1 00:00:00,500 --> 00:00:06,170 PROFESSOR: So I'm going to do today this identical particles. 2 00:00:06,170 --> 00:00:08,330 We'll continue with identical particles. 3 00:00:08,330 --> 00:00:12,020 We'll get almost to the resolution of the issue 4 00:00:12,020 --> 00:00:14,330 of exchange degeneracy. 5 00:00:14,330 --> 00:00:20,970 In order to do this identical particles properly, 6 00:00:20,970 --> 00:00:26,810 you have to do a bit of group theory or understanding 7 00:00:26,810 --> 00:00:30,090 what permutation operators are well. 8 00:00:30,090 --> 00:00:34,470 So it allows you to think about these matters more clearly. 9 00:00:34,470 --> 00:00:37,110 So that's what we have to do today. 10 00:00:37,110 --> 00:00:39,020 So let me review. 11 00:00:39,020 --> 00:00:41,480 So we were doing identical particles. 12 00:00:41,480 --> 00:00:46,130 And we spoke of exchange degeneracy. 13 00:00:46,130 --> 00:00:51,500 We had, say, for the state of two spins-- 14 00:00:51,500 --> 00:00:56,480 identical spins-- we said could be represented by this state 15 00:00:56,480 --> 00:01:05,300 or by this state in which you exchange the pluses 16 00:01:05,300 --> 00:01:05,925 and minuses. 17 00:01:08,550 --> 00:01:12,150 So the problem of exchange degeneracy 18 00:01:12,150 --> 00:01:14,520 is that, for identical particles, 19 00:01:14,520 --> 00:01:19,190 you would almost say that these two states are equivalent. 20 00:01:19,190 --> 00:01:23,030 One particle is in plus 1 particle, is in minus. 21 00:01:23,030 --> 00:01:24,240 They're identical. 22 00:01:24,240 --> 00:01:26,910 So you cannot tell whether it's the first or the second. 23 00:01:26,910 --> 00:01:30,770 So you would say these two states are equivalent. 24 00:01:30,770 --> 00:01:35,450 But we saw that that leads to a contradiction. 25 00:01:35,450 --> 00:01:38,630 If these two states are equivalent, 26 00:01:38,630 --> 00:01:42,140 any linear superposition of them should be equivalent. 27 00:01:42,140 --> 00:01:45,320 And then when we ask the question 28 00:01:45,320 --> 00:01:48,500 of the probability of finding these states 29 00:01:48,500 --> 00:01:53,630 in a particular state that was along the x direction-- both 30 00:01:53,630 --> 00:01:56,480 spins along the x direction-- we found 31 00:01:56,480 --> 00:02:00,620 that it depended on what linear combination you took here. 32 00:02:00,620 --> 00:02:04,220 So it's just not consistent with the principles 33 00:02:04,220 --> 00:02:06,860 of quantum mechanics to say that these two 34 00:02:06,860 --> 00:02:09,990 states are equivalent. 35 00:02:09,990 --> 00:02:11,570 So they're not. 36 00:02:11,570 --> 00:02:18,590 And we better understand how we can solve this problem. 37 00:02:18,590 --> 00:02:23,750 Now, I think the solution, in a sense, you've heard before. 38 00:02:23,750 --> 00:02:27,230 And you know how you're supposed to work 39 00:02:27,230 --> 00:02:28,700 with identical particles. 40 00:02:28,700 --> 00:02:33,110 But I think going through this in detail 41 00:02:33,110 --> 00:02:38,540 and seeing what is the structure of operators and Hilbert spaces 42 00:02:38,540 --> 00:02:40,820 that leads to that solution or why 43 00:02:40,820 --> 00:02:44,820 that solution is the unique solution to this problem, 44 00:02:44,820 --> 00:02:46,250 it's interesting. 45 00:02:46,250 --> 00:02:50,660 So we begin with permutation operators. 46 00:02:56,020 --> 00:02:56,760 Operators. 47 00:03:01,570 --> 00:03:05,980 So we consider a two-particle system. 