 00:29:12,300 --> 00:29:15,706 No wonder they're correlated. 395 00:29:15,706 --> 00:29:17,870 You get plus, gets minus. 396 00:29:17,870 --> 00:29:19,580 Get minus, get plus. 397 00:29:19,580 --> 00:29:22,170 What is the probability that you get plus? 398 00:29:22,170 --> 00:29:23,060 50%. 399 00:29:23,060 --> 00:29:24,700 What is the probability you get minus? 400 00:29:24,700 --> 00:29:26,170 50%. 401 00:29:26,170 --> 00:29:28,100 Everything is reproduced. 402 00:29:28,100 --> 00:29:29,340 Mystery over. 403 00:29:29,340 --> 00:29:31,543 No quantum superpositions. 404 00:29:35,092 --> 00:29:35,592 OK? 405 00:29:38,490 --> 00:29:40,870 There's no mistake here. 406 00:29:40,870 --> 00:29:45,210 No mathematical mistake. 407 00:29:45,210 --> 00:29:48,120 You're saying that the particle has a definite spin. 408 00:29:48,120 --> 00:29:49,710 You maybe not know it. 409 00:29:49,710 --> 00:29:52,150 But we say, look, your particles in fact 410 00:29:52,150 --> 00:29:57,050 have a definite spin, z-plus and z-minus and those things. 411 00:29:57,050 --> 00:30:02,560 And this is where hidden variables would come along. 412 00:30:02,560 --> 00:30:07,890 You would say, well, if you have the particle 1, it's spin 413 00:30:07,890 --> 00:30:11,210 is a function of some hidden variable that you don't know. 414 00:30:11,210 --> 00:30:12,710 But if you knew it, you would know 415 00:30:12,710 --> 00:30:17,180 what the spin is because it has a definite spin. 416 00:30:17,180 --> 00:30:19,760 You don't know what is the hidden variable. 417 00:30:19,760 --> 00:30:22,150 But as a function of the hidden variable 418 00:30:22,150 --> 00:30:26,120 the spin is known, definite, not a superposition, 419 00:30:26,120 --> 00:30:28,120 nothing like that. 420 00:30:28,120 --> 00:30:33,960 It's a very aggressive attack on quantum mechanics 421 00:30:33,960 --> 00:30:36,780 and something that troubled people. 422 00:30:36,780 --> 00:30:40,220 And in fact fascinates people even up to now 423 00:30:40,220 --> 00:30:43,350 because the idea, these kind of things 424 00:30:43,350 --> 00:30:45,740 are really absolutely wrong. 425 00:30:45,740 --> 00:30:48,460 It's very shocking. 426 00:30:48,460 --> 00:30:53,800 Perhaps the second even more shocking because, by now 427 00:30:53,800 --> 00:30:57,590 may be you're accustomed to all kinds of variables 428 00:30:57,590 --> 00:31:00,310 that are a little more detached from reality. 429 00:31:00,310 --> 00:31:03,010 You have electromagnetic fields. 430 00:31:03,010 --> 00:31:05,670 And they forced you to learn about potentials 431 00:31:05,670 --> 00:31:09,450 that seem to be a little more abstract. 432 00:31:09,450 --> 00:31:11,530 And similarly here. 433 00:31:11,530 --> 00:31:16,550 So no problem at all. 434 00:31:16,550 --> 00:31:20,980 So people say, OK, this is a simple situation. 435 00:31:20,980 --> 00:31:25,900 But it may be that we're going to do more measurements. 436 00:31:25,900 --> 00:31:29,605 And we're going to consider two directions that are different. 437 00:31:32,306 --> 00:31:39,180 Maybe Alice has two Stern-Gerlach machines, 438 00:31:39,180 --> 00:31:43,130 one that measures about z and one the measures about x. 439 00:31:43,130 --> 00:31:46,210 And Bob has also to Stern-Gerlach machines, 440 00:31:46,210 --> 00:31:49,720 one of the measures about z and one that measures about x. 441 00:31:49,720 --> 00:31:54,340 And they are going to ask different questions. 442 00:31:54,340 --> 00:31:57,290 Because we know the spin, how it transforms. 443 00:31:57,290 --> 00:32:00,180 So getting those results right with two directions 444 00:32:00,180 --> 00:32:02,760 is going to be a little more interesting. 445 00:32:02,760 --> 00:32:07,410 So we're going to try measuring in two possible directions. 446 00:32:07,410 --> 00:32:23,420 Both A and B, Alice and Bob, can measure in two directions, z 447 00:32:23,420 --> 00:32:26,310 and x. 448 00:32:26,310 --> 00:32:30,965 And Einstein would say, look, you can measure in z and in x. 449 00:32:33,740 --> 00:32:38,160 To avoid confusion, let's not talk about one measurement 450 00:32:38,160 --> 00:32:39,100 after another. 451 00:32:39,100 --> 00:32:44,810 Particle comes, you measure in z or you measure in x. 452 00:32:44,810 --> 00:32:48,670 And realism says that you don't know 453 00:32:48,670 --> 00:32:52,000 what you will get because maybe there are hidden variables. 454 00:32:52,000 --> 00:32:55,940 But each particle has a definite answer 455 00:32:55,940 --> 00:32:59,790 if you ask what is the z direction of the spin 456 00:32:59,790 --> 00:33:03,800 and has a definite answer if you ask 457 00:33:03,800 --> 00:33:07,400 what is the x-component of the spin. 458 00:33:07,400 --> 00:33:11,540 Definite answers, real answers, realism again. 459 00:33:11,540 --> 00:33:14,360 So I'm going to label the particles. 460 00:33:14,360 --> 00:33:26,800 For example, this is a label for a particle z-plus, x-minus. 461 00:33:26,800 --> 00:33:33,100 This particle labeled like that would be such 462 00:33:33,100 --> 00:33:38,740 that if you measure its spin in the z direction, it's plus. 463 00:33:38,740 --> 00:33:43,890 If you measure the spin in the x direction, it's minus. 464 00:33:43,890 --> 00:33:57,030 So measured in z, it is up. 465 00:33:57,030 --> 00:34:02,900 Measured in x, it is down. 466 00:34:07,290 --> 00:34:11,885 These are the kind of particles that EPR would say exist. 467 00:34:14,830 --> 00:34:18,090 It's not that you got a particle out 468 00:34:18,090 --> 00:34:22,020 and it's in some strange entangled state. 469 00:34:22,020 --> 00:34:24,630 These particles are flying away. 470 00:34:24,630 --> 00:34:27,030 They're not talking to each other anymore. 471 00:34:27,030 --> 00:34:31,120 And this particle, since you can measure in two directions, 472 00:34:31,120 --> 00:34:33,530 there's some reality, and the measurements 473 00:34:33,530 --> 00:34:36,250 correspond to reality, so there are attributes. 474 00:34:36,250 --> 00:34:40,449 And this particle is classified by having these attributes. 475 00:34:40,449 --> 00:34:42,389 If you measure z, plus. 476 00:34:42,389 --> 00:34:46,409 If you measure x, minus. 477 00:34:46,409 --> 00:34:52,380 So how about this situation. 478 00:34:52,380 --> 00:34:54,429 Well, let me make a list now. 479 00:34:54,429 --> 00:35:00,090 Particles 1 and particle 2. 480 00:35:00,090 --> 00:35:03,540 And this would be the list that EPR would do for you. 481 00:35:03,540 --> 00:35:10,630 EPR comes along and says, look, here's what you're doing. 482 00:35:10,630 --> 00:35:16,540 Particle 1, suppose it's a z-plus, x-plus. 483 00:35:16,540 --> 00:35:22,940 Well, in your beams, actually, when particle 1 is that, 484 00:35:22,940 --> 00:35:29,810 your other particles are z-minus, x-minus. 