1 00:00:00,060 --> 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 --> 00:00:10,130 to offer high quality educational resources for free. 5 00:00:10,130 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:18,105 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:18,105 --> 00:00:18,730 at ocw.mit.edu. 8 00:00:26,200 --> 00:00:29,860 PROFESSOR: Last time we spoke about the Stern-Gerlach 9 00:00:29,860 --> 00:00:34,840 experiment, and how you could have a sequence 10 00:00:34,840 --> 00:00:39,800 of Stern-Gerlach boxes that allow you to understand 11 00:00:39,800 --> 00:00:44,050 the type of states and properties of the physics 12 00:00:44,050 --> 00:00:46,750 having to do with spin-1/2. 13 00:00:46,750 --> 00:00:51,740 So the key thing in the Stern-Gerlach machine 14 00:00:51,740 --> 00:00:57,130 was that a beam of silver atoms, each of which 15 00:00:57,130 --> 00:01:01,880 is really like an electron with a magnetic moment, 16 00:01:01,880 --> 00:01:06,870 was placed in an inhomogeneous strong magnetic field, 17 00:01:06,870 --> 00:01:11,820 and that would classically mean that you would get a deflection 18 00:01:11,820 --> 00:01:16,750 proportional to the z-component of the magnetic moment. 19 00:01:16,750 --> 00:01:19,100 What was a surprise was that by the time 20 00:01:19,100 --> 00:01:21,670 you put the screen on the right side, 21 00:01:21,670 --> 00:01:24,430 it really split into two different beams, 22 00:01:24,430 --> 00:01:29,260 as if the magnetic moments could either 23 00:01:29,260 --> 00:01:33,110 be all the way pointing in the z-direction or all the way 24 00:01:33,110 --> 00:01:35,910 down, pointing opposite to the z-direction, 25 00:01:35,910 --> 00:01:37,660 and nothing in between. 26 00:01:37,660 --> 00:01:40,190 A very surprising result. 27 00:01:40,190 --> 00:01:45,590 So after looking at a few of those boxes, 28 00:01:45,590 --> 00:01:52,680 we decided that we would try to model the spin-1/2 particle 29 00:01:52,680 --> 00:01:57,880 as a two-dimensional complex vector space. 30 00:01:57,880 --> 00:02:01,090 What is the two-dimensional complex vector space? 31 00:02:01,090 --> 00:02:07,980 It's the possible space of states of a spin-1/2 particle. 32 00:02:07,980 --> 00:02:11,980 So our task today to go into detail into that, 33 00:02:11,980 --> 00:02:17,460 and set up the whole machinery of spin-1/2. 34 00:02:17,460 --> 00:02:23,300 So we will do so, even though we haven't quite yet discussed 35 00:02:23,300 --> 00:02:26,090 all the important concepts of linear algebra 36 00:02:26,090 --> 00:02:27,800 that we're going to need. 37 00:02:27,800 --> 00:02:31,000 So today, I'm going to assume that at least you 38 00:02:31,000 --> 00:02:38,050 have some vague notions of linear algebra 39 00:02:38,050 --> 00:02:39,740 reasonably well understood. 40 00:02:39,740 --> 00:02:43,340 And if you don't, well, take them on faith today. 41 00:02:43,340 --> 00:02:46,150 We're going to go through them slowly 42 00:02:46,150 --> 00:02:49,650 in the next couple of lectures, and then 43 00:02:49,650 --> 00:02:51,870 as you will reread this material, 44 00:02:51,870 --> 00:02:54,410 it will make more sense. 45 00:02:54,410 --> 00:02:58,140 So what did we have? 46 00:02:58,140 --> 00:03:03,300 We said that the spin states, or the possible states 47 00:03:03,300 --> 00:03:08,860 of this silver atom, that really correspond to an election, 48 00:03:08,860 --> 00:03:25,340 could be described by states z comma plus and z colon minus So 49 00:03:25,340 --> 00:03:27,950 these are the two states. 50 00:03:27,950 --> 00:03:34,620 This state we say corresponds to an angular momentum Sz hat. 51 00:03:37,200 --> 00:03:42,460 Sz-- I can put it like that-- of h-bar over 2, 52 00:03:42,460 --> 00:03:50,000 and this corresponds to Sz equals minus h-bar over 2. 53 00:03:50,000 --> 00:03:54,100 And those are our two states. 54 00:03:54,100 --> 00:04:00,290 The z label indicates that we've passed, presumably, 55 00:04:00,290 --> 00:04:03,150 these atoms through a filter in the z-direction, 56 00:04:03,150 --> 00:04:05,380 so that we know for certain we're 57 00:04:05,380 --> 00:04:10,410 talking about the z-component of angular momentum of this state. 58 00:04:10,410 --> 00:04:15,540 It is positive, and the values here again 59 00:04:15,540 --> 00:04:20,250 have the label z to remind us that we're talking about states 60 00:04:20,250 --> 00:04:23,660 that have been organized using the z-component of angular 61 00:04:23,660 --> 00:04:24,160 momentum. 62 00:04:27,040 --> 00:04:32,160 You could ask whether this state has some angular momentum-- 63 00:04:32,160 --> 00:04:34,700 spin angular momentum-- in the x-direction 64 00:04:34,700 --> 00:04:36,540 or in the y-direction, and we will 65 00:04:36,540 --> 00:04:42,530 be able to answer that question in an hour from now. 66 00:04:42,530 --> 00:04:49,960 So mathematically, we say that this statement, that 67 00:04:49,960 --> 00:04:54,730 this state, has Sz equals h-bar over 2 68 00:04:54,730 --> 00:05:10,240 means that there is an operator, Sz hat-- hat for operators. 69 00:05:10,240 --> 00:05:21,230 And this operator, we say, acts on this state 70 00:05:21,230 --> 00:05:24,990 to give h-bar over 2 times this state. 71 00:05:29,140 --> 00:05:33,660 So when we have a measurement in quantum mechanics, 72 00:05:33,660 --> 00:05:36,190 we end up talking about operators. 73 00:05:36,190 --> 00:05:38,880 So this case is no exception. 74 00:05:38,880 --> 00:05:44,330 We think of the operator, Sz, that acts in this state 75 00:05:44,330 --> 00:05:46,390 and gives h-bar over 2. 76 00:05:46,390 --> 00:05:53,870 And that same operator, Sz, acts on the other state 77 00:05:53,870 --> 00:05:57,960 and gives you minus h-bar over 2 times the state. 78 00:06:01,000 --> 00:06:06,020 You see, an operator on a state must give a state. 79 00:06:06,020 --> 00:06:09,470 So in this equation, we have a state on the right, 80 00:06:09,470 --> 00:06:12,890 and the nice thing is that the same state 81 00:06:12,890 --> 00:06:14,640 appears on the right. 82 00:06:14,640 --> 00:06:17,220 When that happens, you say that the state 83 00:06:17,220 --> 00:06:20,140 is an eigenstate of the operator. 84 00:06:20,140 --> 00:06:23,250 And, therefore, the states z plus, minus 85 00:06:23,250 --> 00:06:28,620 are eigenstates of the operator Sz with eigenvalues-- 86 00:06:28,620 --> 00:06:36,470 the number that appears here-- equal to plus, minus h over 2. 87 00:06:36,470 --> 00:06:41,770 So the relevant physical assumption 88 00:06:41,770 --> 00:06:47,330 here is the following, that these two states, in a sense, 89 00:06:47,330 --> 00:06:49,790 suffice. 90 00:06:49,790 --> 00:06:52,660 Now, what does that mean? 91 00:06:52,660 --> 00:06:57,300 We could do the experiment again with some Stern-Gerlach machine 92 00:06:57,300 --> 00:06:59,640 that is along the x-axis, and say, oh, 93 00:06:59,640 --> 00:07:04,030 now we've got states x plus and x minus 94 00:07:04,030 --> 00:07:05,520 and we should add them there. 95 00:07:05,520 --> 00:07:10,490 They are also part of the possible states of the system. 96 00:07:10,490 --> 00:07:11,220 Kind of. 97 00:07:11,220 --> 00:07:14,400 They are parts of the possible states of the system. 98 00:07:14,400 --> 00:07:16,850 They are possible states of the system, 99 00:07:16,850 --> 00:07:20,550 but we shouldn't add them to this one. 100 00:07:20,550 --> 00:07:24,570 These will be thought as basis states. 101 00:07:24,570 --> 00:07:27,350 Just like any vector is the superposition 102 00:07:27,350 --> 00:07:31,840 of a number times the x-unit vector 103 00:07:31,840 --> 00:07:33,890 plus a number times the y-unit vector 104 00:07:33,890 --> 00:07:36,410 and a number times the z-unit vector, 105 00:07:36,410 --> 00:07:39,460 we are going to postulate, or try 106 00:07:39,460 --> 00:07:43,420 to construct the theory of spin, based on the idea 107 00:07:43,420 --> 00:07:47,730 that all possible spin states of an electron 108 00:07:47,730 --> 00:07:53,140 are obtained by suitable linear superposition of these two 109 00:07:53,140 --> 00:07:55,200 vectors. 110 00:07:55,200 --> 00:07:59,760 So, , in fact, what we're going to say is that these two 111 00:07:59,760 --> 00:08:05,900 vectors are the basis of a two-dimensional vector space, 112 00:08:05,900 --> 00:08:10,890 such that every possible state is a linear superposition. 113 00:08:10,890 --> 00:08:23,400 So psi, being any possible spin state, 114 00:08:23,400 --> 00:08:35,725 can be written as some constant, C1 times z plus plus C2 times z 115 00:08:35,725 --> 00:08:43,940 minus where these constants, C1 and C2 belong 116 00:08:43,940 --> 00:08:45,210 to the complex numbers. 117 00:08:47,820 --> 00:08:51,690 And by this, we mean that if any possible state is 118 00:08:51,690 --> 00:08:57,160 a superposition like that, the set of all possible states 119 00:08:57,160 --> 00:09:03,670 are the general vectors in a two-dimensional complex vector 120 00:09:03,670 --> 00:09:05,620 space. 121 00:09:05,620 --> 00:09:11,650 Complex vector space, because the coefficients are complex, 122 00:09:11,650 --> 00:09:17,220 and two-dimensional, because there's two basis vectors. 123 00:09:17,220 --> 00:09:19,770 Now this doesn't quite look like a vector. 124 00:09:19,770 --> 00:09:23,190 It looks like those things called kets. 125 00:09:23,190 --> 00:09:25,830 But kets are really vectors, and we're 126 00:09:25,830 --> 00:09:28,160 going to make the correspondence very clear. 127 00:09:32,310 --> 00:09:35,680 So this can be called the first basis 128 00:09:35,680 --> 00:09:44,560 state and the second basis state. 129 00:09:49,950 --> 00:09:53,910 And I want you to realize that the fact that we're 130 00:09:53,910 --> 00:09:56,830 talking about the complex vector space really 131 00:09:56,830 --> 00:09:59,530 means these coefficients are complex. 132 00:09:59,530 --> 00:10:07,080 There's no claim that the vector is complex in any sense, 133 00:10:07,080 --> 00:10:08,300 or this one. 134 00:10:08,300 --> 00:10:11,030 They're just vectors. 135 00:10:11,030 --> 00:10:13,950 This is a vector, and it's not that we say, 136 00:10:13,950 --> 00:10:15,971 oh this vector is complex. 137 00:10:15,971 --> 00:10:16,470 No. 138 00:10:16,470 --> 00:10:20,170 A complex vector space, we think of as a set of vectors, 139 00:10:20,170 --> 00:10:22,540 and then we're allowed to multiply them 140 00:10:22,540 --> 00:10:23,715 by complex numbers. 