48 00:03:05,980 --> 00:03:07,750 We'll begin with two particles. 49 00:03:07,750 --> 00:03:11,290 And then we'll eventually generalize to n particles. 50 00:03:11,290 --> 00:03:13,930 We'll do everything for two particles first. 51 00:03:13,930 --> 00:03:20,320 So the particles, they will live on a vector space, V, 52 00:03:20,320 --> 00:03:30,420 each one so that the set of two particles lives on V tensor V. 53 00:03:30,420 --> 00:03:34,080 So each particle has a set of possible states 54 00:03:34,080 --> 00:03:37,860 that are the vectors in V. And the two particles have 55 00:03:37,860 --> 00:03:43,410 a set of possible states that are the vectors in V tensor V. 56 00:03:43,410 --> 00:03:47,700 Now, for the moment, whether they're identical or not 57 00:03:47,700 --> 00:03:49,545 identical will play no role. 58 00:03:49,545 --> 00:03:53,220 It will play no role throughout this lecture, 59 00:03:53,220 --> 00:03:57,860 where we understand those operators in Hilbert spaces. 60 00:03:57,860 --> 00:04:01,600 So it will not matter. 61 00:04:01,600 --> 00:04:04,230 So the particles may be distinguishable, 62 00:04:04,230 --> 00:04:06,420 may not be distinguishable. 63 00:04:06,420 --> 00:04:16,829 And we will write states here in V tensor V, based as states, 64 00:04:16,829 --> 00:04:22,255 for example, as ui1 tensor u2-- 65 00:04:26,970 --> 00:04:27,750 I'm sorry, uj2. 66 00:04:30,580 --> 00:04:33,240 So with little i and little j, they're 67 00:04:33,240 --> 00:04:36,450 noting different integers. 68 00:04:36,450 --> 00:04:38,760 Those are different basis vectors. 69 00:04:38,760 --> 00:04:44,460 And this is a set of basis vectors in V tensor V-- 70 00:04:44,460 --> 00:04:53,630 so basis vectors for all i and j. 71 00:04:53,630 --> 00:04:57,350 Now, sometimes we get a little lazy, 72 00:04:57,350 --> 00:05:00,560 and we stop writing the 1 and 2. 73 00:05:00,560 --> 00:05:06,710 And we write it as ui tensor uj. 74 00:05:06,710 --> 00:05:13,160 Or even we can get more lazy and just write ui uj. 75 00:05:17,060 --> 00:05:21,950 So if we do that, you should know you really 76 00:05:21,950 --> 00:05:23,070 mean this whole thing. 77 00:05:26,320 --> 00:05:29,470 So is a permutation operator? 78 00:05:29,470 --> 00:05:35,140 It's a linear operator that does some funny thing on the state. 79 00:05:35,140 --> 00:05:36,610 So let's look into it. 80 00:05:36,610 --> 00:05:43,590 So a permutation operator, P2,1 That's a name I will give it. 81 00:05:43,590 --> 00:05:54,880 P2,1 is a linear operator acting on the space V tensor V-- 82 00:05:54,880 --> 00:05:59,010 so linear operator. 83 00:05:59,010 --> 00:06:01,500 And this is called the permutation operator. 84 00:06:01,500 --> 00:06:04,630 So to know what this permutation operator is, 85 00:06:04,630 --> 00:06:07,740 you need to know how it acts on a basis state. 86 00:06:07,740 --> 00:06:16,510 So let's put one of these basis states, ui1 tensor uj2. 87 00:06:20,478 --> 00:06:25,720 And this permutation operator, you just 88 00:06:25,720 --> 00:06:28,420 need to know how it acts in the basis state. 89 00:06:28,420 --> 00:06:33,370 That's the safest way always to know that you 90 00:06:33,370 --> 00:06:35,050 have a linear operator. 91 00:06:35,050 --> 00:06:37,010 If you know how it acts on a basis state, 92 00:06:37,010 --> 00:06:41,290 then on any sum of basis states, you know, because it's linear. 93 00:06:41,290 --> 00:06:43,750 On the other hand, if somebody tells you, 94 00:06:43,750 --> 00:06:47,090 this operator does this to all vectors-- 95 00:06:47,090 --> 00:06:52,060 something-- maybe it's not a linear operator. 