1 00:00:00,050 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,810 under a Creative Commons license. 3 00:00:03,810 --> 00:00:06,540 Your support will help MIT OpenCourseWare continue 4 00:00:06,540 --> 00:00:10,110 to offer high quality educational resources for free. 5 00:00:10,110 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,590 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,590 --> 00:00:17,305 at ocw.mit.edu. 8 00:00:28,050 --> 00:00:32,729 PROFESSOR: Before we get started, 9 00:00:32,729 --> 00:00:36,180 let me ask you guys if you have any questions, 10 00:00:36,180 --> 00:00:39,630 pragmatic or otherwise, about the course so far. 11 00:00:43,051 --> 00:00:43,550 Seriously? 12 00:00:47,330 --> 00:00:49,370 To those of you reading newspapers, 13 00:00:49,370 --> 00:00:51,560 I encourage you to find a slightly different time 14 00:00:51,560 --> 00:00:52,230 to do so. 15 00:00:57,842 --> 00:01:00,050 I really encourage you find a slightly different time 16 00:01:00,050 --> 00:01:00,810 to do so, thanks. 17 00:01:06,740 --> 00:01:10,065 So far we've done basic rules of quantum mechanics. 18 00:01:10,065 --> 00:01:11,730 We've done solids. 19 00:01:11,730 --> 00:01:15,355 We understand a lot about electrons in atoms, 20 00:01:15,355 --> 00:01:21,385 the periodic table, and why diamond is transparent. 21 00:01:25,220 --> 00:01:27,614 One thing we did along the way is we talked about spin. 22 00:01:27,614 --> 00:01:29,780 We found that when we looked at the angular momentum 23 00:01:29,780 --> 00:01:33,100 commutation relations, these guys. 24 00:01:33,100 --> 00:01:35,420 We found that the commutators are the same. 25 00:01:35,420 --> 00:01:39,630 For these commutators, we can get total angular momentum, l l 26 00:01:39,630 --> 00:01:42,900 plus 1 times h bar squared for l squared. 27 00:01:42,900 --> 00:01:46,730 And h bar m for a constant and integer m for angular momentum 28 00:01:46,730 --> 00:01:49,970 in a particular direction, which we conventionally called z. 29 00:01:49,970 --> 00:01:52,540 But we also found that there were half integer values 30 00:01:52,540 --> 00:01:54,250 of the total spin and of the spin 31 00:01:54,250 --> 00:02:01,710 in a particular direction which, for example, with Spin 32 00:02:01,710 --> 00:02:05,430 s as 1/2, 3/2, 5/2, et cetera. 33 00:02:05,430 --> 00:02:10,330 And we discovered that riding a wave function on a sphere that 34 00:02:10,330 --> 00:02:11,980 interpreted these states as states 35 00:02:11,980 --> 00:02:15,560 with definite probability to be at a particular position 36 00:02:15,560 --> 00:02:20,270 on a sphere, a function of theta and phi, was inconsistent. 37 00:02:20,270 --> 00:02:23,900 In order for the wave function to satisfy those properties 38 00:02:23,900 --> 00:02:26,400 that have those eigenvalues and, in particular, half integer 39 00:02:26,400 --> 00:02:27,780 eigenvalues, we discovered that the wave function 40 00:02:27,780 --> 00:02:29,210 had to be doubly valued. 41 00:02:29,210 --> 00:02:31,940 And thus it would equal to minus itself, it equaled 0. 42 00:02:31,940 --> 00:02:33,480 So the rest of today and tomorrow 43 00:02:33,480 --> 00:02:36,100 is going to be an exploration of spin, or tomorrow-- 44 00:02:36,100 --> 00:02:38,550 next lecture, is going to be an exploration of spin 45 00:02:38,550 --> 00:02:41,260 and the consequences of these spin 1/2 states. 46 00:02:41,260 --> 00:02:42,917 Exactly what they are, we already 47 00:02:42,917 --> 00:02:44,750 saw that they're important for understanding 48 00:02:44,750 --> 00:02:45,810 the structure of the periodic table. 49 00:02:45,810 --> 00:02:46,930 So we know they're there. 50 00:02:46,930 --> 00:02:49,740 And they're present from the experiment, 51 00:02:49,740 --> 00:02:51,760 the Stern-Gerlach experiment that we've 52 00:02:51,760 --> 00:02:54,450 discussed many times. 53 00:02:54,450 --> 00:02:57,030 But before I get onto that, I want 54 00:02:57,030 --> 00:02:59,692 to improve on the last experiment we did. 55 00:02:59,692 --> 00:03:01,400 So in particular, in this last experiment 56 00:03:01,400 --> 00:03:04,770 we talked about the effective mass of an object interacting 57 00:03:04,770 --> 00:03:06,740 with a fluid or an object interacting 58 00:03:06,740 --> 00:03:07,790 with its environment. 59 00:03:07,790 --> 00:03:10,680 And why the mass of the object that's moving 60 00:03:10,680 --> 00:03:12,740 is not the same as the mass of the object 61 00:03:12,740 --> 00:03:15,440 when you put it on a balance. 62 00:03:15,440 --> 00:03:17,710 And that's this basic idea of renormalization. 63 00:03:17,710 --> 00:03:18,890 And we demonstrated that. 64 00:03:18,890 --> 00:03:22,450 I demonstrated that with a beaker of water. 65 00:03:22,450 --> 00:03:24,700 So I had a beaker of water and pulled a ping pong ball 66 00:03:24,700 --> 00:03:25,080 under water. 67 00:03:25,080 --> 00:03:26,996 We calculated that it should accelerate upward 68 00:03:26,996 --> 00:03:30,400 when released at 20 times the acceleration of gravity, 69 00:03:30,400 --> 00:03:32,870 depending on the numbers you use, very, very rapidly. 70 00:03:32,870 --> 00:03:33,840 And in fact it went glug, glug, glug, 71 00:03:33,840 --> 00:03:35,390 but it wasn't a terribly satisfying experiment 72 00:03:35,390 --> 00:03:37,348 because it's very hard to get the timing right. 73 00:03:37,348 --> 00:03:41,070 And the basic issue there is that the time scales involved 74 00:03:41,070 --> 00:03:41,990 were very short. 75 00:03:41,990 --> 00:03:43,120 How long did it take for a ping pong 76 00:03:43,120 --> 00:03:44,703 ball to drop from here to the surface? 77 00:03:44,703 --> 00:03:45,540 Not much time. 78 00:03:45,540 --> 00:03:48,110 And to rise through the water, not a whole lot of time. 79 00:03:48,110 --> 00:03:50,350 So I did that experiment, it was sort of comical. 80 00:03:50,350 --> 00:03:51,600 But I wanted to improve on it. 81 00:03:51,600 --> 00:03:53,900 So over the weekend, I went down to my basement 82 00:03:53,900 --> 00:03:56,800 and tweaked the experiment a little bit. 83 00:03:56,800 --> 00:03:59,850 And I called up a couple of my friends, 84 00:03:59,850 --> 00:04:02,615 and we did an improved version of this experiment. 85 00:04:06,340 --> 00:04:09,136 So this is a diver, I think this one is Kathy. 86 00:04:09,136 --> 00:04:10,885 Oh, shoot, we need to turn off the lights. 87 00:04:13,940 --> 00:04:15,000 Sorry. 88 00:04:15,000 --> 00:04:17,860 I totally forgot. 89 00:04:17,860 --> 00:04:21,100 good You'll never see this if we don't. 90 00:04:24,270 --> 00:04:26,620 You can do it. 91 00:04:26,620 --> 00:04:27,770 There we go. 92 00:04:27,770 --> 00:04:28,810 All right. 93 00:04:28,810 --> 00:04:31,290 So here you see my friend, I think this one's, actually, 94 00:04:31,290 --> 00:04:31,910 is it Kathy? 95 00:04:31,910 --> 00:04:33,440 I'm not sure. 96 00:04:33,440 --> 00:04:38,270 It's hard to tell when they have their marks on. 97 00:04:38,270 --> 00:04:41,570 And we're in the tank at the New England Aquarium, 98 00:04:41,570 --> 00:04:44,169 and she's going to perform this experiment for us. 99 00:04:44,169 --> 00:04:45,960 And we're going to film it, as you can see, 100 00:04:45,960 --> 00:04:47,334 the bubbles moving rather slowly, 101 00:04:47,334 --> 00:04:51,520 with a high speed camera filming at 1,200 frames per second 102 00:04:51,520 --> 00:04:54,540 with which we'll be able to analyze the data that results. 103 00:04:54,540 --> 00:04:58,490 The camera cost as much as a nice house. 104 00:04:58,490 --> 00:05:00,960 And it's not mine, but it's important to have 105 00:05:00,960 --> 00:05:02,630 friends who trust you. 106 00:05:02,630 --> 00:05:03,710 OK, so here we are. 107 00:05:03,710 --> 00:05:06,404 You might notice something in the background. 108 00:05:06,404 --> 00:05:07,820 Before we get started, I just want 109 00:05:07,820 --> 00:05:12,042 to emphasize that one should never take casually 110 00:05:12,042 --> 00:05:13,500 the dangers of doing an experiment. 111 00:05:13,500 --> 00:05:16,474 When you plan an experiment you must, ahead of time, 112 00:05:16,474 --> 00:05:18,890 design the experiment, design the experimental parameters. 113 00:05:18,890 --> 00:05:19,810 We designed the lighting. 114 00:05:19,810 --> 00:05:20,870 We designed everything. 115 00:05:20,870 --> 00:05:22,310 We set it up, but there are always 116 00:05:22,310 --> 00:05:23,320 variables you haven't accounted for. 117 00:05:23,320 --> 00:05:24,694 And a truly great experimentalist 118 00:05:24,694 --> 00:05:26,930 is one who has taken account of all the variables. 119 00:05:26,930 --> 00:05:28,638 And I just want to emphasize that I'm not 120 00:05:28,638 --> 00:05:29,870 a great experimentalist. 121 00:05:29,870 --> 00:05:33,780 So here, for example, is a moment. 122 00:07:21,901 --> 00:07:23,650 I probably should have thanked the sharks, 123 00:07:23,650 --> 00:07:26,590 it just occurred to me. 124 00:07:26,590 --> 00:07:29,540 Anyone who wants to take this experimental data, which, 125 00:07:29,540 --> 00:07:32,040 as you can probably guess, filmed for different purposes, 126 00:07:32,040 --> 00:07:36,720 I just manage to get the ping pong ball into the tank. 127 00:07:36,720 --> 00:07:40,330 Anyone who wants to get this and actually take the data, 128 00:07:40,330 --> 00:07:42,667 come to me, I'll give you the raw footage. 129 00:07:42,667 --> 00:07:44,125 And you can read off the positions, 130 00:07:44,125 --> 00:07:46,070 and hopefully for next lecture we'll 131 00:07:46,070 --> 00:07:48,130 have the actual plot of the acceleration 132 00:07:48,130 --> 00:07:50,990 as a function of time. 133 00:07:50,990 --> 00:08:00,762 So with that said and done, the moral of the story 134 00:08:00,762 --> 00:08:02,470 is you have to account for all variables. 135 00:08:02,470 --> 00:08:06,190 The other moral of the story is that you saw this very vividly. 136 00:08:06,190 --> 00:08:09,240 In the motion of the ping pong ball up, when it was released, 137 00:08:09,240 --> 00:08:13,120 there is that very rapid moment of acceleration 138 00:08:13,120 --> 00:08:15,560 when it bursts up very, very rapidly. 139 00:08:15,560 --> 00:08:18,045 But it quickly slows in its acceleration. 140 00:08:18,045 --> 00:08:19,274 Its acceleration slows down. 141 00:08:19,274 --> 00:08:20,940 In fact, a slightly funny thing happens. 142 00:08:20,940 --> 00:08:22,671 If you look carefully, and again, 143 00:08:22,671 --> 00:08:24,170 anyone who wants this can get access 144 00:08:24,170 --> 00:08:26,890 to the video, what you'll see is that the ping pong 145 00:08:26,890 --> 00:08:29,092 ball accelerates and then it sort of slows down. 146 00:08:29,092 --> 00:08:30,550 It literally decreases in velocity, 147 00:08:30,550 --> 00:08:31,900 accelerates and slows down. 148 00:08:31,900 --> 00:08:34,025 Can anyone think what's going on in that situation? 149 00:08:36,802 --> 00:08:38,260 AUDIENCE: Boundary layer formation. 150 00:08:38,260 --> 00:08:38,644 PROFESSOR: Sorry? 151 00:08:38,644 --> 00:08:39,919 AUDIENCE: Boundary layer formation. 152 00:08:39,919 --> 00:08:40,585 PROFESSOR: Good. 153 00:08:40,585 --> 00:08:42,506 Say that in slightly more words. 154 00:08:42,506 --> 00:08:44,971 AUDIENCE: It's starting to pick up 155 00:08:44,971 --> 00:08:48,284 more and more water [INAUDIBLE]. 156 00:08:48,284 --> 00:08:48,950 PROFESSOR: Yeah. 157 00:08:48,950 --> 00:08:51,283 Exactly. so what's going on is as this guy starts slowly 158 00:08:51,283 --> 00:08:53,816 moving along, it's pulling along more and more water, 159 00:08:53,816 --> 00:08:55,440 each bit of water around it is starting 160 00:08:55,440 --> 00:08:57,310 to drag along the layers of water nearby, 161 00:08:57,310 --> 00:08:58,930 and it builds up a sheath of water. 162 00:08:58,930 --> 00:09:02,250 Now that water starts accelerating, and the ping pong 163 00:09:02,250 --> 00:09:07,007 ball and the bubble of water that it's dragging along 164 00:09:07,007 --> 00:09:08,840 need to come to equilibrium with each other. 165 00:09:08,840 --> 00:09:11,140 They need to settle down smoothly to a nice uniform 166 00:09:11,140 --> 00:09:11,880 velocity. 167 00:09:11,880 --> 00:09:13,710 But it takes a while for that equilibrium to happen. 168 00:09:13,710 --> 00:09:15,345 And what actually happens is that the ping pong 169 00:09:15,345 --> 00:09:16,090 ball drives up. 170 00:09:16,090 --> 00:09:17,930 It pulls up the water. 171 00:09:17,930 --> 00:09:19,930 Which then drags with the ping pong ball. 172 00:09:19,930 --> 00:09:21,860 So you can see that in the acceleration, which 173 00:09:21,860 --> 00:09:24,660 is oscillatory with a slight little oscillation. 174 00:09:24,660 --> 00:09:27,550 So it's a damped but not overdamped harmonic 175 00:09:27,550 --> 00:09:29,080 oscillator motion. 176 00:09:29,080 --> 00:09:31,190 Any other questions about the effective mass 177 00:09:31,190 --> 00:09:33,120 of an electron and solid before moving-- Yeah? 178 00:09:33,120 --> 00:09:35,480 AUDIENCE: Why does it speed up after? 179 00:09:35,480 --> 00:09:37,840 That explains why it slows down because it's 180 00:09:37,840 --> 00:09:38,790 forming that sheath-- 181 00:09:38,790 --> 00:09:39,880 PROFESSOR: Right. 182 00:09:39,880 --> 00:09:42,240 Why it speeds back up is it's sort of like a slingshot. 183 00:09:42,240 --> 00:09:45,377 As this guy gets going a little ahead of the pack of water, 184 00:09:45,377 --> 00:09:47,210 the pack of water has an extra driving force 185 00:09:47,210 --> 00:09:48,349 on top of the buoyancy. 186 00:09:48,349 --> 00:09:50,390 It has the fact that it's a little bit displaced. 187 00:09:50,390 --> 00:09:52,690 So it catches up, but it's going slightly greater velocity 188 00:09:52,690 --> 00:09:54,690 than it would be if there were uniform velocity. 189 00:09:54,690 --> 00:09:56,710 So this guy catches up with the ping pong ball. 190 00:10:02,090 --> 00:10:05,310 OK so this is an of course of course in fluid mechanics, 191 00:10:05,310 --> 00:10:13,840 but I guess we don't actually need this anymore. 192 00:10:13,840 --> 00:10:16,490 Picking up on spin. 193 00:10:16,490 --> 00:10:20,000 So the commutation relations for spin are these. 194 00:10:20,000 --> 00:10:22,660 And as we saw last time, we have spin states. 195 00:10:22,660 --> 00:10:26,510 We have we can construct towers of states because from the sx 196 00:10:26,510 --> 00:10:32,530 and s1 we can build s plus-minus is equal to sx plus-minus sy. 197 00:10:40,880 --> 00:10:41,700 Sorry. 198 00:10:41,700 --> 00:10:42,630 i, thank you. 199 00:10:45,380 --> 00:10:48,960 So we can build towers of states using the raising 200 00:10:48,960 --> 00:10:52,990 and lowering operators as plus-minus. 201 00:10:52,990 --> 00:10:55,500 And those states need to end, they need to terminate. 202 00:10:55,500 --> 00:10:58,310 So we find that the spin can have totalling momentum of s 203 00:10:58,310 --> 00:11:04,730 squared h bar squared l l plus 1. 204 00:11:07,310 --> 00:11:09,630 And S in some particular direction, 205 00:11:09,630 --> 00:11:13,280 which we conventionally called z, is h bar m. 206 00:11:23,040 --> 00:11:24,250 I don't want to call this l. 207 00:11:24,250 --> 00:11:27,440 I want to call this s. 208 00:11:27,440 --> 00:11:29,430 For orbital angular momentum this would be l 209 00:11:29,430 --> 00:11:30,720 and this would be an integer. 