485 00:35:29,810 --> 00:35:32,410 So if particle 1 is of this type, 486 00:35:32,410 --> 00:35:34,130 particle 2 is of this type. 487 00:35:34,130 --> 00:35:41,790 That way, EPR protect themselves because they say, look, 488 00:35:41,790 --> 00:35:45,110 if you measure z-plus and you measure z-minus, 489 00:35:45,110 --> 00:35:46,450 you get correlation. 490 00:35:46,450 --> 00:35:51,240 If you get plus and plus, you get correlation as well. 491 00:35:51,240 --> 00:35:57,150 Now there can be particles in z-plus, x-minus. 492 00:35:57,150 --> 00:36:01,150 This will go with a z-minus, x-plus. 493 00:36:01,150 --> 00:36:05,385 There could be a particle z-minus, x-plus, 494 00:36:05,385 --> 00:36:10,270 and this as z-plus, x-minus. 495 00:36:10,270 --> 00:36:15,940 And there could be particle z-minus, x-minus, 496 00:36:15,940 --> 00:36:19,200 z-plus, x-plus. 497 00:36:19,200 --> 00:36:21,730 So there are four cases, four types 498 00:36:21,730 --> 00:36:26,610 of particles they say have been produced. 499 00:36:26,610 --> 00:36:32,630 And 25% of pairs are of this form. 500 00:36:32,630 --> 00:36:38,400 25% of pairs are of this, 25 of this, and 25 of this. 501 00:36:44,750 --> 00:36:47,990 So you could ask some questions to EPR. 502 00:36:47,990 --> 00:36:56,760 What is the probability that you get z-plus for 1 503 00:36:56,760 --> 00:37:01,770 and z-minus on 2. 504 00:37:01,770 --> 00:37:07,502 Well, z-plus on 1, it's these two cases. 505 00:37:07,502 --> 00:37:11,860 And z-minus on 2, well, those two cases. 506 00:37:11,860 --> 00:37:14,955 So this is 50% of the times. 507 00:37:17,630 --> 00:37:20,740 So it's 1/2 that probability. 508 00:37:20,740 --> 00:37:22,250 It's correct. 509 00:37:22,250 --> 00:37:26,650 That's what we would predict from an entangled state 510 00:37:26,650 --> 00:37:28,350 viewpoint. 511 00:37:28,350 --> 00:37:30,770 But let's ask something mixed. 512 00:37:30,770 --> 00:37:32,500 Let's see if we get in trouble. 513 00:37:32,500 --> 00:37:42,925 P of z-plus on 1, and let's say x-plus on 2. 514 00:37:46,620 --> 00:37:53,070 Well, z-plus on 1 and x-plus on 2, that's not it. 515 00:37:53,070 --> 00:37:54,130 This case. 516 00:37:54,130 --> 00:37:58,830 z-plus on one, x-plus on 2. 517 00:37:58,830 --> 00:38:02,330 25% of the times, one 1/4. 518 00:38:05,450 --> 00:38:07,490 This was the probability we calculated. 519 00:38:10,260 --> 00:38:13,365 So you tried with one direction. 520 00:38:17,220 --> 00:38:20,240 You don't need quantum mechanics to produce a result. 521 00:38:20,240 --> 00:38:24,440 You try with two directions, you don't need quantum mechanics 522 00:38:24,440 --> 00:38:27,820 to get the result. 523 00:38:27,820 --> 00:38:33,020 So people got stuck and they said, well, maybe it's 524 00:38:33,020 --> 00:38:33,960 undecidable. 525 00:38:33,960 --> 00:38:35,810 Maybe this is philosophy. 526 00:38:35,810 --> 00:38:37,780 Maybe this is something. 527 00:38:37,780 --> 00:38:40,820 And people had to wait for Bell. 528 00:38:40,820 --> 00:38:44,340 He said I'm going to try three directions. 529 00:38:44,340 --> 00:38:48,600 Now, three directions makes a big difference. 530 00:38:48,600 --> 00:38:51,910 This is the first time it really goes wrong. 531 00:38:51,910 --> 00:38:57,590 So it's kind of surprising, perhaps, at some level. 532 00:38:57,590 --> 00:39:00,850 But this is subtle stuff. 533 00:39:00,850 --> 00:39:04,770 So it takes a while before you find something wrong. 534 00:39:04,770 --> 00:39:08,520 You're talking about showing Einstein it's wrong, 535 00:39:08,520 --> 00:39:09,745 so that's not so easy. 536 00:39:13,420 --> 00:39:16,305 So three directions. 537 00:39:25,280 --> 00:39:31,450 So particles are going to be of types-- EPR would say, look, 538 00:39:31,450 --> 00:39:34,180 I'm going to use the same strategy. 539 00:39:34,180 --> 00:39:38,330 I'm going to say that particles have three attributes now. 540 00:39:38,330 --> 00:39:39,385 They're all physical. 541 00:39:39,385 --> 00:39:41,160 They correspond to reality. 542 00:39:41,160 --> 00:39:44,400 Because if you measure in either of three directions, 543 00:39:44,400 --> 00:39:47,120 they have to have an answer for that. 544 00:39:47,120 --> 00:39:49,650 So here's a label for a particle. 545 00:39:55,688 --> 00:40:02,240 And for example, a-plus, b-minus, c-plus. 546 00:40:02,240 --> 00:40:07,100 So if you measure, it would give spin in a direction 547 00:40:07,100 --> 00:40:09,440 plus h bar over 2. 548 00:40:09,440 --> 00:40:14,530 Spin in the b direction minus-- well, in the b direction 549 00:40:14,530 --> 00:40:17,190 would give you minus h bar over 2. 550 00:40:17,190 --> 00:40:22,880 Spin in the c direction would give you plus h bar over 2 . 551 00:40:25,850 --> 00:40:28,710 So we're not measuring simultaneously. 552 00:40:28,710 --> 00:40:31,690 We're just asking, well, you take a particle, 553 00:40:31,690 --> 00:40:33,850 do a measurement, and see what you get. 554 00:40:37,200 --> 00:40:40,440 And we're always going to be asking for probabilities 555 00:40:40,440 --> 00:40:44,380 of this kind, probably that the first particle is doing this 556 00:40:44,380 --> 00:40:47,980 and the second is doing that. 557 00:40:47,980 --> 00:40:53,080 Well, EPR would start now with particles 558 00:40:53,080 --> 00:40:58,403 again, particle 1, particle 2. 559 00:41:01,820 --> 00:41:03,120 And populations. 560 00:41:03,120 --> 00:41:05,400 So let's list quickly the particles. 561 00:41:05,400 --> 00:41:09,040 a-plus, b-plus, c-plus. 562 00:41:09,040 --> 00:41:15,970 Then you will go a-plus, b-plus, c-minus. 563 00:41:15,970 --> 00:41:18,540 Then you've done the c-plus, c-minus here. 564 00:41:18,540 --> 00:41:20,770 You go for two more. 565 00:41:20,770 --> 00:41:25,900 A-plus, b-minus, c-plus. 566 00:41:25,900 --> 00:41:28,750 a-plus, b-minus, c-minus. 567 00:41:33,440 --> 00:41:37,450 I was supposed to fit four more there. 568 00:41:37,450 --> 00:41:38,240 Can I? 569 00:41:38,240 --> 00:41:41,210 Well, I will try. a-minus. 570 00:41:41,210 --> 00:41:43,306 Now you've done all the four a-pluses 571 00:41:43,306 --> 00:41:44,430 so you need four a-minuses. 572 00:41:49,000 --> 00:41:57,810 Then you have b-plus, b-plus, b-minus, b-minus, c-plus, 573 00:41:57,810 --> 00:42:01,100 c-minus, c-plus, c-minus. 574 00:42:01,100 --> 00:42:02,545 I got all, I think. 575 00:42:05,350 --> 00:42:08,480 And particle two of course is correlated. 576 00:42:08,480 --> 00:42:10,800 You would say, well, I don't need to write it. 577 00:42:10,800 --> 00:42:15,800 But it helps seeing what's going on so I'll write it. a-minus, 578 00:42:15,800 --> 00:42:21,980 b-minus, c-minus, a-minus, b-minus, 579 00:42:21,980 --> 00:42:29,810 c-plus, a-minus-- let's use bars here-- a-minus, b-plus, 580 00:42:29,810 --> 00:42:38,410 c-minus, a-minus, b-plus, c-plus, a-plus, b-minus, 581 00:42:38,410 --> 00:42:47,430 c-minus, a-plus, b-minus, c-plus, a-plus, b-plus, 582 00:42:47,430 --> 00:42:54,730 c-minus, a-plus, b-minus, c-- this one is minus 583 00:42:54,730 --> 00:42:56,750 so it's plus here. 584 00:42:56,750 --> 00:42:58,080 We're done. 585 00:42:58,080 --> 00:43:01,560 Lots of labels. 