141 00:10:26,280 --> 00:10:38,840 OK, so we have this, and this way of thinking of the vectors 142 00:10:38,840 --> 00:10:40,730 is quite all right. 143 00:10:40,730 --> 00:10:44,230 But we want to be more concrete. 144 00:10:44,230 --> 00:10:46,200 For that, we're going to use what 145 00:10:46,200 --> 00:10:47,800 is called a representation. 146 00:10:47,800 --> 00:10:51,270 So I will use the word representation 147 00:10:51,270 --> 00:10:58,010 to mean some way of exhibiting a vector 148 00:10:58,010 --> 00:11:02,050 or state in a more concrete way. 149 00:11:02,050 --> 00:11:07,160 As something that any one of us would call a vector. 150 00:11:07,160 --> 00:11:14,080 So as a matter of notation, this being the first basis state 151 00:11:14,080 --> 00:11:18,230 is sometimes written as a ket with a 1. 152 00:11:18,230 --> 00:11:20,370 Like that. 153 00:11:20,370 --> 00:11:23,700 And this being this second basis state 154 00:11:23,700 --> 00:11:27,720 is sometimes written this way. 155 00:11:27,720 --> 00:11:31,340 But here is the real issue of what 156 00:11:31,340 --> 00:11:33,890 we were calling a representation. 157 00:11:33,890 --> 00:11:38,580 If this is a two-dimensional vector space, 158 00:11:38,580 --> 00:11:41,820 you're accustomed to three-dimensional vector space. 159 00:11:41,820 --> 00:11:43,040 What are vectors? 160 00:11:43,040 --> 00:11:45,380 They're triplets of numbers. 161 00:11:45,380 --> 00:11:46,330 Three numbers. 162 00:11:46,330 --> 00:11:48,110 That's a vector. 163 00:11:48,110 --> 00:11:51,600 Column vectors, it's perhaps easier to think about them. 164 00:11:51,600 --> 00:11:52,990 So column vectors. 165 00:11:52,990 --> 00:11:55,570 So here's what we're going to say. 166 00:11:55,570 --> 00:11:58,790 We have this state z plus. 167 00:11:58,790 --> 00:12:01,720 It's also called 1. 168 00:12:01,720 --> 00:12:04,000 It's just a name, but we're going 169 00:12:04,000 --> 00:12:10,200 to represent it as a column vector. 170 00:12:10,200 --> 00:12:12,060 And as a column vector, I'm going 171 00:12:12,060 --> 00:12:14,750 to represent it as the column vector 1, 0. 172 00:12:22,160 --> 00:12:25,160 And this is why I put this double arrow. 173 00:12:25,160 --> 00:12:27,540 I'm not saying it's the same thing-- 174 00:12:27,540 --> 00:12:30,830 although the really it is-- it's just 175 00:12:30,830 --> 00:12:37,220 a way of thinking about it as some vector in what we would 176 00:12:37,220 --> 00:12:39,080 call canonically a vector space. 177 00:12:39,080 --> 00:12:40,018 Yes 178 00:12:40,018 --> 00:12:42,208 AUDIENCE: So do the components of the column vector 179 00:12:42,208 --> 00:12:45,940 there have any correspondence to the actual. 180 00:12:45,940 --> 00:12:48,750 Does it have any basis in the actual physical process going 181 00:12:48,750 --> 00:12:49,250 on? 182 00:12:49,250 --> 00:12:52,780 Or, what is their connection to the actual physical [INAUDIBLE] 183 00:12:52,780 --> 00:12:53,860 represented here? 184 00:12:53,860 --> 00:12:56,040 PROFESSOR: Well, we'll see it in a second. 185 00:12:56,040 --> 00:12:59,240 It will become a little clearer. 186 00:12:59,240 --> 00:13:03,740 But this is like saying, I have a two-dimensional two0 vector 187 00:13:03,740 --> 00:13:07,770 space, so I'm going to think of the first state as this vector. 188 00:13:07,770 --> 00:13:09,490 But how do I write this vector? 189 00:13:09,490 --> 00:13:12,110 Well, it's the vector ex. 190 00:13:12,110 --> 00:13:15,260 Well, if I would write them in components, 191 00:13:15,260 --> 00:13:20,310 I would say, for a vector, I can put two numbers here, a and b. 192 00:13:20,310 --> 00:13:23,120 And this is the a-component and b-component. 193 00:13:23,120 --> 00:13:28,085 So here it is, ex would be 1, 0. 194 00:13:28,085 --> 00:13:33,740 And ey would be 0, 1. 195 00:13:33,740 --> 00:13:36,837 If I have this notation then the point a, 196 00:13:36,837 --> 00:13:41,500 b is represented by a and b as a column vector. 197 00:13:41,500 --> 00:13:45,580 So at this moment, it's just a way 198 00:13:45,580 --> 00:13:51,370 of associating a vector in the two-dimensional canonical 199 00:13:51,370 --> 00:13:52,320 vector space. 200 00:13:52,320 --> 00:13:54,770 It's just the column here. 201 00:13:54,770 --> 00:13:58,520 So the other state, minus-- it's also 202 00:13:58,520 --> 00:14:03,470 called 2-- will be represented by 0 1 1. 203 00:14:03,470 --> 00:14:12,840 1 And therefore, this state, psi, 204 00:14:12,840 --> 00:14:22,480 which is C1 z plus plus C2 z minus 205 00:14:22,480 --> 00:14:31,870 will be represented as C1 times the first vector plus C2 times 206 00:14:31,870 --> 00:14:33,800 the second vector. 207 00:14:33,800 --> 00:14:39,220 Or multiplying, in C1, C2. 208 00:14:42,810 --> 00:14:49,890 So this state can be written as a linear superposition 209 00:14:49,890 --> 00:14:52,590 of these two basis vectors in this way-- 210 00:14:52,590 --> 00:14:54,720 you can write it this way. 211 00:14:54,720 --> 00:14:57,500 You want to save some writing, then 212 00:14:57,500 --> 00:15:00,010 you can write them with 1 and 2. 213 00:15:00,010 --> 00:15:04,432 But as a vector, it's represented by a column vector 214 00:15:04,432 --> 00:15:05,265 with two components. 215 00:15:08,690 --> 00:15:10,865 That's our state. 216 00:15:13,550 --> 00:15:17,650 Now in doing this, I want to emphasize, 217 00:15:17,650 --> 00:15:20,090 we're introducing the physical assumption 218 00:15:20,090 --> 00:15:23,720 that this will be enough to describe all possible spin 219 00:15:23,720 --> 00:15:27,020 states, which is far from obvious at this stage. 220 00:15:29,560 --> 00:15:35,230 Nevertheless, let's use some of the ideas 221 00:15:35,230 --> 00:15:38,020 from the experiment, the Stern-Gerlach experiment. 222 00:15:38,020 --> 00:15:44,590 We did one example of a box that filtered the plus z states, 223 00:15:44,590 --> 00:15:48,160 and then put it against another z machine, 224 00:15:48,160 --> 00:15:51,470 and then all the states went through the up. 225 00:15:51,470 --> 00:15:57,050 Which is to say that plus states have no amplitude, 226 00:15:57,050 --> 00:15:59,980 no probability to be in the minus states. 227 00:15:59,980 --> 00:16:02,980 They all went through the plus. 228 00:16:02,980 --> 00:16:06,150 So when we're going to introduce now the physical translation 229 00:16:06,150 --> 00:16:10,620 of this fact, as saying that these states are 230 00:16:10,620 --> 00:16:13,890 orthogonal to each other. 231 00:16:13,890 --> 00:16:19,950 So, this will require the whole framework, in detail, 232 00:16:19,950 --> 00:16:24,310 of bras and kets to say really, precisely-- 233 00:16:24,310 --> 00:16:26,910 but we're going to do that now and explain 234 00:16:26,910 --> 00:16:30,580 the minimum necessary for you to understand it. 235 00:16:30,580 --> 00:16:32,840 But we'll come back to it later. 236 00:16:32,840 --> 00:16:36,320 So this physical statement will be 237 00:16:36,320 --> 00:16:42,250 stated as z minus with z plus. 238 00:16:42,250 --> 00:16:48,870 The overlap, the bra-ket of this, is 0. 239 00:16:48,870 --> 00:16:53,460 The fact that all particles went through and went out 240 00:16:53,460 --> 00:16:59,610 through the plus output will state to us, well, 241 00:16:59,610 --> 00:17:02,200 these states are well normalized. 242 00:17:02,200 --> 00:17:11,819 So z plus, z plus is 1 243 00:17:11,819 --> 00:17:15,150 Similarly, you could have blocked the other input, 244 00:17:15,150 --> 00:17:16,965 and you would have concluded that the minus 245 00:17:16,965 --> 00:17:19,240 state is orthogonal to the plus. 246 00:17:19,240 --> 00:17:24,839 So we also say that these, too, are orthogonal, 247 00:17:24,839 --> 00:17:29,020 and the minus states are well normalized. 248 00:17:34,360 --> 00:17:36,870 Now here we had to write four equations. 249 00:17:40,192 --> 00:17:45,460 And the notation, one and two becomes handy, 250 00:17:45,460 --> 00:17:48,620 because we can summarize all these statements 251 00:17:48,620 --> 00:17:56,840 by the equation Ij equals delta Ij. 252 00:17:56,840 --> 00:18:04,730 Look, if this equation is 2 with 1 equals 0. 253 00:18:07,500 --> 00:18:11,240 The bra 2, the ket 1. 254 00:18:11,240 --> 00:18:15,220 This is 1 with 1 is equal to 1. 255 00:18:15,220 --> 00:18:22,020 Here is is 1 with 2 is equal to 0, and 2 with 2 is equal to 1. 256 00:18:22,020 --> 00:18:24,850 So this is exactly what we have here. 257 00:18:28,720 --> 00:18:33,340 Now, I didn't define for you these so-called bras. 258 00:18:33,340 --> 00:18:36,790 So by completeness, I will define them now. 259 00:18:41,480 --> 00:18:47,120 And the way I will define them is as follows. 260 00:18:47,120 --> 00:18:56,390 I will say that while for the one vector basis state 261 00:18:56,390 --> 00:19:04,460 you associate at 1, 0, you will associate to one bra, the row 262 00:19:04,460 --> 00:19:08,030 vector 1, 0. 263 00:19:08,030 --> 00:19:11,590 I sometimes tend to write equal, but-- equal 264 00:19:11,590 --> 00:19:13,760 is all right-- but it's a little clearer 265 00:19:13,760 --> 00:19:17,470 to say that there's arrows here. 266 00:19:17,470 --> 00:19:24,440 So we're going to associate to 1, 1, 0-- we did it before-- 267 00:19:24,440 --> 00:19:28,260 but now to the bra, we think of the rho vector. 268 00:19:28,260 --> 00:19:29,500 Like this. 269 00:19:29,500 --> 00:19:34,220 Similarly, I can do the same with 2. 270 00:19:34,220 --> 00:19:37,670 2 was the vector 0, 1. 271 00:19:37,670 --> 00:19:41,050 It's a column vector, so 2 was a bra. 272 00:19:41,050 --> 00:19:47,960 We will think of it as the row vector 0, 1. 273 00:20:00,440 --> 00:20:08,080 We're going to do this now a little more generally. 274 00:20:08,080 --> 00:20:11,990 So, suppose you have state, alpha, 275 00:20:11,990 --> 00:20:19,810 which is alpha 1, 1 plus alpha 2, 2. 276 00:20:19,810 --> 00:20:26,100 Well, to this, you would associate the column vector 277 00:20:26,100 --> 00:20:29,500 alpha 1, alpha 2. 278 00:20:29,500 --> 00:20:37,640 Suppose you have a beta state, beta 1, 1 plus beta 2, 2. 279 00:20:37,640 --> 00:20:42,360 You would associate beta 1, beta 2 as their representations. 