210 00:11:30,720 --> 00:11:32,330 But for spinning angular momentum 211 00:11:32,330 --> 00:11:33,996 these are all the states we could build, 212 00:11:33,996 --> 00:11:40,310 all the towers we could build, which were 2n plus 1 over 2, 213 00:11:40,310 --> 00:11:43,240 which were not expressible in terms of wave functions, 214 00:11:43,240 --> 00:11:45,370 functions of a position on a sphere. 215 00:11:45,370 --> 00:11:46,950 These are all the 1/2 integer states. 216 00:11:46,950 --> 00:11:49,790 So s could be 1/2, 3/2, 5/2, and so on. 217 00:11:49,790 --> 00:11:53,780 And then m sub s is going to go from minus s to s 218 00:11:53,780 --> 00:12:00,690 in integer steps, just like m for the orbital angular 219 00:12:00,690 --> 00:12:02,450 momentum, l. 220 00:12:02,450 --> 00:12:03,950 So I want to talk about these states 221 00:12:03,950 --> 00:12:08,860 in some detail over the rest of this lecture and the next one. 222 00:12:08,860 --> 00:12:13,645 So the first thing to talk about is how we describe spin. 223 00:12:13,645 --> 00:12:14,770 In principle, this is easy. 224 00:12:14,770 --> 00:12:16,020 What we want, is we want to describe 225 00:12:16,020 --> 00:12:18,650 the state of a particle that carries this intrinsic angular 226 00:12:18,650 --> 00:12:19,860 momentum spin. 227 00:12:19,860 --> 00:12:20,700 So that's easy. 228 00:12:20,700 --> 00:12:23,600 The particle sits at some point, but the problem 229 00:12:23,600 --> 00:12:27,670 is it could be sitting at some point with angular momentum 230 00:12:27,670 --> 00:12:33,640 with spin in the z direction, say, plus h bar over 2. 231 00:12:33,640 --> 00:12:37,319 And let's focus on the case s is equal to 1/2. 232 00:12:37,319 --> 00:12:38,860 So I'll be focusing, for the lecture, 233 00:12:38,860 --> 00:12:41,910 for simplicity on the case, total spin 234 00:12:41,910 --> 00:12:44,480 is 1/2, which is the two-state ladder, 235 00:12:44,480 --> 00:12:46,220 but all of this generalizes naturally. 236 00:12:46,220 --> 00:12:49,220 In fact, that's a very good test of your understanding. 237 00:12:49,220 --> 00:12:51,350 So for s as 1/2, we have two states, 238 00:12:51,350 --> 00:12:55,160 which I will conventionally call the up in the z direction 239 00:12:55,160 --> 00:12:57,711 and the down in the z direction. 240 00:12:57,711 --> 00:12:59,210 And I will often omit the subscript. 241 00:12:59,210 --> 00:13:01,730 If I omit the subscript it's usually z 242 00:13:01,730 --> 00:13:06,830 unless from context you see that it's something else. 243 00:13:06,830 --> 00:13:10,732 So the wave function tells us the state of the system. 244 00:13:10,732 --> 00:13:12,690 But we need to know now for a spinning particle 245 00:13:12,690 --> 00:13:18,110 whether it's in the spin 1/2 up or spin 1/2 down state. 246 00:13:18,110 --> 00:13:21,360 And so we could write that as there's some amplitude 247 00:13:21,360 --> 00:13:26,370 that it's in the plus 1/2 state of x and at position x. 248 00:13:26,370 --> 00:13:28,920 And there's some amplitude that it's at position x 249 00:13:28,920 --> 00:13:32,270 and it's in the minus 1/2 state or the down state. 250 00:13:35,820 --> 00:13:43,192 And we again need that the total probability is one. 251 00:13:43,192 --> 00:13:45,150 Another way to say this is that the probability 252 00:13:45,150 --> 00:13:50,234 that we find the particle to be at x with plus or minus h bar 253 00:13:50,234 --> 00:13:53,380 upon 2 being the spin in the z direction. 254 00:13:53,380 --> 00:13:57,660 So at x, spin in the z direction is h bar upon 2 plus or minus 255 00:13:57,660 --> 00:14:04,145 is equal to norm squared of psi plus or minus of x squared. 256 00:14:04,145 --> 00:14:06,020 So this is one way you could talk about spin, 257 00:14:06,020 --> 00:14:08,630 and you could develop the theory of spin nicely here. 258 00:14:08,630 --> 00:14:11,632 But it's a somewhat cumbersome formalism. 259 00:14:11,632 --> 00:14:13,090 The formulation I want to introduce 260 00:14:13,090 --> 00:14:15,900 is one which involves matrices and which 261 00:14:15,900 --> 00:14:18,680 presages the study of matrix mechanics 262 00:14:18,680 --> 00:14:20,505 which you'll be using in 805. 263 00:14:20,505 --> 00:14:22,505 So instead, I want to take these two components, 264 00:14:22,505 --> 00:14:25,420 and what we see already is that we can't use a single wave 265 00:14:25,420 --> 00:14:27,870 function to describe a particle at spin 1/2. 266 00:14:27,870 --> 00:14:30,320 We need to use two functions. 267 00:14:30,320 --> 00:14:32,070 And I want to organize them in a nice way. 268 00:14:32,070 --> 00:14:34,300 I'm going to write them as psi is 269 00:14:34,300 --> 00:14:38,940 equal to-- and I'll call this capital psi of x-- 270 00:14:38,940 --> 00:14:42,630 is a two component vector, or so-called spinner, 271 00:14:42,630 --> 00:14:46,115 psi up of x and psi down of x. 272 00:14:46,115 --> 00:14:50,560 So it's a two component object. 273 00:14:50,560 --> 00:14:54,190 It's got a top and a bottom component. 274 00:14:54,190 --> 00:14:57,720 And notice that its conjugate, or its adjoint, psi dagger, 275 00:14:57,720 --> 00:15:00,570 is going to be equal to psi up, complex 276 00:15:00,570 --> 00:15:04,790 conjugate psi down, a row vector, or a row spinner. 277 00:15:10,740 --> 00:15:15,860 And for normalization we'll need that the total probability 278 00:15:15,860 --> 00:15:19,400 is 1 which says that psi capital dagger 279 00:15:19,400 --> 00:15:34,000 psi, psi capital with psi is equal to 1. 280 00:15:34,000 --> 00:15:36,266 But this is going to be equal to the integral dx. 281 00:15:36,266 --> 00:15:38,432 And now we have to take the inner product of the two 282 00:15:38,432 --> 00:15:40,120 vectors. 283 00:15:40,120 --> 00:15:49,740 So integral dx of psi up squared plus psi down squared. 284 00:15:49,740 --> 00:15:50,740 Cool? 285 00:15:50,740 --> 00:15:51,724 Yep? 286 00:15:51,724 --> 00:15:55,157 AUDIENCE: What's the coordinate x representing here? 287 00:15:55,157 --> 00:15:57,740 PROFESSOR: The coordinate x is just representing the position. 288 00:15:57,740 --> 00:15:59,030 So what I'm saying here is I have, 289 00:15:59,030 --> 00:16:01,113 again, we're in one dimension just for simplicity, 290 00:16:01,113 --> 00:16:05,920 it's saying, look if I have a particle that carries spin 1/2, 291 00:16:05,920 --> 00:16:06,912 it could be anywhere. 292 00:16:06,912 --> 00:16:08,120 Let's say it's at position x. 293 00:16:08,120 --> 00:16:10,369 So what's the amplitude at position x and spinning up, 294 00:16:10,369 --> 00:16:12,500 and I'm not going to indicate spinning down. 295 00:16:12,500 --> 00:16:13,000 OK? 296 00:16:16,750 --> 00:16:17,500 I like my coffee. 297 00:16:20,050 --> 00:16:22,090 So that's what the of x indicates, and I've just 298 00:16:22,090 --> 00:16:23,145 been dropping the of x. 299 00:16:25,957 --> 00:16:28,290 So there's some probability that it's at any given point 300 00:16:28,290 --> 00:16:29,940 and either spin up or spin down. 301 00:16:29,940 --> 00:16:31,670 Now, again, it's important, although I'm going to do this, 302 00:16:31,670 --> 00:16:33,930 and we conventionally do this spin up and down, 303 00:16:33,930 --> 00:16:35,592 this spin is pointing in a vector 304 00:16:35,592 --> 00:16:36,800 space that's two dimensional. 305 00:16:36,800 --> 00:16:42,334 It's either plus 1/2 or minus 1/2 h bar upon 2 h bar. 306 00:16:42,334 --> 00:16:43,875 So it's not like the spin is an arrow 307 00:16:43,875 --> 00:16:45,920 in three dimensional space that points. 308 00:16:45,920 --> 00:16:47,930 Rather, what it is, it's saying, if I measure 309 00:16:47,930 --> 00:16:52,090 the spin along some axis, it can take one of two values. 310 00:16:52,090 --> 00:16:54,520 And that was shown in the Stern-Gerlach experiment, where 311 00:16:54,520 --> 00:16:59,970 if we have a gradient of the magnetic field, dbz, 312 00:16:59,970 --> 00:17:06,220 in the z direction, and this is our Stern-Gerlach box, 313 00:17:06,220 --> 00:17:08,680 and we send an electron in, the electron always 314 00:17:08,680 --> 00:17:12,140 comes out in one of two positions. 315 00:17:12,140 --> 00:17:13,920 OK. 316 00:17:13,920 --> 00:17:17,020 Now, this is not saying there is a vector associated with this, 317 00:17:17,020 --> 00:17:20,160 that the electron has a angular momentum vector that 318 00:17:20,160 --> 00:17:21,910 points in some particular direction. 319 00:17:21,910 --> 00:17:24,900 Rather, it's saying that there are two possible values, 320 00:17:24,900 --> 00:17:26,650 and we're measuring along the z direction. 321 00:17:26,650 --> 00:17:27,680 Cool? 322 00:17:27,680 --> 00:17:29,430 So it's important not to make that mistake 323 00:17:29,430 --> 00:17:33,460 to think of this as some three dimensional vector. 324 00:17:33,460 --> 00:17:36,760 It's very explicitly a vector in a two dimensional vector space. 325 00:17:36,760 --> 00:17:38,460 It's not related to regular rotations. 326 00:17:38,460 --> 00:17:39,433 Yeah? 327 00:17:39,433 --> 00:17:43,377 AUDIENCE: Where you write psi of x equals psi plus 1 328 00:17:43,377 --> 00:17:44,856 plus psi [INAUDIBLE]. 329 00:17:44,856 --> 00:17:45,842 PROFESSOR: Yes. 330 00:17:45,842 --> 00:17:47,978 AUDIENCE: Do we need a 1 over square root of 2 331 00:17:47,978 --> 00:17:49,310 in front of that thing? 332 00:17:49,310 --> 00:17:52,280 PROFESSOR: Yeah, I haven't assumed their normalization, 333 00:17:52,280 --> 00:17:55,050 but each one could be independently normalized 334 00:17:55,050 --> 00:17:56,000 appropriately. 335 00:17:56,000 --> 00:17:57,132 AUDIENCE: So [INAUDIBLE]. 336 00:17:57,132 --> 00:17:57,840 PROFESSOR: Right. 337 00:17:57,840 --> 00:17:59,900 The whole thing has to be properly normalized, 338 00:17:59,900 --> 00:18:02,951 and writing it this way, this is just the [INAUDIBLE]. 339 00:18:02,951 --> 00:18:03,450 Good. 340 00:18:06,510 --> 00:18:09,080 So there's another nice bit of notation 341 00:18:09,080 --> 00:18:11,890 for this which is often used, which 342 00:18:11,890 --> 00:18:14,230 is probably the most common notation. 343 00:18:14,230 --> 00:18:21,060 Which is to write psi of x is equal to psi up 344 00:18:21,060 --> 00:18:26,650 at x times the vector 1, 0 plus psi down of x times the vector 345 00:18:26,650 --> 00:18:27,150 0, 1. 346 00:18:30,540 --> 00:18:32,550 So this is what I'm going to refer to as the up 347 00:18:32,550 --> 00:18:34,052 vector in the z direction. 348 00:18:34,052 --> 00:18:35,510 And this is what I'm going to refer 349 00:18:35,510 --> 00:18:37,840 to as the down in the z direction vector. 350 00:18:45,120 --> 00:18:51,330 And that's going to allow me to write all the operations we're 351 00:18:51,330 --> 00:18:54,020 going to need to study spin in terms of simple two 352 00:18:54,020 --> 00:18:54,900 by two matrices. 353 00:18:54,900 --> 00:18:55,400 Yeah? 354 00:18:55,400 --> 00:18:57,392 AUDIENCE: Will those two psi's not be the same? 355 00:18:57,392 --> 00:18:59,100 PROFESSOR: Yeah, in general, they're not. 356 00:18:59,100 --> 00:19:02,065 So for example, here's a situation, a configuration, 357 00:19:02,065 --> 00:19:03,570 a quantum system could be in. 358 00:19:03,570 --> 00:19:05,611 The quantum system could be in the configuration, 359 00:19:05,611 --> 00:19:07,361 the particle is here and it's spinning up. 360 00:19:07,361 --> 00:19:08,860 And it could be in the configuration 361 00:19:08,860 --> 00:19:11,070 the particles over here, and it's spinning down. 362 00:19:11,070 --> 00:19:12,800 And given that it could be in those two configurations 363 00:19:12,800 --> 00:19:14,410 it could also be in the superposition 364 00:19:14,410 --> 00:19:17,090 over here and up and over here and down. 365 00:19:17,090 --> 00:19:17,669 Right? 366 00:19:17,669 --> 00:19:19,710 So that would be different spatial wave functions 367 00:19:19,710 --> 00:19:21,330 multiplying the different spin wave 368 00:19:21,330 --> 00:19:23,330 functions, spin part of the wave function. 369 00:19:23,330 --> 00:19:24,091 Make sense? 370 00:19:24,091 --> 00:19:24,590 OK. 371 00:19:24,590 --> 00:19:26,267 So these are not the same function. 372 00:19:26,267 --> 00:19:27,600 They could be the same function. 373 00:19:27,600 --> 00:19:31,020 It could be that you could be either spin up or spin down 374 00:19:31,020 --> 00:19:34,152 at any given point with some funny distribution, 375 00:19:34,152 --> 00:19:35,610 but they don't need to be the same. 376 00:19:35,610 --> 00:19:38,130 That's the crucial thing. 377 00:19:38,130 --> 00:19:39,380 Other questions? 378 00:19:39,380 --> 00:19:40,734 Yeah. 379 00:19:40,734 --> 00:19:42,542 AUDIENCE: I know that [INAUDIBLE] 380 00:19:42,542 --> 00:19:44,285 is a way to call them. 381 00:19:44,285 --> 00:19:47,930 So like [INAUDIBLE] something, are they 382 00:19:47,930 --> 00:19:51,524 anti-parallel, perpendicular, or are they something? 383 00:19:51,524 --> 00:19:52,190 PROFESSOR: Yeah. 384 00:19:52,190 --> 00:19:54,100 So here's what we can say. 385 00:19:54,100 --> 00:19:56,880 We know that an electron which carries sz 386 00:19:56,880 --> 00:20:00,530 is plus h bar upon 2, and a state corresponding 387 00:20:00,530 --> 00:20:02,960 to minus h bar upon 2 are orthogonal because those 388 00:20:02,960 --> 00:20:07,080 are two different eigenvectors of the same operator, sz. 389 00:20:07,080 --> 00:20:09,070 So these guys are orthogonal. 390 00:20:09,070 --> 00:20:10,710 Up in the z direction and down to the z 391 00:20:10,710 --> 00:20:12,070 directions are orthogonal. 392 00:20:12,070 --> 00:20:13,680 And thank you for this question, it's 393 00:20:13,680 --> 00:20:15,264 a good way to think about how wrong it 394 00:20:15,264 --> 00:20:17,930 is to think of up and down being up and down in the z direction. 395 00:20:17,930 --> 00:20:19,100 Are these guys orthogonal? 396 00:20:19,100 --> 00:20:20,409 These vectors in space? 397 00:20:20,409 --> 00:20:20,950 AUDIENCE: No. 398 00:20:20,950 --> 00:20:24,900 PROFESSOR: No, they happen to be parallel with a minus 1 399 00:20:24,900 --> 00:20:26,180 in the overlap, right? 400 00:20:26,180 --> 00:20:29,830 So sz as up sc as down are orthogonal vectors, 401 00:20:29,830 --> 00:20:31,850 but this is clearly not sz as down, right? 402 00:20:31,850 --> 00:20:34,640 So the direction that they're pointing, the up and down, 403 00:20:34,640 --> 00:20:36,510 should not be thought of as a direction 404 00:20:36,510 --> 00:20:38,143 in three-dimensional space. 405 00:20:38,143 --> 00:20:38,910 AUDIENCE: It's just [INAUDIBLE]. 406 00:20:38,910 --> 00:20:40,535 PROFESSOR: It's just a different thing. 407 00:20:40,535 --> 00:20:44,080 What it does tell you, is if you rotate the system by a given 408 00:20:44,080 --> 00:20:46,450 amount, how does the phase of the wave function change. 409 00:20:53,760 --> 00:20:56,950 But what it does tell you is how the spin operations act on it. 410 00:20:56,950 --> 00:20:59,211 In particular, sz acts with a plus 1/2 or minus 1/2. 411 00:20:59,211 --> 00:21:00,460 They're just different states. 412 00:21:00,460 --> 00:21:01,724 Yeah? 413 00:21:01,724 --> 00:21:03,215 AUDIENCE: Is this similar to what 414 00:21:03,215 --> 00:21:05,700 happens when polarizations [INAUDIBLE]? 415 00:21:05,700 --> 00:21:08,380 PROFESSOR: It's similar to the story of polarizations 416 00:21:08,380 --> 00:21:10,960 except polarizations are vectors not spinners. 417 00:21:10,960 --> 00:21:14,324 It's similar in the sense that they look and smell 418 00:21:14,324 --> 00:21:15,990 like vectors in three-dimensional space, 419 00:21:15,990 --> 00:21:17,470 but they mean slightly-- technically-- slightly 420 00:21:17,470 --> 00:21:18,430 different things. 