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 --> 00:00:10,120 to offer high-quality educational resources for free. 5 00:00:10,120 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,275 at ocw.mit.edu. 8 00:00:21,400 --> 00:00:27,970 PROFESSOR: Now, a theorem that was quite powerful and applied 9 00:00:27,970 --> 00:00:33,060 to complex vector spaces for old V longing 10 00:00:33,060 --> 00:00:40,560 to V complex vector space. 11 00:00:40,560 --> 00:00:46,340 This implied that the operator was zero, 12 00:00:46,340 --> 00:00:51,010 and it's not true for real vector spaces. 13 00:00:51,010 --> 00:00:54,620 And we gave a proof that made it clear that indeed, 14 00:00:54,620 --> 00:00:59,970 a proof wouldn't work for a real vector space. 15 00:00:59,970 --> 00:01:04,489 I will ask you in the homework to do something that presumably 16 00:01:04,489 --> 00:01:07,020 would be the first thing you would do if you had 17 00:01:07,020 --> 00:01:11,800 to try to understand why this is true-- take two by two 18 00:01:11,800 --> 00:01:16,420 matrices, and just see why it has to be 0 in one case, 19 00:01:16,420 --> 00:01:19,410 and it doesn't have to be 0 in the other case. 20 00:01:19,410 --> 00:01:22,140 And I think that will give you a better 21 00:01:22,140 --> 00:01:25,210 perspective on why this happens. 22 00:01:25,210 --> 00:01:27,700 And then once you do it for a two by two, 23 00:01:27,700 --> 00:01:32,600 and you see how it works, you can do it for n by n matrices, 24 00:01:32,600 --> 00:01:34,250 and it will be clear as well. 25 00:01:34,250 --> 00:01:37,220 So we'll use this theorem. 26 00:01:37,220 --> 00:01:42,320 Our immediate application of this theorem 27 00:01:42,320 --> 00:01:50,150 was a well-known result that we could prove now 28 00:01:50,150 --> 00:01:55,680 rigorously-- that t is equal to t dagger, which 29 00:01:55,680 --> 00:01:59,200 is to say the operator is equal to the adjoined. 30 00:01:59,200 --> 00:02:03,740 And I think I will call this-- and in the notes 31 00:02:03,740 --> 00:02:07,570 you will see always the adjoint, as opposed to Hermitian 32 00:02:07,570 --> 00:02:08,780 congregate. 33 00:02:08,780 --> 00:02:10,789 And I will say whenever the operator 34 00:02:10,789 --> 00:02:14,860 is equal to the adjoint, that it is Hermitian. 35 00:02:14,860 --> 00:02:19,150 So a Hermitian operator is equivalent to saying 36 00:02:19,150 --> 00:02:29,480 that v Tv is a real number for all v. And we proved that. 37 00:02:34,760 --> 00:02:39,450 In other words, physically, Hermitian operators 38 00:02:39,450 --> 00:02:43,320 have real expectation values. 39 00:02:43,320 --> 00:02:47,420 This is an expectation value, because-- as you remember 40 00:02:47,420 --> 00:02:51,480 in the bracket notation-- v Tv, you 41 00:02:51,480 --> 00:02:57,230 can write it v T v-- the same thing. 42 00:02:57,230 --> 00:03:01,480 So it is an expectation value, so it's an important thing, 43 00:03:01,480 --> 00:03:04,890 because we usually deal with Hermitian operators, 44 00:03:04,890 --> 00:03:08,240 and we want expectation values of Hermitian operators 45 00:03:08,240 --> 00:03:10,770 to be real. 46 00:03:10,770 --> 00:03:13,770 Now that we're talking about Hermitian operators, 47 00:03:13,770 --> 00:03:20,360 I delay a complete discussion of diagonalization, 48 00:03:20,360 --> 00:03:23,170 and diagonalization of several operators 49 00:03:23,170 --> 00:03:26,440 simultaneously, for the next lecture. 