280 00:20:45,500 --> 00:20:49,440 Now here comes the definition for which 281 00:20:49,440 --> 00:20:52,630 this is just a special case. 282 00:20:52,630 --> 00:20:56,790 And it's a definition of the general bra. 283 00:20:56,790 --> 00:21:11,050 So the general bra here, alpha, is defined to be alpha 1*, 284 00:21:11,050 --> 00:21:19,770 bra of the first, plus alpha 2*, bra of the second. 285 00:21:19,770 --> 00:21:26,710 So this is alpha 1* times the first bra, 286 00:21:26,710 --> 00:21:35,400 which we think of it as 1, 0, plus alpha 2* times the second 287 00:21:35,400 --> 00:21:38,190 bra, which is 0, 1. 288 00:21:38,190 --> 00:21:42,410 So this whole thing is represented by alpha 1*, 289 00:21:42,410 --> 00:21:43,890 alpha 2*. 290 00:21:51,600 --> 00:22:02,200 So, here we've had a column vector representation 291 00:22:02,200 --> 00:22:07,320 of a state, and the bra is the row vector representation 292 00:22:07,320 --> 00:22:11,120 of the state in which this is constructed 293 00:22:11,120 --> 00:22:14,670 with complex conjugation. 294 00:22:14,670 --> 00:22:19,740 Now these kind of definitions will 295 00:22:19,740 --> 00:22:23,410 be discussed in more detail and more axiomatically 296 00:22:23,410 --> 00:22:25,800 early very soon, so that you see where you're going. 297 00:22:25,800 --> 00:22:28,860 But the intuition that you're going to get from this 298 00:22:28,860 --> 00:22:30,640 is quite valuable. 299 00:22:30,640 --> 00:22:32,286 So what is the bra-ket? 300 00:22:34,800 --> 00:22:39,090 Alpha-beta is the so-called bra-ket. 301 00:22:39,090 --> 00:22:42,130 And this is a number. 302 00:22:42,130 --> 00:22:45,280 And the reason for complex conjugation 303 00:22:45,280 --> 00:22:50,250 is, ultimately, that when these two things are the same, 304 00:22:50,250 --> 00:22:52,270 it should give a positive number. 305 00:22:52,270 --> 00:22:54,880 It's like the length squared. 306 00:22:54,880 --> 00:22:59,230 So that's the reason for complex conjugation, eventually. 307 00:22:59,230 --> 00:23:04,720 But, for now, you are supposed to get a number from here. 308 00:23:04,720 --> 00:23:08,160 And the a reasonable way to get a number, which 309 00:23:08,160 --> 00:23:11,620 is a definition, is that you get a number 310 00:23:11,620 --> 00:23:16,090 by a matrix multiplication of the representatives. 311 00:23:16,090 --> 00:23:22,800 So you take the representative of alpha, which is alpha 1*, 312 00:23:22,800 --> 00:23:25,140 alpha 2*. 313 00:23:25,140 --> 00:23:28,480 And do the matrix product with the representative 314 00:23:28,480 --> 00:23:32,970 of beta, which is beta 1, beta 2. 315 00:23:32,970 --> 00:23:41,480 And that's alpha 1*, beta 1 plus alpha 2*, beta 2. 316 00:23:41,480 --> 00:23:45,670 And that's the number called the inner product, 317 00:23:45,670 --> 00:23:48,760 or bra-ket product. 318 00:23:48,760 --> 00:23:55,300 And this is the true meaning of relations of this kind. 319 00:23:55,300 --> 00:23:57,670 If you're given an arbitrary states, 320 00:23:57,670 --> 00:24:01,180 you compute the inner product this way. 321 00:24:01,180 --> 00:24:08,280 And vectors that satisfy this are called orthonormal 322 00:24:08,280 --> 00:24:11,770 because they're orthogonal and normal with respect 323 00:24:11,770 --> 00:24:14,490 to each other in the sense of the bra and ket. 324 00:24:17,810 --> 00:24:31,060 So this definition, as you can see, 325 00:24:31,060 --> 00:24:34,970 is also consistent with what you have up there, 326 00:24:34,970 --> 00:24:36,020 and you can check it. 327 00:24:36,020 --> 00:24:43,380 If you take I with j, 1, say, with 2-- like this-- 328 00:24:43,380 --> 00:24:46,290 you do the inner product, and you get 0. 329 00:24:46,290 --> 00:24:50,270 And similarly for all the other states. 330 00:24:50,270 --> 00:24:57,980 So let's then complete the issue of representations. 331 00:24:57,980 --> 00:25:01,600 We had representations of the states 332 00:25:01,600 --> 00:25:06,560 as column vectors-- two by two column vectors or row vectors. 333 00:25:06,560 --> 00:25:11,370 Now let's talk about this operator we started with. 334 00:25:11,370 --> 00:25:16,220 If this is an operator, acting on states, 335 00:25:16,220 --> 00:25:20,330 now I want to think of its representation, which 336 00:25:20,330 --> 00:25:24,310 would be the way it acts on these two component vectors. 337 00:25:24,310 --> 00:25:28,700 So it must be a two by two matrix, because only a two 338 00:25:28,700 --> 00:25:34,510 by two matrix acts naturally on two component vectors. 339 00:25:34,510 --> 00:25:38,970 So here is the claim that we have. 340 00:25:38,970 --> 00:25:46,920 Claim, that Sz hat is represented-- 341 00:25:46,920 --> 00:25:50,973 but we'll just put equal-- by this matrix. 342 00:25:59,050 --> 00:26:00,790 You see, it was an operator. 343 00:26:00,790 --> 00:26:03,790 We never talked about matrices. 344 00:26:03,790 --> 00:26:10,570 But once we start talking about the basis vectors as column 345 00:26:10,570 --> 00:26:13,810 vectors, then you can ask if this is correct. 346 00:26:13,810 --> 00:26:18,470 So for example, I'm supposed to find 347 00:26:18,470 --> 00:26:23,980 that Sz hat acting on this state 1 348 00:26:23,980 --> 00:26:28,270 is supposed to be h-bar over 2 times the state 1. 349 00:26:28,270 --> 00:26:28,850 You see? 350 00:26:28,850 --> 00:26:30,220 True. 351 00:26:30,220 --> 00:26:35,060 Then you say, oh let's put the representation, h-bar over 2, 352 00:26:35,060 --> 00:26:38,280 1 minus 1, 0, 0. 353 00:26:38,280 --> 00:26:40,980 State one, what's its representation? 354 00:26:40,980 --> 00:26:43,510 1, 0. 355 00:26:43,510 --> 00:26:45,520 OK, let's act on it. 356 00:26:45,520 --> 00:26:49,150 So, this gives me h-bar over 2. 357 00:26:49,150 --> 00:26:52,850 I do the first product, I get a 1. 358 00:26:52,850 --> 00:26:56,220 I do the second product, I get a 0. 359 00:26:56,220 --> 00:26:58,970 Oh, that seems right, because this 360 00:26:58,970 --> 00:27:03,090 is h over 2 times the representation of the state 1. 361 00:27:06,100 --> 00:27:12,470 And if I check this, and as well that Sz on 2 362 00:27:12,470 --> 00:27:18,740 is equal minus h-bar over 2, 2-- which can also be checked-- I 363 00:27:18,740 --> 00:27:21,070 need to check no more. 364 00:27:21,070 --> 00:27:25,320 Because it suffices that this operator does 365 00:27:25,320 --> 00:27:29,020 what it's supposed to do of the basis vectors. 366 00:27:29,020 --> 00:27:30,690 And it will do what it's supposed 367 00:27:30,690 --> 00:27:34,260 to do on arbitrary vectors. 368 00:27:34,260 --> 00:27:35,470 So we're done. 369 00:27:35,470 --> 00:27:40,460 This is the operator Sx, and we seem 370 00:27:40,460 --> 00:27:45,390 to have put together a lot of the ideas of the experiment 371 00:27:45,390 --> 00:27:48,910 into a mathematical framework. 372 00:27:48,910 --> 00:27:54,040 But we're not through because we have this question, 373 00:27:54,040 --> 00:27:58,380 so what if you align and operate the machine along x? 374 00:27:58,380 --> 00:28:02,055 What are the possible spin states along the x-direction? 375 00:28:02,055 --> 00:28:03,930 How do you know that all that the spins state 376 00:28:03,930 --> 00:28:09,370 that points along x can be described in this vector space? 377 00:28:09,370 --> 00:28:13,770 How do I know there exists a number C1, C2 so 378 00:28:13,770 --> 00:28:17,130 that this linear combination is a spin 379 00:28:17,130 --> 00:28:21,130 state that points along x. 380 00:28:21,130 --> 00:28:28,430 Well, at this moment, you really have to invent something. 381 00:28:28,430 --> 00:28:35,020 And the process of invention is never a very linear one. 382 00:28:35,020 --> 00:28:37,780 You use analogies-- you use whatever 383 00:28:37,780 --> 00:28:40,950 you can-- to invent what you need. 384 00:28:40,950 --> 00:28:43,940 So, given that that's a possibility, 385 00:28:43,940 --> 00:28:48,520 we could follow what Feynman does in his Feynman lectures, 386 00:28:48,520 --> 00:28:51,900 of discussing how to begin rotating 387 00:28:51,900 --> 00:28:56,080 Stern-Gerlach machines, and doing all kinds of things. 388 00:28:56,080 --> 00:29:02,040 It's an interesting argument, and it's a little hard 389 00:29:02,040 --> 00:29:07,240 to follow, a little tedious at points. 390 00:29:07,240 --> 00:29:10,940 And we're going to follow a different route. 391 00:29:10,940 --> 00:29:14,540 I'm going to assume that you remember a little about angular 392 00:29:14,540 --> 00:29:19,670 momentum, and I think you do remember this much. 393 00:29:19,670 --> 00:29:23,193 I want to say, well, this is spin angular momentum. 394 00:29:27,190 --> 00:29:30,780 Well, let's compare it with orbital angular momentum, 395 00:29:30,780 --> 00:29:32,130 and see where we are. 396 00:29:32,130 --> 00:29:37,260 You see, another way of asking the question would be, well, 397 00:29:37,260 --> 00:29:42,970 what are the operators Sx and Sy. 398 00:29:42,970 --> 00:29:44,180 Where do I get them? 399 00:29:44,180 --> 00:29:48,190 Well, the reason I want to bring in the angular momentum 400 00:29:48,190 --> 00:29:54,330 is because there you have Lz, but you also have Lx and Ly. 401 00:29:54,330 --> 00:29:59,590 So angular momentum had Lz, just like we had here, 402 00:29:59,590 --> 00:30:07,905 but also Lx and Ly. 403 00:30:10,690 --> 00:30:14,920 Now these spin things look a lot more mysterious, a lot more 404 00:30:14,920 --> 00:30:27,650 basic, because, like Lz, it was xpy minus ypx. 405 00:30:27,650 --> 00:30:32,513 So you knew how this operator acts on wave functions. 406 00:30:35,100 --> 00:30:38,620 You know, it multiplies by y, takes an x derivative, 407 00:30:38,620 --> 00:30:40,460 or it's a dd phi. 408 00:30:40,460 --> 00:30:48,130 It has a nice thing, but Sz on the other hand, there's no x, 409 00:30:48,130 --> 00:30:49,470 there's no derivatives. 410 00:30:49,470 --> 00:30:51,810 It's a different space. 411 00:30:51,810 --> 00:30:53,840 It's working in a totally different space, 412 00:30:53,840 --> 00:30:57,270 in the space of a two-dimensional complex vector 413 00:30:57,270 --> 00:31:00,170 space of column vectors with two numbers. 414 00:31:00,170 --> 00:31:01,860 That's where it acts. 415 00:31:01,860 --> 00:31:06,900 I'm sorry there's no dd x, nothing familiar about it. 416 00:31:06,900 --> 00:31:11,920 But that's what we have been handed. 417 00:31:11,920 --> 00:31:14,910 So this thing acts on wave functions, 418 00:31:14,910 --> 00:31:16,400 and thus natural things. 