421 00:21:18,430 --> 00:21:21,737 In the case of polarizations of light, 422 00:21:21,737 --> 00:21:23,570 those really are honest vectors, and there's 423 00:21:23,570 --> 00:21:27,000 a sharp relationship between rotations in space. 424 00:21:27,000 --> 00:21:28,270 But that's a sort of quirk. 425 00:21:28,270 --> 00:21:31,640 They're both spin and vectors. 426 00:21:31,640 --> 00:21:33,990 These are not. 427 00:21:33,990 --> 00:21:37,750 OK so here's a notation I'm going to use. 428 00:21:37,750 --> 00:21:40,550 Just to alert you, a common notation 429 00:21:40,550 --> 00:21:42,440 that people use in Dirac notation 430 00:21:42,440 --> 00:21:49,130 is to say that the wave function is equal to psi up at x times 431 00:21:49,130 --> 00:21:56,890 the state up plus psi down at x times the state down. 432 00:21:56,890 --> 00:22:00,370 So for those of you who speak Dirac notation at this point, 433 00:22:00,370 --> 00:22:04,259 then this means the same thing as this. 434 00:22:04,259 --> 00:22:06,300 For those of you who don't, then this means this. 435 00:22:11,030 --> 00:22:13,690 What I want to do is I want to develop a theory of the spin 436 00:22:13,690 --> 00:22:15,340 operators, and I want to understand 437 00:22:15,340 --> 00:22:17,595 what it means to be a spin 1/2 state. 438 00:22:17,595 --> 00:22:20,220 Now, in particular, what I mean by develop a theory of the spin 439 00:22:20,220 --> 00:22:21,980 operators, if I was talking about four orbital angular 440 00:22:21,980 --> 00:22:23,730 momentum, say the orbital angular momentum 441 00:22:23,730 --> 00:22:26,080 in the z direction, I know what operator this is. 442 00:22:26,080 --> 00:22:29,860 If you hand me a wave function, I can act on it with lz 443 00:22:29,860 --> 00:22:31,239 and tell you what the result is. 444 00:22:31,239 --> 00:22:33,530 And that means that I can construct the eigenfunctions. 445 00:22:33,530 --> 00:22:36,790 And that means I can construct the allowed eigenvalues, 446 00:22:36,790 --> 00:22:38,610 and I can talk about probabilities. 447 00:22:38,610 --> 00:22:39,194 Right? 448 00:22:39,194 --> 00:22:40,610 But in order to do all that I need 449 00:22:40,610 --> 00:22:43,060 to know how the operator acts. 450 00:22:43,060 --> 00:22:44,800 And we know how this operator acts. 451 00:22:44,800 --> 00:22:50,790 It acts as h bar upon i dd phi where phi is the angular 452 00:22:50,790 --> 00:22:53,580 coordinate around the equator. 453 00:22:53,580 --> 00:22:56,160 And so given any wave function, a function of x, y, and z 454 00:22:56,160 --> 00:22:58,640 or r, theta, and phi, I can act with this operator, 455 00:22:58,640 --> 00:23:00,870 know how the operator acts. 456 00:23:00,870 --> 00:23:08,152 And it's true, that again, lx with lx is equal to i h bar lz. 457 00:23:08,152 --> 00:23:10,110 It satisfies the same time computation relation 458 00:23:10,110 --> 00:23:10,642 as the spin. 459 00:23:10,642 --> 00:23:13,100 But we know the spin operators cannot be expressed in terms 460 00:23:13,100 --> 00:23:15,569 of derivatives along a sphere. 461 00:23:15,569 --> 00:23:16,860 I've harped on that many times. 462 00:23:16,860 --> 00:23:18,651 So what I want to know is what's the analog 463 00:23:18,651 --> 00:23:20,390 of this equation for spin? 464 00:23:20,390 --> 00:23:23,330 What is a representation of the spin operators 465 00:23:23,330 --> 00:23:26,450 acting on the spinners, acting on states 466 00:23:26,450 --> 00:23:27,660 that carry half integer spin? 467 00:23:27,660 --> 00:23:29,368 We know it's not going to be derivatives. 468 00:23:29,368 --> 00:23:30,730 What is it going to be? 469 00:23:30,730 --> 00:23:32,010 Everyone cool with the goal? 470 00:23:36,480 --> 00:23:38,670 In order to do that, we need to first decide 471 00:23:38,670 --> 00:23:41,620 just some basic definition of spin in the z direction. 472 00:23:41,620 --> 00:23:44,240 So what do the angular momentum operators do? 473 00:23:44,240 --> 00:23:47,080 Well whatever else is true of the spin in the z direction 474 00:23:47,080 --> 00:23:57,100 operator, sz acting on a state up is equal to h bar upon 2 up. 475 00:23:57,100 --> 00:24:02,620 And sz actually on a state down is 476 00:24:02,620 --> 00:24:06,830 equal to h bar minus upon 2 down. 477 00:24:11,600 --> 00:24:16,780 And similarly s squared on up or down-- 478 00:24:16,780 --> 00:24:19,510 Oh, by the way, I'm going to relatively casually 479 00:24:19,510 --> 00:24:25,820 oscillate between the notations up and plus, up or down, 480 00:24:25,820 --> 00:24:27,095 and plus or minus. 481 00:24:31,820 --> 00:24:35,180 So sometimes I will write plus for up and minus for down. 482 00:24:35,180 --> 00:24:37,570 So I apologize for the sin of this. 483 00:24:37,570 --> 00:24:39,750 s squared on plus, this is the plus 1/2 state, 484 00:24:39,750 --> 00:24:45,440 is equal to h bar squared ss plus 1, but s is 1/2, 485 00:24:45,440 --> 00:24:53,330 so 1/2 times 1/2 plus 1 is 3/4 h bar squared times 3 over 4. 486 00:24:53,330 --> 00:24:56,860 And we get the same thing up, ditto down. 487 00:25:00,420 --> 00:25:04,710 Because s squared acts the same way on all states in a tower. 488 00:25:04,710 --> 00:25:07,110 Going up and down through a tower of angular momentum 489 00:25:07,110 --> 00:25:08,693 states, raising and lowering, does not 490 00:25:08,693 --> 00:25:10,410 change the total angular momentum 491 00:25:10,410 --> 00:25:12,360 because s plus and s minus commute with s 492 00:25:12,360 --> 00:25:15,600 squared because they have exactly the same commutation 493 00:25:15,600 --> 00:25:19,330 relations as the angular momentum. 494 00:25:19,330 --> 00:25:22,197 That's an awesome sound. 495 00:25:22,197 --> 00:25:23,780 So I want to know what these look like 496 00:25:23,780 --> 00:25:26,849 in terms of this vector space notation, up and down. 497 00:25:26,849 --> 00:25:28,890 And for the moment I'm going to dispense entirely 498 00:25:28,890 --> 00:25:29,960 with the spatial dependence. 499 00:25:29,960 --> 00:25:31,190 I'm going to treat the spatial dependence 500 00:25:31,190 --> 00:25:32,585 as an overall constant. 501 00:25:32,585 --> 00:25:35,070 So we're equally likely to be in all positions. 502 00:25:35,070 --> 00:25:41,460 So we can focus just on the spin part of the state. 503 00:25:41,460 --> 00:25:46,680 So again I want to replace up by 1, 0 and down by 0, 1. 504 00:25:46,680 --> 00:25:48,510 And I want to think about how this looks. 505 00:25:48,510 --> 00:25:51,995 So what this looks like is sz acting on [INAUDIBLE] 1, 506 00:25:51,995 --> 00:25:56,400 0 is equal to h bar upon 2, 1, 0. 507 00:25:56,400 --> 00:26:01,250 And sz on 0, 1 is h bar upon 2, 0, 1. 508 00:26:04,470 --> 00:26:13,150 And s squared on 1, 0 is equal to 3 h bar squared on 4, 1, 0. 509 00:26:13,150 --> 00:26:18,950 And ditto for 0, 1. 510 00:26:18,950 --> 00:26:20,590 Yeah? 511 00:26:20,590 --> 00:26:22,840 AUDIENCE: That [INAUDIBLE] should have a line over it. 512 00:26:22,840 --> 00:26:23,300 PROFESSOR: Oh, thank you. 513 00:26:23,300 --> 00:26:24,240 Yes, it really should. 514 00:26:24,240 --> 00:26:26,955 AUDIENCE: So how do you get 3/4 there? 515 00:26:26,955 --> 00:26:27,830 PROFESSOR: 3/4, good. 516 00:26:27,830 --> 00:26:29,670 Where that came from is that remember 517 00:26:29,670 --> 00:26:32,250 that when we constructed the eigenfunctions of l squared, 518 00:26:32,250 --> 00:26:37,340 l squared acting on a state lm was 519 00:26:37,340 --> 00:26:42,250 equal to h bar squared l l plus 1. 520 00:26:42,250 --> 00:26:45,654 l plus 1 on [? file m. ?] Now if we do exactly the same logic, 521 00:26:45,654 --> 00:26:47,070 which we actually did at the time. 522 00:26:47,070 --> 00:26:49,460 We did in full generality whether the total angular 523 00:26:49,460 --> 00:26:53,040 momentum was an integer or a half integer. 524 00:26:53,040 --> 00:26:54,486 We found that if we took, I'm just 525 00:26:54,486 --> 00:26:56,920 going to use for the half integer states the symbol s, 526 00:26:56,920 --> 00:26:58,870 but it's exactly the same calculation. 527 00:26:58,870 --> 00:27:03,030 s squared on phi and again sm sub s is equal to h bar 528 00:27:03,030 --> 00:27:09,730 squared s s plus 1 phi s m sub s. 529 00:27:09,730 --> 00:27:13,660 OK and so for s equals 1/2 then s, 530 00:27:13,660 --> 00:27:20,160 s plus 1, is equal to 1/2 times 3/2, which is 3/4. 531 00:27:20,160 --> 00:27:20,660 Yeah? 532 00:27:20,660 --> 00:27:22,160 AUDIENCE: [INAUDIBLE] s equals [INAUDIBLE]. 533 00:27:22,160 --> 00:27:23,160 Isn't that [INAUDIBLE]? 534 00:27:23,160 --> 00:27:26,780 PROFESSOR: Ah, but remember, does l go negative? 535 00:27:26,780 --> 00:27:27,280 Great. 536 00:27:27,280 --> 00:27:28,330 Does s go negative? 537 00:27:28,330 --> 00:27:29,920 No. s is just labeling the tower. 538 00:27:29,920 --> 00:27:31,980 So s is 0, it's 1, it's 2. 539 00:27:31,980 --> 00:27:35,960 And so for example, here, these are states where the s is 1/2 540 00:27:35,960 --> 00:27:39,850 and the s in the z direction can be plus 1/2 or minus 1/2. 541 00:27:39,850 --> 00:27:43,330 s is 3/2 and then s in the z direction can be m 542 00:27:43,330 --> 00:27:47,820 is 3/2, 1/2, minus 1/2, minus 3/2. 543 00:27:47,820 --> 00:27:49,110 OK. 544 00:27:49,110 --> 00:27:52,010 Other questions? 545 00:27:52,010 --> 00:27:53,798 OK, yeah. 546 00:27:53,798 --> 00:27:56,280 AUDIENCE: What about the lowering and raising of those? 547 00:27:56,280 --> 00:27:58,652 PROFESSOR: Good, we're going to have to construct them, 548 00:27:58,652 --> 00:27:59,610 because, what are they? 549 00:27:59,610 --> 00:28:01,110 Well, they lower and raise, so we're 550 00:28:01,110 --> 00:28:03,591 going to have to build the states that lower and raise. 551 00:28:03,591 --> 00:28:06,007 AUDIENCE: And would lowering on the up will give you down, 552 00:28:06,007 --> 00:28:08,579 but raising on up--? 553 00:28:08,579 --> 00:28:09,370 PROFESSOR: Awesome. 554 00:28:09,370 --> 00:28:11,709 So what did it mean that we had towers? 555 00:28:11,709 --> 00:28:12,750 Let me do that back here. 556 00:28:12,750 --> 00:28:14,329 So the question is, look, we're going 557 00:28:14,329 --> 00:28:16,370 to have to use the raising and lowering operators 558 00:28:16,370 --> 00:28:17,995 at the end of the day, but what happens 559 00:28:17,995 --> 00:28:20,790 if I raise the bottom state-- what if I raise down? 560 00:28:20,790 --> 00:28:23,010 I'll get up. 561 00:28:23,010 --> 00:28:25,940 And what happens if I raise up? 562 00:28:25,940 --> 00:28:26,670 You get 0. 563 00:28:26,670 --> 00:28:30,270 That's the statement that the tower ends. 564 00:28:30,270 --> 00:28:33,810 On the other hand, if s is 3/2, what happens if I raise 1/2? 565 00:28:33,810 --> 00:28:34,800 I get 3/2. 566 00:28:34,800 --> 00:28:38,070 And if I raise 3/2, I get nothing, I get 0, identically. 567 00:28:38,070 --> 00:28:39,950 And that's the statement that the tower ends. 568 00:28:39,950 --> 00:28:42,360 So for every tower labeled by s, we 569 00:28:42,360 --> 00:28:47,390 have a set of states labeled by m which goes from minus s to s 570 00:28:47,390 --> 00:28:48,750 in integer steps. 571 00:28:48,750 --> 00:28:50,430 The raising operator raises us by 1, 572 00:28:50,430 --> 00:28:52,100 the lowering operator lowers by 1. 573 00:28:52,100 --> 00:28:54,300 The lowering operator annihilates the bottom state, 574 00:28:54,300 --> 00:28:57,140 the raising operator kills the top state. 575 00:28:57,140 --> 00:28:58,510 Cool? 576 00:28:58,510 --> 00:28:59,254 Yeah. 577 00:28:59,254 --> 00:29:01,110 AUDIENCE: Do states like 3/2 and 5/2 578 00:29:01,110 --> 00:29:03,430 have anything akin to up and down? 579 00:29:03,430 --> 00:29:05,110 PROFESSOR: Yeah, so-- do they have 580 00:29:05,110 --> 00:29:06,460 anything akin to up and down. 581 00:29:06,460 --> 00:29:08,097 Up and down is just a name. 582 00:29:08,097 --> 00:29:09,930 It doesn't really communicate anything other 583 00:29:09,930 --> 00:29:14,750 than it's shorthand for spin in the z direction 1/2. 584 00:29:14,750 --> 00:29:17,300 So the question could be translated as, 585 00:29:17,300 --> 00:29:19,350 are there convenient and illuminating names 586 00:29:19,350 --> 00:29:21,190 for the spin 3/2 states? 587 00:29:21,190 --> 00:29:24,010 And I don't really know. 588 00:29:24,010 --> 00:29:26,059 I don't know. 589 00:29:26,059 --> 00:29:27,100 I mean, the states exist. 590 00:29:27,100 --> 00:29:28,400 So we can build nuclear particles that 591 00:29:28,400 --> 00:29:30,691 have angular momentum 3/2, or 5/2, all sorts of things. 592 00:29:30,691 --> 00:29:32,940 But I don't know of a useful name. 593 00:29:32,940 --> 00:29:34,728 For the most part, we simplify our life 594 00:29:34,728 --> 00:29:35,936 by focusing on the 1/2 state. 595 00:29:35,936 --> 00:29:39,740 And as you'll discover in 8.05, the spin 1/2 states, 596 00:29:39,740 --> 00:29:41,320 if you know them very, very well, 597 00:29:41,320 --> 00:29:43,611 you can use everything you know about them to construct 598 00:29:43,611 --> 00:29:47,700 all of the spin 5/2 8/2-- well, 8/2 is stupid, but-- 9/2, 599 00:29:47,700 --> 00:29:49,410 all those from the spin 1/2. 600 00:29:49,410 --> 00:29:52,290 So it turns out spin 1/2 is sort of Ur-- 601 00:29:52,290 --> 00:29:53,940 in a way that can be made very precise, 602 00:29:53,940 --> 00:29:55,010 and that's the theory of Lie algebras. 603 00:29:55,010 --> 00:29:55,509 Yeah. 604 00:29:58,089 --> 00:29:59,755 AUDIENCE: Can you just elaborate on what 605 00:29:59,755 --> 00:30:02,171 you meant by, you can't really think of spin as an angular 606 00:30:02,171 --> 00:30:03,200 momentum vector? 607 00:30:03,200 --> 00:30:04,340 PROFESSOR: Yeah. 608 00:30:04,340 --> 00:30:05,030 So OK, good. 609 00:30:05,030 --> 00:30:07,654 So the question is, elaborate a little bit on what you mean by, 610 00:30:07,654 --> 00:30:10,160 spin can't be thought of as an angular momentum vector. 611 00:30:10,160 --> 00:30:13,590 Spin certainly can be thought of as an angular momentum, 612 00:30:13,590 --> 00:30:17,620 because the whole point here was that if you have a charged 613 00:30:17,620 --> 00:30:23,980 particle and it carries spin, then it has a magnetic moment. 614 00:30:23,980 --> 00:30:25,840 And a magnetic moment is the charge 615 00:30:25,840 --> 00:30:28,950 times the angular momentum. 616 00:30:28,950 --> 00:30:31,250 So if it carries spin, and it carries charge, 617 00:30:31,250 --> 00:30:33,334 and thus it carries magnetic moment-- 618 00:30:33,334 --> 00:30:35,500 that's pretty much what we mean by angular momentum. 619 00:30:35,500 --> 00:30:38,470 That's as good a diagnostic as any. 620 00:30:38,470 --> 00:30:42,230 Meanwhile, on top of satisfying that experimental property, 621 00:30:42,230 --> 00:30:44,300 this, just as a set of commutation relations, 622 00:30:44,300 --> 00:30:46,674 these commutation relations are the commutation relations 623 00:30:46,674 --> 00:30:47,581 of angular momentum. 624 00:30:47,581 --> 00:30:50,080 It just turns out that we can have states with total angular 625 00:30:50,080 --> 00:30:54,360 momentum little s, which is 1/2 integral-- 1/2, 3/2, et cetera. 626 00:30:54,360 --> 00:30:56,940 Now, the things that I want to emphasize are twofold. 627 00:30:56,940 --> 00:30:58,390 First off, something I've harped on over and over again, 628 00:30:58,390 --> 00:31:00,690 so I'll attempt to limit my uses of this phrase. 