586 00:43:01,560 --> 00:43:03,860 And you could say, well, maybe you 587 00:43:03,860 --> 00:43:06,710 want to put 1/8 in each of them. 588 00:43:06,710 --> 00:43:09,290 But actually the argument is more interesting. 589 00:43:09,290 --> 00:43:13,700 It doesn't need you have to try to put fractions. 590 00:43:13,700 --> 00:43:18,020 So let's consider that there's a total number of particles 591 00:43:18,020 --> 00:43:24,030 N, which is N1 plus up to N8. 592 00:43:24,030 --> 00:43:29,370 And here are N1 of this, N2 of this, N3 of this, 593 00:43:29,370 --> 00:43:35,670 N4 of this, N5, N6, N7, and N8. 594 00:43:44,630 --> 00:43:46,444 Let's see how-- 595 00:43:49,860 --> 00:43:51,640 All right, so we have that. 596 00:43:55,050 --> 00:43:57,700 Well, it takes imagination to see 597 00:43:57,700 --> 00:44:03,370 how you're going to run into some contradiction. 598 00:44:03,370 --> 00:44:05,930 So what is the basis for the contradiction? 599 00:44:05,930 --> 00:44:12,840 Somehow this formula, which is really quantum mechanical 600 00:44:12,840 --> 00:44:17,820 must eventually go wrong with all these attempts 601 00:44:17,820 --> 00:44:21,090 to deny that the world is quantum mechanical. 602 00:44:21,090 --> 00:44:27,870 So we could split again those particles into equal fractions. 603 00:44:27,870 --> 00:44:29,710 But there's no need to do that. 604 00:44:29,710 --> 00:44:31,950 And it's clearer if you don't. 605 00:44:31,950 --> 00:44:36,710 So you try to combine the three directions into one equation. 606 00:44:36,710 --> 00:44:40,150 So one way to do that would be to say, OK, 607 00:44:40,150 --> 00:44:44,830 what is the probability that you get a-plus and b-plus? 608 00:44:44,830 --> 00:44:47,700 So this is for the particle number 1. 609 00:44:47,700 --> 00:44:50,180 And this is particle number 2. 610 00:44:50,180 --> 00:44:53,810 So then you must look at the table 611 00:44:53,810 --> 00:44:56,000 and say which one's do that. 612 00:44:56,000 --> 00:44:57,930 Well, you need a-plus in the first 613 00:44:57,930 --> 00:45:00,720 so it's one of the first four rows. 614 00:45:00,720 --> 00:45:02,770 And b-plus in the second. 615 00:45:02,770 --> 00:45:16,600 So actually, it's these two cases, N3 and N4, over N. 616 00:45:16,600 --> 00:45:18,970 We want to involve three directions. 617 00:45:18,970 --> 00:45:21,660 So let's go for another one. 618 00:45:21,660 --> 00:45:27,180 P of a-plus with c-plus. 619 00:45:27,180 --> 00:45:28,900 Let's see how much is that. 620 00:45:28,900 --> 00:45:31,320 Again, this is for the first particle. 621 00:45:31,320 --> 00:45:34,500 This is for the second particle. 622 00:45:34,500 --> 00:45:39,310 So I must look at the first four rows. 623 00:45:39,310 --> 00:45:43,380 And see that you have an a-plus. 624 00:45:43,380 --> 00:45:46,530 First part is in a-plus for the first four rows. 625 00:45:46,530 --> 00:45:48,590 But the second should be in c-plus 626 00:45:48,590 --> 00:45:52,860 and that is cases N2 and N4. 627 00:45:52,860 --> 00:46:00,950 So we get N2 plus N4 here over N. 628 00:46:00,950 --> 00:46:06,276 Well, we've involved this a with b, a with c. 629 00:46:06,276 --> 00:46:10,880 How about involving b with c? 630 00:46:10,880 --> 00:46:13,835 So I'll put c-plus, b-plus, for example. 631 00:46:18,190 --> 00:46:22,310 OK c-plus, b-plus. 632 00:46:22,310 --> 00:46:26,260 I must look at c-pluses, and b-pluses, no. 633 00:46:26,260 --> 00:46:27,950 C-plus and b-plus, yes. 634 00:46:27,950 --> 00:46:31,840 N3 is there, which is good because it already was there. 635 00:46:34,690 --> 00:46:36,510 And which else? 636 00:46:36,510 --> 00:46:39,620 c-plus and b-plus. 637 00:46:39,620 --> 00:46:41,240 c-plus, no. 638 00:46:41,240 --> 00:46:42,851 c-plus here b-plus, yes. 639 00:46:42,851 --> 00:46:43,350 N7. 640 00:46:53,820 --> 00:47:15,870 Now N3 plus N4 is less than N3 plus N7 plus N4 plus N2. 641 00:47:15,870 --> 00:47:19,060 You see you have N3 and N4. 642 00:47:19,060 --> 00:47:24,410 And now I add whatever N7 and N2 are. 643 00:47:24,410 --> 00:47:29,110 And that's then an inequality because it's 644 00:47:29,110 --> 00:47:31,130 going to be more cases. 645 00:47:31,130 --> 00:47:40,780 Now I divide by N. So you obtain an inequality 646 00:47:40,780 --> 00:47:49,630 that P of a-plus, b-plus is less than 647 00:47:49,630 --> 00:47:56,550 or equal than N3 plus N7 and N4 plus N2. 648 00:47:56,550 --> 00:48:03,770 I'll right the second first, P of a-plus, c-plus 649 00:48:03,770 --> 00:48:07,600 plus P of c-plus, b-plus. 650 00:48:15,190 --> 00:48:19,700 So I didn't put specific populations 1/3, 1/4. 651 00:48:19,700 --> 00:48:25,360 But in general whatever populations you choose, 652 00:48:25,360 --> 00:48:28,450 this inequality must hold. 653 00:48:28,450 --> 00:48:31,040 So that's the more clever strategy. 654 00:48:31,040 --> 00:48:34,730 Because suppose you choose some populations 655 00:48:34,730 --> 00:48:37,580 and you don't get in trouble, well, 656 00:48:37,580 --> 00:48:39,230 maybe with some other populations 657 00:48:39,230 --> 00:48:40,480 you would get in trouble. 658 00:48:40,480 --> 00:48:43,380 Maybe it's not so easy to get the relative factors. 659 00:48:43,380 --> 00:48:47,210 So here is something that must be 660 00:48:47,210 --> 00:48:50,810 true whatever populations you choose. 661 00:48:50,810 --> 00:48:56,030 And now that is Bell's inequality. 662 00:48:56,030 --> 00:49:02,190 So the achievement of Bell is to somehow 663 00:49:02,190 --> 00:49:08,960 translate this assumption of realism into an inequality. 664 00:49:08,960 --> 00:49:17,180 And now quantum mechanics has a formula for these things, 665 00:49:17,180 --> 00:49:18,450 for this probability. 666 00:49:18,450 --> 00:49:22,260 So we can test whether this is true. 667 00:49:22,260 --> 00:49:25,030 So let's do that. 668 00:49:25,030 --> 00:49:28,200 This is the so-called Bell inequality. 669 00:49:28,200 --> 00:49:34,605 So if quantum mechanics is true, the following should hold. 670 00:49:42,630 --> 00:49:43,865 Let's see that. 671 00:49:48,590 --> 00:50:02,340 If QM is true, well, there should 672 00:50:02,340 --> 00:50:04,600 be a problem with this inequality. 673 00:50:04,600 --> 00:50:07,280 So let's see what happens. 674 00:50:07,280 --> 00:50:09,490 Let's see if it's true, this inequality. 675 00:50:13,900 --> 00:50:16,050 Is it really true? 676 00:50:16,050 --> 00:50:19,980 Well, the left hand side would be, 677 00:50:19,980 --> 00:50:23,760 given the formula that we had, 1/2 678 00:50:23,760 --> 00:50:29,250 of the sine squared of theta ab over 2. 679 00:50:33,750 --> 00:50:37,540 So let me just emphasize, this was 680 00:50:37,540 --> 00:50:40,085 derived using local realism. 