96 00:06:52,060 --> 00:06:55,220 It may be a more complicated thing. 97 00:06:55,220 --> 00:06:58,630 So here, what it will do, it will 98 00:06:58,630 --> 00:07:03,400 put the state that was in 2 in position 1, 99 00:07:03,400 --> 00:07:08,350 and put the state that was in 1 in position 2. 100 00:07:08,350 --> 00:07:16,210 So it will take you j1 tensor ui2. 101 00:07:19,390 --> 00:07:21,640 So the state in 2-- 102 00:07:21,640 --> 00:07:24,550 this first thing that says that the state in 2-- 103 00:07:24,550 --> 00:07:26,800 should be put in position 1. 104 00:07:26,800 --> 00:07:28,400 So the j label-- 105 00:07:28,400 --> 00:07:29,320 it was uj. 106 00:07:29,320 --> 00:07:31,630 Now it's in position 1. 107 00:07:31,630 --> 00:07:35,350 And the state that was in 1 should go to position 2. 108 00:07:37,990 --> 00:07:42,050 The one thing we never do is flip these things. 109 00:07:42,050 --> 00:07:44,890 The first particle goes before the second particle. 110 00:07:44,890 --> 00:07:46,810 That's our order of things. 111 00:07:46,810 --> 00:07:50,390 But the state has moved. 112 00:07:50,390 --> 00:07:50,890 OK. 113 00:07:50,890 --> 00:07:56,420 So this is the prescription of what this does. 114 00:07:56,420 --> 00:07:59,920 And then you realize already one property-- 115 00:07:59,920 --> 00:08:07,540 that P2,1, if you let it act twice in a state, 116 00:08:07,540 --> 00:08:11,380 it will flip the ij the first time. 117 00:08:11,380 --> 00:08:14,240 And it will flip them again the second time. 118 00:08:14,240 --> 00:08:18,910 So this is the unit operator. 119 00:08:18,910 --> 00:08:20,860 So it's inverse. 120 00:08:24,720 --> 00:08:27,170 So that's a nice thing. 121 00:08:27,170 --> 00:08:31,490 Now, perhaps a little less obvious, 122 00:08:31,490 --> 00:08:38,059 but kind of still simple, is that P2,1 is Hermition. 123 00:08:48,890 --> 00:08:51,640 How do you check that an operator is Hermition? 124 00:08:51,640 --> 00:09:01,470 An operator, M, is hermission if you have, for example, M alpha 125 00:09:01,470 --> 00:09:05,880 beta is equal to alpha and beta. 126 00:09:08,530 --> 00:09:11,080 The M operator moves from this position 127 00:09:11,080 --> 00:09:12,950 to the other with an M dagger. 128 00:09:12,950 --> 00:09:15,840 But if M is Hermition, it will work like that. 129 00:09:19,280 --> 00:09:24,130 So let's try this with the operator P2,1. 130 00:09:24,130 --> 00:09:29,460 So we'll try a P2,1 on the state alpha, 131 00:09:29,460 --> 00:09:31,890 and then [INAUDIBLE] with a state beta. 132 00:09:31,890 --> 00:09:35,840 So we need a number of letters. 133 00:09:35,840 --> 00:09:37,030 So let's do it here. 134 00:09:37,030 --> 00:09:51,180 P2,1 on a state alpha, which will be ui1 uj2, 135 00:09:51,180 --> 00:09:58,130 [INAUDIBLE] with a state beta uk1 ul2. 136 00:10:00,550 --> 00:10:01,050 OK. 137 00:10:01,050 --> 00:10:02,180 Here it is. 138 00:10:02,180 --> 00:10:06,300 P2,1 acting on some state, alpha, 139 00:10:06,300 --> 00:10:08,410 [INAUDIBLE] with a state beta. 140 00:10:08,410 --> 00:10:10,640 We'll evaluate it. 141 00:10:10,640 --> 00:10:14,630 Well, we can never write it quickly, I think. 142 00:10:14,630 --> 00:10:19,770 This flips the j and i so the j is now 143 00:10:19,770 --> 00:10:23,230 on the first particle and the case in the first particle. 