629 00:31:00,690 --> 00:31:05,530 But you cannot think of these states with s is 1/2 as wave 630 00:31:05,530 --> 00:31:07,564 functions determining position on a sphere. 631 00:31:07,564 --> 00:31:08,980 So that's the first sense in which 632 00:31:08,980 --> 00:31:11,350 you can't think of it as equivalent to orbital angular 633 00:31:11,350 --> 00:31:11,850 momentum. 634 00:31:11,850 --> 00:31:12,800 But there's a second sense, which 635 00:31:12,800 --> 00:31:15,169 is that up and down should not be thought of as spin 636 00:31:15,169 --> 00:31:17,710 in the z direction being up and spin in the z direction being 637 00:31:17,710 --> 00:31:20,152 down meaning a vector in three dimensions pointing up 638 00:31:20,152 --> 00:31:22,110 and a vector in three dimensions pointing down, 639 00:31:22,110 --> 00:31:23,550 because those states are orthogonal. 640 00:31:23,550 --> 00:31:25,341 Whereas these two three-dimensional vectors 641 00:31:25,341 --> 00:31:27,230 are not orthogonal-- they're parallel. 642 00:31:27,230 --> 00:31:29,310 They have a non-zero inner product. 643 00:31:29,310 --> 00:31:34,730 So up and down, the names we give these spin 1/2 states, 644 00:31:34,730 --> 00:31:37,310 should not be confused with pointing up in the z direction 645 00:31:37,310 --> 00:31:39,070 and down in the z direction. 646 00:31:39,070 --> 00:31:43,710 It's just a formal name we give to the plus 1/2 and minus 1/2 647 00:31:43,710 --> 00:31:46,070 angular momentum in z direction states. 648 00:31:46,070 --> 00:31:47,575 Does that answer your question? 649 00:31:47,575 --> 00:31:48,952 AUDIENCE: Yeah, so, when you make 650 00:31:48,952 --> 00:31:50,452 a measurement of the value of spin-- 651 00:31:50,452 --> 00:31:52,517 so perhaps you do a Stern-Gerlach experiment-- 652 00:31:52,517 --> 00:31:55,580 and you get what the spin is, can you 653 00:31:55,580 --> 00:31:59,895 not then say, all right, this is spin plus 1/2, spin minus? 654 00:31:59,895 --> 00:32:04,244 It's as z is positive 1/2 as z is minus 1/2? 655 00:32:04,244 --> 00:32:05,410 PROFESSOR: Yeah, absolutely. 656 00:32:05,410 --> 00:32:07,076 So if you do a Stern-Gerlach experiment, 657 00:32:07,076 --> 00:32:09,594 you can identify those electrons that had spin plus 1/2 658 00:32:09,594 --> 00:32:11,010 and those that had spin minus 1/2, 659 00:32:11,010 --> 00:32:12,650 and they come out in different places. 660 00:32:12,650 --> 00:32:13,920 That's absolutely true. 661 00:32:13,920 --> 00:32:16,369 I just want to emphasize that the up vector does not 662 00:32:16,369 --> 00:32:17,785 mean that they're somehow attached 663 00:32:17,785 --> 00:32:21,840 to the electronic vector that's pointing in the z direction. 664 00:32:21,840 --> 00:32:22,340 Good. 665 00:32:22,340 --> 00:32:22,840 Yeah. 666 00:32:25,050 --> 00:32:26,000 Go ahead, whichever. 667 00:32:26,000 --> 00:32:29,580 AUDIENCE: How do we verify that uncharged particles have spin? 668 00:32:29,580 --> 00:32:31,580 PROFESSOR: Yeah, that's an interesting question. 669 00:32:31,580 --> 00:32:33,690 So the question is, how do we know 670 00:32:33,690 --> 00:32:35,720 if an uncharged particle has spin? 671 00:32:35,720 --> 00:32:37,150 And there are many ways to answer 672 00:32:37,150 --> 00:32:39,877 this question, one of which we're 673 00:32:39,877 --> 00:32:41,960 going to come to later which has to do with Bell's 674 00:32:41,960 --> 00:32:44,180 inequality, which is a sort of slick way to do it. 675 00:32:44,180 --> 00:32:46,640 But a very coarse way is this way. 676 00:32:46,640 --> 00:32:48,772 We believe, in a deep and fundamental way, 677 00:32:48,772 --> 00:32:50,730 that the total angular momentum of the universe 678 00:32:50,730 --> 00:32:52,990 is conserved, in the following sense. 679 00:32:52,990 --> 00:32:56,757 There's no preferred axis in the universe. 680 00:32:56,757 --> 00:32:58,340 If you're a cosmologist, just stay out 681 00:32:58,340 --> 00:32:59,881 of the room for the next few minutes. 682 00:32:59,881 --> 00:33:02,260 So there's no preferred axis in the universe 683 00:33:02,260 --> 00:33:05,259 and the law of physics should be invariant under rotation. 684 00:33:05,259 --> 00:33:07,550 Now, if you take a system that has a bunch of particles 685 00:33:07,550 --> 00:33:10,960 with known angular momentum-- let me give you an example. 686 00:33:10,960 --> 00:33:12,215 Take a neutron. 687 00:33:12,215 --> 00:33:14,990 A neutron has spin 1/2. 688 00:33:14,990 --> 00:33:17,090 Wait, how did I know that? 689 00:33:17,090 --> 00:33:20,960 We can do that experiment by doing the following thing. 690 00:33:20,960 --> 00:33:23,610 We can take a neutron and bind it to a proton 691 00:33:23,610 --> 00:33:27,170 and see that the resulting object has spin 1. 692 00:33:27,170 --> 00:33:30,910 So let me try to think of a way that doesn't involve a neutron. 693 00:33:30,910 --> 00:33:38,020 Grant me for the moment that you know that a neutron has 694 00:33:38,020 --> 00:33:39,610 spin 1/2. 695 00:33:39,610 --> 00:33:43,340 So let's just imagine that we knew that, by hook or by crook. 696 00:33:43,340 --> 00:33:46,270 We then do the following experiment. 697 00:33:46,270 --> 00:33:47,420 We wait. 698 00:33:47,420 --> 00:33:49,180 Take a neutron, let it sit in empty space. 699 00:33:49,180 --> 00:33:51,430 When that neutron decays, it does a very cool thing. 700 00:33:51,430 --> 00:33:54,340 It decays relatively quickly into your proton 701 00:33:54,340 --> 00:33:56,218 and an electron that you see. 702 00:33:56,218 --> 00:33:58,676 You see them go flying away the proton has positive charge, 703 00:33:58,676 --> 00:34:01,359 and the electron has negative charge and it goes flying away. 704 00:34:01,359 --> 00:34:02,400 But you've got a problem. 705 00:34:02,400 --> 00:34:06,285 Because you knew that the neutron had spin 1/2, 706 00:34:06,285 --> 00:34:07,160 which is [INAUDIBLE]. 707 00:34:07,160 --> 00:34:09,219 And then you decay one into a proton and an electron. 708 00:34:09,219 --> 00:34:10,719 And the total angular momentum there 709 00:34:10,719 --> 00:34:13,150 is 1/2 plus 1/2 or 1/2 minus 1/2. 710 00:34:13,150 --> 00:34:14,850 It's either 1 or 0. 711 00:34:14,850 --> 00:34:16,726 And you've got a problem. 712 00:34:16,726 --> 00:34:18,350 Angular momentum hasn't been conserved. 713 00:34:18,350 --> 00:34:20,929 So what do you immediately deduce? 714 00:34:20,929 --> 00:34:23,560 That another particle must have also been emitted 715 00:34:23,560 --> 00:34:25,690 that had 1/2 integer angular momentum 716 00:34:25,690 --> 00:34:27,150 to conserve angular momentum. 717 00:34:27,150 --> 00:34:30,107 And it couldn't carry any charge because the electron 718 00:34:30,107 --> 00:34:32,440 and the proton were neutral, and the neutron is neutral. 719 00:34:32,440 --> 00:34:34,497 So things like this you can always 720 00:34:34,497 --> 00:34:36,330 deduce from conservation of angular momentum 721 00:34:36,330 --> 00:34:37,204 one way or the other. 722 00:34:37,204 --> 00:34:39,219 But the best way to do it is going 723 00:34:39,219 --> 00:34:42,151 to be some version of addition of angular momentum 724 00:34:42,151 --> 00:34:44,484 where you have some object like an electron and a proton 725 00:34:44,484 --> 00:34:46,670 and you allow them to stick together 726 00:34:46,670 --> 00:34:48,380 and you discover it has total spin 1. 727 00:34:51,590 --> 00:34:52,260 Yeah. 728 00:34:52,260 --> 00:34:53,889 We can talk about that in more detail afterwards. 729 00:34:53,889 --> 00:34:55,449 That's a particularly nice way to do the experiment. 730 00:34:55,449 --> 00:34:55,949 Yeah. 731 00:34:58,374 --> 00:35:00,779 AUDIENCE: Angular momentum [INAUDIBLE] weird vector 732 00:35:00,779 --> 00:35:07,032 since if you reflect your system through [INAUDIBLE]. 733 00:35:07,032 --> 00:35:07,977 How does that work? 734 00:35:07,977 --> 00:35:09,060 PROFESSOR: Yeah, OK, good. 735 00:35:09,060 --> 00:35:10,500 I don't want to get into this in too much detail, 736 00:35:10,500 --> 00:35:12,958 but it's a really good question, so come to my office hours 737 00:35:12,958 --> 00:35:14,850 and ask or go to recitations and ask. 738 00:35:14,850 --> 00:35:16,016 It's a really good question. 739 00:35:16,016 --> 00:35:18,360 The question is this-- angular momentum 740 00:35:18,360 --> 00:35:21,790 has a funny property under parity, under reflection. 741 00:35:21,790 --> 00:35:25,720 So if you look in a mirror this way, here's angular momentum 742 00:35:25,720 --> 00:35:27,390 and it's got-- right-hand rule, it's 743 00:35:27,390 --> 00:35:29,473 got angular momentum up-- if I look in the mirror, 744 00:35:29,473 --> 00:35:31,270 it's going this way. 745 00:35:31,270 --> 00:35:35,520 So it would appear to have right angular momentum down. 746 00:35:35,520 --> 00:35:38,110 That's what it looks like if you reflect in a mirror. 747 00:35:38,110 --> 00:35:39,060 Other direction. 748 00:35:39,060 --> 00:35:41,974 So that's a funny property of angular momentum. 749 00:35:41,974 --> 00:35:43,890 It's also a true property of angular momentum. 750 00:35:43,890 --> 00:35:45,460 It's fine. 751 00:35:45,460 --> 00:35:47,832 And what about spin, is the question. 752 00:35:47,832 --> 00:35:50,040 Does spin also have this funny property under parity, 753 00:35:50,040 --> 00:35:52,160 is that basically the question? 754 00:35:52,160 --> 00:35:53,380 Yeah, and it does. 755 00:35:53,380 --> 00:35:55,710 And working out exactly how to show that 756 00:35:55,710 --> 00:35:57,210 is a sort of entertaining exercise. 757 00:35:57,210 --> 00:35:59,470 So again, it's beyond the scope of the lecture, 758 00:35:59,470 --> 00:36:02,630 so come ask me in office hours and we can talk about that. 759 00:36:02,630 --> 00:36:03,420 Yeah, one more. 760 00:36:03,420 --> 00:36:06,348 AUDIENCE: For orbital angular momentum, say for l equals 1, 761 00:36:06,348 --> 00:36:09,139 we had states like m equals plus 1 and minus 1. 762 00:36:09,139 --> 00:36:09,764 PROFESSOR: Yes. 763 00:36:09,764 --> 00:36:14,400 AUDIENCE: And we did think of those as angular momentum 764 00:36:14,400 --> 00:36:15,140 vectors. 765 00:36:15,140 --> 00:36:15,495 PROFESSOR: Absolutely. 766 00:36:15,495 --> 00:36:17,778 AUDIENCE: But those states are also orthogonal, are they not? 767 00:36:17,778 --> 00:36:19,195 PROFESSOR: Yeah, those states are also orthogonal. 768 00:36:19,195 --> 00:36:21,794 AUDIENCE: So even though the angular momentum vectors 769 00:36:21,794 --> 00:36:23,210 aren't orthogonal, they're still-- 770 00:36:23,210 --> 00:36:24,497 it's just a different sense. 771 00:36:24,497 --> 00:36:25,830 PROFESSOR: That's exactly right. 772 00:36:25,830 --> 00:36:28,497 So again, even in the case of integer angular momentum, 773 00:36:28,497 --> 00:36:30,080 you've got to be careful about talking 774 00:36:30,080 --> 00:36:32,440 about the top state and the bottom state 775 00:36:32,440 --> 00:36:34,960 corresponding to pointing in some direction, 776 00:36:34,960 --> 00:36:36,680 because they're orthogonal states. 777 00:36:36,680 --> 00:36:39,180 However, they do correspond to a particular angular momentum 778 00:36:39,180 --> 00:36:40,770 vector in three dimensional space. 779 00:36:40,770 --> 00:36:42,980 They correspond to a distribution on the sphere. 780 00:36:42,980 --> 00:36:45,070 So there's a sense in which they do correspond 781 00:36:45,070 --> 00:36:47,356 to real rotations, real eigenfunctions on a sphere, 782 00:36:47,356 --> 00:36:49,230 and there's also a sense in which they don't, 783 00:36:49,230 --> 00:36:50,605 because they're still orthogonal. 784 00:36:50,605 --> 00:36:51,810 That's exactly right. 785 00:36:51,810 --> 00:36:52,580 So let me move on. 786 00:36:52,580 --> 00:36:55,190 I'm going to stop questions at this point. 787 00:36:55,190 --> 00:37:00,150 So good. 788 00:37:00,150 --> 00:37:01,770 So these are the properties that need 789 00:37:01,770 --> 00:37:04,140 to be satisfied by our operators. 790 00:37:04,140 --> 00:37:08,360 And it's pretty easy to see in this basis what 791 00:37:08,360 --> 00:37:09,850 these operators must be. 792 00:37:09,850 --> 00:37:12,660 Sz has eigenvectors 1, 0 and 0, 1. 793 00:37:12,660 --> 00:37:18,640 So Sz should be equal to h bar upon 2 1, 0, 0, minus 1. 794 00:37:18,640 --> 00:37:22,480 So let's just check this on 1, 0 gives me 1, 795 00:37:22,480 --> 00:37:26,550 0, so it gives me the same thing back times h bar upon 2. 796 00:37:26,550 --> 00:37:27,050 Cool. 797 00:37:27,050 --> 00:37:29,460 And acting on 0, 1, or the down state, 798 00:37:29,460 --> 00:37:32,270 we get h bar upon 2 times 0 minus 1. 799 00:37:32,270 --> 00:37:35,410 That could be minus 1. 800 00:37:35,410 --> 00:37:40,142 Oh, sorry, 1, 0 gives me 0 and 0 minus 1 on 1 gives me minus 1, 801 00:37:40,142 --> 00:37:41,350 which is 1 with a minus sign. 802 00:37:41,350 --> 00:37:43,870 That's a minus sign. 803 00:37:43,870 --> 00:37:45,620 So this works out like a champ. 804 00:37:45,620 --> 00:37:47,790 And S squared, meanwhile, is equal to-- well, 805 00:37:47,790 --> 00:37:50,810 it's got to give me h bar squared times 3/4 806 00:37:50,810 --> 00:37:52,715 for both of these vectors. 807 00:37:52,715 --> 00:37:54,340 So h bar squared-- and meanwhile, these 808 00:37:54,340 --> 00:37:58,730 are eigenstates-- h bar squared times 3/4 times 1, 0, 0, 1. 809 00:38:04,330 --> 00:38:10,240 So we know one other fact, which was brought up 810 00:38:10,240 --> 00:38:14,320 just a minute ago, which was that if we take S plus 811 00:38:14,320 --> 00:38:16,820 and you act on the state 0, 1, what should you get? 812 00:38:20,510 --> 00:38:23,150 If you raise your 1-- 1, 0. 813 00:38:23,150 --> 00:38:24,000 Great. 814 00:38:24,000 --> 00:38:27,609 So we also worked out the normalization coefficient 815 00:38:27,609 --> 00:38:28,400 on the problem set. 816 00:38:28,400 --> 00:38:31,860 And that normalization coefficient turns out to be 1. 817 00:38:31,860 --> 00:38:34,600 And let's be careful-- we've got an h 818 00:38:34,600 --> 00:38:38,020 bar, for dimensional analysis reasons. 819 00:38:38,020 --> 00:38:41,450 So meanwhile, S minus, similarly, on 0, 820 00:38:41,450 --> 00:38:45,220 1, is equal to 0. 821 00:38:45,220 --> 00:38:50,305 And S plus on 0, 1, is equal to-- oh, 822 00:38:50,305 --> 00:38:51,430 sorry, we already did that. 823 00:38:51,430 --> 00:38:55,390 We want S minus on 1, 0. 824 00:38:55,390 --> 00:38:58,660 Let's see-- S minus on-- we want S plus on 1, 825 00:38:58,660 --> 00:39:02,740 0 is equal to-- right, 0. 826 00:39:02,740 --> 00:39:10,560 And S minus on 1, 0 is equal to h bar 0, 1. 827 00:39:10,560 --> 00:39:12,640 OK, so putting all this together, 828 00:39:12,640 --> 00:39:14,960 you can pretty quickly get that S plus 829 00:39:14,960 --> 00:39:20,870 is equal to-- we need an h bar and we 830 00:39:20,870 --> 00:39:28,300 need it to raise the lower one and kill the top state. 831 00:39:28,300 --> 00:39:31,520 So on 1, 0, what does S plus do? 832 00:39:31,520 --> 00:39:33,880 That gives us 0 that gives us 0. 833 00:39:33,880 --> 00:39:34,920 Good. 834 00:39:34,920 --> 00:39:37,409 And on the lowered state, 0, 1, that 835 00:39:37,409 --> 00:39:38,950 gives me a 1 up top and that gives me 836 00:39:38,950 --> 00:39:40,810 a 0 downstairs, so it works out like this. 837 00:39:40,810 --> 00:39:41,650 So we've got h bar. 838 00:39:41,650 --> 00:39:48,500 Similarly, S minus is equal to h bar times 0, 0, 1, 0. 