419 00:31:16,400 --> 00:31:19,410 Well, the other one acts on column vectors. 420 00:31:19,410 --> 00:31:23,160 Two-by-two-- two component column vectors, 421 00:31:23,160 --> 00:31:24,860 and that's all right. 422 00:31:24,860 --> 00:31:28,540 But we also know that Lz is Hermitian. 423 00:31:28,540 --> 00:31:31,550 And that was good, because it actually 424 00:31:31,550 --> 00:31:33,950 meant that this is good observable. 425 00:31:33,950 --> 00:31:36,310 You can measure it. 426 00:31:36,310 --> 00:31:39,340 Is Sz Hermitian? 427 00:31:39,340 --> 00:31:41,170 Well, yes it is. 428 00:31:41,170 --> 00:31:44,460 Hermeticity of a matrix-- as we'll discuss it 429 00:31:44,460 --> 00:31:49,310 in a lot of detail, maybe more than you want-- 430 00:31:49,310 --> 00:31:52,760 means you can transpose it complex conjugated, 431 00:31:52,760 --> 00:31:56,380 and you get the same matrix. 432 00:31:56,380 --> 00:31:58,430 Well that matrix is Hermitian. 433 00:31:58,430 --> 00:31:59,920 So that's nice. 434 00:31:59,920 --> 00:32:02,260 That maybe is important. 435 00:32:02,260 --> 00:32:04,740 So what other operators do we have? 436 00:32:04,740 --> 00:32:05,910 Lx and Ly. 437 00:32:05,910 --> 00:32:18,550 And if we think of Lx as L1, Ly as L2, and Lz as L3, 438 00:32:18,550 --> 00:32:22,740 you had a basic computation relation. 439 00:32:22,740 --> 00:32:33,530 Li with Lj was equal to i epsilon ijk Lk-hat-- oops 440 00:32:33,530 --> 00:32:34,030 i-hbar. 441 00:32:36,730 --> 00:32:41,780 And this was called the algebra of angular momentum. 442 00:32:41,780 --> 00:32:46,350 These three operators satisfy these identities. 443 00:32:46,350 --> 00:32:53,170 i and j are here, k is supposed to be summed over-- repeated 444 00:32:53,170 --> 00:32:56,690 in this is our sum from 1, 2, and 3. 445 00:32:56,690 --> 00:33:01,250 And epsilon ijk is totally anti-symmetric with epsilon 1, 446 00:33:01,250 --> 00:33:05,270 2, 3 equal to plus 1. 447 00:33:05,270 --> 00:33:09,180 You may or may not know this epsilon. 448 00:33:09,180 --> 00:33:12,870 You will get some practice on that very soon. 449 00:33:12,870 --> 00:33:15,410 Now for all intents and purposes, 450 00:33:15,410 --> 00:33:19,200 we might as well write the explicit formulas 451 00:33:19,200 --> 00:33:23,880 between Lx, Ly equal i-hbar Lz. 452 00:33:26,710 --> 00:33:32,700 Ly Lz equals i-hbar Lx. 453 00:33:32,700 --> 00:33:42,765 And Lz Lx-- there are hats all over-- equal i-hbar Ly. 454 00:33:52,680 --> 00:33:55,970 So we had this for orbital angular momentum, 455 00:33:55,970 --> 00:34:00,630 or for angular momentum in general. 456 00:34:00,630 --> 00:34:03,420 So what we're going to do now is we're 457 00:34:03,420 --> 00:34:16,790 going to try to figure out what are Sx and Sy by trying 458 00:34:16,790 --> 00:34:19,550 to find a complete analogy. 459 00:34:19,550 --> 00:34:24,570 We're going to declare that S is going to be angular momentum. 460 00:34:24,570 --> 00:34:36,900 So we're going to want that Sx with Sy will be i-hbar Sz. 461 00:34:36,900 --> 00:34:41,900 Sy with Sz will be i-hbar Sx. 462 00:34:44,530 --> 00:34:52,775 And finally, Sz with Sx is i-hbar Sy. 463 00:34:55,389 --> 00:35:01,820 And we're going to try that these things be Hermitian. 464 00:35:01,820 --> 00:35:02,700 Sx and Sy. 465 00:35:06,230 --> 00:35:11,240 So let me break for a second and ask if there are questions. 466 00:35:11,240 --> 00:35:15,650 We're aiming to complete the theory 467 00:35:15,650 --> 00:35:21,460 by taking S to be angular momentum, and see what we get. 468 00:35:21,460 --> 00:35:24,650 Can we invent operators Sx and Sy 469 00:35:24,650 --> 00:35:26,160 that will do the right thing? 470 00:35:26,160 --> 00:35:27,654 Yes. 471 00:35:27,654 --> 00:35:30,546 AUDIENCE: What's the name for the epsilon ijk? 472 00:35:30,546 --> 00:35:34,570 I know there's a special name for the [INAUDIBLE]. 473 00:35:34,570 --> 00:35:35,836 PROFESSOR: What's the name? 474 00:35:35,836 --> 00:35:38,316 AUDIENCE: There's a [INAUDIBLE] tensor. 475 00:35:38,316 --> 00:35:40,280 PROFESSOR: That's right. 476 00:35:40,280 --> 00:35:41,655 Let [INAUDIBLE] to be the tensor. 477 00:35:44,300 --> 00:35:46,145 It can be used for cross products. 478 00:35:46,145 --> 00:35:48,790 It's very useful for cross products. 479 00:35:48,790 --> 00:35:51,600 It's a really useful tensor. 480 00:35:51,600 --> 00:35:53,030 Other questions. 481 00:35:53,030 --> 00:35:55,060 More questions about what we're going 482 00:35:55,060 --> 00:35:58,230 to try to do, or this so far. 483 00:36:02,260 --> 00:36:03,060 Yes. 484 00:36:03,060 --> 00:36:05,590 AUDIENCE: When you use the term representation, 485 00:36:05,590 --> 00:36:07,884 is that like the technical mathematical term 486 00:36:07,884 --> 00:36:10,046 of representation, like in algebra? 487 00:36:10,046 --> 00:36:10,670 PROFESSOR: Yes. 488 00:36:10,670 --> 00:36:15,210 It's representation of operators in vector spaces. 489 00:36:15,210 --> 00:36:19,900 So we've used the canonical vector space 490 00:36:19,900 --> 00:36:25,800 with column vectors represented by entries one and numbers. 491 00:36:25,800 --> 00:36:28,730 And then the operators become matrices, 492 00:36:28,730 --> 00:36:32,980 so whenever an operator is viewed as a matrix, 493 00:36:32,980 --> 00:36:34,845 we think of it as a representation. 494 00:36:37,970 --> 00:36:38,760 Other questions. 495 00:36:38,760 --> 00:36:39,966 Yes. 496 00:36:39,966 --> 00:36:41,830 AUDIENCE: Will we talk about later 497 00:36:41,830 --> 00:36:46,020 why we can make an analogy between L and S? 498 00:36:46,020 --> 00:36:49,450 Or is it [INAUDIBLE]? 499 00:36:52,670 --> 00:36:57,990 PROFESSOR: Well you see, this is a very strong analogy, 500 00:36:57,990 --> 00:37:00,800 but there will be big differences 501 00:37:00,800 --> 00:37:04,930 from orbital angular momentum and spin angular momentum. 502 00:37:04,930 --> 00:37:07,990 And basically having to do with the fact 503 00:37:07,990 --> 00:37:10,400 that the eigenvalues of these operators 504 00:37:10,400 --> 00:37:13,320 are plus minus h-bar over 2. 505 00:37:13,320 --> 00:37:16,040 And in the orbital case they tend 506 00:37:16,040 --> 00:37:19,460 to be plus minus integer values of h-bar. 507 00:37:19,460 --> 00:37:24,535 So this is a very deep statement about the algebra 508 00:37:24,535 --> 00:37:28,840 of these operators that still allows the physics of them 509 00:37:28,840 --> 00:37:30,700 to be quite different. 510 00:37:30,700 --> 00:37:34,820 But this is probably the only algebra that makes sense. 511 00:37:34,820 --> 00:37:36,160 It's angular momentum. 512 00:37:36,160 --> 00:37:40,280 So we're going to try to develop that algebra like that, 513 00:37:40,280 --> 00:37:41,040 as well here. 514 00:37:41,040 --> 00:37:43,180 You could take it to be an assumption. 515 00:37:43,180 --> 00:37:48,280 And as I said, an experiment doesn't tell you the unique way 516 00:37:48,280 --> 00:37:49,820 to invent the mathematics. 517 00:37:49,820 --> 00:37:52,540 You try to invent the consistent mathematics 518 00:37:52,540 --> 00:37:55,530 and see if it coincides with the experiment. 519 00:37:55,530 --> 00:38:03,720 And this is a very natural thing to try to invent 520 00:38:03,720 --> 00:38:05,960 So what are we facing? 521 00:38:05,960 --> 00:38:10,340 We're facing a slightly nontrivial problem 522 00:38:10,340 --> 00:38:13,010 of figuring out these operators. 523 00:38:13,010 --> 00:38:16,250 And they should be Hermitian. 524 00:38:16,250 --> 00:38:20,880 So let's try to think of Hermitian two-by-two matrices. 525 00:38:26,110 --> 00:38:34,620 So here is a Hermitian two-by-two matrix. 526 00:38:34,620 --> 00:38:38,850 I can put an arbitrary constant here because it 527 00:38:38,850 --> 00:38:43,580 should be invariant on their transposition, which 528 00:38:43,580 --> 00:38:47,170 doesn't change this diagonal value in complex conjugation. 529 00:38:47,170 --> 00:38:49,980 So c should be real. 530 00:38:49,980 --> 00:38:51,335 d should be real. 531 00:38:57,440 --> 00:39:01,050 For the matrix to be Hermitian, two-by-two matrix, 532 00:39:01,050 --> 00:39:04,300 I could put an a here. 533 00:39:04,300 --> 00:39:09,500 And then this a would have to appear here as well. 534 00:39:09,500 --> 00:39:15,440 I can put minus ib, and then I would have plus ib here. 535 00:39:15,440 --> 00:39:20,310 So when I transpose a complex conjugate, I get this one. 536 00:39:20,310 --> 00:39:30,180 So this matrix with abc and d real is Hermitian. 537 00:39:39,850 --> 00:39:42,860 Hermiticity is some sort of reality condition. 538 00:39:42,860 --> 00:39:47,940 Now, for convenience, I would put a 2c and a 2d here. 539 00:39:50,650 --> 00:39:53,340 It doesn't change things too much. 540 00:39:53,340 --> 00:39:55,780 Now to look at what we're talking about. 541 00:39:55,780 --> 00:40:00,673 We're talking about this set of Hermitian matrices. 542 00:40:04,420 --> 00:40:10,210 Funnily, you can think of that again as a vector space. 543 00:40:10,210 --> 00:40:11,350 Why a vector space? 544 00:40:11,350 --> 00:40:14,310 Well, we'll think about it, and in a few seconds, 545 00:40:14,310 --> 00:40:15,500 it will become clear. 546 00:40:15,500 --> 00:40:18,830 But let me just try to do something here 547 00:40:18,830 --> 00:40:22,450 that might help us. 548 00:40:25,720 --> 00:40:29,640 We're trying to identify Sx and Sy from here 549 00:40:29,640 --> 00:40:33,280 so that this commutation relations hold. 550 00:40:33,280 --> 00:40:39,840 Well, if Sx and Sy have anything to do with the identity matrix, 551 00:40:39,840 --> 00:40:41,510 they would commute with everything 552 00:40:41,510 --> 00:40:45,050 and would do nothing for you. 553 00:40:45,050 --> 00:40:50,070 So, I will remove from this matrices then trying 554 00:40:50,070 --> 00:40:54,300 to understand something having to do with the identity. 555 00:40:54,300 --> 00:40:58,670 So I'll remove a Hermitian matrix, 556 00:40:58,670 --> 00:41:03,960 which is c plus d times the identity-- 557 00:41:03,960 --> 00:41:07,500 the two-by-two identity matrix. 