681 00:50:44,190 --> 00:50:46,790 Local realism gives that. 682 00:50:46,790 --> 00:50:50,910 So you do the experiment, get these probabilities. 683 00:50:50,910 --> 00:50:55,350 And if realism is true, this should hold. 684 00:50:55,350 --> 00:50:58,270 Let's see what quantum mechanics has to say. 685 00:50:58,270 --> 00:51:03,390 Let's plug-in the values that you get from quantum mechanics. 686 00:51:03,390 --> 00:51:05,500 Now we calculated this probability. 687 00:51:05,500 --> 00:51:06,850 We put the first term. 688 00:51:06,850 --> 00:51:12,950 Here is sine squared of theta ac over 2 and 1/2 689 00:51:12,950 --> 00:51:17,130 of sine squared theta bc over 2. 690 00:51:21,600 --> 00:51:23,810 So does that work? 691 00:51:23,810 --> 00:51:25,290 Does that always work? 692 00:51:25,290 --> 00:51:31,310 Can I orient this axis in such a way to disprove EPR? 693 00:51:31,310 --> 00:51:36,150 And in fact, it turns out to be quite easy to do that. 694 00:51:36,150 --> 00:51:41,280 So you choose three vectors like this. 695 00:51:41,280 --> 00:51:50,610 ac, bc, so c here, a here, and b here, I believe. 696 00:51:50,610 --> 00:51:52,000 Yep. 697 00:51:52,000 --> 00:51:55,770 So put an angle theta here. 698 00:51:55,770 --> 00:51:58,550 An angle theta here. 699 00:51:58,550 --> 00:52:00,650 And then what do you have? 700 00:52:00,650 --> 00:52:05,970 Theta ab would be 2 theta. 701 00:52:05,970 --> 00:52:15,210 Theta ac would be equal to theta bc equal to theta. 702 00:52:15,210 --> 00:52:20,300 That's a pretty nice simple choice of angles. 703 00:52:20,300 --> 00:52:23,260 If you choose these angles now, let's 704 00:52:23,260 --> 00:52:26,060 see what happens with our inequality. 705 00:52:26,060 --> 00:52:33,250 So you get 1/2 sine squared theta ab over 2 would be theta. 706 00:52:33,250 --> 00:52:37,140 Is it less than or equal to this? 707 00:52:37,140 --> 00:52:39,120 Well, these two become the same. 708 00:52:39,120 --> 00:52:44,090 So you get sine squared of theta over 2. 709 00:52:48,080 --> 00:52:50,680 Violated or not violated? 710 00:52:50,680 --> 00:52:55,530 Is it true or false, this, for all theta? 711 00:52:55,530 --> 00:52:56,330 What do you say? 712 00:53:02,150 --> 00:53:04,094 Yes? 713 00:53:04,094 --> 00:53:09,670 STUDENT: If theta is less than pi, that's not true. 714 00:53:09,670 --> 00:53:11,770 PROFESSOR: Close. 715 00:53:11,770 --> 00:53:14,440 It's not true for a small theta. 716 00:53:14,440 --> 00:53:19,640 So if you're this, and you're desperate to know, 717 00:53:19,640 --> 00:53:21,290 the thing you have to do is assume 718 00:53:21,290 --> 00:53:23,030 theta is very, very small. 719 00:53:23,030 --> 00:53:24,530 See if you get in trouble. 720 00:53:24,530 --> 00:53:25,920 How much is this? 721 00:53:25,920 --> 00:53:27,843 1/2 theta squared. 722 00:53:31,715 --> 00:53:33,531 Let's see. 723 00:53:33,531 --> 00:53:34,900 Half of this one? 724 00:53:34,900 --> 00:53:36,280 Yeah. 725 00:53:36,280 --> 00:53:37,595 And how much is this? 726 00:53:37,595 --> 00:53:40,063 This is theta squared over 4. 727 00:53:40,063 --> 00:53:43,010 Sorry, I was not seeing it. 728 00:53:43,010 --> 00:53:46,780 Sine theta for small theta is roughly theta. 729 00:53:46,780 --> 00:53:49,060 So here is theta over 2 squared. 730 00:53:49,060 --> 00:53:51,820 But here is 1/2 theta squared. 731 00:53:51,820 --> 00:53:54,590 And it's false. 732 00:53:54,590 --> 00:53:57,280 The 1/2 theta squared is not smaller 733 00:53:57,280 --> 00:53:59,800 than one quarter theta squared. 734 00:53:59,800 --> 00:54:04,260 And in fact, for theta equal pi over 2, 735 00:54:04,260 --> 00:54:07,640 I think this is an equality. 736 00:54:07,640 --> 00:54:11,260 Because theta equal pi over 2, you 737 00:54:11,260 --> 00:54:14,210 get on the left hand side 1/2 less than 738 00:54:14,210 --> 00:54:22,030 or equal than sine of 45 degrees, which is correct. 739 00:54:22,030 --> 00:54:25,430 So it's an equality of pi over 2. 740 00:54:25,430 --> 00:54:27,770 Fails below. 741 00:54:27,770 --> 00:54:31,760 So it was a shock. 742 00:54:31,760 --> 00:54:37,370 That if you could do an experiment in quantum mechanics 743 00:54:37,370 --> 00:54:45,290 and experiment with correlated, entangled particles, 744 00:54:45,290 --> 00:54:48,580 that you could to measure these probabilities, 745 00:54:48,580 --> 00:54:50,060 these correlations. 746 00:54:50,060 --> 00:54:54,740 And you would obtain a result that actually contradicts 747 00:54:54,740 --> 00:54:57,710 for certain alignments of your experiments 748 00:54:57,710 --> 00:55:00,440 the assumptions of local realism. 749 00:55:00,440 --> 00:55:05,360 So it was a great result of Bell to show 750 00:55:05,360 --> 00:55:09,630 that quantum mechanics is in contradiction 751 00:55:09,630 --> 00:55:12,230 with local realism. 752 00:55:12,230 --> 00:55:18,600 There's no way to keep the ideas of quantum mechanics 753 00:55:18,600 --> 00:55:21,550 and put hidden variables and assume 754 00:55:21,550 --> 00:55:24,120 that there's real values for things 755 00:55:24,120 --> 00:55:27,190 and that there's no effect at a distance. 756 00:55:27,190 --> 00:55:30,330 It would all be contradicted by an experiment. 757 00:55:30,330 --> 00:55:34,620 That was done later and the definite version 758 00:55:34,620 --> 00:55:39,200 of the experiments around 1980 or '82 759 00:55:39,200 --> 00:55:42,000 by Alan Aspect and others. 760 00:55:42,000 --> 00:55:46,300 Very clever experiments worth reading and understanding. 761 00:55:46,300 --> 00:55:51,780 But confirmed that the quantum mechanical result 762 00:55:51,780 --> 00:55:54,730 is really true by measuring correlated pairs. 763 00:55:54,730 --> 00:55:57,493 And this inequality is violated. 764 00:55:57,493 --> 00:55:57,993 Yes? 765 00:55:57,993 --> 00:56:00,132 STUDENT: So what about David Bohm's theory 766 00:56:00,132 --> 00:56:01,746 of hidden variables quantum mechanics? 767 00:56:01,746 --> 00:56:02,245 So-- 768 00:56:02,245 --> 00:56:05,000 PROFESSOR: David Bohm's theory of what? 769 00:56:05,000 --> 00:56:07,820 STUDENT: His hidden variable quantum mechanics theory 770 00:56:07,820 --> 00:56:10,594 allegedly reproduces the same results as quantum mechanics 771 00:56:10,594 --> 00:56:13,024 but it's still a hidden variables theory. 772 00:56:13,024 --> 00:56:16,140 PROFESSOR: I don't think there's any hidden variable theory that 773 00:56:16,140 --> 00:56:17,630 works. 774 00:56:17,630 --> 00:56:19,640 David Bohm, I think, actually was 775 00:56:19,640 --> 00:56:24,050 credited by rewriting EPR, who essentially talked 776 00:56:24,050 --> 00:56:28,580 about position and momenta to talk about spins. 777 00:56:28,580 --> 00:56:30,500 And he might have been the first one that 778 00:56:30,500 --> 00:56:33,090 began to try to do hidden variable theory. 