839 00:39:52,610 --> 00:39:55,990 So we've got these guys-- so much 840 00:39:55,990 --> 00:40:00,606 from just the definitions of raising and lowering. 841 00:40:00,606 --> 00:40:01,980 And by taking inner products, you 842 00:40:01,980 --> 00:40:05,540 can just derive those two lines from these. 843 00:40:05,540 --> 00:40:16,140 But notice that Sx is equal to S plus plus S minus upon 2, 844 00:40:16,140 --> 00:40:23,220 and Sy is equal to S plus minus S minus upon 2i. 845 00:40:26,100 --> 00:40:33,470 So this tells us that Sx is equal to h bar upon 2 times 846 00:40:33,470 --> 00:40:36,040 S plus-- we're going to get a 1 here-- plus S 847 00:40:36,040 --> 00:40:39,930 minus-- we're going to get a 1 here-- 0, 1, 1, 0. 848 00:40:39,930 --> 00:40:47,125 And Sy is equal to, again, upon 2i times h bar, h 849 00:40:47,125 --> 00:40:52,590 bar upon 2i, times S plus, which is going to give me 1 and minus 850 00:40:52,590 --> 00:40:55,540 S minus which is going to give me minus 1, 0, 0. 851 00:40:55,540 --> 00:41:01,458 But we can pull this i in, so 1/i is like minus i. 852 00:41:01,458 --> 00:41:07,160 So minus i times minus 1 is going to give me i 853 00:41:07,160 --> 00:41:09,580 and minus i times 1 is going to give me i. 854 00:41:12,460 --> 00:41:14,952 So-- 855 00:41:14,952 --> 00:41:16,422 AUDIENCE: Shouldn't it be minus i? 856 00:41:16,422 --> 00:41:17,130 PROFESSOR: Sorry? 857 00:41:17,130 --> 00:41:18,074 Yeah, did I write i? 858 00:41:18,074 --> 00:41:19,240 That should've been minus i. 859 00:41:19,240 --> 00:41:19,739 Thank you. 860 00:41:23,390 --> 00:41:27,850 So now we have a nice representation of these spin 861 00:41:27,850 --> 00:41:32,380 operations, of the spin operators. 862 00:41:32,380 --> 00:41:42,730 And explicitly we have that Sx is equal to h bar upon 2 0, 863 00:41:42,730 --> 00:41:50,990 1, 1, 0, Sy is equal to h bar upon 2 0, minus i, i, 0. 864 00:41:50,990 --> 00:41:57,090 And Sz is equal to h bar upon 2 1, 0, 0, minus 1. 865 00:42:00,440 --> 00:42:02,325 So why is Sz the only one that's diagonal? 866 00:42:05,970 --> 00:42:07,650 Is it something special about Sz? 867 00:42:10,460 --> 00:42:15,670 AUDIENCE: I mean, we've chosen z as the axis along which 868 00:42:15,670 --> 00:42:17,370 to project S squared. 869 00:42:17,370 --> 00:42:18,320 PROFESSOR: Exactly. 870 00:42:18,320 --> 00:42:19,930 So the thing that's special about Sz 871 00:42:19,930 --> 00:42:22,130 is that at the very beginning of this, 872 00:42:22,130 --> 00:42:25,860 we decided to work in a basis of eigenstates of Sz, 873 00:42:25,860 --> 00:42:28,220 with definite values of Sz. 874 00:42:28,220 --> 00:42:31,250 So if they have definite values, then acting with Sz 875 00:42:31,250 --> 00:42:32,740 is just going to give you a number. 876 00:42:32,740 --> 00:42:34,490 That's what it is to be a diagonal matrix. 877 00:42:34,490 --> 00:42:38,300 You act on a basis vector, you just get a number out. 878 00:42:38,300 --> 00:42:45,300 So we started out by working in the eigenbasis of Sz. 879 00:42:45,300 --> 00:42:55,515 And as a consequence, we find that Sz is diagonal. 880 00:42:55,515 --> 00:42:57,140 And this is a general truth that you'll 881 00:42:57,140 --> 00:43:06,320 discover in matrix mechanics when 882 00:43:06,320 --> 00:43:08,830 you work in the eigenbasis of an operator, 883 00:43:08,830 --> 00:43:11,980 that operator is represented by a diagonal matrix. 884 00:43:11,980 --> 00:43:14,600 And so we often say, rather than to work in an eigenbasis, 885 00:43:14,600 --> 00:43:15,900 we often say, to diagonalize. 886 00:43:15,900 --> 00:43:17,066 Yeah. 887 00:43:17,066 --> 00:43:19,360 AUDIENCE: Are the signs right for Sy? 888 00:43:19,360 --> 00:43:24,206 Because if we had h bar over 2i, and as we initially had a 1 889 00:43:24,206 --> 00:43:27,045 in the top right and minus 1 in the bottom left, 890 00:43:27,045 --> 00:43:28,400 shouldn't we just multiply by i? 891 00:43:28,400 --> 00:43:30,316 PROFESSOR: I'm pretty sure-- so originally, we 892 00:43:30,316 --> 00:43:32,369 had a downstairs i, right? 893 00:43:32,369 --> 00:43:34,160 So let's think about what this looked like. 894 00:43:34,160 --> 00:43:36,161 This was 1 and minus 1, right? 895 00:43:36,161 --> 00:43:36,660 Agreed? 896 00:43:36,660 --> 00:43:38,850 So this is minus 1 over i. 897 00:43:38,850 --> 00:43:39,760 So we pull in the i. 898 00:43:39,760 --> 00:43:43,230 So that we go from minus 1 to minus 1 over i. 899 00:43:43,230 --> 00:43:46,430 And we go from 1 to 1 over i. 900 00:43:46,430 --> 00:43:49,280 And I claim that one over i is minus i. 901 00:43:49,280 --> 00:43:50,090 AUDIENCE: Oh, OK. 902 00:43:50,090 --> 00:43:50,673 PROFESSOR: OK? 903 00:43:50,673 --> 00:43:52,400 And minus 1 over i is i. 904 00:43:52,400 --> 00:43:53,003 That cool? 905 00:43:53,003 --> 00:43:53,830 Good. 906 00:43:53,830 --> 00:44:00,860 OK, so this is i and minus i. 907 00:44:00,860 --> 00:44:02,380 I always get that screwy, but it's 908 00:44:02,380 --> 00:44:04,740 useful to memorize these matrices. 909 00:44:04,740 --> 00:44:07,210 You might think it's a little silly to memorize matrices. 910 00:44:07,210 --> 00:44:09,630 But these turn out to be ridiculously useful 911 00:44:09,630 --> 00:44:10,880 and they come up all the time. 912 00:44:10,880 --> 00:44:13,140 This is called sigma x. 913 00:44:13,140 --> 00:44:14,986 This is called sigma y. 914 00:44:14,986 --> 00:44:16,110 And this is called sigma z. 915 00:44:16,110 --> 00:44:19,820 And different people decide whether to put the 1/2 in there 916 00:44:19,820 --> 00:44:22,020 or not, the h bar does not go in there. 917 00:44:22,020 --> 00:44:24,722 Some people put in the 2, some people don't put in the 1/2, 918 00:44:24,722 --> 00:44:25,680 it's a matter of taste. 919 00:44:25,680 --> 00:44:30,150 Just be careful and be consistent, as usual. 920 00:44:30,150 --> 00:44:33,550 And these are called the Pauli matrices because A, 921 00:44:33,550 --> 00:44:36,822 we really like Pauli and B, Pauli introduced them. 922 00:44:36,822 --> 00:44:39,480 Although he didn't actually introduce them in some sense-- 923 00:44:39,480 --> 00:44:41,890 this mathematical structure was introduced 924 00:44:41,890 --> 00:44:43,380 ages and ages and ages ago. 925 00:44:43,380 --> 00:44:46,370 But physicists cite the physicist, not 926 00:44:46,370 --> 00:44:48,850 the mathematician. 927 00:44:48,850 --> 00:44:51,310 OK I'm not saying that's good. 928 00:44:51,310 --> 00:44:53,600 I'm just saying it happens. 929 00:44:53,600 --> 00:44:56,255 So notice a consequence of these. 930 00:44:56,255 --> 00:44:58,380 An important consequence of these-- the whole point 931 00:44:58,380 --> 00:45:02,700 here was to build a representation of the spin 932 00:45:02,700 --> 00:45:03,620 operators. 933 00:45:03,620 --> 00:45:05,490 Now whatever else the spin operators do, 934 00:45:05,490 --> 00:45:07,620 they had better satisfy that computation relation, 935 00:45:07,620 --> 00:45:09,180 otherwise they're not really spin operators. 936 00:45:09,180 --> 00:45:11,200 That's what we mean by being spin operators. 937 00:45:11,200 --> 00:45:12,200 So let's check. 938 00:45:12,200 --> 00:45:17,892 Is it true that Sx commutator with Sy is equal to i h bar Sz? 939 00:45:17,892 --> 00:45:19,330 So this is a question mark. 940 00:45:22,590 --> 00:45:23,270 And let's check. 941 00:45:23,270 --> 00:45:24,660 Let's do the commutator. 942 00:45:24,660 --> 00:45:27,900 From the Sx, we're going to get an h bar upon 2. 943 00:45:27,900 --> 00:45:30,530 From the Sy, from each Sy, we're going to get a factor of h bar 944 00:45:30,530 --> 00:45:33,155 upon 2, so I'll just write that as h bar upon 2 squared-- times 945 00:45:33,155 --> 00:45:35,690 the commutator of two matrices-- 0, 1, 1, 946 00:45:35,690 --> 00:45:39,730 0 commutator with 0, minus i, i, 0. 947 00:45:42,250 --> 00:45:45,990 This is equal to h bar squared upon 4 times-- 948 00:45:45,990 --> 00:45:46,990 let's write this out. 949 00:45:46,990 --> 00:45:50,250 The first term is going to be this matrix times this matrix. 950 00:45:50,250 --> 00:45:54,690 That's going to be, again, a matrix-- 0, 1, 0, 1. 951 00:45:54,690 --> 00:45:56,030 So that first one is a 1. 952 00:45:56,030 --> 00:45:59,360 0, 1, minus i-- oh, sorry, that's an i. 953 00:45:59,360 --> 00:46:01,890 0, 1, that's an i, that's a minus i. 954 00:46:01,890 --> 00:46:04,056 So 0, 1, 0, i gives me an i. 955 00:46:04,056 --> 00:46:06,770 0, 1, minus i, 0 gives me a 0. 956 00:46:06,770 --> 00:46:09,380 Second row-- 1, 0, 0, i gives me 0. 957 00:46:09,380 --> 00:46:15,180 And 1, 0, minus i, 0 gives me minus i. 958 00:46:15,180 --> 00:46:17,790 And then the second term is the flipped order, right? 959 00:46:17,790 --> 00:46:18,680 The commutator term. 960 00:46:18,680 --> 00:46:23,680 So we get minus the commutator term, which is going to be 0, 961 00:46:23,680 --> 00:46:25,180 minus i, 0, 1. 962 00:46:25,180 --> 00:46:28,500 That gives me minus i. 963 00:46:28,500 --> 00:46:32,620 0, minus i, 1, 0, that gives me 0. 964 00:46:32,620 --> 00:46:36,830 Bottom row-- i, 0, 0, 1-- 0. 965 00:46:36,830 --> 00:46:39,430 And i, 0, 1, 0 give me i. 966 00:46:39,430 --> 00:46:41,150 OK, so notice what we get this is 967 00:46:41,150 --> 00:46:45,040 equal to h bar squared upon 4. 968 00:46:45,040 --> 00:46:47,650 And both of those matrices are the same thing. 969 00:46:47,650 --> 00:46:52,170 Those matrices are both i, 0, 0, minus i with minus i, 0, 0, 970 00:46:52,170 --> 00:47:02,190 i, giving us i, 0, 0, minus i times 2 from the two terms. 971 00:47:02,190 --> 00:47:04,370 The 2's cancel, and this gives me 972 00:47:04,370 --> 00:47:13,600 h bar squared upon 2 times i, 0, 0, minus i. 973 00:47:13,600 --> 00:47:23,050 But this is also known as-- pulling out an i and an h bar-- 974 00:47:23,050 --> 00:47:27,520 times h bar upon 2 1, 0, 0, minus 1. 975 00:47:27,520 --> 00:47:29,610 This is equal to i h bar Sz. 976 00:47:34,580 --> 00:47:37,710 So these matrices represent the angular momentum 977 00:47:37,710 --> 00:47:39,460 commutators quite nicely. 978 00:47:42,544 --> 00:47:44,460 And in fact, if you check, all the commutators 979 00:47:44,460 --> 00:47:47,510 work out beautifully. 980 00:47:47,510 --> 00:47:50,110 So quickly, just as a reminder, what 981 00:47:50,110 --> 00:47:51,690 are the possible measurable values 982 00:47:51,690 --> 00:47:54,020 of Sz for the spin 1/2 system? 983 00:47:56,834 --> 00:47:58,250 What possible values could you get 984 00:47:58,250 --> 00:48:03,050 if you measured Sz, spin in the z direction? 985 00:48:03,050 --> 00:48:05,720 Yeah, plus or minus h bar upon 2. 986 00:48:05,720 --> 00:48:10,050 Now what about the eigenvectors? 987 00:48:10,050 --> 00:48:11,870 What are they-- of Sz? 988 00:48:14,492 --> 00:48:16,200 In this notation, there are these states. 989 00:48:16,200 --> 00:48:17,991 There's one eigenvector, there's the other. 990 00:48:17,991 --> 00:48:19,970 But let's ask the same question about Sx. 991 00:48:19,970 --> 00:48:22,150 So for Sx, what are the allowed eigenvalues? 992 00:48:25,530 --> 00:48:27,295 Well, we can answer this in two ways. 993 00:48:27,295 --> 00:48:29,420 The first way we can answer this is by saying look, 994 00:48:29,420 --> 00:48:30,235 there's nothing deep about z. 995 00:48:30,235 --> 00:48:32,100 It was just the stupid direction we started with. 996 00:48:32,100 --> 00:48:34,490 We could have started by working with the eigenbasis of Sx 997 00:48:34,490 --> 00:48:36,364 and we would've found exactly the same story. 998 00:48:36,364 --> 00:48:38,100 So it must be plus or minus h bar upon 2. 999 00:48:38,100 --> 00:48:40,230 But the reason you make that argument is A, 1000 00:48:40,230 --> 00:48:42,330 it's slick and B, it's the only one 1001 00:48:42,330 --> 00:48:45,680 you can make without knowing something else about how 1002 00:48:45,680 --> 00:48:47,330 Sx acts. 1003 00:48:47,330 --> 00:48:48,620 But now we know what Sx is. 1004 00:48:48,620 --> 00:48:51,220 Sx is that operator. 1005 00:48:51,220 --> 00:48:54,570 So now we can ask, what are the eigenvalues of that operator? 1006 00:48:54,570 --> 00:48:58,920 And if you compute the eigenvalues of that operator, 1007 00:48:58,920 --> 00:49:00,700 you find that there are two eigenvalues 1008 00:49:00,700 --> 00:49:07,970 Sx is equal to h bar upon 2 and Sx 1009 00:49:07,970 --> 00:49:09,746 is equal to minus h bar upon 2. 1010 00:49:24,910 --> 00:49:27,744 And now I can ask, well, what are the eigenvectors? 1011 00:49:27,744 --> 00:49:29,160 Now, we know what the eigenvectors 1012 00:49:29,160 --> 00:49:30,534 are because we can just construct 1013 00:49:30,534 --> 00:49:32,080 the eigenvectors of Sx plus. 1014 00:49:32,080 --> 00:49:36,370 And if you construct the eigenvectors of Sx plus-- 1015 00:49:36,370 --> 00:49:37,560 should I take the time? 1016 00:49:37,560 --> 00:49:40,460 How many people want me to do the eigenvectors of Sx 1017 00:49:40,460 --> 00:49:41,210 explicitly? 1018 00:49:41,210 --> 00:49:41,880 Yeah, that's kind of what I figured. 1019 00:49:41,880 --> 00:49:42,560 OK, good. 1020 00:49:42,560 --> 00:49:45,520 So the eigenvectors of Sx are, for example, on 1, 1021 00:49:45,520 --> 00:49:55,550 1 is equal to-- well, Sx on 1, 1, the first term, 1022 00:49:55,550 --> 00:49:57,547 that 0, 1, is going to give me a 1, the 1, 1023 00:49:57,547 --> 00:49:58,630 0 is going to give me a 1. 1024 00:49:58,630 --> 00:50:03,130 So this is h bar upon 2 coefficient of Sx on 1, 1. 1025 00:50:03,130 --> 00:50:08,560 And Sx on 1, minus 1 is going to give me h bar upon 2 minus 1, 1026 00:50:08,560 --> 00:50:11,380 minus 1, because all Sx does is swap 1027 00:50:11,380 --> 00:50:14,120 the first and second components. 1028 00:50:14,120 --> 00:50:16,610 So it gives me minus 1, takes it to the top, 1029 00:50:16,610 --> 00:50:18,200 but that's just an overall minus sign. 1030 00:50:18,200 --> 00:50:20,480 So again, we have the correct eigenvalues, plus and minus 1031 00:50:20,480 --> 00:50:22,290 h bar upon 2, and now we know the eigenvector. 1032 00:50:22,290 --> 00:50:23,410 So what does this tell us? 1033 00:50:23,410 --> 00:50:26,720 What does it tell us that up in the x direction 1034 00:50:26,720 --> 00:50:32,410 is equal to 1 over root 2 if I normalize things properly. 1035 00:50:32,410 --> 00:50:35,384 Up in the z direction plus down in the z direction. 1036 00:50:35,384 --> 00:50:36,675 That's what this is telling me. 1037 00:50:42,780 --> 00:50:45,720 This vector is equal to up in the z direction-- that's 1038 00:50:45,720 --> 00:50:48,280 this guy-- plus down in the z direction. 1039 00:50:51,630 --> 00:50:54,307 But this isn't properly normalized, 1040 00:50:54,307 --> 00:50:56,515 and properly normalizing it gives us this expression. 1041 00:51:00,010 --> 00:51:01,514 So what does this tell us? 1042 00:51:01,514 --> 00:51:03,930 This tells us that if we happen to know that the system is 1043 00:51:03,930 --> 00:51:07,990 in the state with angular momentum, or spin, 1044 00:51:07,990 --> 00:51:11,590 angular momentum in the x direction being plus 1/2, then 1045 00:51:11,590 --> 00:51:14,640 the probability to measure up in the z direction 1046 00:51:14,640 --> 00:51:17,210 in a subsequent measurement is 1/2. 1047 00:51:17,210 --> 00:51:20,610 And the probability to measure down is 1/2. 1048 00:51:20,610 --> 00:51:22,630 If you know it's up in the x direction, 1049 00:51:22,630 --> 00:51:26,190 the probability of measuring up in the z or down in the z 1050 00:51:26,190 --> 00:51:26,822 are equal. 