558 00:41:07,500 --> 00:41:11,080 This is a Hermitian matrix, as well. 559 00:41:11,080 --> 00:41:14,670 And I can remove it, and then this matrix is still Hermitian, 560 00:41:14,670 --> 00:41:19,760 and this piece that I've removed doesn't change commutators 561 00:41:19,760 --> 00:41:22,410 as they appear on the left hand side. 562 00:41:22,410 --> 00:41:25,390 So if you have an Sx and an xy here, 563 00:41:25,390 --> 00:41:27,110 and you're trying to do a computation, 564 00:41:27,110 --> 00:41:30,510 it would not contribute, so you might as well just 565 00:41:30,510 --> 00:41:32,570 get rid of them. 566 00:41:32,570 --> 00:41:36,460 So if we remove this, we are left 567 00:41:36,460 --> 00:41:40,250 with-- you're subtracting c plus d from the diagonal. 568 00:41:40,250 --> 00:41:44,920 So here you'll have c minus d. 569 00:41:44,920 --> 00:41:54,520 Here you'll get b minus c, a minus ib, and a plus ib. 570 00:41:54,520 --> 00:41:59,810 And we should keep searching for Sx and Sy among these matrices. 571 00:42:02,810 --> 00:42:08,690 But then you say, look, I already got Sz, 572 00:42:08,690 --> 00:42:11,300 and that was Hermitian. 573 00:42:11,300 --> 00:42:15,380 And Sz was Hermitian, and it had a number, 574 00:42:15,380 --> 00:42:18,920 and the opposite number on the other diagonal entry. 575 00:42:21,700 --> 00:42:27,380 If Sx and Sy have a little bit of Sz, I don't care. 576 00:42:27,380 --> 00:42:30,930 I don't want these to be independent matrices. 577 00:42:30,930 --> 00:42:33,740 I don't want to confuse the situation. 578 00:42:33,740 --> 00:42:39,980 So if this thing has something along Sz, I want it out. 579 00:42:39,980 --> 00:42:45,400 So since precisely this number is opposite to this one, 580 00:42:45,400 --> 00:42:50,270 I can add to this matrix some multiple of Sz 581 00:42:50,270 --> 00:42:52,890 and kill these things in the diagonal. 582 00:42:56,420 --> 00:43:06,760 So add the multiple and Sz multiple, 583 00:43:06,760 --> 00:43:09,140 and we finally get this matrix. 584 00:43:09,140 --> 00:43:17,240 0, a minus ib, a plus ib, and 0. 585 00:43:23,530 --> 00:43:25,680 So we've made quite some progress. 586 00:43:29,720 --> 00:43:32,450 Let's see now what we have. 587 00:43:35,190 --> 00:43:45,870 Well, that matrix could be written as a times 0, 1, 1, 588 00:43:45,870 --> 00:43:54,050 0 plus b times 0, minus i, i, 0. 589 00:44:00,840 --> 00:44:05,780 Which is to say that it's this Hermitian matrix 590 00:44:05,780 --> 00:44:08,640 times a real number, and this Hermitian matrix 591 00:44:08,640 --> 00:44:10,140 times a real number. 592 00:44:10,140 --> 00:44:14,060 And that makes sense because if you take a Hermitian matrix 593 00:44:14,060 --> 00:44:18,610 and multiply by a real number, the matrix is still Hermitian. 594 00:44:18,610 --> 00:44:21,560 So this is still Hermitian because these are real. 595 00:44:21,560 --> 00:44:24,760 This is still Hermitian because a is real, 596 00:44:24,760 --> 00:44:28,520 and if you add Hermitian matrices, it's still Hermitian. 597 00:44:28,520 --> 00:44:35,320 So in some sense, the set of Hermitian matrices, 598 00:44:35,320 --> 00:44:41,630 two-by-two Hermitian matrices, is a real vector space 599 00:44:41,630 --> 00:44:44,970 with four basis vectors. 600 00:44:44,970 --> 00:44:50,220 One basis vector is this, another basis vector is this, 601 00:44:50,220 --> 00:44:55,620 the third basis vector is the Sz part, 602 00:44:55,620 --> 00:44:58,399 and the fourth basis vector is the identity 603 00:44:58,399 --> 00:44:59,190 that we subtracted. 604 00:45:02,880 --> 00:45:09,090 And I'm listing the other two that we got rid 605 00:45:09,090 --> 00:45:11,990 of because physically we're not that interested 606 00:45:11,990 --> 00:45:15,660 given that we want Sx and Sz. 607 00:45:15,660 --> 00:45:18,370 So, Sx and Sy. 608 00:45:18,370 --> 00:45:19,800 But here it is. 609 00:45:19,800 --> 00:45:24,280 These four two-by-two matrices are 610 00:45:24,280 --> 00:45:28,795 sort of the linearly independent Hermitian matrices. 611 00:45:31,780 --> 00:45:36,060 You can think of them as vectors, four basis vectors. 612 00:45:36,060 --> 00:45:39,380 You multiply by real numbers, and now you add them, 613 00:45:39,380 --> 00:45:43,030 and you got the most general Hermitian matrix. 614 00:45:43,030 --> 00:45:48,870 So this is part of the subtlety of this whole idea of vector 615 00:45:48,870 --> 00:45:51,500 spaces of matrices, which can be thought 616 00:45:51,500 --> 00:45:54,920 of as vectors sometimes, as well. 617 00:45:54,920 --> 00:45:58,880 So that's why these matrices are quite famous. 618 00:45:58,880 --> 00:46:04,070 But before we just discuss why they are so famous, 619 00:46:04,070 --> 00:46:05,040 let's think of this. 620 00:46:05,040 --> 00:46:08,890 Where we're looking for Sx and Sy, 621 00:46:08,890 --> 00:46:12,950 and we actually seem to have two matrices here 622 00:46:12,950 --> 00:46:19,660 that could do the job, as two independent Hermitian 623 00:46:19,660 --> 00:46:22,830 two-by-two matrices. 624 00:46:22,830 --> 00:46:28,290 But we must add a little extra information. 625 00:46:28,290 --> 00:46:30,390 We don't know what the scale is. 626 00:46:30,390 --> 00:46:36,270 Should I multiply this by 5 and call that Sx? 627 00:46:36,270 --> 00:46:38,830 Or this by 3? 628 00:46:38,830 --> 00:46:41,240 We're missing a little more physics. 629 00:46:41,240 --> 00:46:42,660 What is the physics? 630 00:46:42,660 --> 00:46:49,240 The eigenvalues of Sx should also be plus minus h over 2. 631 00:46:49,240 --> 00:46:54,240 And the eigenvalues of Sy should also be plus minus h over 2. 632 00:46:54,240 --> 00:46:56,120 Just like for Sz. 633 00:46:56,120 --> 00:46:59,300 you could have started the whole Stern-Gerlach things 634 00:46:59,300 --> 00:47:00,810 thinking of x, and you would have 635 00:47:00,810 --> 00:47:04,020 obtained plus minus h over 2. 636 00:47:04,020 --> 00:47:06,110 So that is the physical constraint. 637 00:47:08,850 --> 00:47:10,990 I have to figure out those numbers. 638 00:47:10,990 --> 00:47:15,730 Maybe Sx is this one, as y is this one. 639 00:47:15,730 --> 00:47:18,300 And you can say, oh, you never told us 640 00:47:18,300 --> 00:47:21,640 if you're going to get the unique answer here. 641 00:47:21,640 --> 00:47:24,080 And yes, I did tell you, and you're not 642 00:47:24,080 --> 00:47:26,110 going to get a unique answer. 643 00:47:26,110 --> 00:47:28,820 There are some sign notations and some other things, 644 00:47:28,820 --> 00:47:32,020 but any answer is perfectly good. 645 00:47:32,020 --> 00:47:35,054 So once you get an answer, it's perfectly good. 646 00:47:35,054 --> 00:47:36,720 Of course, we're going to get the answer 647 00:47:36,720 --> 00:47:38,640 that everybody likes. 648 00:47:38,640 --> 00:47:41,520 And the convention is that happily that everybody 649 00:47:41,520 --> 00:47:43,920 uses this same convention. 650 00:47:43,920 --> 00:47:44,891 Questions. 651 00:47:44,891 --> 00:47:46,516 AUDIENCE: So I have a related question, 652 00:47:46,516 --> 00:47:50,200 because at the beginning we could have chosen the top right 653 00:47:50,200 --> 00:47:53,880 entry to be a plus ib and the bottom left to be a minus ib 654 00:47:53,880 --> 00:47:56,980 and that would have yielded a different basis matrix. 655 00:47:56,980 --> 00:48:00,206 PROFESSOR: Right, I would have called this plus and minus. 656 00:48:00,206 --> 00:48:00,706 Yes. 657 00:48:00,706 --> 00:48:04,110 AUDIENCE: Are we going to show that this is the correct form? 658 00:48:04,110 --> 00:48:06,210 PROFESSOR: No, it's not the correct form. 659 00:48:06,210 --> 00:48:11,110 It is a correct form, and it's equivalent to any other form 660 00:48:11,110 --> 00:48:12,540 you could find. 661 00:48:12,540 --> 00:48:14,360 That's what we can show. 662 00:48:14,360 --> 00:48:16,590 In fact, I will show that there's 663 00:48:16,590 --> 00:48:19,990 an obvious ambiguity here. 664 00:48:19,990 --> 00:48:23,080 Well, in fact, maybe I can tell it do you, I think. 665 00:48:23,080 --> 00:48:36,260 If you let Sx go to minus Sy, and Sy goes to plus Sx, 666 00:48:36,260 --> 00:48:38,340 nothing changes in these equations. 667 00:48:38,340 --> 00:48:41,870 They become the same equations. 668 00:48:41,870 --> 00:48:44,350 You know, Sx would become minus Sy, 669 00:48:44,350 --> 00:48:46,940 and this Sx-- this is not changed. 670 00:48:46,940 --> 00:48:49,870 But, in fact, if you put minus Sy and Sx 671 00:48:49,870 --> 00:48:53,330 as the same commutator then this one 672 00:48:53,330 --> 00:48:55,180 will become actually this commutator, 673 00:48:55,180 --> 00:48:56,700 and this one will become that. 674 00:48:56,700 --> 00:49:00,230 So I could change whatever I get for Sx, 675 00:49:00,230 --> 00:49:02,765 change it from minus Sy, for example, 676 00:49:02,765 --> 00:49:05,240 and get the same thing. 677 00:49:05,240 --> 00:49:07,460 So there are many changes you can do. 678 00:49:07,460 --> 00:49:11,770 The only thing we need is one answer that works. 679 00:49:11,770 --> 00:49:13,520 And I'm going to write, of course, the one 680 00:49:13,520 --> 00:49:14,980 that everybody likes. 681 00:49:14,980 --> 00:49:17,630 But don't worry about that. 682 00:49:17,630 --> 00:49:21,550 So let's think of eigenvectors and eigenvalues now. 683 00:49:21,550 --> 00:49:24,850 I don't know how much you remember that, 684 00:49:24,850 --> 00:49:29,000 but we'll just take it at this moment that you do. 685 00:49:29,000 --> 00:49:33,910 So 0, 1, 1, 0 has two eigenvalues, 686 00:49:33,910 --> 00:49:39,950 and lambda equals 1, with eigenvector 1 687 00:49:39,950 --> 00:49:42,265 over square root of 2, 1, 1. 688 00:49:45,000 --> 00:49:49,420 And a lambda equals minus 1 with eigenvector 1 689 00:49:49,420 --> 00:49:52,075 over square root of 2, 1, minus 1. 690 00:50:03,230 --> 00:50:17,000 The other one, it's equally easy to do. 691 00:50:17,000 --> 00:50:21,970 We'll discuss eigenvectors and eigenvalues later. 692 00:50:21,970 --> 00:50:30,490 Minus i, i, 0, 0, plus a lambda equals one eigenvector, 693 00:50:30,490 --> 00:50:36,000 with components 1 over square root of 2, 1, and i. 694 00:50:36,000 --> 00:50:39,860 I'm pretty sure it's 1 and i. 695 00:50:39,860 --> 00:50:43,690 Yes, and a lambda equals minus 1, 696 00:50:43,690 --> 00:50:49,320 with components 1 over square root 2, 1, minus i. 697 00:50:49,320 --> 00:50:52,820 Now I put the 1 over square root of 2 698 00:50:52,820 --> 00:50:55,250 because I wanted them to be normalized. 699 00:50:55,250 --> 00:50:58,540 Remember how you're supposed to normalize these things. 700 00:50:58,540 --> 00:51:02,485 You're supposed to take the row vector, complex conjugate, 701 00:51:02,485 --> 00:51:03,880 and multiply. 