50 00:03:26,440 --> 00:03:31,600 Today, I want to move forward a bit and do some other things. 51 00:03:31,600 --> 00:03:36,310 And this way we spread out a little more the math, 52 00:03:36,310 --> 00:03:41,070 and we can begin to look more at physical issues, 53 00:03:41,070 --> 00:03:42,720 and how they apply here. 54 00:03:42,720 --> 00:03:46,750 But at any rate, we're just here already, 55 00:03:46,750 --> 00:03:50,200 and we can prove two basic things that 56 00:03:50,200 --> 00:03:55,010 are the kinds of things that an 805 student should 57 00:03:55,010 --> 00:03:57,630 be able to prove at any time. 58 00:03:57,630 --> 00:03:59,960 They're really very simple. 59 00:03:59,960 --> 00:04:05,020 And it's a kind of proof that is very short-- couple of lines-- 60 00:04:05,020 --> 00:04:08,760 something you should be able to reproduce at any time. 61 00:04:08,760 --> 00:04:16,070 So the first theorem says the eigenvalues 62 00:04:16,070 --> 00:04:22,600 of Hermitian operators-- H is for Hermitian-- are real. 63 00:04:25,430 --> 00:04:34,080 And I will do a little bit of notation, in which I will start 64 00:04:34,080 --> 00:04:39,030 with an expression, and evaluate it sort of to the left 65 00:04:39,030 --> 00:04:40,130 and to the right. 66 00:04:40,130 --> 00:04:42,820 So when you have an equation, you start here, 67 00:04:42,820 --> 00:04:44,380 and you start evaluating there. 68 00:04:44,380 --> 00:04:51,270 So I will start with this-- consider v Tv. 69 00:04:51,270 --> 00:04:55,040 And I will box it as being the origin, 70 00:04:55,040 --> 00:04:57,830 and I will start evaluating it. 71 00:04:57,830 --> 00:05:12,230 Now if v is an eigenvector-- so let v be an eigenvector so 72 00:05:12,230 --> 00:05:17,530 that Tv is equal to lambda v. 73 00:05:17,530 --> 00:05:22,830 And now we say consider that expression in the box. 74 00:05:22,830 --> 00:05:24,590 And you try to evaluate it. 75 00:05:24,590 --> 00:05:27,600 So one way to evaluate it-- I evaluate it to the left, 76 00:05:27,600 --> 00:05:29,320 and then evaluate to the right. 77 00:05:29,320 --> 00:05:32,830 Is the naive evaluation-- t on v is lambda v, 78 00:05:32,830 --> 00:05:34,940 so substitute it there. 79 00:05:34,940 --> 00:05:38,690 v, lambda v-- we know it. 80 00:05:38,690 --> 00:05:42,740 And then by homogeneity, this lambda goes out, 81 00:05:42,740 --> 00:05:48,230 and therefore it's lambda v v. 82 00:05:48,230 --> 00:05:55,730 On the other hand, we have that we can go to the right. 83 00:05:55,730 --> 00:06:00,970 And what is the way you move an operator to the first position? 84 00:06:00,970 --> 00:06:02,580 By putting a dagger. 85 00:06:02,580 --> 00:06:04,480 That's a definition. 86 00:06:04,480 --> 00:06:05,610 So this is by definition. 87 00:06:10,140 --> 00:06:15,190 Now, we use that the operator is Hermitian. 88 00:06:15,190 --> 00:06:26,700 So this is equal to Tv v. And this is by T Hermitian. 89 00:06:26,700 --> 00:06:30,960 Then you can apply again the equation of the eigenvalues, 90 00:06:30,960 --> 00:06:39,220 so this is lambda v v. And by conjugate homogeneity 91 00:06:39,220 --> 00:06:46,760 of the first input, this is lambda star v v. 92 00:06:46,760 --> 00:06:48,550 So at the end of the day, you have 93 00:06:48,550 --> 00:06:52,300 something on the extreme left, and something 94 00:06:52,300 --> 00:06:53,405 on the extreme right. 