779 00:56:33,090 --> 00:56:36,610 But no hidden variable theory works at this moment. 780 00:56:36,610 --> 00:56:38,780 And this shows it. 781 00:56:38,780 --> 00:56:42,680 So people say that actually this assumes 782 00:56:42,680 --> 00:56:47,130 that there's local hidden variable theories 783 00:56:47,130 --> 00:56:49,140 and there's non-local hidden variable 784 00:56:49,140 --> 00:56:51,290 theories and all kinds of strange things. 785 00:56:51,290 --> 00:56:53,760 But it's more and more unnatural. 786 00:56:53,760 --> 00:56:56,440 So it doesn't seem to do something very interesting. 787 00:56:56,440 --> 00:56:57,690 Yes, there are many questions. 788 00:56:57,690 --> 00:56:59,586 Aaron, maybe you want to say something. 789 00:56:59,586 --> 00:57:02,442 STUDENT: Let's see, I think Bohm has a non-local hidden variable 790 00:57:02,442 --> 00:57:03,400 theory solution. 791 00:57:03,400 --> 00:57:04,610 It's kind of awful looking. 792 00:57:04,610 --> 00:57:09,696 But I guess violates the second principle rather than 793 00:57:09,696 --> 00:57:10,694 the first one. 794 00:57:10,694 --> 00:57:14,851 It also doesn't extend to-- this doesn't really 795 00:57:14,851 --> 00:57:15,684 work for everything. 796 00:57:15,684 --> 00:57:19,343 We don't know how to make it work for a spin-1/2 particle 797 00:57:19,343 --> 00:57:23,169 to find [? a dimension. ?] So [INTERPOSING VOICES] really 798 00:57:23,169 --> 00:57:26,670 believe to be a true theory of [? equivalence. ?] 799 00:57:26,670 --> 00:57:28,610 PROFESSOR: Thank you. 800 00:57:28,610 --> 00:57:30,810 More questions. 801 00:57:30,810 --> 00:57:32,244 Steve? 802 00:57:32,244 --> 00:57:36,334 STUDENT: In the case with EPR, is it a problem that we can 803 00:57:36,334 --> 00:57:40,070 have scenarios where the spin is greater than a spin-1/2 804 00:57:40,070 --> 00:57:41,480 particle would have? 805 00:57:41,480 --> 00:57:43,350 If we had the state a-plus, b-plus, c-plus, 806 00:57:43,350 --> 00:57:47,260 we would have 3 h bar over 2? 807 00:57:47,260 --> 00:57:48,240 PROFESSOR: No. 808 00:57:48,240 --> 00:57:52,770 The statement that is done here is not 809 00:57:52,770 --> 00:57:59,140 that-- well, this is a label for a particle. 810 00:57:59,140 --> 00:58:03,110 EPR just assumed that if you measured 811 00:58:03,110 --> 00:58:05,700 a you would be able to get this. 812 00:58:05,700 --> 00:58:09,080 If you measure along b, you would get this. 813 00:58:09,080 --> 00:58:12,090 And if you measured along c, you would get this. 814 00:58:12,090 --> 00:58:14,210 So this is a single particle. 815 00:58:14,210 --> 00:58:18,080 Any measurement gives some results this side of the list. 816 00:58:18,080 --> 00:58:22,252 There's no sense in which these are added. 817 00:58:22,252 --> 00:58:25,280 STUDENT: But even when you measure one or the other, 818 00:58:25,280 --> 00:58:28,710 the other values still exist for those measurements. 819 00:58:28,710 --> 00:58:32,740 PROFESSOR: Well, I believe there's 820 00:58:32,740 --> 00:58:34,830 no need to discuss that. 821 00:58:34,830 --> 00:58:37,730 So they don't talk about the statement 822 00:58:37,730 --> 00:58:41,530 of doing subsequent measurements within this statement. 823 00:58:41,530 --> 00:58:44,180 You just take this particle and you decide. 824 00:58:44,180 --> 00:58:47,320 You measure a or measure b or measure c. 825 00:58:47,320 --> 00:58:49,640 You don't try to measure simultaneously. 826 00:58:49,640 --> 00:58:52,650 You don't try to measure one after another. 827 00:58:52,650 --> 00:58:54,680 You just do one measurement. 828 00:58:54,680 --> 00:58:58,070 And that already, which is the minimum you can do, 829 00:58:58,070 --> 00:58:59,590 gets you in trouble. 830 00:58:59,590 --> 00:59:05,230 So I'm not sure how EPR would phrase subsequent measurements 831 00:59:05,230 --> 00:59:07,000 after they've done the first measurement 832 00:59:07,000 --> 00:59:07,890 or things like that. 833 00:59:07,890 --> 00:59:13,070 But they're not necessary for this stage. 834 00:59:13,070 --> 00:59:17,440 OK, so look, it's a very interesting thing. 835 00:59:17,440 --> 00:59:22,560 There's lots to discuss here, but it's best if you read it. 836 00:59:22,560 --> 00:59:23,730 James had a question. 837 00:59:23,730 --> 00:59:26,050 Just let's take one quick question. 838 00:59:26,050 --> 00:59:28,845 STUDENT: I was just wondering if there was any other extension 839 00:59:28,845 --> 00:59:31,024 beyond three directions for Bell's inequalities. 840 00:59:31,024 --> 00:59:33,886 Is there a N-direction of Bell's inequalities 841 00:59:33,886 --> 00:59:35,317 or some type of form of it? 842 00:59:35,317 --> 00:59:36,760 Or is there [INAUDIBLE] 843 00:59:36,760 --> 00:59:41,110 PROFESSOR: There are other forms of Bell inequalities. 844 00:59:41,110 --> 00:59:43,860 I'm not sure if it's popular with four 845 00:59:43,860 --> 00:59:45,950 directions or anything. 846 00:59:45,950 --> 00:59:49,760 But certainly Weinberg, for example, discuss other ways. 847 00:59:49,760 --> 00:59:52,740 There are alternative ways to phrase it. 848 00:59:52,740 --> 00:59:57,010 I've talked about here probabilities 849 00:59:57,010 --> 00:59:58,860 to observe results. 850 00:59:58,860 --> 01:00:01,520 There is a more, perhaps, [? common ?] way 851 01:00:01,520 --> 01:00:06,640 talking about expectation values or correlation functions. 852 01:00:06,640 --> 01:00:09,645 This is something you'll do in the homework. 853 01:00:09,645 --> 01:00:13,810 The sort of game that is done in the homework that 854 01:00:13,810 --> 01:00:17,030 was a suggestion by Aaron to put it in the homework-- 855 01:00:17,030 --> 01:00:21,340 this game in which with quantum strategy and entangled pair 856 01:00:21,340 --> 01:00:25,400 you beat the system-- is yet another formulation of the Bell 857 01:00:25,400 --> 01:00:27,850 inequalities as well. 858 01:00:27,850 --> 01:00:29,710 So lot to do. 859 01:00:29,710 --> 01:00:36,900 But I think it's better now that we stop and talked 860 01:00:36,900 --> 01:00:43,050 about angular momentum from now until the end of the semester. 861 01:00:43,050 --> 01:00:45,300 This was about spins, but we now have 862 01:00:45,300 --> 01:00:49,540 to put together angular momentum which is orbital 863 01:00:49,540 --> 01:00:52,350 and spin and all the various kinds. 864 01:00:52,350 --> 01:00:54,965 So we'll begin with angular momentum. 865 01:01:01,290 --> 01:01:05,500 So there are notes on the web on that 866 01:01:05,500 --> 01:01:09,190 that I wrote and modified a little this time. 867 01:01:09,190 --> 01:01:12,640 And lots of little exercises. 868 01:01:12,640 --> 01:01:15,910 So what I want to do now is guide you 869 01:01:15,910 --> 01:01:19,210 through the things that happen there so that you 870 01:01:19,210 --> 01:01:21,770 get a view of what we're going to do. 871 01:01:21,770 --> 01:01:28,210 We're going to work with this thing in a elegant way using 872 01:01:28,210 --> 01:01:30,640 vector notation for operators is going 873 01:01:30,640 --> 01:01:33,560 to help us understand things better. 