702 00:51:03,880 --> 00:51:06,320 Well, you would get 1 for the length 703 00:51:06,320 --> 00:51:10,680 of this, 1 for the length of this. 704 00:51:10,680 --> 00:51:12,630 You would get one for the length of this, 705 00:51:12,630 --> 00:51:15,130 but remember, you have to complex conjugate, 706 00:51:15,130 --> 00:51:17,860 otherwise you'll get 0. 707 00:51:17,860 --> 00:51:20,270 Also, you will get one for the length of this. 708 00:51:20,270 --> 00:51:25,070 So these are our eigenvalues. 709 00:51:25,070 --> 00:51:29,930 So actually, with eigenvalues lambda equals 1 and minus 1 710 00:51:29,930 --> 00:51:32,910 for these two, we're in pretty good shape. 711 00:51:32,910 --> 00:51:43,020 We could try Sx to be h-bar over 2 times 0, 1, 1, 0. 712 00:51:43,020 --> 00:51:55,310 And Sy to be h-bar over 2, 0, minus i, i, 0. 713 00:51:55,310 --> 00:51:58,140 These would have the right eigenvalues 714 00:51:58,140 --> 00:52:00,930 because if you multiply a matrix by a number, 715 00:52:00,930 --> 00:52:03,430 the eigenvalue gets multiplied by this number, 716 00:52:03,430 --> 00:52:10,140 so the plus minus 1s become plus minus h-bar over 2. 717 00:52:10,140 --> 00:52:13,480 But what are we was supposed to check? 718 00:52:13,480 --> 00:52:15,218 If this is to work, we're supposed 719 00:52:15,218 --> 00:52:16,343 to check these commutators. 720 00:52:18,940 --> 00:52:23,060 So let's do one, at least. 721 00:52:23,060 --> 00:52:27,480 Sx commutator with Sy. 722 00:52:27,480 --> 00:52:33,580 So what do we get? h-bar over 2, h-bar over 2-- two of them-- 723 00:52:33,580 --> 00:52:40,150 then the first matrix, 0, 1, 1, 0 times 0, minus i, i, 724 00:52:40,150 --> 00:52:48,980 0, minus 0, minus i, i, 0 times 0, 1, 1, 0. 725 00:52:48,980 --> 00:53:03,720 Which is h-bar over 2 times h-bar over 2, times i, 0, 0, 726 00:53:03,720 --> 00:53:13,860 minus i, minus minus i, 0, 0, i. 727 00:53:20,410 --> 00:53:21,805 And we're almost there. 728 00:53:34,430 --> 00:53:36,190 What do we have? 729 00:53:36,190 --> 00:53:40,630 Well, we have h-bar over 2, h-bar over 2. 730 00:53:40,630 --> 00:53:44,150 And we've got 2i and minus 2i. 731 00:53:47,860 --> 00:53:58,710 So this is h-bar over 2 times h-bar over 2, times 2i times 1, 732 00:53:58,710 --> 00:54:01,360 minus 1. 733 00:54:01,360 --> 00:54:07,610 And this whole thing is i h-bar, and the other part 734 00:54:07,610 --> 00:54:14,410 is h-bar over 2, 1, minus 1, which is i h-bar as z-hat. 735 00:54:19,480 --> 00:54:20,440 Good, it works. 736 00:54:24,840 --> 00:54:27,080 You know, the only thing that could 737 00:54:27,080 --> 00:54:30,240 have gone wrong-- you could have identified 1 with a minus, 738 00:54:30,240 --> 00:54:33,540 or something like that It would have been equally good. 739 00:54:33,540 --> 00:54:36,310 Once you have these operators, we're fine. 740 00:54:36,310 --> 00:54:41,170 So one has to check that the other ones work, and they do. 741 00:54:41,170 --> 00:54:44,540 I will leave them for you to check. 742 00:54:44,540 --> 00:54:49,820 And therefore, we've got the three matrices. 743 00:54:49,820 --> 00:54:55,360 It's a very important result-- the Sx Sy, and Sz. 744 00:54:58,210 --> 00:55:04,130 I will not rewrite them, but they should be boxed nicely, 745 00:55:04,130 --> 00:55:07,900 the three of them together, with that 746 00:55:07,900 --> 00:55:10,015 one there on top of the blackboard. 747 00:55:12,790 --> 00:55:17,270 And of course by construction, they're Hermitian. 748 00:55:17,270 --> 00:55:19,490 They're famous enough that people 749 00:55:19,490 --> 00:55:22,090 have defined the following object. 750 00:55:22,090 --> 00:55:27,680 Si is defined to be h-bar over 2, 751 00:55:27,680 --> 00:55:32,050 sigma i, the power of the matrix sigmas. 752 00:55:32,050 --> 00:55:38,310 And these are poly matrices, sigma one is 0, 1, 1, 0. 753 00:55:38,310 --> 00:55:43,140 Sigma two is 0, minus i, i, 0. 754 00:55:43,140 --> 00:55:48,340 And sigma three is equal to 1, minus 1, 0, 0. 755 00:55:55,170 --> 00:55:58,820 OK, so in principle-- yes, question. 756 00:55:58,820 --> 00:56:00,236 AUDIENCE: Is it at all significant 757 00:56:00,236 --> 00:56:05,980 that the poly matrices are all squared [INAUDIBLE]? 758 00:56:05,980 --> 00:56:09,100 PROFESSOR: Yes, it is significant. 759 00:56:09,100 --> 00:56:13,300 We'll use it, but at this moment, it's not urgent for us. 760 00:56:17,470 --> 00:56:20,686 We'll have no application of that property 761 00:56:20,686 --> 00:56:22,690 for a little while, but it will help 762 00:56:22,690 --> 00:56:27,508 us do a lot of the algebra of the poly matrices. 763 00:56:27,508 --> 00:56:30,980 AUDIENCE: [INAUDIBLE] eigenvalues, right? 764 00:56:30,980 --> 00:56:31,972 PROFESSOR: Sorry? 765 00:56:31,972 --> 00:56:35,940 AUDIENCE: Doesn't that follow from the eigenvalue properties 766 00:56:35,940 --> 00:56:40,256 that we've [INAUDIBLE] plus or minus one. 767 00:56:44,030 --> 00:56:46,910 Because those were both squared. [INAUDIBLE]. 768 00:56:50,280 --> 00:56:51,390 PROFESSOR: That's right. 769 00:56:51,390 --> 00:56:54,340 I think so. 770 00:56:54,340 --> 00:57:05,110 Our eigenvalues-- yes, it's true. 771 00:57:05,110 --> 00:57:09,100 That the fact that the eigenvalues are plus minus 1 772 00:57:09,100 --> 00:57:13,390 will imply that these matrices squared themselves. 773 00:57:13,390 --> 00:57:17,280 So it's incorporated into our analysis. 774 00:57:17,280 --> 00:57:21,210 The thing that I will say is that you don't need it 775 00:57:21,210 --> 00:57:23,680 in the expression of the commutators. 776 00:57:23,680 --> 00:57:26,470 So in the commentators, it didn't 777 00:57:26,470 --> 00:57:28,480 play a role to begin with. 778 00:57:28,480 --> 00:57:31,800 Put it as an extra condition. 779 00:57:31,800 --> 00:57:35,990 Now what is the next thing we really want to understand? 780 00:57:35,990 --> 00:57:39,540 Is that in terms of plain states, 781 00:57:39,540 --> 00:57:44,940 we now have the answer for most of the experiments we could do. 782 00:57:44,940 --> 00:57:48,520 So in particular, remember that we 783 00:57:48,520 --> 00:57:53,560 said that we would have Sx, for example, having states 784 00:57:53,560 --> 00:57:59,880 x plus minus, which are h-bar over 2 785 00:57:59,880 --> 00:58:05,930 plus minus, x comma plus minus. 786 00:58:05,930 --> 00:58:11,660 The states along the x-direction referred 787 00:58:11,660 --> 00:58:16,010 like that would be the eigenstates of the Sx operator. 788 00:58:16,010 --> 00:58:20,370 But we've calculated the states of the Sx operator-- 789 00:58:20,370 --> 00:58:23,040 they're here. 790 00:58:23,040 --> 00:58:27,180 The Sx operator is h-bar over 2 times this matrix. 791 00:58:27,180 --> 00:58:29,200 And we have those things. 792 00:58:29,200 --> 00:58:33,606 So the plus eigenvalue and the minus eigenvalue 793 00:58:33,606 --> 00:58:36,170 will just show up here. 794 00:58:36,170 --> 00:58:44,780 So let me write them, and explain, in plain language, 795 00:58:44,780 --> 00:58:47,490 what these states are. 796 00:58:47,490 --> 00:58:53,630 So the eigenstate with lambda equal 797 00:58:53,630 --> 00:59:00,720 1-- that would correspond to h-bar over two-- so the x 798 00:59:00,720 --> 00:59:04,890 plus corresponds to this vector. 799 00:59:07,790 --> 00:59:10,070 So what is that state? 800 00:59:10,070 --> 00:59:15,980 It's that vector which, if you want more explicitly, 801 00:59:15,980 --> 00:59:24,070 it's the z plus, plus z minus. 802 00:59:24,070 --> 00:59:30,080 This is the state 1 over square root of 2, 1, 1. 803 00:59:30,080 --> 00:59:39,423 The x minus is z plus, minus z minus. 804 00:59:42,200 --> 00:59:47,800 As you see on that blackboard, it's 1 minus 1. 805 00:59:47,800 --> 00:59:49,150 So here it is. 806 00:59:49,150 --> 00:59:55,010 The states that you were looking for, that are aligned along x-- 807 00:59:55,010 --> 00:59:57,860 plus x or minus x-- are not new states 808 00:59:57,860 --> 01:00:00,340 that have you to add to the state space. 809 01:00:00,340 --> 01:00:05,160 They are linear combinations of the states you've got. 810 01:00:05,160 --> 01:00:14,070 We can invert this formula and write, for example, 811 01:00:14,070 --> 01:00:22,445 that z plus is 1 over square root of 2, x plus, 812 01:00:22,445 --> 01:00:29,470 plus 1 over square root of 2, x minus. 813 01:00:29,470 --> 01:00:39,330 And z minus is 1 over square root of 2, x plus, minus 1 814 01:00:39,330 --> 01:00:43,130 over square root of-- minus the square root of 2 815 01:00:43,130 --> 01:00:49,740 is already out, I'm sorry-- minus x minus. 816 01:00:52,680 --> 01:00:58,170 So actually, this answers the question that you had. 817 01:00:58,170 --> 01:01:02,170 For example, you put a z plus state, 818 01:01:02,170 --> 01:01:07,250 and you put an x filter-- what amplitude 819 01:01:07,250 --> 01:01:17,060 do you have to find a state in the x plus, given that you 820 01:01:17,060 --> 01:01:20,240 start with a state on the z plus? 821 01:01:24,680 --> 01:01:28,030 Well, you put an x plus from here. 822 01:01:28,030 --> 01:01:31,560 You get 1 from this and 0 from this one 823 01:01:31,560 --> 01:01:34,360 because the states are always orthogonal. 824 01:01:34,360 --> 01:01:37,860 The states are orthogonal-- you should check that. 825 01:01:37,860 --> 01:01:41,270 And therefore, this is 1 over square root of 2. 826 01:01:41,270 --> 01:01:47,200 If you ask for x minus with respect to z plus, 827 01:01:47,200 --> 01:01:50,100 that's also 1 over square root of 2. 828 01:01:50,100 --> 01:01:52,430 And these are the amplitudes for this state 829 01:01:52,430 --> 01:01:56,040 to be found in this, for this state to be found in them. 830 01:01:56,040 --> 01:01:56,890 They're equal. 831 01:01:56,890 --> 01:01:59,610 The probabilities are 1/2. 832 01:01:59,610 --> 01:02:01,180 And that's good. 833 01:02:01,180 --> 01:02:03,570 Our whole theory of angular momentum 834 01:02:03,570 --> 01:02:06,460 has given us something that is perfectly 835 01:02:06,460 --> 01:02:10,040 consistent with the Stern-Gerlach experiment, 836 01:02:10,040 --> 01:02:14,440 and it gives you these probabilities. 837 01:02:14,440 --> 01:02:17,695 You can construct in the same way the y states. 