144 00:10:23,230 --> 00:10:28,250 So the inner product, assuming this is an orthonormal basis, 145 00:10:28,250 --> 00:10:30,560 is just delta jk. 146 00:10:35,190 --> 00:10:40,830 And here, this operator moves the i label to position 2. 147 00:10:40,830 --> 00:10:44,110 And here you have an l in position 2. 148 00:10:44,110 --> 00:10:46,050 Therefore, it's a delta il. 149 00:10:49,200 --> 00:10:57,305 On the other hand, we can calculate this ui1 uj2, 150 00:10:57,305 --> 00:11:05,730 and now put the P2,1 operator here between the uk1 151 00:11:05,730 --> 00:11:06,305 and the ul2. 152 00:11:14,360 --> 00:11:18,650 This time you flip the k and l, so the l ends up 153 00:11:18,650 --> 00:11:21,455 being the first Hilbert space, where there's an i. 154 00:11:21,455 --> 00:11:26,410 So you get delta li. 155 00:11:26,410 --> 00:11:29,720 And the key, ends up in the second Hilbert 156 00:11:29,720 --> 00:11:32,100 space, where you have a j here. 157 00:11:35,410 --> 00:11:37,990 So you have a j with a k. 158 00:11:40,690 --> 00:11:44,680 And these two are the same. 159 00:11:44,680 --> 00:11:48,730 And intuition-- of course, I've written all these equations. 160 00:11:48,730 --> 00:11:51,880 But intuition is, if you have an inner product-- 161 00:11:51,880 --> 00:11:56,210 this chronicle delta sign-- you exchange two here. 162 00:11:56,210 --> 00:11:59,950 The result is the same as if you exchange two here. 163 00:11:59,950 --> 00:12:01,700 It will do the same thing. 164 00:12:01,700 --> 00:12:05,760 So it's a Hermitian operator. 165 00:12:05,760 --> 00:12:10,360 So operator P2,1 is Hermitian. 166 00:12:10,360 --> 00:12:18,320 That means that the relation P2,1 P2,1 equal 1. 167 00:12:18,320 --> 00:12:20,330 Since this is Hermitian-- 168 00:12:20,330 --> 00:12:22,210 this P2,1 dagger-- 169 00:12:22,210 --> 00:12:24,950 P2,1 equal 1. 170 00:12:24,950 --> 00:12:28,450 So we learn that also the operator is unitary. 171 00:12:32,650 --> 00:12:37,840 P2,1 is unitary. 172 00:12:37,840 --> 00:12:41,320 That's nice, because that means that could 173 00:12:41,320 --> 00:12:47,230 be a symmetry in quantum mechanics. 174 00:12:47,230 --> 00:12:49,060 Operators in quantum mechanics that 175 00:12:49,060 --> 00:12:51,990 preserve the norm of states can be symmetries. 176 00:12:51,990 --> 00:12:54,580 And P2,1 is unitary. 177 00:12:59,530 --> 00:13:00,080 OK. 178 00:13:00,080 --> 00:13:05,040 So we have our operator P2,1. 179 00:13:05,040 --> 00:13:07,690 Let's do things with it. 180 00:13:07,690 --> 00:13:12,920 So we can define two things with P2,1. 181 00:13:12,920 --> 00:13:23,520 Define this operator, S, which will be 1/2 1 plus P2,1, 182 00:13:23,520 --> 00:13:32,930 and the operator, A, which is 1/2 1 minus P2,1. 183 00:13:32,930 --> 00:13:35,780 And here comes the claim. 184 00:13:35,780 --> 00:13:42,236 These two operators are orthogonal projectors. 185 00:13:42,236 --> 00:13:44,150 They're orthogonal projectors. 186 00:13:50,640 --> 00:13:51,520 Projectors. 187 00:13:55,220 --> 00:13:58,555 So let's review what that means. 188 00:14:05,020 --> 00:14:12,800 So basically, let me say what it means. 189 00:14:12,800 --> 00:14:18,020 An operator is an orthogonal projector. 190 00:14:18,020 --> 00:14:22,160 The test is that the operator squared is equal to itself. 191 00:14:22,160 --> 00:14:24,710 That's the name of a projector. 192 00:14:24,710 --> 00:14:28,880 And the other thing is that it must be Hermitian. 193 00:14:28,880 --> 00:14:31,400 These two things must happen for something 194 00:14:31,400 --> 00:14:35,330 to be an orthogonal projector. 