874 01:01:33,560 --> 01:01:38,050 So we've seen angular momentum before. 875 01:01:38,050 --> 01:01:43,250 And let me summarize simple things that we know about it. 876 01:01:43,250 --> 01:01:49,390 If you have angular momentum, and let's begin with orbital. 877 01:01:49,390 --> 01:01:56,950 This is what we have when we take Lx to be y Pz minus z Py. 878 01:01:59,960 --> 01:02:04,768 Ly equals z Px minus x Pz. 879 01:02:08,330 --> 01:02:14,432 And Lz to be equal to x Py minus y Pz. 880 01:02:18,010 --> 01:02:22,240 These are the angular momentum, orbital angular momentum 881 01:02:22,240 --> 01:02:24,760 operators. 882 01:02:24,760 --> 01:02:28,270 Now, it's better for many things to use labels 883 01:02:28,270 --> 01:02:32,900 like x, y, and z, those operators. 884 01:02:32,900 --> 01:02:36,460 Call them x1 hat, x2 hat, x3 hat. 885 01:02:40,360 --> 01:02:50,165 And Px, Py, Pz, P1, P2, P3. 886 01:02:53,220 --> 01:02:57,740 In that way you can write commutation relations like xi, 887 01:02:57,740 --> 01:03:04,160 Pj equal i h bar delta ij. 888 01:03:04,160 --> 01:03:13,180 xi with xj equal pi with pj equal 0. 889 01:03:13,180 --> 01:03:16,640 So it allows you to write things more quickly. 890 01:03:19,570 --> 01:03:22,110 In fact, the angular momentum operators 891 01:03:22,110 --> 01:03:24,490 become also a little simpler. 892 01:03:24,490 --> 01:03:33,360 It's sort of x2 P3 minus x3 P2. 893 01:03:33,360 --> 01:03:38,900 And you'll have the x, y, z labels and all that. 894 01:03:38,900 --> 01:03:43,920 So we want to use vector notation. 895 01:03:43,920 --> 01:03:46,820 Now, vector notation, you can do it in two ways. 896 01:03:46,820 --> 01:03:48,220 You can talk about triplets. 897 01:03:48,220 --> 01:03:51,670 Those are vectors. 898 01:03:51,670 --> 01:03:55,440 Or you can form the vectors themselves. 899 01:03:55,440 --> 01:03:57,270 Now if you form the vectors, you get 900 01:03:57,270 --> 01:04:00,320 objects that sometimes are a little disconcerting. 901 01:04:00,320 --> 01:04:03,480 But we try that they not be so. 902 01:04:03,480 --> 01:04:10,970 So here is the r vector operator. 903 01:04:10,970 --> 01:04:14,250 You could think of it as the triplet of x, y, and z. 904 01:04:14,250 --> 01:04:19,700 But let's call it like this, x thing times the first basis 905 01:04:19,700 --> 01:04:25,510 vector plus y operator times the second basis vector 906 01:04:25,510 --> 01:04:29,470 plus z operator times the third basis vector. 907 01:04:29,470 --> 01:04:32,050 And we've done things like that. 908 01:04:32,050 --> 01:04:35,490 And we understand that these basis vectors really 909 01:04:35,490 --> 01:04:38,910 don't talk with these operators. 910 01:04:38,910 --> 01:04:40,820 They can be moved across each other. 911 01:04:40,820 --> 01:04:43,980 The basis vectors are things that 912 01:04:43,980 --> 01:04:47,455 help you write expressions, rather, 913 01:04:47,455 --> 01:04:50,430 at talking about triplets. 914 01:04:50,430 --> 01:04:51,765 Same thing for momentum. 915 01:04:55,030 --> 01:04:56,200 Let's do it this way. 916 01:04:56,200 --> 01:05:05,370 P1 e1 plus P2 e2 plus P3 e3. 917 01:05:05,370 --> 01:05:10,420 And finally, well, angular momentum, the vector operator-- 918 01:05:10,420 --> 01:05:14,930 you've done a lot of the angular momentum vector operator 919 01:05:14,930 --> 01:05:16,810 for spin. 920 01:05:16,810 --> 01:05:30,590 So here you would put Lx or L1 e1 plus L2 e2 plus L3 e3. 921 01:05:30,590 --> 01:05:37,510 So those are ways to write equations 922 01:05:37,510 --> 01:05:42,850 that carry all the operators and treat them as vectors, even 923 01:05:42,850 --> 01:05:44,200 though they're operators. 924 01:05:44,200 --> 01:05:46,460 So they're unusual vectors. 925 01:05:46,460 --> 01:05:52,690 They're vectors whose components are operators, are not numbers. 926 01:05:52,690 --> 01:06:00,520 So the obvious question is, what changes then? 927 01:06:00,520 --> 01:06:03,520 So we're going to define dot product and cross 928 01:06:03,520 --> 01:06:06,770 products as we had before. 929 01:06:06,770 --> 01:06:12,380 But we have to be a little aware that when we write these things 930 01:06:12,380 --> 01:06:15,360 we could make a mistake unless we're not careful. 931 01:06:15,360 --> 01:06:18,560 So here are two vector operators. 932 01:06:25,590 --> 01:06:29,990 What is the dot product of these two vector operators? 933 01:06:29,990 --> 01:06:34,870 Well, you know it's supposed to be the first component of this 934 01:06:34,870 --> 01:06:37,850 and the first component of that, second second, third third. 935 01:06:37,850 --> 01:06:45,970 So it should be ai bi, summed. 936 01:06:45,970 --> 01:06:49,050 Repeated in this is our [? summed. ?] 937 01:06:49,050 --> 01:06:55,110 Now, I should not write bi ai with a reverse order 938 01:06:55,110 --> 01:06:59,430 because this thing, the components, are now operators. 939 01:06:59,430 --> 01:07:01,840 And maybe they don't commute. 940 01:07:01,840 --> 01:07:06,573 So I've defined this once and for all to be this. 941 01:07:06,573 --> 01:07:14,490 And a cross b, the i-th component of this thing 942 01:07:14,490 --> 01:07:18,040 is going to be defined once and for all 943 01:07:18,040 --> 01:07:24,454 to be epsilon ijk aj bk. 944 01:07:28,250 --> 01:07:29,675 Definition. 945 01:07:29,675 --> 01:07:32,046 The a to the left of the b. 946 01:07:35,238 --> 01:07:42,770 And with this, we can check our usual rules 947 01:07:42,770 --> 01:07:44,730 of manipulation of operators. 948 01:07:44,730 --> 01:07:46,270 So one more definition. 949 01:07:46,270 --> 01:07:53,790 a squared is going to be aa, and it's going to be ai ai. 950 01:08:00,190 --> 01:08:09,050 Simplest calculation is a dot b equals to b dot a. 951 01:08:13,910 --> 01:08:15,100 Yes or not? 952 01:08:15,100 --> 01:08:16,710 No, they're operators. 953 01:08:16,710 --> 01:08:18,950 So let's calculate the difference. 954 01:08:18,950 --> 01:08:22,822 Let's get a little practice calculating differences. 955 01:08:29,750 --> 01:08:38,840 So I write a dot b is equal to ai bi, its sum. 956 01:08:38,840 --> 01:08:47,149 So then I say, that's ai bi plus bi ai. 957 01:08:47,149 --> 01:08:52,400 You see the commutator is ai bi minus bi ai. 958 01:08:52,400 --> 01:08:55,399 And I add it back. 959 01:08:55,399 --> 01:09:01,149 But this thing, bi ai is b dot a. 960 01:09:01,149 --> 01:09:03,319 So here I've got a formula. 961 01:09:03,319 --> 01:09:12,910 a dot b is equal, actually, to b dot a plus this commutator, 962 01:09:12,910 --> 01:09:16,220 ai bi. 963 01:09:16,220 --> 01:09:23,420 And now you've got a new formula for operator vector analysis. 964 01:09:23,420 --> 01:09:26,090 a dot b and b dot a are not the same 965 01:09:26,090 --> 01:09:28,850 but they differ by this thing. 