838 01:02:21,750 --> 01:02:26,980 So the y states are the eigenstates 839 01:02:26,980 --> 01:02:31,845 of that second matrix, Sy, that we wrote on the left. 840 01:02:37,270 --> 01:02:45,500 So this matrix is Sy, so its eigenstates-- I'm sorry, 841 01:02:45,500 --> 01:02:47,430 Sy is there. 842 01:02:47,430 --> 01:02:50,060 Sy is there. 843 01:02:50,060 --> 01:02:52,850 The eigenstates are those, so immediately 844 01:02:52,850 --> 01:03:00,390 you translate that to say that Sy has eigenstates y 845 01:03:00,390 --> 01:03:03,770 plus minus, whose eigenvalues are 846 01:03:03,770 --> 01:03:08,880 plus minus h-bar over 2y plus minus. 847 01:03:08,880 --> 01:03:16,790 And y plus is equal 1 over square root of 2, 848 01:03:16,790 --> 01:03:29,150 z plus-- and look at the first eigenvector-- plus iz minus. 849 01:03:29,150 --> 01:03:33,650 And, in fact, they can put one formula for both. 850 01:03:33,650 --> 01:03:34,640 Here they are. 851 01:03:38,520 --> 01:03:43,600 So, it's kind of neat that the x1s were 852 01:03:43,600 --> 01:03:46,580 found by linear combinations, and they're orthogonal. 853 01:03:46,580 --> 01:03:49,100 Now, if you didn't have complex numbers, 854 01:03:49,100 --> 01:03:53,130 you could not form another linear combination 855 01:03:53,130 --> 01:03:55,300 of this orthogonal. 856 01:03:55,300 --> 01:03:57,100 But thanks to these complex numbers, 857 01:03:57,100 --> 01:04:00,880 you can put an i there-- there's no i in the x ones-- 858 01:04:00,880 --> 01:04:04,430 and the states are orthogonal, something 859 01:04:04,430 --> 01:04:06,850 that you should check. 860 01:04:06,850 --> 01:04:09,170 So again, you can invert and find 861 01:04:09,170 --> 01:04:11,550 the z states in terms of y, and you 862 01:04:11,550 --> 01:04:14,860 would conclude that the amplitudes are really 863 01:04:14,860 --> 01:04:18,060 the same up to signs, or maybe complex numbers, 864 01:04:18,060 --> 01:04:22,360 but the probabilities are identical. 865 01:04:22,360 --> 01:04:25,990 So we've gotten a long way. 866 01:04:25,990 --> 01:04:28,650 We basically have a theory that seems 867 01:04:28,650 --> 01:04:33,460 to describe the whole result of the Stern-Gerlach experiment, 868 01:04:33,460 --> 01:04:37,690 but now your theory can do more for you. 869 01:04:37,690 --> 01:04:39,600 Now, in the last few minutes, we're 870 01:04:39,600 --> 01:04:42,550 going to calculate the states that 871 01:04:42,550 --> 01:04:44,960 are along arbitrary directions. 872 01:04:44,960 --> 01:04:52,410 So here I produced a state that is along 873 01:04:52,410 --> 01:04:57,180 the x-direction plus, and along the x-direction minus. 874 01:04:57,180 --> 01:05:02,020 What I would like to construct, to finish this story, 875 01:05:02,020 --> 01:05:06,150 is a state that is along some arbitrary direction. 876 01:05:09,910 --> 01:05:16,010 So the state that points along some unit vector n. 877 01:05:16,010 --> 01:05:22,560 So here is space, and here's a unit vector n with components 878 01:05:22,560 --> 01:05:26,700 nx, , ny, and nz. 879 01:05:26,700 --> 01:05:39,775 Or you can write the vector n as nx ex plus ny ey plus nz ez. 880 01:05:43,530 --> 01:05:45,660 And I would like to understand how 881 01:05:45,660 --> 01:05:49,710 I can construct, in general, a spin state that 882 01:05:49,710 --> 01:05:53,110 could be said to be in the n direction. 883 01:05:53,110 --> 01:05:56,610 We have the ones along the z, x, and y, 884 01:05:56,610 --> 01:06:00,560 but let's try to get something more general, the most general 885 01:06:00,560 --> 01:06:03,490 one. 886 01:06:03,490 --> 01:06:09,740 So for this, we think of the triplet 887 01:06:09,740 --> 01:06:17,350 of operators S, which would be Sx, Sy, and Sz. 888 01:06:19,930 --> 01:06:27,020 Now you can, if you wish, write this as Sx-hat ex vector, 889 01:06:27,020 --> 01:06:33,635 plus Sy-hat ey vector, plus Sz hat ez vector. 890 01:06:36,800 --> 01:06:39,580 But this object, if you write it like that, 891 01:06:39,580 --> 01:06:43,350 is really a strange object. 892 01:06:43,350 --> 01:06:44,150 Think of it. 893 01:06:44,150 --> 01:06:49,430 It's matrices, or operators, multiplied by unit vectors. 894 01:06:49,430 --> 01:06:52,790 These vectors have nothing to do with the space 895 01:06:52,790 --> 01:06:56,210 in which the matrices act. 896 01:06:56,210 --> 01:07:00,430 The matrices act in an abstract, two-dimensional vector space, 897 01:07:00,430 --> 01:07:04,540 while these vectors are sort of for accounting purposes. 898 01:07:04,540 --> 01:07:06,430 That's why we sometimes don't write them, 899 01:07:06,430 --> 01:07:09,300 and say we have a triplet. 900 01:07:09,300 --> 01:07:12,850 So this product means almost nothing. 901 01:07:12,850 --> 01:07:14,260 They're just sitting together. 902 01:07:14,260 --> 01:07:18,660 You could put the e to the left of the x or to the right. 903 01:07:18,660 --> 01:07:19,560 It's a vector. 904 01:07:19,560 --> 01:07:23,130 You're not supposed to put the vector inside the matrix, 905 01:07:23,130 --> 01:07:24,480 either. 906 01:07:24,480 --> 01:07:25,930 They don't talk to each. 907 01:07:25,930 --> 01:07:29,650 It's an accounting procedure. 908 01:07:29,650 --> 01:07:33,680 It is useful sometimes; we will use it to derive identities 909 01:07:33,680 --> 01:07:37,110 soon, but it's an accounting procedure. 910 01:07:37,110 --> 01:07:39,670 So here's what I want to define. 911 01:07:39,670 --> 01:07:45,000 So this is a crazy thing, some sort of vector valued operator, 912 01:07:45,000 --> 01:07:46,110 or something like that. 913 01:07:46,110 --> 01:07:48,450 But what we really need is what we'll 914 01:07:48,450 --> 01:07:57,430 call S-hat n, which will be defined as n dot S. Where 915 01:07:57,430 --> 01:08:01,750 we take naively what a dot product is supposed to mean. 916 01:08:01,750 --> 01:08:04,160 This component times this component, 917 01:08:04,160 --> 01:08:05,560 which happens to be an operator. 918 01:08:05,560 --> 01:08:07,950 This times this, this times that. 919 01:08:07,950 --> 01:08:18,100 nx Sx plus ny Sy, plus nz Sz. 920 01:08:18,100 --> 01:08:20,830 And this thing is something very intuitive. 921 01:08:20,830 --> 01:08:22,800 It is just an operator. 922 01:08:22,800 --> 01:08:25,850 It doesn't have anymore a vector with it. 923 01:08:25,850 --> 01:08:29,484 So it's a single operator. 924 01:08:29,484 --> 01:08:33,170 If your vector points in the z-direction, 925 01:08:33,170 --> 01:08:37,734 nx and ny z, and you have Sz because it's a unit vector. 926 01:08:42,800 --> 01:08:46,859 If the vector points in the x-direction, you get Sx. 927 01:08:46,859 --> 01:08:51,960 If the vector points in the y-direction, you get Sy. 928 01:08:51,960 --> 01:08:56,149 In general, this we call the spin operator 929 01:08:56,149 --> 01:09:00,180 in the direction of the vector n-- 930 01:09:00,180 --> 01:09:10,220 spin operator in the direction of n. 931 01:09:14,040 --> 01:09:19,540 OK, so what about that spin operator? 932 01:09:19,540 --> 01:09:25,779 Well, it had eigenvalues plus minus h-bar over 2 along z, x, 933 01:09:25,779 --> 01:09:29,880 and y-- probably does still have those eigenvalues-- 934 01:09:29,880 --> 01:09:34,630 but we have to make this a little clearer. 935 01:09:34,630 --> 01:09:40,450 So for that we'll take nx and ny and nz 936 01:09:40,450 --> 01:09:43,050 to be the polar coordinate things. 937 01:09:43,050 --> 01:09:49,229 So this vector is going to have a theta here 938 01:09:49,229 --> 01:09:52,510 on the azimuthal angle phi over here. 939 01:09:52,510 --> 01:09:59,550 So nz is cosine theta. 940 01:09:59,550 --> 01:10:02,575 nx and ny have sine theta. 941 01:10:05,420 --> 01:10:09,370 And nx cosine phi, and this one has sine phi. 942 01:10:13,770 --> 01:10:19,940 So what is the operator Sn vector hat? 943 01:10:19,940 --> 01:10:26,080 Well, it's nx times Sx. 944 01:10:26,080 --> 01:10:29,450 So, I'll put a h-bar over 2 in front, 945 01:10:29,450 --> 01:10:39,590 so we'll have nx sigma x, or sigma1, plus ny sigma2, , 946 01:10:39,590 --> 01:10:42,510 plus nz sigma3. 947 01:10:42,510 --> 01:10:47,460 Remember the spin operators are proportional h-bar 948 01:10:47,460 --> 01:10:53,480 over 2 times the sigmas-- so sigma1, sigma2, sigma3. 949 01:10:53,480 --> 01:10:58,590 And look what we get. h-bar over 2. 950 01:10:58,590 --> 01:11:04,160 Sigma1 has an nx here, nx. 951 01:11:04,160 --> 01:11:10,100 Sigma2 has minus iny plus iny. 952 01:11:10,100 --> 01:11:14,260 And sigma3, we have a nz minus nz. 953 01:11:18,680 --> 01:11:27,612 So this is h-bar over 2, nz is cosine theta, nx minus iny-- 954 01:11:27,612 --> 01:11:30,130 you'd say, oh it's a pretty awful thing, 955 01:11:30,130 --> 01:11:37,760 but it's very simple-- nx minus iny is sine theta times e 956 01:11:37,760 --> 01:11:39,630 to the minus i phi. 957 01:11:45,430 --> 01:11:51,310 Here it would be sine theta, e to the i phi, 958 01:11:51,310 --> 01:11:53,786 and here we'll have minus cosine theta. 959 01:11:56,360 --> 01:12:00,760 So this is the whole matrix, Sn-hat, like that. 960 01:12:03,810 --> 01:12:06,580 Well, in the last couple of minutes, 961 01:12:06,580 --> 01:12:10,985 let's calculate the eigenvectors and eigenvalues. 962 01:12:15,990 --> 01:12:17,350 So what do we get? 963 01:12:17,350 --> 01:12:19,410 Well, for the eigenvalues, remember 964 01:12:19,410 --> 01:12:23,980 what is the computation of an eigenvalue of a matrix. 965 01:12:23,980 --> 01:12:27,710 An eigenvalue for matrix a, you write 966 01:12:27,710 --> 01:12:34,325 that by solving the determinant of a minus lambda 1 equals 0. 967 01:12:37,660 --> 01:12:40,800 So for any matrix a, if we want to find 968 01:12:40,800 --> 01:12:44,330 the eigenvalues of this matrix, we 969 01:12:44,330 --> 01:12:49,760 would have to write eigenvalues of Sn-hat. 970 01:12:52,960 --> 01:12:55,840 We have to ride the determinant of this, 971 01:12:55,840 --> 01:13:04,190 minus lambda i, so the determinant of h-bar over 2 972 01:13:04,190 --> 01:13:09,770 cosine theta, minus lambda, minus h-bar over 2 cosine 973 01:13:09,770 --> 01:13:12,220 theta, minus lambda. 974 01:13:12,220 --> 01:13:22,420 And here, it's sine theta, e to the minus i phi, sine theta 975 01:13:22,420 --> 01:13:29,630 e to the i phi, the determinant of this being 0. 976 01:13:29,630 --> 01:13:32,560 It's not as bad as it looks. 977 01:13:32,560 --> 01:13:35,750 It's actually pretty simple. 