1051 00:51:26,822 --> 00:51:27,530 You're at chance. 1052 00:51:27,530 --> 00:51:29,299 You're at even odds. 1053 00:51:29,299 --> 00:51:30,340 Everyone agree with that? 1054 00:51:30,340 --> 00:51:33,530 That's the meaning of this expression. 1055 00:51:33,530 --> 00:51:36,580 And similarly, down in the x direction is equal to 1 1056 00:51:36,580 --> 00:51:39,820 over root 2, up in the z direction 1057 00:51:39,820 --> 00:51:42,130 minus down in the z direction. 1058 00:51:42,130 --> 00:51:44,730 And we get that from here. 1059 00:51:44,730 --> 00:51:48,810 This state is explicitly, by construction, the eigenstate 1060 00:51:48,810 --> 00:51:51,200 of Sx as we've constructed Sx. 1061 00:51:51,200 --> 00:51:52,940 And we have a natural expression in terms 1062 00:51:52,940 --> 00:51:55,602 of the z eigenvectors up and down. 1063 00:51:55,602 --> 00:51:57,310 That's what this expression is giving us. 1064 00:51:57,310 --> 00:52:00,910 It gives us an expression of the Sx eigenvector 1065 00:52:00,910 --> 00:52:02,400 in the basis of Sz eigenvectors. 1066 00:52:06,410 --> 00:52:08,590 So for example, this tells you that the probability 1067 00:52:08,590 --> 00:52:12,940 to measure up in the z direction, given 1068 00:52:12,940 --> 00:52:15,272 that we measured down in the x direction 1069 00:52:15,272 --> 00:52:17,230 first-- so this is the conditional probability. 1070 00:52:17,230 --> 00:52:19,470 Suppose I first measured down in the x direction, 1071 00:52:19,470 --> 00:52:21,219 what's the probability that I subsequently 1072 00:52:21,219 --> 00:52:22,580 measure up in the z direction? 1073 00:52:22,580 --> 00:52:25,450 This is equal to-- well, it's the norm 1074 00:52:25,450 --> 00:52:27,640 squared of the expansion coefficient. 1075 00:52:27,640 --> 00:52:31,910 So first down in the x direction and the probability 1076 00:52:31,910 --> 00:52:37,424 that we're up in the z direction is 1 upon root 2 squared. 1077 00:52:37,424 --> 00:52:38,840 Usual rules of quantum mechanics-- 1078 00:52:38,840 --> 00:52:41,090 take the expansion coefficient, take its norm squared, 1079 00:52:41,090 --> 00:52:42,470 that's the probability-- 1/2. 1080 00:52:48,875 --> 00:52:54,090 And we can do exactly the same thing for Sy. 1081 00:52:57,190 --> 00:53:01,196 So let's do the same thing for Sy 1082 00:53:01,196 --> 00:53:03,070 without actually working out all the details. 1083 00:53:11,060 --> 00:53:17,920 So doing the same thing for Sy, up in the y direction 1084 00:53:17,920 --> 00:53:24,070 is equal to 1 upon root 2 times up in the z direction plus i 1085 00:53:24,070 --> 00:53:27,460 down in the z direction. 1086 00:53:27,460 --> 00:53:32,642 And down in the y direction is equal to 1 upon root 2 1087 00:53:32,642 --> 00:53:38,400 up in the z direction minus i down in the z direction. 1088 00:53:38,400 --> 00:53:40,790 And I encourage you to check your knowledge 1089 00:53:40,790 --> 00:53:43,339 by deriving these eigenvectors, which 1090 00:53:43,339 --> 00:53:45,130 you can do given our representations of Sy. 1091 00:53:47,770 --> 00:53:51,020 Now here's a nice thing that we're going to use later. 1092 00:53:51,020 --> 00:53:52,160 Consider the following. 1093 00:53:52,160 --> 00:53:56,470 Consider the operator S theta, which 1094 00:53:56,470 --> 00:54:02,984 I'm going to define as cosine theta Sz plus-- oh, sorry. 1095 00:54:02,984 --> 00:54:04,650 I'm not even going to write it that way. 1096 00:54:04,650 --> 00:54:09,800 So what I mean by S theta is equal to-- take 1097 00:54:09,800 --> 00:54:15,380 our spherical directions and consider an angle in the-- this 1098 00:54:15,380 --> 00:54:22,910 is x, let's say, y, x, z-- consider an angle theta down 1099 00:54:22,910 --> 00:54:31,700 in the zx plane, and we can ask, what is the spin 1100 00:54:31,700 --> 00:54:34,330 operator along the direction theta? 1101 00:54:34,330 --> 00:54:37,170 What's the angular momentum in the direction theta? 1102 00:54:37,170 --> 00:54:41,469 If theta, for example, is equal to pi/2, this is Sx. 1103 00:54:41,469 --> 00:54:43,010 So I'm just defining a spin operator, 1104 00:54:43,010 --> 00:54:45,426 which is the angular momentum along a particular direction 1105 00:54:45,426 --> 00:54:47,610 theta in the xz plane. 1106 00:54:47,610 --> 00:54:49,025 Everyone cool with that? 1107 00:54:49,025 --> 00:54:51,150 This is going to turn out to be very useful for us, 1108 00:54:51,150 --> 00:54:53,389 and I encourage you to derive the following. 1109 00:54:53,389 --> 00:54:55,180 And if you don't derive the following, then 1110 00:54:55,180 --> 00:54:57,760 hopefully it will be done in your recitations. 1111 00:54:57,760 --> 00:55:00,910 Well, I will chat with your recitation instructors. 1112 00:55:00,910 --> 00:55:02,970 And if you do this, then what are 1113 00:55:02,970 --> 00:55:05,050 we going to get for the eigenvalues of S theta? 1114 00:55:08,530 --> 00:55:10,510 What possible eigenvalues could S theta have? 1115 00:55:17,915 --> 00:55:18,790 [? AUDIENCE: None. ?] 1116 00:55:18,790 --> 00:55:19,022 PROFESSOR: Good. 1117 00:55:19,022 --> 00:55:19,610 Why? 1118 00:55:19,610 --> 00:55:21,960 AUDIENCE: Because it can't have any other redirection 1119 00:55:21,960 --> 00:55:24,387 [INAUDIBLE] no matter where you start. 1120 00:55:24,387 --> 00:55:25,220 PROFESSOR: Fabulous. 1121 00:55:25,220 --> 00:55:25,720 OK. 1122 00:55:25,720 --> 00:55:28,280 So the answer that was given is that it's 1123 00:55:28,280 --> 00:55:30,690 the same, plus or minus h bar upon 2 1124 00:55:30,690 --> 00:55:33,719 as Sz, or indeed as Sx or Sy, because it can't possibly 1125 00:55:33,719 --> 00:55:35,760 matter what direction you chose at the beginning. 1126 00:55:35,760 --> 00:55:37,950 I could have called this direction theta z. 1127 00:55:37,950 --> 00:55:39,300 How do you stop me? 1128 00:55:39,300 --> 00:55:41,080 We could have done that. 1129 00:55:41,080 --> 00:55:42,184 We didn't. 1130 00:55:42,184 --> 00:55:43,600 That was our first wave answering. 1131 00:55:43,600 --> 00:55:45,260 Second wave answering is what? 1132 00:55:45,260 --> 00:55:48,230 Well, construct the operator S theta 1133 00:55:48,230 --> 00:55:52,115 and find its eigenvectors and eigenvalues. 1134 00:55:52,115 --> 00:55:53,365 So I encourage you to do that. 1135 00:55:53,365 --> 00:55:58,290 And what you find is that, of course, 1136 00:55:58,290 --> 00:56:01,110 the eigenvalues are plus or minus h bar upon 2 1137 00:56:01,110 --> 00:56:09,140 and up at the angle theta is equal to cosine of theta upon 2 1138 00:56:09,140 --> 00:56:13,020 up in the z direction plus sine of theta 1139 00:56:13,020 --> 00:56:17,310 upon 2 down in the z direction. 1140 00:56:17,310 --> 00:56:21,540 And down theta is equal to cosine theta 1141 00:56:21,540 --> 00:56:27,930 upon 2 up in the z direction minus sine of theta 1142 00:56:27,930 --> 00:56:35,050 over 2 down in the z direction. 1143 00:56:35,050 --> 00:56:35,872 Oops, no. 1144 00:56:35,872 --> 00:56:36,580 I got that wrong. 1145 00:56:36,580 --> 00:56:39,530 This is sine, and this is minus cosine. 1146 00:56:39,530 --> 00:56:40,624 Good. 1147 00:56:40,624 --> 00:56:41,540 That makes more sense. 1148 00:56:45,020 --> 00:56:45,520 OK. 1149 00:56:45,520 --> 00:56:46,822 So let's just sanity check. 1150 00:56:46,822 --> 00:56:48,530 These guys should be properly normalized. 1151 00:56:48,530 --> 00:56:51,190 So if we take the norm squared of this guy, 1152 00:56:51,190 --> 00:56:53,175 the cross terms vanish because up z and down z 1153 00:56:53,175 --> 00:56:54,366 are orthogonal states. 1154 00:56:54,366 --> 00:56:56,740 So we're going to get a co squared plus the sine squared. 1155 00:56:56,740 --> 00:56:57,240 That's 1. 1156 00:56:57,240 --> 00:56:58,551 So that's properly normalized. 1157 00:56:58,551 --> 00:56:59,550 Same thing for this guy. 1158 00:56:59,550 --> 00:57:00,570 The minus doesn't change anything 1159 00:57:00,570 --> 00:57:01,569 because we norm squared. 1160 00:57:01,569 --> 00:57:03,271 Now, I'll check that they're orthogonal. 1161 00:57:03,271 --> 00:57:05,770 If we take this guy dotted into this guy, everything's real. 1162 00:57:05,770 --> 00:57:09,700 So we get a cosine from up up, we're 1163 00:57:09,700 --> 00:57:11,022 going to get a cosine sine. 1164 00:57:11,022 --> 00:57:13,480 And from down down, we're going to get a minus cosine sine. 1165 00:57:13,480 --> 00:57:14,287 So that gives us 0. 1166 00:57:14,287 --> 00:57:15,870 So these guys are orthogonal, and they 1167 00:57:15,870 --> 00:57:18,630 satisfy all the nice properties we want. 1168 00:57:18,630 --> 00:57:25,245 So this is a good check-- do this-- of your knowledge. 1169 00:57:25,245 --> 00:57:27,120 And if we had problem sets allowed this week, 1170 00:57:27,120 --> 00:57:30,080 I would give you this in your problem set, but we don't. 1171 00:57:35,879 --> 00:57:36,830 Yeah, OK. 1172 00:57:44,737 --> 00:57:45,695 Suppose that I measure. 1173 00:57:45,695 --> 00:57:47,880 So I want to use these states for something. 1174 00:57:47,880 --> 00:57:58,330 Suppose that I measure Sz and find 1175 00:57:58,330 --> 00:58:02,690 Sz is equal to 1/2 h bar plus h bar upon 2, 1176 00:58:02,690 --> 00:58:06,010 OK, at some moment in time. 1177 00:58:10,232 --> 00:58:11,190 First question is easy. 1178 00:58:11,190 --> 00:58:14,190 What's the state of the system subsequent to that measurement? 1179 00:58:17,004 --> 00:58:19,350 AUDIENCE: [INAUDIBLE] 1180 00:58:19,350 --> 00:58:21,777 PROFESSOR: Man, you all are so quiet today. 1181 00:58:21,777 --> 00:58:24,110 What's the state of the system subsequent to measurement 1182 00:58:24,110 --> 00:58:26,695 that Sz is plus h bar upon 2? 1183 00:58:26,695 --> 00:58:28,406 AUDIENCE: Up z. 1184 00:58:28,406 --> 00:58:29,080 PROFESSOR: Up z. 1185 00:58:29,080 --> 00:58:29,580 Good. 1186 00:58:29,580 --> 00:58:36,700 So our state psi is up z upon measurement, 1187 00:58:36,700 --> 00:58:38,915 OK, after measurement. 1188 00:58:48,540 --> 00:58:50,805 And I need new chalk. 1189 00:58:50,805 --> 00:58:51,305 OK. 1190 00:58:54,480 --> 00:59:04,316 Now, if I measure Sx, so 1, 2, measure Sx, what will I get? 1191 00:59:04,316 --> 00:59:07,250 AUDIENCE: [INAUDIBLE] 1192 00:59:07,250 --> 00:59:08,300 PROFESSOR: OK. 1193 00:59:08,300 --> 00:59:11,726 What values will I observe with what probabilities? 1194 00:59:11,726 --> 00:59:13,600 Well, first off, what are the possible values 1195 00:59:13,600 --> 00:59:14,607 that you can measure? 1196 00:59:14,607 --> 00:59:16,010 AUDIENCE: Plus or minus h bar over 2. 1197 00:59:16,010 --> 00:59:17,801 PROFESSOR: Right, the possible eigenvalues, 1198 00:59:17,801 --> 00:59:20,760 so which is plus or minus h bar upon 2, but with what 1199 00:59:20,760 --> 00:59:21,720 probability? 1200 00:59:21,720 --> 00:59:22,595 AUDIENCE: [INAUDIBLE] 1201 00:59:22,595 --> 00:59:23,386 PROFESSOR: Exactly. 1202 00:59:23,386 --> 00:59:25,200 So we know this from the eigenstate of Sx. 1203 00:59:25,200 --> 00:59:28,040 If we know that we're in the state up z, 1204 00:59:28,040 --> 00:59:30,400 we can take linear combinations of this guy 1205 00:59:30,400 --> 00:59:35,810 to show that up z is equal to 1 over root 2. 1206 00:59:35,810 --> 00:59:37,090 So let's just check. 1207 00:59:37,090 --> 00:59:40,460 If we add these two together, up x and down x, 1208 00:59:40,460 --> 00:59:43,240 if we add them together, we'll get 1/2, 1209 00:59:43,240 --> 00:59:50,810 1/2, 2/2, a root 2 times up z, is up x plus down x, up x 1210 00:59:50,810 --> 00:59:56,060 plus down x, dividing through by the 1/2. 1211 00:59:56,060 --> 00:59:58,290 So here we've expressed up in the z direction 1212 00:59:58,290 --> 01:00:01,810 in a basis of x and y, which is what we're supposed to do. 1213 01:00:01,810 --> 01:00:04,920 So the probability that I measure a plus 1/2 1214 01:00:04,920 --> 01:00:11,016 as x is equal to plus 1/2 h bar upon 2 1215 01:00:11,016 --> 01:00:14,290 is equal to 1 over root 2 squared, so 1/2. 1216 01:00:14,290 --> 01:00:19,780 And ditto, Sx is equal to minus h bar upon 2. 1217 01:00:19,780 --> 01:00:21,271 Same probability. 1218 01:00:21,271 --> 01:00:21,770 OK? 1219 01:00:21,770 --> 01:00:22,740 What about Sy? 1220 01:00:28,680 --> 01:00:30,310 Again, we get one of two values. 1221 01:00:30,310 --> 01:00:33,770 Oops, plus or minus h bar upon 2. 1222 01:00:33,770 --> 01:00:38,070 But the probability of measuring plus is equal to 1/2, 1223 01:00:38,070 --> 01:00:47,050 and the probability that we measure minus 1/2 is 1/2. 1224 01:00:47,050 --> 01:00:49,310 That's h bar upon 2. 1225 01:00:49,310 --> 01:00:49,810 OK? 1226 01:00:49,810 --> 01:00:51,018 So this should look familiar. 1227 01:00:54,920 --> 01:00:57,040 Going back to the very first lecture, 1228 01:00:57,040 --> 01:01:00,260 hardness was spin in the z direction, 1229 01:01:00,260 --> 01:01:05,624 and color was spin in the x direction, I guess. 1230 01:01:05,624 --> 01:01:07,290 And we added, at one point, a third one, 1231 01:01:07,290 --> 01:01:09,230 which I think I called whimsy, which 1232 01:01:09,230 --> 01:01:12,740 is equal to spin in the y direction. 1233 01:01:12,740 --> 01:01:13,240 OK? 1234 01:01:13,240 --> 01:01:15,085 And now all of the box operation that we 1235 01:01:15,085 --> 01:01:17,210 used in that very first lecture, you can understand 1236 01:01:17,210 --> 01:01:19,140 is nothing other than stringing together 1237 01:01:19,140 --> 01:01:22,100 chains of Stern-Gerlach experiments 1238 01:01:22,100 --> 01:01:24,630 doing Sx, Sy, and Sz. 1239 01:01:24,630 --> 01:01:27,182 So let's be explicit about that. 1240 01:01:27,182 --> 01:01:29,510 Let's make that concrete. 1241 01:01:29,510 --> 01:01:32,760 Actually, let's do that here. 1242 01:01:32,760 --> 01:01:38,900 So for example, suppose we put a random electron 1243 01:01:38,900 --> 01:01:43,691 into an Sz color box. 1244 01:01:43,691 --> 01:01:44,190 OK. 1245 01:01:44,190 --> 01:01:46,620 Some are going to come out up, and some 1246 01:01:46,620 --> 01:01:51,520 are going to come out down in the z direction. 1247 01:01:51,520 --> 01:01:57,210 And if we send this into now an Sx color box, 1248 01:01:57,210 --> 01:01:59,610 this is going to give us either up in the x direction 1249 01:01:59,610 --> 01:02:02,070 or down in the x direction. 1250 01:02:02,070 --> 01:02:06,200 And what we'll get out is 50-50, right? 1251 01:02:06,200 --> 01:02:06,700 OK. 1252 01:02:06,700 --> 01:02:10,040 So let's take the ones that came out down. 1253 01:02:10,040 --> 01:02:15,450 And if we send those back into an Sz, what do we get? 1254 01:02:15,450 --> 01:02:16,770 AUDIENCE: 50-50. 1255 01:02:16,770 --> 01:02:21,450 PROFESSOR: Yeah, 50-50, because down x is 1 over 2 up z 1256 01:02:21,450 --> 01:02:23,091 plus 1 over 2 down z. 1257 01:02:23,091 --> 01:02:25,340 What was going on in that very first experiment, where 1258 01:02:25,340 --> 01:02:28,200 we did hardness, color, hardness? 1259 01:02:28,200 --> 01:02:29,650 Superposition. 1260 01:02:29,650 --> 01:02:33,680 And what superposition was a hard electron? 1261 01:02:33,680 --> 01:02:37,320 It was 1 over a 2, white plus black. 1262 01:02:37,320 --> 01:02:40,130 We know precisely which superposition, and here it is. 1263 01:02:40,130 --> 01:02:40,890 OK? 1264 01:02:40,890 --> 01:02:42,990 And now, let's do that last experiment, where 1265 01:02:42,990 --> 01:02:51,510 we take these guys, Sx down, and I'm turning this upside down, 1266 01:02:51,510 --> 01:02:52,140 beam joiners. 