195 00:14:35,330 --> 00:14:38,580 Basically, what does it mean to be an orthogonal projector? 196 00:14:38,580 --> 00:14:44,180 It's an operator that projects to some space. 197 00:14:44,180 --> 00:14:47,030 So an operator has a set of states 198 00:14:47,030 --> 00:14:50,930 that are in its image in the range of the operator. 199 00:14:50,930 --> 00:14:53,420 That's what it projects into. 200 00:14:53,420 --> 00:14:55,670 And it has a null space-- 201 00:14:55,670 --> 00:14:58,730 the things that are killed by the projector. 202 00:14:58,730 --> 00:15:01,700 A projector has precisely that kind of thing. 203 00:15:05,890 --> 00:15:09,310 So the thing that makes it an orthogonal projector is that 204 00:15:09,310 --> 00:15:13,270 the range of the projector-- the vectors that you get-- 205 00:15:13,270 --> 00:15:15,790 are orthogonal to the vectors that 206 00:15:15,790 --> 00:15:17,650 are killed by the projector. 207 00:15:20,920 --> 00:15:24,310 The whole vector space splits into two parts-- 208 00:15:24,310 --> 00:15:27,340 what do you get from the projector and the rest. 209 00:15:27,340 --> 00:15:31,210 And the rest is things that are killed by the projector. 210 00:15:31,210 --> 00:15:33,070 And they're orthogonal to it. 211 00:15:33,070 --> 00:15:35,590 So let me say it here. 212 00:15:35,590 --> 00:15:50,540 A projector P sub U to U subset of V is orthogonal-- 213 00:15:50,540 --> 00:15:53,900 so this is 805 type stuff-- 214 00:15:53,900 --> 00:15:58,040 if the vector space can be written 215 00:15:58,040 --> 00:16:03,680 as the null space of PU. 216 00:16:03,680 --> 00:16:18,880 Plus the range of PU with null or PU 217 00:16:18,880 --> 00:16:25,653 perpendicular, or orthogonal, to range of PU. 218 00:16:30,290 --> 00:16:35,870 So this is the definition of an orthogonal projector. 219 00:16:35,870 --> 00:16:39,260 And here is, of course, a simple claim. 220 00:16:39,260 --> 00:16:56,370 A Hermitian operator, P such that P squared is equal to P, 221 00:16:56,370 --> 00:16:59,610 is an orthogonal projector. 222 00:17:03,090 --> 00:17:03,900 Projector. 223 00:17:07,391 --> 00:17:07,890 OK. 224 00:17:07,890 --> 00:17:14,240 So that's second claim. 225 00:17:14,240 --> 00:17:16,069 So let's see again. 226 00:17:16,069 --> 00:17:19,339 What we want to show is that these are projectors, 227 00:17:19,339 --> 00:17:21,330 and they're orthogonal projectors. 228 00:17:24,619 --> 00:17:26,240 And this is what it means. 229 00:17:26,240 --> 00:17:29,570 But practically speaking, all we have to do 230 00:17:29,570 --> 00:17:31,930 is check that this operator satisfies 231 00:17:31,930 --> 00:17:36,250 this and our Hermitian. 232 00:17:36,250 --> 00:17:37,730 That's all you need to do. 233 00:17:37,730 --> 00:17:42,350 Let me explain why this is the case. 234 00:17:42,350 --> 00:17:45,070 So whenever you have-- 235 00:17:45,070 --> 00:17:48,620 let's do a little proof. 236 00:17:48,620 --> 00:17:51,740 So if you have a vector, a vector 237 00:17:51,740 --> 00:17:57,260 can be written as the projector times the vector plus 1 238 00:17:57,260 --> 00:18:00,215 minus the projector acting on the vector. 239 00:18:03,970 --> 00:18:08,830 This-- the projector on the vector-- 240 00:18:08,830 --> 00:18:14,595 is in the range of P, because the range of P 241 00:18:14,595 --> 00:18:23,380 are the vectors obtained by acting with P. Moreover, 242 00:18:23,380 --> 00:18:29,530 this is in the null of P. Why? 243 00:18:29,530 --> 00:18:32,170 Because P kills these vectors. 