95 00:06:57,240 --> 00:07:00,230 v-- if there is an eigenvector, v 96 00:07:00,230 --> 00:07:03,810 can be assumed to be non-zero. 97 00:07:03,810 --> 00:07:06,250 The way we are saying things in a sense, 98 00:07:06,250 --> 00:07:09,340 0-- we also think of it as an eigenvector, 99 00:07:09,340 --> 00:07:11,700 but it's a trivial one. 100 00:07:11,700 --> 00:07:14,690 But the fact that there's an eigenvalue 101 00:07:14,690 --> 00:07:17,570 means there's a non-zero v that solves this equation. 102 00:07:17,570 --> 00:07:21,000 So we're using that non-zero v. And therefore, this 103 00:07:21,000 --> 00:07:24,670 is a number that is non-zero. 104 00:07:24,670 --> 00:07:28,510 You bring one to the other side, and you 105 00:07:28,510 --> 00:07:35,640 have lambda minus lambda star times v v equals 0. 106 00:07:35,640 --> 00:07:37,490 This is different from 0. 107 00:07:40,140 --> 00:07:42,480 This is different from 0. 108 00:07:42,480 --> 00:07:45,340 And therefore, lambda is equal to lambda star. 109 00:07:49,030 --> 00:07:57,170 So it's a classic proof-- relatively straightforward. 110 00:07:57,170 --> 00:08:02,190 The second theorem is as simple to prove. 111 00:08:02,190 --> 00:08:05,300 And it's already interesting. 112 00:08:05,300 --> 00:08:09,660 And it states that different eigenvectors of a Hermitian 113 00:08:09,660 --> 00:08:14,490 operator-- well, different eigenvalues 114 00:08:14,490 --> 00:08:16,380 of Hermitian operators correspond 115 00:08:16,380 --> 00:08:19,630 to orthogonal eigenfunctions, or eigenvectors. 116 00:08:19,630 --> 00:08:35,140 So different eigenvalues of Hermitian ops correspond 117 00:08:35,140 --> 00:08:43,080 to orthogonal eigenfunctions-- eigenvectors, I'm sorry. 118 00:08:47,510 --> 00:08:50,080 So what are we saying here? 119 00:08:50,080 --> 00:08:55,340 We're saying that suppose you have a v1 120 00:08:55,340 --> 00:08:59,480 that [INAUDIBLE] T gives you a lambda1 v1. 121 00:08:59,480 --> 00:09:02,160 That's one eigenvalue. 122 00:09:02,160 --> 00:09:08,110 You have another one-- v2 is equal to lambda 2 v2, 123 00:09:08,110 --> 00:09:11,430 and lambda 1 is different from lambda 2. 124 00:09:14,960 --> 00:09:19,990 Now, just focusing on a fact that is going to show up later, 125 00:09:19,990 --> 00:09:24,630 is going to make life interesting, 126 00:09:24,630 --> 00:09:28,460 is that some eigenvalues may have 127 00:09:28,460 --> 00:09:32,180 a multiplicity of eigenvectors. 128 00:09:32,180 --> 00:09:37,190 In other words, If a vector v is an eigenvector, 129 00:09:37,190 --> 00:09:40,530 minus v is an eigenvector, square root of three v 130 00:09:40,530 --> 00:09:45,240 is an eigenvector, but that's a one-dimensional subspace. 131 00:09:45,240 --> 00:09:48,180 But sometimes for a given eigenvalue, 132 00:09:48,180 --> 00:09:50,510 there may be a higher dimensional 133 00:09:50,510 --> 00:09:53,850 subspace of eigenvectors. 134 00:09:53,850 --> 00:09:57,970 That's a problem of degeneracy, and it's very interesting-- 135 00:09:57,970 --> 00:10:00,840 makes life really interesting in quantum mechanics. 