1267 01:02:52,140 --> 01:02:56,059 We take the up in the x direction and the down 1268 01:02:56,059 --> 01:02:58,100 in the x direction, and we combine them together, 1269 01:02:58,100 --> 01:03:02,772 and we put them back into Sz, what do we get? 1270 01:03:02,772 --> 01:03:03,730 Well, what's the state? 1271 01:03:03,730 --> 01:03:09,600 1 over root 2 down x plus 1 over root 2 up x? 1272 01:03:09,600 --> 01:03:12,030 AUDIENCE: [INAUDIBLE] 1273 01:03:12,030 --> 01:03:14,580 PROFESSOR: It's up z. 1274 01:03:14,580 --> 01:03:18,640 Up z into an Sz box, what do we get? 1275 01:03:18,640 --> 01:03:19,285 Up z with 100%. 1276 01:03:22,220 --> 01:03:25,660 That's white with 100%, or I'm sorry, hard with 100%. 1277 01:03:25,660 --> 01:03:27,942 AUDIENCE: [INAUDIBLE] down z with a down z. 1278 01:03:27,942 --> 01:03:28,650 PROFESSOR: Sorry. 1279 01:03:28,650 --> 01:03:29,462 AUDIENCE: You put in down. 1280 01:03:29,462 --> 01:03:30,420 PROFESSOR: Oh, I put in down. 1281 01:03:30,420 --> 01:03:31,128 Shoot, I'm sorry. 1282 01:03:31,128 --> 01:03:32,910 Well, yeah, indeed, I meant to put in up. 1283 01:03:32,910 --> 01:03:35,510 Yes, down z with 100% confidence. 1284 01:03:35,510 --> 01:03:37,230 If we remove the mirror, 50-50. 1285 01:03:37,230 --> 01:03:38,932 If we add in the mirror, 100%. 1286 01:03:38,932 --> 01:03:40,390 And the difference is whether we're 1287 01:03:40,390 --> 01:03:42,740 taking one component of the wave function, 1288 01:03:42,740 --> 01:03:46,054 or whether we're superposing them back together. 1289 01:03:46,054 --> 01:03:47,010 All right? 1290 01:03:47,010 --> 01:03:48,970 Imagine we know we have a system in this state, 1291 01:03:48,970 --> 01:03:52,590 and I say, look, this component is also coincidentally very far 1292 01:03:52,590 --> 01:03:55,190 away, and I'm going to not look at them. 1293 01:03:55,190 --> 01:03:58,656 So of the ones that I look at, I have 1 over root 2 up x. 1294 01:03:58,656 --> 01:04:00,030 But if I look at the full system, 1295 01:04:00,030 --> 01:04:03,370 the superimposed system, that adds together 1296 01:04:03,370 --> 01:04:05,890 to be an eigenstate of Sz. 1297 01:04:05,890 --> 01:04:08,498 These are our color boxes. 1298 01:04:08,498 --> 01:04:08,998 Yeah? 1299 01:04:08,998 --> 01:04:09,736 AUDIENCE: But how do you know, when 1300 01:04:09,736 --> 01:04:11,458 you put the two beams to the beam joiner, 1301 01:04:11,458 --> 01:04:13,545 it serves to add their two states together? 1302 01:04:13,545 --> 01:04:14,670 PROFESSOR: Yeah, excellent. 1303 01:04:14,670 --> 01:04:15,870 This is a very subtle point. 1304 01:04:15,870 --> 01:04:18,420 So here we have to decide what we mean by the beam splitter. 1305 01:04:18,420 --> 01:04:20,420 And what I'm going to mean by the beam splitter, 1306 01:04:20,420 --> 01:04:21,510 by the mirrors and beam joiners-- 1307 01:04:21,510 --> 01:04:23,150 so the question is, how do we know that it does this 1308 01:04:23,150 --> 01:04:24,860 without changing the superposition? 1309 01:04:24,860 --> 01:04:27,600 And what I want to do is define this thing 1310 01:04:27,600 --> 01:04:30,410 as the object that takes the two incident wave functions, 1311 01:04:30,410 --> 01:04:31,780 and it just adds them together. 1312 01:04:31,780 --> 01:04:33,530 It should give me the direct superposition 1313 01:04:33,530 --> 01:04:35,570 with the appropriate phases and coefficients. 1314 01:04:35,570 --> 01:04:39,870 So if this was plus up x, then it stays plus up x. 1315 01:04:39,870 --> 01:04:43,470 If it's minus up x, it's going to be minus up x. 1316 01:04:43,470 --> 01:04:46,700 Whatever the phase is of this state, when it gets here 1317 01:04:46,700 --> 01:04:48,780 it just adds together the two components. 1318 01:04:48,780 --> 01:04:50,570 That's my definition of that adding box. 1319 01:04:50,570 --> 01:04:54,687 AUDIENCE: So realistically, what does that look like? 1320 01:04:54,687 --> 01:04:55,520 PROFESSOR: Oh, what? 1321 01:04:55,520 --> 01:04:56,895 You think I'm an experimentalist? 1322 01:04:56,895 --> 01:04:57,630 [LAUGHTER] 1323 01:04:57,630 --> 01:04:59,600 Look, every time I try an experiment, 1324 01:04:59,600 --> 01:05:00,910 I get hit by a shark. 1325 01:05:00,910 --> 01:05:01,410 OK? 1326 01:05:01,410 --> 01:05:04,320 [LAUGHTER] 1327 01:05:04,320 --> 01:05:05,500 Yeah, no. 1328 01:05:05,500 --> 01:05:07,480 How you actually implement that in real systems 1329 01:05:07,480 --> 01:05:08,890 is a more complicated story. 1330 01:05:08,890 --> 01:05:11,330 So you should direct that question 1331 01:05:11,330 --> 01:05:14,720 to Matt, who's a very good experimentalist. 1332 01:05:14,720 --> 01:05:16,350 OK. 1333 01:05:16,350 --> 01:05:19,570 So finally, let's go back to the Stern-Gerlach experiment, 1334 01:05:19,570 --> 01:05:22,080 and let's actually run the Stern-Gerlach experiment. 1335 01:05:22,080 --> 01:05:24,241 I guess I'll do that here. 1336 01:05:24,241 --> 01:05:26,740 So let's think about what the Stern-Gerlach experiment looks 1337 01:05:26,740 --> 01:05:29,010 like in this notation, and not just in this notation, 1338 01:05:29,010 --> 01:05:32,590 in the honest language of spin. 1339 01:05:32,590 --> 01:05:35,934 And I'm going to do a slightly abbreviated version of this 1340 01:05:35,934 --> 01:05:37,600 because you guys can fill in the details 1341 01:05:37,600 --> 01:05:40,981 with your knowledge of 802 and 803. 1342 01:05:40,981 --> 01:05:41,480 OK. 1343 01:05:41,480 --> 01:05:43,104 So here's the Stern-Gerlach experiment. 1344 01:05:43,104 --> 01:05:46,635 We have a gradient in the magnetic field. 1345 01:05:46,635 --> 01:05:47,635 This is the z direction. 1346 01:05:47,635 --> 01:05:52,516 And we have a gradient where B in the z direction 1347 01:05:52,516 --> 01:06:01,492 has some B0, a constant plus beta z. 1348 01:06:01,492 --> 01:06:02,300 OK? 1349 01:06:02,300 --> 01:06:06,601 So it's got a constant piece and a small gradient. 1350 01:06:06,601 --> 01:06:07,600 Everyone cool with that? 1351 01:06:07,600 --> 01:06:09,495 It's just the magnetic field gets stronger and stronger 1352 01:06:09,495 --> 01:06:11,160 in the z direction, there's a constant, 1353 01:06:11,160 --> 01:06:13,500 and then there's a rate of increase in the z direction. 1354 01:06:13,500 --> 01:06:17,310 And I'm going to send my electron through. 1355 01:06:17,310 --> 01:06:21,090 Now remember, my electron has a wave function, psi electron, 1356 01:06:21,090 --> 01:06:23,820 is equal to-- well, it's got some amplitude to be up, 1357 01:06:23,820 --> 01:06:28,500 a up z, plus some amplitude to be down, b down z. 1358 01:06:28,500 --> 01:06:30,050 And if this is a random electron, 1359 01:06:30,050 --> 01:06:31,120 then its state is going to be random, 1360 01:06:31,120 --> 01:06:33,120 and a and b are going to be random numbers whose 1361 01:06:33,120 --> 01:06:35,710 norm squared add up to 1, proper normalization. 1362 01:06:35,710 --> 01:06:37,247 Cool? 1363 01:06:37,247 --> 01:06:38,705 So here's our random initial state, 1364 01:06:38,705 --> 01:06:40,250 and we send it into this region where 1365 01:06:40,250 --> 01:06:41,750 we've got a magnetic field gradient. 1366 01:06:41,750 --> 01:06:43,690 And what happens? 1367 01:06:43,690 --> 01:06:51,720 Well, we know that the energy of an electron that 1368 01:06:51,720 --> 01:06:57,170 carries some angular momentum is a constant, mu naught, 1369 01:06:57,170 --> 01:06:59,740 times its angular momentum dotted 1370 01:06:59,740 --> 01:07:03,230 into any ambient magnetic field. 1371 01:07:03,230 --> 01:07:04,810 Whoops, sorry, with a minus sign. 1372 01:07:04,810 --> 01:07:06,768 This is saying that magnets want to anti-align. 1373 01:07:09,514 --> 01:07:10,930 Now, in particular, here we've got 1374 01:07:10,930 --> 01:07:13,460 a magnetic field in the z direction. 1375 01:07:13,460 --> 01:07:19,280 So this is minus mu 0 Sz Bz. 1376 01:07:19,280 --> 01:07:28,210 And Bz was a constant, beta 0-- sorry, B0 plus beta z. 1377 01:07:28,210 --> 01:07:30,090 So the energy has two terms. 1378 01:07:30,090 --> 01:07:32,220 It has a constant term, which just depends on Sz, 1379 01:07:32,220 --> 01:07:33,720 and then there's a term that depends 1380 01:07:33,720 --> 01:07:36,435 on z as well as depending on Sz [INAUDIBLE]. 1381 01:07:36,435 --> 01:07:42,625 So we can write this as a-- so Sz, remember what Sz is. 1382 01:07:42,625 --> 01:07:48,520 Sz is equal to h bar upon 2, 1, 0, 0, minus 1. 1383 01:07:48,520 --> 01:07:57,244 So this is a matrix with some coefficient up a-- 1384 01:07:57,244 --> 01:07:58,410 do I want to write this out? 1385 01:08:02,220 --> 01:08:04,830 Yeah, I guess I don't really need to write this out. 1386 01:08:04,830 --> 01:08:08,749 But this is a matrix, and this is our energy operator. 1387 01:08:08,749 --> 01:08:11,290 And it acts on any given state to give us another state back. 1388 01:08:11,290 --> 01:08:13,676 It's an operator. 1389 01:08:13,676 --> 01:08:14,175 OK. 1390 01:08:17,189 --> 01:08:19,240 And importantly, I want this to be 1391 01:08:19,240 --> 01:08:25,637 only-- this is in some region where we're 1392 01:08:25,637 --> 01:08:28,220 doing the experiment, where we have a magnetic field gradient. 1393 01:08:28,220 --> 01:08:30,609 Then outside of this region, we have no magnetic field 1394 01:08:30,609 --> 01:08:33,160 and no magnetic field gradient. 1395 01:08:33,160 --> 01:08:35,979 So it's 0 to the left and 0 to the right. 1396 01:08:41,310 --> 01:08:44,470 So as we've talked about before, the electron 1397 01:08:44,470 --> 01:08:47,649 feels a force due to this gradient to the magnetic field. 1398 01:08:47,649 --> 01:08:49,507 The energy depends on z, so the derivative 1399 01:08:49,507 --> 01:08:51,090 of the energy with respect to z, which 1400 01:08:51,090 --> 01:08:54,279 is the force in the z direction, is non-zero. 1401 01:08:54,279 --> 01:08:56,784 But for the moment that's a bit of a red herring. 1402 01:08:56,784 --> 01:08:58,950 Instead of worrying about the center of mass motion, 1403 01:08:58,950 --> 01:09:00,899 let's just focus on the overall phase. 1404 01:09:03,420 --> 01:09:11,140 So let's take our initial electron with this initial wave 1405 01:09:11,140 --> 01:09:17,750 function a up z plus b down z, and let's 1406 01:09:17,750 --> 01:09:23,069 note that in this time-- so what does this matrix look like? 1407 01:09:23,069 --> 01:09:23,569 OK. 1408 01:09:23,569 --> 01:09:26,479 So fine, let's actually look at this matrix. 1409 01:09:26,479 --> 01:09:30,910 So Sz is the matrix 1, 0, 0, minus 1 times h bar upon 2. 1410 01:09:30,910 --> 01:09:39,580 So we have h bar upon 2 minus mu 0. 1411 01:09:39,580 --> 01:09:41,465 And then B0 plus beta z, which I'll just 1412 01:09:41,465 --> 01:09:47,779 write as B, which is equal to some constant C, 0, 0, 1413 01:09:47,779 --> 01:09:54,480 some constant minus C. Everyone agree with that? 1414 01:09:54,480 --> 01:09:56,630 Where a constant, I just mean it's a number, 1415 01:09:56,630 --> 01:09:59,696 but it does depend on z, because the z is in here. 1416 01:10:04,626 --> 01:10:05,750 Now, here's the nice thing. 1417 01:10:05,750 --> 01:10:14,390 When we expand the energy on up z, this is equal to C of z 1418 01:10:14,390 --> 01:10:19,180 up z, because there's our energy operator. 1419 01:10:19,180 --> 01:10:26,685 And energy on down z is equal to minus Cz of z down to z. 1420 01:10:26,685 --> 01:10:28,060 So up and down in the z direction 1421 01:10:28,060 --> 01:10:31,250 are still eigenfunctions of the energy operator. 1422 01:10:31,250 --> 01:10:32,970 We've chosen an interaction, we've 1423 01:10:32,970 --> 01:10:35,100 chosen a potential which is already diagonal, 1424 01:10:35,100 --> 01:10:36,390 so the energy is diagonal. 1425 01:10:36,390 --> 01:10:39,450 It's already in its eigenbasis. 1426 01:10:39,450 --> 01:10:44,760 So as a consequence, this is the energy, energy of the up state 1427 01:10:44,760 --> 01:10:48,270 is equal to, in the z direction, is equal to plus C. 1428 01:10:48,270 --> 01:10:53,180 And energy in the down state, energy of the down state, 1429 01:10:53,180 --> 01:10:56,654 is equal to minus C of z. 1430 01:10:56,654 --> 01:10:58,080 Everyone agree with that? 1431 01:10:58,080 --> 01:10:59,760 So what this magnetic field does it 1432 01:10:59,760 --> 01:11:02,050 splits the degeneracy of the up and down states. 1433 01:11:02,050 --> 01:11:03,424 The up and down states originally 1434 01:11:03,424 --> 01:11:04,520 had no energy splitting. 1435 01:11:04,520 --> 01:11:05,990 They were both zero energy. 1436 01:11:05,990 --> 01:11:07,865 We turn on this magnetic field, and one state 1437 01:11:07,865 --> 01:11:09,531 has positive energy, and the other state 1438 01:11:09,531 --> 01:11:10,480 has negative energy. 1439 01:11:10,480 --> 01:11:12,570 So that degeneracy has been split. 1440 01:11:12,570 --> 01:11:16,410 Where did that original degeneracy come from? 1441 01:11:16,410 --> 01:11:18,410 Why did we have a degeneracy in the first place? 1442 01:11:18,410 --> 01:11:19,310 AUDIENCE: [INAUDIBLE] 1443 01:11:19,310 --> 01:11:20,460 PROFESSOR: Spherical symmetry. 1444 01:11:20,460 --> 01:11:22,300 And we turn on the magnetic field, which picks out 1445 01:11:22,300 --> 01:11:24,049 a direction, and we break that degeneracy, 1446 01:11:24,049 --> 01:11:26,080 and we lift the splitting. 1447 01:11:26,080 --> 01:11:27,090 OK? 1448 01:11:27,090 --> 01:11:29,090 So here we see that the splitting is lifted. 1449 01:11:29,090 --> 01:11:30,970 And now we want to ask, how does the wave function evolve 1450 01:11:30,970 --> 01:11:31,260 in time? 1451 01:11:31,260 --> 01:11:32,920 So this was our initial wave function. 1452 01:11:35,450 --> 01:11:39,089 But we know from the Schrodinger equation that psi of t 1453 01:11:39,089 --> 01:11:41,130 is going to be equal to-- well, those are already 1454 01:11:41,130 --> 01:11:52,600 eigenvectors-- so a e to the minus i e up t upon h bar up z 1455 01:11:52,600 --> 01:12:07,009 plus b e to the minus i e down t upon h bar down z. 1456 01:12:07,009 --> 01:12:07,508 All right? 1457 01:12:07,508 --> 01:12:23,260 But we can write this as equals a e to the i mu 0 b0 t over 2 1458 01:12:23,260 --> 01:12:39,450 plus i mu 0 beta t z upon 2 up z plus b same e to the minus 1459 01:12:39,450 --> 01:12:41,520 i--oops, that should be, oh, no, that's plus-- 1460 01:12:41,520 --> 01:12:45,855 that's e to the i mu 0 b0 t over 2. 1461 01:12:49,250 --> 01:12:51,240 And if we write this as two exponentials, 1462 01:12:51,240 --> 01:13:00,130 e to the minus i mu 0 beta t z upon 2-- oh no, 1463 01:13:00,130 --> 01:13:02,081 that's a minus, good, OK-- down z. 1464 01:13:02,081 --> 01:13:02,580 OK. 1465 01:13:02,580 --> 01:13:05,090 So what is this telling us? 1466 01:13:05,090 --> 01:13:06,840 So what this tells us is that we start off 1467 01:13:06,840 --> 01:13:09,430 in a state, which has some amplitude to be up 1468 01:13:09,430 --> 01:13:13,085 and some amplitude to be down, a and b. 1469 01:13:13,085 --> 01:13:16,420 And at a later time t, sine of t, 1470 01:13:16,420 --> 01:13:20,360 what we find after we run it through this apparatus 1471 01:13:20,360 --> 01:13:22,760 is that this is the amplitude. 1472 01:13:22,760 --> 01:13:23,760 What do you notice? 1473 01:13:23,760 --> 01:13:25,260 Well, we notice two things. 1474 01:13:25,260 --> 01:13:28,200 The first is that the system evolves 1475 01:13:28,200 --> 01:13:31,190 with some overall energy. 