244 00:18:32,170 --> 00:18:33,530 How do you check that? 245 00:18:33,530 --> 00:18:38,950 P, acting on that vector-- 246 00:18:38,950 --> 00:18:46,930 P with this is P. But P squared is equal to P. So you get 0. 247 00:18:46,930 --> 00:18:48,940 So here it is. 248 00:18:48,940 --> 00:18:56,290 The vector space decomposes into the range of P plus the null 249 00:18:56,290 --> 00:19:03,340 of P. So that's the first claim for a orthogonal projector. 250 00:19:03,340 --> 00:19:06,370 The second claim is that these are orthogonal. 251 00:19:06,370 --> 00:19:10,510 So if I take PV-- 252 00:19:10,510 --> 00:19:14,650 something in the range of P-- 253 00:19:14,650 --> 00:19:20,290 and 1 minus PV tilde-- 254 00:19:20,290 --> 00:19:25,330 something in the null space of P-- 255 00:19:25,330 --> 00:19:28,930 this should be orthogonal. 256 00:19:28,930 --> 00:19:31,660 And this is clear, because I can move now 257 00:19:31,660 --> 00:19:35,220 this operator to the other side. 258 00:19:35,220 --> 00:19:43,600 And it will be V P dagger 1 minus P V tilde. 259 00:19:43,600 --> 00:19:47,780 And since P is Hermitian-- that's where you use that P is 260 00:19:47,780 --> 00:19:49,620 Hermitian-- 261 00:19:49,620 --> 00:19:56,600 this is V P1 minus P V tilde. 262 00:19:56,600 --> 00:20:01,720 And that's equal to 0, because P times 1 minus P already is 0. 263 00:20:04,306 --> 00:20:07,600 So OK. 264 00:20:10,990 --> 00:20:15,580 These are our conditions. 265 00:20:15,580 --> 00:20:18,190 Orthogonal projectors are very nice. 266 00:20:18,190 --> 00:20:23,560 These are the really good things. 267 00:20:23,560 --> 00:20:29,470 You have a projector that decomposes into two parts, 268 00:20:29,470 --> 00:20:31,930 just by having P squared equal P. 269 00:20:31,930 --> 00:20:36,370 But to have orthogonal, you need that they'd be Hermitian. 270 00:20:36,370 --> 00:20:41,680 So what is our situation with those operators? 271 00:20:41,680 --> 00:20:43,115 They are Hermitian. 272 00:20:46,340 --> 00:20:57,460 So S and A are Hermitian, because one is Hermitian, 273 00:20:57,460 --> 00:21:01,600 and P2,1 is Hermitian. 274 00:21:01,600 --> 00:21:04,750 So the operators are Hermitian. 275 00:21:04,750 --> 00:21:09,220 And they are projectors, because, well, you check them. 276 00:21:09,220 --> 00:21:11,780 You multiply them by themselves. 277 00:21:11,780 --> 00:21:15,170 So we can do the two of them simultaneously. 278 00:21:15,170 --> 00:21:16,940 This is with a plus sign. 279 00:21:16,940 --> 00:21:20,260 That's S. With a minus sign, it's A. 280 00:21:20,260 --> 00:21:24,940 So we need to show that S squared is equal to S 281 00:21:24,940 --> 00:21:31,915 and A squared is equal to A. So here we have both cases. 282 00:21:35,600 --> 00:21:45,020 So the product is 1/4, then plus minus 2 P2,1, 283 00:21:45,020 --> 00:21:48,860 and then the product of these operators with themselves. 284 00:21:48,860 --> 00:21:52,390 Plus minus and plus minus always gives you a plus. 285 00:21:52,390 --> 00:21:59,860 And P2,1 times P2,1, we showed that that's equal to 1. 286 00:21:59,860 --> 00:22:02,770 So happily, this 1 and 1 give a 2. 287 00:22:02,770 --> 00:22:03,760 There's another 2. 288 00:22:03,760 --> 00:22:09,850 Divides here-- this is 1/2 1 plus minus P2,1. 289 00:22:09,850 --> 00:22:14,530 So that's what we wanted to show with a top sign, 290 00:22:14,530 --> 00:22:19,050 S times S is equal to S, with a bottom sign, 291 00:22:19,050 --> 00:22:22,240 A times A is equal to A.