966 01:09:28,850 --> 01:09:35,540 And a very important corollary, very famous corollary, 967 01:09:35,540 --> 01:09:39,800 what is r dot p? 968 01:09:39,800 --> 01:09:43,410 Is it the same as p dot r? 969 01:09:43,410 --> 01:09:45,649 If you're working quantum mechanics 970 01:09:45,649 --> 01:09:48,970 you maybe tempted to, oh r dot p and p dot r are the same, 971 01:09:48,970 --> 01:09:52,489 but r and p don't commute, so what is the difference? 972 01:09:52,489 --> 01:10:01,095 R dot p is equal to p dot r plus xi pi. 973 01:10:05,250 --> 01:10:07,910 And how much is that? 974 01:10:07,910 --> 01:10:09,670 [INAUDIBLE] 975 01:10:09,670 --> 01:10:11,145 Sorry? 976 01:10:11,145 --> 01:10:12,260 STUDENT: i h bar. 977 01:10:12,260 --> 01:10:13,400 PROFESSOR: i h bar? 978 01:10:13,400 --> 01:10:14,080 STUDENT: 3. 979 01:10:14,080 --> 01:10:15,720 PROFESSOR: 3 i h bar. 980 01:10:15,720 --> 01:10:17,910 Yes, don't forget the sum. 981 01:10:17,910 --> 01:10:19,620 This is supposed to be summed. 982 01:10:19,620 --> 01:10:24,920 So it says x1 commutative, p1, x2 going to 3 i h bar. 983 01:10:24,920 --> 01:10:28,010 So here's a famous formula. 984 01:10:28,010 --> 01:10:37,758 rp differs from pr plus three h bar i-- i h bar, people write. 985 01:10:37,758 --> 01:10:38,650 i h bar. 986 01:10:43,840 --> 01:10:51,350 Another formula that you would be curious to know. 987 01:10:51,350 --> 01:10:55,500 Well, the dot product was supposed to be symmetric. 988 01:10:55,500 --> 01:10:57,240 It's not. 989 01:10:57,240 --> 01:11:03,230 The cross product is supposed to be antisymmetric. 990 01:11:03,230 --> 01:11:04,750 Is it or not? 991 01:11:04,750 --> 01:11:14,244 a cross b sub i is equal to epsilon ijk aj bk. 992 01:11:16,980 --> 01:11:18,530 What do I do next? 993 01:11:18,530 --> 01:11:21,780 I want to move the a and b's around. 994 01:11:21,780 --> 01:11:26,970 So I'm going to replace this by a commutator 995 01:11:26,970 --> 01:11:30,550 plus the other ordering, ijk. 996 01:11:30,550 --> 01:11:36,130 And I put here-- well, this would be a parentheses-- aj bk 997 01:11:36,130 --> 01:11:40,860 commutator plus bk aj. 998 01:11:47,300 --> 01:11:52,290 Now, what do we get? 999 01:11:52,290 --> 01:11:55,870 Well, you have to look at the second term. 1000 01:11:55,870 --> 01:12:01,060 Let's put the first term here because that's-- pretty much, 1001 01:12:01,060 --> 01:12:04,370 we're not going to be able to do much with it. 1002 01:12:04,370 --> 01:12:05,564 aj bk. 1003 01:12:08,640 --> 01:12:11,400 And the first term, I would write it like this. 1004 01:12:11,400 --> 01:12:24,510 Minus epsilon, flip these two, ikj bk aj. 1005 01:12:24,510 --> 01:12:28,360 If you do it like that, then it sort of 1006 01:12:28,360 --> 01:12:32,100 fits nicely with the definition of the cross product. 1007 01:12:32,100 --> 01:12:37,320 Because in the cross product, the first label those here, 1008 01:12:37,320 --> 01:12:41,490 the second label goes with the last labeled of the epsilon. 1009 01:12:41,490 --> 01:12:50,400 So this thing is minus b cross a. 1010 01:12:50,400 --> 01:12:52,270 And that was a cross b. 1011 01:12:55,460 --> 01:13:01,690 Plus epsilon ijk aj bk. 1012 01:13:07,330 --> 01:13:11,520 So that's your formula for of how the cross product. 1013 01:13:11,520 --> 01:13:13,860 Now fails to be antisymmetric. 1014 01:13:16,700 --> 01:13:19,770 It's not necessarily antisymmetric 1015 01:13:19,770 --> 01:13:23,460 unless you're lucky. 1016 01:13:23,460 --> 01:13:33,700 So here is a property that you should try to think about. 1017 01:13:33,700 --> 01:13:36,950 How about r cross r? 1018 01:13:40,040 --> 01:13:44,095 Is it 0 or not 0? 1019 01:13:44,095 --> 01:13:45,770 r cross r. 1020 01:13:45,770 --> 01:13:49,970 You could say, what can I do here? 1021 01:13:49,970 --> 01:13:57,720 Well, r cross r minus r cross r-- 1022 01:13:57,720 --> 01:14:00,440 two r across r's should be equal to this. 1023 01:14:00,440 --> 01:14:03,740 But both components are x's. 1024 01:14:03,740 --> 01:14:06,355 So r cross r is 0. 1025 01:14:09,230 --> 01:14:20,060 p cross p, that's also 0. 1026 01:14:20,060 --> 01:14:25,145 And therefore, say, L cross L, is it 0? 1027 01:14:28,840 --> 01:14:29,340 Maybe. 1028 01:14:29,340 --> 01:14:29,946 Is that right? 1029 01:14:32,780 --> 01:14:34,492 No. 1030 01:14:34,492 --> 01:14:38,790 L cross L, we'll see it a little later, 1031 01:14:38,790 --> 01:14:44,070 but it's not 0 because one L with another L don't commute. 1032 01:14:47,740 --> 01:14:52,250 Lights, high. 1033 01:14:52,250 --> 01:14:52,830 OK. 1034 01:14:52,830 --> 01:14:58,330 So L cross L is actually not 0 because this thing is not 0. 1035 01:14:58,330 --> 01:15:03,010 We'll talk about it a little later. 1036 01:15:03,010 --> 01:15:06,480 But actually this is a very famous one. 1037 01:15:06,480 --> 01:15:11,050 L cross L is proportional to L with an h bar. 1038 01:15:11,050 --> 01:15:14,660 It's a lovely formula. 1039 01:15:14,660 --> 01:15:18,080 So another interesting thing. 1040 01:15:18,080 --> 01:15:25,520 Well, what is r cross p? 1041 01:15:25,520 --> 01:15:34,380 r cross p, from this formula we would minus p cross r. 1042 01:15:34,380 --> 01:15:36,120 And how about the other term? 1043 01:15:36,120 --> 01:15:37,860 Is it 0 or not 0? 1044 01:15:37,860 --> 01:15:42,380 Is r cross p equal to minus p cross r, or does that fail? 1045 01:15:46,200 --> 01:15:47,816 Well, r and p don't commute. 1046 01:15:50,390 --> 01:15:53,570 Actually this formula-- I should have-- somebody 1047 01:15:53,570 --> 01:15:55,050 should have complained. 1048 01:15:55,050 --> 01:15:58,690 This should be the i-th component, the i-th component, 1049 01:15:58,690 --> 01:16:00,050 and here is the i-th component. 1050 01:16:04,210 --> 01:16:08,120 So i-th component the i-th component. 1051 01:16:08,120 --> 01:16:09,990 But then let's look at here. 1052 01:16:09,990 --> 01:16:19,580 Epsilon ijk xj pk. 1053 01:16:19,580 --> 01:16:26,670 But xj and pk is i h bar delta jk. 1054 01:16:26,670 --> 01:16:30,060 Now, this delta jk is symmetric. 1055 01:16:30,060 --> 01:16:31,110 This is antisymmetric. 1056 01:16:34,260 --> 01:16:40,760 You should get accustomed to the idea that that is 0. 1057 01:16:40,760 --> 01:16:44,730 If this is antisymmetric and this is symmetric, this is 0. 1058 01:16:44,730 --> 01:16:47,770 Let me do it one more time. 1059 01:16:47,770 --> 01:16:50,620 Maybe this is not completely obvious. 1060 01:16:50,620 --> 01:16:56,460 But epsilon ijk delta jk. 1061 01:16:56,460 --> 01:16:58,810 The intuition is also obvious. 1062 01:16:58,810 --> 01:17:01,470 Epsilon must have the three numbers different 1063 01:17:01,470 --> 01:17:03,659 and this forces them to be the same. 1064 01:17:03,659 --> 01:17:04,575 But it's more general. 