1476 01:13:31,190 --> 01:13:32,820 So the phase rotates as usual. 1477 01:13:32,820 --> 01:13:34,700 These are energy eigenstates, but the amount 1478 01:13:34,700 --> 01:13:35,991 of phase rotation depends on z. 1479 01:13:35,991 --> 01:13:40,640 So in particular, this is e to the i some number times z. 1480 01:13:40,640 --> 01:13:42,550 And what is e to the i some number times z? 1481 01:13:46,430 --> 01:13:48,870 If you know you have a state that's of the form e to the i 1482 01:13:48,870 --> 01:13:52,101 some number, which I will, I don't know, call kz times z, 1483 01:13:52,101 --> 01:13:53,850 what is this telling you about the system? 1484 01:14:00,085 --> 01:14:01,710 This is an eigenstate of what operator? 1485 01:14:01,710 --> 01:14:02,580 AUDIENCE: Momentum. 1486 01:14:02,580 --> 01:14:03,540 PROFESSOR: Momentum in the z direction. 1487 01:14:03,540 --> 01:14:04,800 It carries what momentum? 1488 01:14:04,800 --> 01:14:05,591 AUDIENCE: h bar kz. 1489 01:14:05,591 --> 01:14:07,280 PROFESSOR: h bar kz, exactly. 1490 01:14:07,280 --> 01:14:08,310 So what about this? 1491 01:14:08,310 --> 01:14:10,319 If I have a system in this state, 1492 01:14:10,319 --> 01:14:12,610 what can you say about its momentum in the z direction? 1493 01:14:16,019 --> 01:14:17,480 AUDIENCE: [INAUDIBLE] 1494 01:14:17,480 --> 01:14:18,660 PROFESSOR: It's non-zero. 1495 01:14:18,660 --> 01:14:19,580 Right here's a z. 1496 01:14:19,580 --> 01:14:20,871 These are a bunch of constants. 1497 01:14:20,871 --> 01:14:23,450 It's beta, t, mu 0, 2, i. 1498 01:14:23,450 --> 01:14:24,790 So it's non-zero. 1499 01:14:24,790 --> 01:14:28,406 In fact, it's got minus some constant times z. 1500 01:14:28,406 --> 01:14:29,140 Right? 1501 01:14:29,140 --> 01:14:32,700 So this is a state with negative momentum. 1502 01:14:32,700 --> 01:14:33,360 Everyone agree? 1503 01:14:33,360 --> 01:14:37,100 It's got momentum z down in the z direction. 1504 01:14:37,100 --> 01:14:39,770 What about this guy? 1505 01:14:39,770 --> 01:14:40,440 Momentum up. 1506 01:14:40,440 --> 01:14:43,842 So the states that are up in the z direction get a kick up. 1507 01:14:43,842 --> 01:14:45,800 And the states that are down in the z direction 1508 01:14:45,800 --> 01:14:46,466 get a kick down. 1509 01:14:46,466 --> 01:14:50,000 They pick up some momentum down in the z direction. 1510 01:14:50,000 --> 01:14:50,500 Yeah? 1511 01:14:50,500 --> 01:14:52,870 AUDIENCE: Where did that big T come from? 1512 01:14:52,870 --> 01:14:56,010 PROFESSOR: Big T should be little t, sorry. 1513 01:14:56,010 --> 01:14:58,960 Sorry, just bad handwriting there. 1514 01:14:58,960 --> 01:14:59,960 That's just t. 1515 01:14:59,960 --> 01:15:00,903 Yeah? 1516 01:15:00,903 --> 01:15:03,801 AUDIENCE: So in this case, the eigenvalues-- 1517 01:15:03,801 --> 01:15:04,767 it's a function of z? 1518 01:15:04,767 --> 01:15:05,733 PROFESSOR: Mhm. 1519 01:15:05,733 --> 01:15:07,864 AUDIENCE: Is that allowable? 1520 01:15:07,864 --> 01:15:08,530 PROFESSOR: Yeah. 1521 01:15:08,530 --> 01:15:11,840 So that seems bad, but remember what I started out doing. 1522 01:15:11,840 --> 01:15:13,367 Oop, did I erase it? 1523 01:15:13,367 --> 01:15:14,200 Shoot, we erased it. 1524 01:15:14,200 --> 01:15:16,540 So we started out saying, look, the wave function is up 1525 01:15:16,540 --> 01:15:19,400 in the z direction, or really rather here, up 1526 01:15:19,400 --> 01:15:21,274 in the z direction times some constant. 1527 01:15:21,274 --> 01:15:23,190 Now, in general, this shouldn't be a constant. 1528 01:15:23,190 --> 01:15:24,799 It should be some function. 1529 01:15:24,799 --> 01:15:26,840 So this should be a function of position times up 1530 01:15:26,840 --> 01:15:27,490 in the z direction. 1531 01:15:27,490 --> 01:15:29,770 This is a function position times down in the z direction. 1532 01:15:29,770 --> 01:15:32,145 So what we've done here is we said, under time evolution, 1533 01:15:32,145 --> 01:15:36,330 that function changes in time, but it 1534 01:15:36,330 --> 01:15:39,020 stays some linear combination up and down. 1535 01:15:39,020 --> 01:15:42,100 So you can reorganize this as-- the time evolution equation 1536 01:15:42,100 --> 01:15:44,455 here is an equation for the coefficients of up z 1537 01:15:44,455 --> 01:15:46,189 and the coefficient of down z. 1538 01:15:46,189 --> 01:15:46,730 AUDIENCE: OK. 1539 01:15:49,880 --> 01:15:52,100 PROFESSOR: Other questions? 1540 01:15:52,100 --> 01:15:52,620 OK. 1541 01:15:52,620 --> 01:15:55,160 So the upshot of all this is that we run this experiment, 1542 01:15:55,160 --> 01:15:57,810 and what we discover is that this component of the state 1543 01:15:57,810 --> 01:16:03,370 that was up z gets a kick in the plus z direction. 1544 01:16:03,370 --> 01:16:06,580 And any electron that came from this term 1545 01:16:06,580 --> 01:16:09,860 in the superposition will be kicked up up z. 1546 01:16:09,860 --> 01:16:13,930 And any electron that came from the superposition down z 1547 01:16:13,930 --> 01:16:15,274 will have down. 1548 01:16:15,274 --> 01:16:17,690 Now, what we really mean is not that an electron does this 1549 01:16:17,690 --> 01:16:19,900 or does that, but rather that the initial stage 1550 01:16:19,900 --> 01:16:23,360 of an electron that's here, with the superposition of z 1551 01:16:23,360 --> 01:16:28,610 up and down, ends up in the state as a superposition of up 1552 01:16:28,610 --> 01:16:31,874 z being up here and down z being down here. 1553 01:16:31,874 --> 01:16:32,417 OK? 1554 01:16:32,417 --> 01:16:34,500 An electron didn't do one, it didn't do the other. 1555 01:16:34,500 --> 01:16:36,950 It ends up in a superposition, so the state at the end. 1556 01:16:36,950 --> 01:16:39,170 So what the Stern-Gerlach experiment has done, 1557 01:16:39,170 --> 01:16:41,530 apparatus has done, is it's correlated 1558 01:16:41,530 --> 01:16:45,540 the position of the electron with its spin. 1559 01:16:45,540 --> 01:16:50,590 So if you find the amplitude, to find it up here and down 1560 01:16:50,590 --> 01:16:51,270 is very small. 1561 01:16:51,270 --> 01:16:53,820 The amplitude to find it up here and up is very large. 1562 01:16:53,820 --> 01:16:57,500 Similarly, the amplitude, to find it down here and up is 0. 1563 01:16:57,500 --> 01:17:02,710 And the amplitude, to find it down here and down is large. 1564 01:17:02,710 --> 01:17:03,390 Cool? 1565 01:17:03,390 --> 01:17:05,610 So this is exactly what we wanted from the boxes. 1566 01:17:05,610 --> 01:17:08,250 We wanted not to do something funny to the spins, 1567 01:17:08,250 --> 01:17:11,690 we just wanted to correlate the position with the spin. 1568 01:17:11,690 --> 01:17:14,480 And so the final state is a superposition of these guys. 1569 01:17:14,480 --> 01:17:15,480 And which superposition? 1570 01:17:15,480 --> 01:17:17,450 It's exactly this superposition. 1571 01:17:17,450 --> 01:17:19,575 So these calculations are gone through in the notes 1572 01:17:19,575 --> 01:17:20,930 that are going to be posted. 1573 01:17:20,930 --> 01:17:22,520 OK. 1574 01:17:22,520 --> 01:17:24,930 So questions at this point? 1575 01:17:24,930 --> 01:17:25,430 Yeah? 1576 01:17:25,430 --> 01:17:27,179 AUDIENCE: And so when you put the electron 1577 01:17:27,179 --> 01:17:28,559 through the Stern-Gerlach device, 1578 01:17:28,559 --> 01:17:30,975 does that count as a measurement of the particle's angular 1579 01:17:30,975 --> 01:17:31,450 momentum? 1580 01:17:31,450 --> 01:17:32,325 PROFESSOR: Excellent. 1581 01:17:32,325 --> 01:17:34,760 When I put the electrons through the Stern-Gerlach device, 1582 01:17:34,760 --> 01:17:36,660 do they come out with a definite position? 1583 01:17:36,660 --> 01:17:37,560 AUDIENCE: No. 1584 01:17:37,560 --> 01:17:37,690 PROFESSOR: No. 1585 01:17:37,690 --> 01:17:39,070 At the end of the experiment, they're 1586 01:17:39,070 --> 01:17:41,160 in a superposition of either being here or being here. 1587 01:17:41,160 --> 01:17:42,826 And they're in a superposition of either 1588 01:17:42,826 --> 01:17:44,140 being up spin or down spin. 1589 01:17:44,140 --> 01:17:46,270 Have we determined the spin through putting it 1590 01:17:46,270 --> 01:17:48,530 through this apparatus? 1591 01:17:48,530 --> 01:17:49,030 No. 1592 01:17:49,030 --> 01:17:50,310 We haven't done any measurement. 1593 01:17:50,310 --> 01:17:52,434 The measurement comes when we now do the following. 1594 01:17:52,434 --> 01:17:54,800 We put a detector here that absorbs an electron. 1595 01:17:54,800 --> 01:17:56,185 We say, ah, yeah, it got hit. 1596 01:17:56,185 --> 01:17:58,060 And then you've measured the angular momentum 1597 01:17:58,060 --> 01:17:59,740 by measuring where it came out. 1598 01:17:59,740 --> 01:18:02,740 If it comes out down here, it will be in the positive. 1599 01:18:02,740 --> 01:18:05,805 So this is a nice example of something called entanglement. 1600 01:18:05,805 --> 01:18:07,930 And this is where we're going to pick up next time. 1601 01:18:07,930 --> 01:18:09,600 Entanglement says the following. 1602 01:18:09,600 --> 01:18:12,230 Suppose I know one property of a particle, 1603 01:18:12,230 --> 01:18:19,870 for example, that it's up here, or suppose I'm in the state psi 1604 01:18:19,870 --> 01:18:30,430 is equal to up, so up here and up in the z direction plus 1605 01:18:30,430 --> 01:18:36,280 down there and down in the z direction. 1606 01:18:36,280 --> 01:18:37,482 OK? 1607 01:18:37,482 --> 01:18:39,940 This means that, at the moment, initially, if I said, look, 1608 01:18:39,940 --> 01:18:42,273 is it going to be up or down with equal probabilities, 1 1609 01:18:42,273 --> 01:18:44,800 over root 2 with equal amplitudes, 1610 01:18:44,800 --> 01:18:46,560 if I measure spin in the z direction, what 1611 01:18:46,560 --> 01:18:47,325 value will I get? 1612 01:18:50,860 --> 01:18:52,430 What values could I get if I measure 1613 01:18:52,430 --> 01:18:54,241 spin in the z direction? 1614 01:18:54,241 --> 01:18:54,990 Plus or minus 1/2. 1615 01:18:54,990 --> 01:18:56,531 And what are the probabilities that I 1616 01:18:56,531 --> 01:18:58,170 measure plus 1/2 or minus 1/2? 1617 01:18:58,170 --> 01:18:59,310 AUDIENCE: [INAUDIBLE] 1618 01:18:59,310 --> 01:19:01,280 PROFESSOR: Even odds, right? 1619 01:19:01,280 --> 01:19:04,330 We've done that experiment. 1620 01:19:04,330 --> 01:19:06,300 On the other hand, if I tell you that I've 1621 01:19:06,300 --> 01:19:10,360 measured it to be up here, what will you then 1622 01:19:10,360 --> 01:19:13,980 deduce about what its spin is? 1623 01:19:13,980 --> 01:19:14,946 AUDIENCE: Plus. 1624 01:19:14,946 --> 01:19:16,070 PROFESSOR: Always plus 1/2. 1625 01:19:16,070 --> 01:19:17,600 Did I have to do the measurement to determine 1626 01:19:17,600 --> 01:19:18,391 that it's plus 1/2. 1627 01:19:21,200 --> 01:19:22,760 No, because I already know the state, 1628 01:19:22,760 --> 01:19:24,920 so I know exactly what I will get if I do the experiment. 1629 01:19:24,920 --> 01:19:26,940 I measure up here, and then the wave function 1630 01:19:26,940 --> 01:19:29,290 is this without the 1 over root 2, upon measurement. 1631 01:19:29,290 --> 01:19:31,498 But as a consequence, if I subsequently measure up z, 1632 01:19:31,498 --> 01:19:34,820 the only possible value is up in the z direction. 1633 01:19:34,820 --> 01:19:36,346 Yeah? 1634 01:19:36,346 --> 01:19:38,660 And this is called entanglement. 1635 01:19:38,660 --> 01:19:46,809 And here it's entanglement of the position with the spin. 1636 01:19:46,809 --> 01:19:48,350 And next time, what we'll do is we'll 1637 01:19:48,350 --> 01:19:51,710 study the EPR experiment, which says the following thing. 1638 01:19:51,710 --> 01:19:54,220 Suppose I take two electrons, OK, take two electrons, 1639 01:19:54,220 --> 01:19:56,720 and I put them in the state one is up and the other is down, 1640 01:19:56,720 --> 01:19:59,840 or the first is up and the second is down. 1641 01:19:59,840 --> 01:20:00,530 All right? 1642 01:20:00,530 --> 01:20:03,610 So up, down plus down, up. 1643 01:20:03,610 --> 01:20:04,120 OK? 1644 01:20:04,120 --> 01:20:07,890 I now take my two electrons, and I send them to distant places. 1645 01:20:07,890 --> 01:20:12,150 Suppose I measure one of them to be up in the x direction. 1646 01:20:12,150 --> 01:20:13,570 Yeah? 1647 01:20:13,570 --> 01:20:16,451 Then I know that the other one is down in that direction. 1648 01:20:16,451 --> 01:20:16,950 Sorry. 1649 01:20:16,950 --> 01:20:18,010 If I measure up in the z direction, 1650 01:20:18,010 --> 01:20:19,590 I determine the other one is down in the z direction. 1651 01:20:19,590 --> 01:20:22,449 But suppose someone over here who's causally disconnected 1652 01:20:22,449 --> 01:20:23,990 measures not spin in the z direction, 1653 01:20:23,990 --> 01:20:25,940 but spin in the x direction. 1654 01:20:25,940 --> 01:20:28,890 They'll measure one of two things, either plus or minus. 1655 01:20:28,890 --> 01:20:31,670 Now, knowing what we know, that it's in the z eigenstate, 1656 01:20:31,670 --> 01:20:36,690 it will be either plus 1/2 in the x direction or minus 1/2 1657 01:20:36,690 --> 01:20:38,210 in the x direction. 1658 01:20:38,210 --> 01:20:40,270 Suppose I get both measurements. 1659 01:20:40,270 --> 01:20:42,630 The distant person over here does the measurement of z 1660 01:20:42,630 --> 01:20:46,180 and says, aha, mine is up, so the other one must be down. 1661 01:20:46,180 --> 01:20:49,830 But the person over here doesn't measure Sz, they measure Sx, 1662 01:20:49,830 --> 01:20:51,254 and they get that it's plus Sx. 1663 01:20:51,254 --> 01:20:53,170 And what they've done as a result of these two 1664 01:20:53,170 --> 01:20:55,340 experiments, Einstein, Podolsky, and Rosen say, 1665 01:20:55,340 --> 01:20:58,460 is this electron has been measured by the distant guy 1666 01:20:58,460 --> 01:21:05,500 to be spin z down, but by this guy to be measured spin x up. 1667 01:21:05,500 --> 01:21:10,139 So I know Sx and Sz definitely. 1668 01:21:10,139 --> 01:21:12,680 But that flies in the face of the uncertainty relation, which 1669 01:21:12,680 --> 01:21:16,250 tells us we can't have spin z and spin x definitely. 1670 01:21:16,250 --> 01:21:18,122 Einstein and Podolsky and Rosen say 1671 01:21:18,122 --> 01:21:20,080 there's something missing in quantum mechanics, 1672 01:21:20,080 --> 01:21:21,496 because I can do these experiments 1673 01:21:21,496 --> 01:21:26,440 and determine that this particle is Sz down and Sx up. 1674 01:21:26,440 --> 01:21:32,840 But what quantum mechanics says is 1675 01:21:32,840 --> 01:21:35,202 that I may have done the measurement of Sz, 1676 01:21:35,202 --> 01:21:36,660 but that hasn't determined anything 1677 01:21:36,660 --> 01:21:37,850 about the state over here. 1678 01:21:37,850 --> 01:21:39,600 There is no predetermined value. 1679 01:21:39,600 --> 01:21:41,620 What we need to do is we need to tease out, 1680 01:21:41,620 --> 01:21:45,130 we need an experimental version of this tension. 1681 01:21:45,130 --> 01:21:47,444 The experimental version of this tension, 1682 01:21:47,444 --> 01:21:48,860 fleshed out in the EPR experiment, 1683 01:21:48,860 --> 01:21:50,010 is called Bell's inequality. 1684 01:21:50,010 --> 01:21:51,670 We studied it in the very first lecture, 1685 01:21:51,670 --> 01:21:54,211 and we're going to show it's a violation in the next lecture. 1686 01:21:54,211 --> 01:21:55,470 See you guys next time. 1687 01:21:55,470 --> 01:21:57,020 [APPLAUSE]