978 01:13:35,750 --> 01:13:38,550 These are a plus b, a minus b. 979 01:13:38,550 --> 01:13:42,100 Here the phases cancel out. 980 01:13:42,100 --> 01:13:49,630 The algebra you can read in the notes, but you do get lambda 981 01:13:49,630 --> 01:13:54,840 equals plus minus h-bar over 2. 982 01:13:54,840 --> 01:14:02,750 Now that is fine, and we now want the eigenvectors. 983 01:14:02,750 --> 01:14:08,450 Those are more non-trivial, so they need a little more work. 984 01:14:08,450 --> 01:14:11,860 So what are you supposed to do to find an eigenvector? 985 01:14:11,860 --> 01:14:17,760 You're supposed to take this a minus lambda i, acting 986 01:14:17,760 --> 01:14:20,310 on a vector, and put it equal to zero. 987 01:14:20,310 --> 01:14:21,435 And that's the eigenvector. 988 01:14:24,350 --> 01:14:28,440 So, for this case, we're going to try 989 01:14:28,440 --> 01:14:34,600 to find the eigenvector n plus. 990 01:14:34,600 --> 01:14:44,410 So this is the one that has Sn on this state-- well, 991 01:14:44,410 --> 01:14:51,485 I'll write it here, plus minus h over 2, n plus minus here. 992 01:14:54,180 --> 01:14:57,210 So let's try to find this one that corresponds 993 01:14:57,210 --> 01:15:03,675 to the eigenvalue equal to plus h-bar over 2. 994 01:15:03,675 --> 01:15:17,890 Now this state is C1 times z plus, plus C2 times z minus. 995 01:15:17,890 --> 01:15:21,420 These are our basis states, so it's a little combination. 996 01:15:21,420 --> 01:15:24,970 Or it's C1, C2. 997 01:15:24,970 --> 01:15:26,790 Think of it as a matrix. 998 01:15:26,790 --> 01:15:32,610 So we want the eigenvalues of that-- the eigenvector 999 01:15:32,610 --> 01:15:36,450 for that-- so what do we have? 1000 01:15:36,450 --> 01:15:47,100 Well, we would have Sn-hat minus h-bar over 2 times 1, 1001 01:15:47,100 --> 01:15:55,560 on this C1, C2 equals 0. 1002 01:15:55,560 --> 01:15:58,680 The eigenvector equation is that this operator 1003 01:15:58,680 --> 01:16:03,280 minus the eigenvalue must give you that. 1004 01:16:03,280 --> 01:16:07,730 So the h-bars over 2, happily, go out, 1005 01:16:07,730 --> 01:16:10,930 and you don't really need to worry about them anymore. 1006 01:16:10,930 --> 01:16:19,700 And you get here cosine theta minus 1, sine theta 1007 01:16:19,700 --> 01:16:27,000 e to the minus i phi, sine theta e to the i phi, 1008 01:16:27,000 --> 01:16:36,720 and minus cosine theta minus 1, C1, C2 equals 0. 1009 01:16:36,720 --> 01:16:44,205 All right, so you have two equations, 1010 01:16:44,205 --> 01:16:47,220 and both relate C1 and C2. 1011 01:16:47,220 --> 01:16:51,050 Happily, and the reason this works is because 1012 01:16:51,050 --> 01:16:57,120 with this eigenvalue that we've used that appears here, 1013 01:16:57,120 --> 01:16:59,220 these two equations are the same. 1014 01:16:59,220 --> 01:17:01,100 So you can take either one, and they 1015 01:17:01,100 --> 01:17:05,900 must imply the same relation between C1 and C2. 1016 01:17:05,900 --> 01:17:07,490 Something you can check. 1017 01:17:07,490 --> 01:17:11,840 So let me write one of them. 1018 01:17:11,840 --> 01:17:19,090 C2 is equal to e to the i phi, 1 minus cosine theta 1019 01:17:19,090 --> 01:17:21,990 over sine theta C1. 1020 01:17:38,930 --> 01:17:42,070 It's from the first line. 1021 01:17:42,070 --> 01:17:49,560 So you have to remember, in order to simplify these things, 1022 01:17:49,560 --> 01:17:51,950 your half angle identities. 1023 01:17:51,950 --> 01:17:52,450 Sorry. 1024 01:17:55,160 --> 01:18:00,930 1 minus cosine theta is 2 sine squared theta over 2, 1025 01:18:00,930 --> 01:18:06,000 and sine theta is 2 sine theta over 2 cosine theta over 2. 1026 01:18:06,000 --> 01:18:15,540 So this becomes e to the i phi sine theta over 2, 1027 01:18:15,540 --> 01:18:19,040 over cosine theta over 2, C1. 1028 01:18:21,780 --> 01:18:24,560 Now we want these things to be well normalized, 1029 01:18:24,560 --> 01:18:32,620 so we want C1 squared plus C2 squared equal to 1. 1030 01:18:32,620 --> 01:18:37,620 So, you know what C2 is, so this gives you 1031 01:18:37,620 --> 01:18:43,430 C1 squared times 1 plus-- and C2 you use this, 1032 01:18:43,430 --> 01:18:46,670 when you square the phase goes away-- 1033 01:18:46,670 --> 01:18:54,100 sine squared theta over 2, cosine squared over 2 1034 01:18:54,100 --> 01:18:56,940 must be equal to 1. 1035 01:18:56,940 --> 01:18:59,930 Well, the numerator is 1, so you learn 1036 01:18:59,930 --> 01:19:08,250 that C1 squared is equal to cosine squared theta over 2. 1037 01:19:08,250 --> 01:19:10,370 Now you have to take the square root, 1038 01:19:10,370 --> 01:19:15,220 and you could put an i or a phase or something. 1039 01:19:15,220 --> 01:19:17,800 But look, whatever phase you choose, 1040 01:19:17,800 --> 01:19:22,200 you could choose C1 to be cosine theta over 2, 1041 01:19:22,200 --> 01:19:24,060 and say, I'm done. 1042 01:19:24,060 --> 01:19:25,370 I want this one. 1043 01:19:25,370 --> 01:19:28,810 Somebody would say, no let's put the phase, e 1044 01:19:28,810 --> 01:19:32,140 so to the i pi over 5. 1045 01:19:32,140 --> 01:19:37,870 So that doesn't look good, but four or even worse, 1046 01:19:37,870 --> 01:19:41,100 this phase will show up in C2 because C2 1047 01:19:41,100 --> 01:19:42,870 is proportional to C1. 1048 01:19:42,870 --> 01:19:45,090 So I can get rid of it. 1049 01:19:45,090 --> 01:19:48,110 I only should put it if I really need it, 1050 01:19:48,110 --> 01:19:51,660 and I don't think I need it, so I won't put it. 1051 01:19:51,660 --> 01:19:53,330 And you can always change your mind 1052 01:19:53,330 --> 01:19:57,150 later-- nobody's going to take your word for this. 1053 01:19:57,150 --> 01:20:02,180 So, in this case, C2 would be sine theta over 2, 1054 01:20:02,180 --> 01:20:05,850 e to the i phi. 1055 01:20:05,850 --> 01:20:08,000 It's nice, but it's [INAUDIBLE]. 1056 01:20:08,000 --> 01:20:11,965 And therefore, we got this state n plus, 1057 01:20:11,965 --> 01:20:19,420 which is supposed to be cosine theta over 2, z plus, 1058 01:20:19,420 --> 01:20:30,060 and plus sine theta over 2, e to the i phi, z minus. 1059 01:20:30,060 --> 01:20:31,365 This is a great result. 1060 01:20:34,470 --> 01:20:38,700 It gives the arbitrarily located spin 1061 01:20:38,700 --> 01:20:42,480 state that point in the n-direction. 1062 01:20:42,480 --> 01:20:46,030 As a linear superposition of your two basis states, 1063 01:20:46,030 --> 01:20:49,240 it answers conclusively the question 1064 01:20:49,240 --> 01:20:52,350 that any spin state in your system 1065 01:20:52,350 --> 01:20:56,860 can be represented in this two-dimensional vector space. 1066 01:20:56,860 --> 01:21:05,030 Now moreover, if I take that theta equals 0, 1067 01:21:05,030 --> 01:21:09,440 I have the z-axis, and it's independent of the angle phi. 1068 01:21:09,440 --> 01:21:12,850 The phi angle becomes singular at the North Pole, 1069 01:21:12,850 --> 01:21:14,160 but that's all right. 1070 01:21:14,160 --> 01:21:18,340 When theta is equal to 0, this term is 0 anyway. 1071 01:21:18,340 --> 01:21:21,300 And therefore, this goes, and when theta is equal to 0, 1072 01:21:21,300 --> 01:21:24,620 you recover the plus state. 1073 01:21:24,620 --> 01:21:27,495 Now you can calculate the minus state. 1074 01:21:30,430 --> 01:21:35,610 And if you follow exactly the same economical procedure, 1075 01:21:35,610 --> 01:21:38,930 you will get the following answer. 1076 01:21:38,930 --> 01:21:41,520 And I think, unless you've done a lot 1077 01:21:41,520 --> 01:21:44,280 of eigenvalue calculations, this is a calculation 1078 01:21:44,280 --> 01:21:47,390 you should just redo. 1079 01:21:47,390 --> 01:21:50,300 So the thing that you get, without thinking much, 1080 01:21:50,300 --> 01:21:58,940 is that n minus is equal to sine theta over 2 plus, minus 1081 01:21:58,940 --> 01:22:06,230 cosine theta over 2, e to the i phi minus. 1082 01:22:06,230 --> 01:22:09,980 At least some way of solving this equation gives you that. 1083 01:22:09,980 --> 01:22:12,970 You could say, this is natural, and this is fine. 1084 01:22:12,970 --> 01:22:17,730 But that is not so nice, actually. 1085 01:22:17,730 --> 01:22:38,870 Take theta equal to pi-- no, I'm sorry. 1086 01:22:38,870 --> 01:22:43,470 Again, you take theta equal to 0. 1087 01:22:43,470 --> 01:22:45,570 Theta equal to 0-- this is supposed 1088 01:22:45,570 --> 01:22:49,050 to be the minus state along the direction. 1089 01:22:49,050 --> 01:22:51,650 So this is supposed to give you the minus state. 1090 01:22:51,650 --> 01:22:56,030 Because the vector n is along up, in the z-direction, 1091 01:22:56,030 --> 01:22:58,050 but you're looking at the minus component. 1092 01:22:58,050 --> 01:23:00,300 So theta equals 0. 1093 01:23:00,300 --> 01:23:03,600 Sure, there's no plus, but theta equals 0, 1094 01:23:03,600 --> 01:23:06,340 and you get the minus state. 1095 01:23:06,340 --> 01:23:11,850 And this is 1, and phi is ill-defined-- it's not so nice, 1096 01:23:11,850 --> 01:23:14,790 therefore-- so, at this moment, it's 1097 01:23:14,790 --> 01:23:20,600 convenient to multiply this state by e to the minus i phi, 1098 01:23:20,600 --> 01:23:23,240 times minus 1. 1099 01:23:23,240 --> 01:23:27,830 Just multiply it by that, so that n minus 1100 01:23:27,830 --> 01:23:34,490 is equal to minus sine theta over 2, e to the minus i phi, 1101 01:23:34,490 --> 01:23:42,680 plus, plus cosine theta over 2, minus. 1102 01:23:42,680 --> 01:23:44,785 And that's a nice definition of the state. 1103 01:23:47,520 --> 01:23:53,130 When theta is equal to 0, you're fine, 1104 01:23:53,130 --> 01:23:57,510 and it's more naturally equivalent to what you know. 1105 01:23:57,510 --> 01:24:01,770 Theta equal to 0 gives you the minus state, or z minus. 1106 01:24:01,770 --> 01:24:04,770 I didn't put the zs here, for laziness. 1107 01:24:04,770 --> 01:24:08,510 And for theta equal to 0, the way the phase phi 1108 01:24:08,510 --> 01:24:09,460 doesn't matter. 1109 01:24:09,460 --> 01:24:10,960 So it's a little nicer. 1110 01:24:10,960 --> 01:24:15,430 You could work with this one, but you might this well 1111 01:24:15,430 --> 01:24:16,520 leave it like that. 1112 01:24:16,520 --> 01:24:22,000 So we have our general states, we've done everything here 1113 01:24:22,000 --> 01:24:26,450 that required some linear algebra without doing 1114 01:24:26,450 --> 01:24:28,560 a review of linear algebra, but that's 1115 01:24:28,560 --> 01:24:31,800 what we'll start to do next time.