1065 01:17:04,575 --> 01:17:07,740 If this is antisymmetric and this is symmetric, 1066 01:17:07,740 --> 01:17:09,200 they should be 0. 1067 01:17:09,200 --> 01:17:16,560 And the way you do that is you say relabel j and k. 1068 01:17:16,560 --> 01:17:19,210 Whatever was called j call it k. 1069 01:17:19,210 --> 01:17:21,790 Whatever was called k call it j. 1070 01:17:21,790 --> 01:17:28,850 So this is epsilon ikj delta kj. 1071 01:17:28,850 --> 01:17:31,900 And then we use the symmetry properties. 1072 01:17:31,900 --> 01:17:35,240 So when you exchange this back you get a minus sign when 1073 01:17:35,240 --> 01:17:37,360 you exchange this back you don't get a sign. 1074 01:17:37,360 --> 01:17:42,690 So minus ijk delta jk. 1075 01:17:42,690 --> 01:17:47,480 So you have shown that this thing is equal to minus itself. 1076 01:17:47,480 --> 01:17:50,440 And therefore it's 0. 1077 01:17:50,440 --> 01:17:52,790 Something that's equal to minus itself is 0. 1078 01:17:52,790 --> 01:17:55,400 So this is 0. 1079 01:17:55,400 --> 01:18:02,910 And you've got that r cross p is really minus p cross r. 1080 01:18:02,910 --> 01:18:07,480 And that's what we call the angular momentum, r cross p. 1081 01:18:07,480 --> 01:18:11,320 But now you know that it's also equal to minus p cross r. 1082 01:18:18,490 --> 01:18:22,840 Let's see, one other one, for example, 1083 01:18:22,840 --> 01:18:24,520 that is classically true. 1084 01:18:24,520 --> 01:18:27,450 Let's see if it's quantum mechanically true. 1085 01:18:32,430 --> 01:18:38,430 r dot L. The angular momentum L is supposed 1086 01:18:38,430 --> 01:18:41,980 to be perpendicular to r and perpendicular 1087 01:18:41,980 --> 01:18:44,290 to p because this is the cross product. 1088 01:18:44,290 --> 01:18:50,440 Is that true quantum mechanically or not true? 1089 01:18:50,440 --> 01:18:56,550 Well, maybe I didn't say-- well, we said L is r cross p. 1090 01:18:56,550 --> 01:19:02,580 So this is r dot r cross p. 1091 01:19:06,510 --> 01:19:17,942 So this would be ri times epsilon ijk-- no, x-- xj pk. 1092 01:19:21,790 --> 01:19:24,250 So what is this? 1093 01:19:24,250 --> 01:19:30,554 It's epsilon ijk xi xj pk. 1094 01:19:35,110 --> 01:19:41,100 Well, this is 0 because these two x's are operators but they 1095 01:19:41,100 --> 01:19:42,130 commute. 1096 01:19:42,130 --> 01:19:46,180 Therefore this object is symmetrical in i and j. 1097 01:19:46,180 --> 01:19:48,070 This is antisymmetric. 1098 01:19:48,070 --> 01:19:49,430 This is 0. 1099 01:19:49,430 --> 01:19:53,540 So r dot L is actually 0. 1100 01:19:56,730 --> 01:20:02,660 How about p dot L. Well, there's two ways 1101 01:20:02,660 --> 01:20:05,470 of doing that, one that looks pretty obvious, 1102 01:20:05,470 --> 01:20:07,700 one that looks a little harder. 1103 01:20:07,700 --> 01:20:09,850 Let's do the one that looks a little harder. 1104 01:20:09,850 --> 01:20:21,400 This would be pi epsilon ijk xj pk. 1105 01:20:27,330 --> 01:20:30,910 So here is the temptation. 1106 01:20:30,910 --> 01:20:32,180 Must be 0. 1107 01:20:32,180 --> 01:20:34,840 Because there is a pi pk. 1108 01:20:34,840 --> 01:20:38,680 Symmetric in i and k. 1109 01:20:38,680 --> 01:20:41,175 And here's antisymmetric in i and k. 1110 01:20:41,175 --> 01:20:44,150 But it's this wrong. 1111 01:20:44,150 --> 01:20:46,030 These are not obviously symmetric 1112 01:20:46,030 --> 01:20:48,310 unless you can move them across. 1113 01:20:48,310 --> 01:20:52,710 And there's an x in the middle laughing at you 1114 01:20:52,710 --> 01:20:55,690 and saying, beware. 1115 01:20:55,690 --> 01:20:56,820 This could go wrong. 1116 01:20:59,630 --> 01:21:02,890 So you have to be a little more careful. 1117 01:21:02,890 --> 01:21:05,880 So let's be careful. 1118 01:21:05,880 --> 01:21:16,490 ijk and you put pi xj pk, all our operators. 1119 01:21:16,490 --> 01:21:23,050 But pi with xj the commutator would be a delta ij and that 1120 01:21:23,050 --> 01:21:23,710 vanishes. 1121 01:21:23,710 --> 01:21:31,240 So this p actually can be moved across the x and therefore show 1122 01:21:31,240 --> 01:21:37,240 that the p is on the other side. 1123 01:21:37,240 --> 01:21:39,350 Because the epsilon is there. 1124 01:21:39,350 --> 01:21:40,840 So it's a lucky thing. 1125 01:21:40,840 --> 01:21:48,490 So pi xj pk is not symmetrical in i and k. 1126 01:21:48,490 --> 01:21:51,240 But if there's an epsilon, you're in better shape. 1127 01:21:51,240 --> 01:22:00,260 So here the p can be moved. ijk xj pi pk. 1128 01:22:00,260 --> 01:22:04,930 And now nobody can stop you from saying this is symmetric. 1129 01:22:04,930 --> 01:22:08,790 They're next to each other, they can be moved across each other. 1130 01:22:08,790 --> 01:22:11,422 It's really symmetric in i and k. 1131 01:22:11,422 --> 01:22:15,370 And therefore i and k antisymmetric, this is 0. 1132 01:22:15,370 --> 01:22:19,790 p dot L is equal to 0 as well. 1133 01:22:19,790 --> 01:22:22,130 So this was a little bit hard way. 1134 01:22:22,130 --> 01:22:28,970 If you had to used that L is actually equal to p cross r, 1135 01:22:28,970 --> 01:22:32,480 then the p would have been next to the p from the beginning. 1136 01:22:32,480 --> 01:22:34,490 And you would have saved a little time. 1137 01:22:34,490 --> 01:22:36,780 The same time that it took you to realize 1138 01:22:36,780 --> 01:22:41,110 that this equality is true. 1139 01:22:41,110 --> 01:22:45,380 So those are true equalities. 1140 01:22:45,380 --> 01:22:47,160 p dot L is 0. 1141 01:22:47,160 --> 01:22:51,320 And it's also equal to L dot p. 1142 01:22:51,320 --> 01:22:54,400 It's not obvious that L and p commute. 1143 01:22:54,400 --> 01:22:58,930 But in this case L dot p is 0. 1144 01:22:58,930 --> 01:23:02,292 If you use this formula for L, it's obvious. 1145 01:23:02,292 --> 01:23:03,940 r dot L is 0. 1146 01:23:03,940 --> 01:23:07,650 It's also equal to L dot r. 1147 01:23:07,650 --> 01:23:09,440 Doesn't make a difference. 1148 01:23:09,440 --> 01:23:11,890 It's also true as well. 1149 01:23:11,890 --> 01:23:17,230 So that's roughly what goes on. 1150 01:23:17,230 --> 01:23:22,150 Look, I would like you to just read those pages. 1151 01:23:22,150 --> 01:23:24,200 It's a continuation of this. 1152 01:23:24,200 --> 01:23:27,230 It's about eight pages. 1153 01:23:27,230 --> 01:23:29,840 I leave exercises to be done. 1154 01:23:29,840 --> 01:23:31,180 That's in the homework. 1155 01:23:31,180 --> 01:23:33,430 They're of this type, playing with it. 1156 01:23:33,430 --> 01:23:36,690 And this is a good thing that you get accustomed to it. 1157 01:23:36,690 --> 01:23:39,880 And the material that you need just 1158 01:23:39,880 --> 01:23:42,690 for the last problem of the homework 1159 01:23:42,690 --> 01:23:44,430 will be covered on Monday. 1160 01:23:44,430 --> 01:23:47,740 And we can talk about it in recitation tomorrow. 1161 01:23:47,740 --> 01:23:50,750 So that's it for today.