136 00:10:00,840 --> 00:10:05,800 So if you have degeneracy, and that set of eigenvectors 137 00:10:05,800 --> 00:10:08,950 form a subspace, and you can choose a basis, 138 00:10:08,950 --> 00:10:12,850 and you could have several vectors here. 139 00:10:12,850 --> 00:10:14,630 Now what do you do in that case? 140 00:10:14,630 --> 00:10:18,880 The theorem doesn't say much, so it means choose any one. 141 00:10:18,880 --> 00:10:21,790 If you had the bases there, choose any one. 142 00:10:21,790 --> 00:10:26,490 The fact remains that if these two eigenvalues are different, 143 00:10:26,490 --> 00:10:30,900 then you will be able to show that the eigenvectors are 144 00:10:30,900 --> 00:10:32,220 orthogonal. 145 00:10:32,220 --> 00:10:39,530 So if you have some space of eigenvectors-- 146 00:10:39,530 --> 00:10:42,320 a [INAUDIBLE] higher-dimensional space of eigenvectors, 147 00:10:42,320 --> 00:10:45,700 one eigenvalue, and another space with another-- 148 00:10:45,700 --> 00:10:50,420 any vector here is orthogonal to any vector there. 149 00:10:50,420 --> 00:10:52,055 So how do you show this? 150 00:10:54,710 --> 00:10:56,150 How do you show this property? 151 00:10:56,150 --> 00:11:01,420 Well you have to involve v1 and v2, 152 00:11:01,420 --> 00:11:05,950 so you're never going to be using the property that 153 00:11:05,950 --> 00:11:08,950 gives Hermitian, unless you have an inner product. 154 00:11:08,950 --> 00:11:12,240 So if you don't have any idea how to prove that, 155 00:11:12,240 --> 00:11:15,810 you presumably at some stage realize 156 00:11:15,810 --> 00:11:18,720 that you probably have to use an inner product. 157 00:11:18,720 --> 00:11:21,760 And we should mix the vectors, so maybe 158 00:11:21,760 --> 00:11:25,100 a V2 inner product with this. 159 00:11:25,100 --> 00:11:30,295 So we'll take a v2 inner product with T v1. 160 00:11:33,050 --> 00:11:37,260 And this is interesting, because we can use it, 161 00:11:37,260 --> 00:11:44,960 that T v1 is lambda 1 v1 to show that this is just lambda 1 162 00:11:44,960 --> 00:11:45,495 v2 v1. 163 00:11:48,700 --> 00:11:51,900 And that already brings all kinds of good things. 164 00:11:51,900 --> 00:11:54,110 You're interested in this inner product. 165 00:11:54,110 --> 00:11:56,420 You want to show it's 0, so it shows up. 166 00:11:56,420 --> 00:11:57,640 So it's a good idea. 167 00:11:57,640 --> 00:12:00,440 So we have evaluated this, and now you 168 00:12:00,440 --> 00:12:04,700 have to think of evaluating it in a different way. 169 00:12:04,700 --> 00:12:07,040 Again, the operator is Hermitian, 170 00:12:07,040 --> 00:12:10,120 so it's asking you to move it to the other side 171 00:12:10,120 --> 00:12:12,200 and exploit to that. 172 00:12:12,200 --> 00:12:14,810 So we'll move it to the other side a little quicker 173 00:12:14,810 --> 00:12:15,430 this time. 174 00:12:15,430 --> 00:12:18,720 It goes as T dagger, but T dagger is equal to T, 175 00:12:18,720 --> 00:12:20,380 because it's Hermitian. 176 00:12:20,380 --> 00:12:24,230 So this is the center of the equation. 177 00:12:24,230 --> 00:12:25,140 We go one way. 178 00:12:25,140 --> 00:12:27,880 We go the other way-- this time down. 179 00:12:27,880 --> 00:12:37,100 So we'll put T v2 v1, and this is equal to lambda-- 180 00:12:37,100 --> 00:12:45,520 let me go a little slow here-- lambda 2 v2 v1. 181 00:12:45,520 --> 00:12:49,930 Your impulse should be it goes out as lambda 2 star, 182 00:12:49,930 --> 00:12:52,980 but the eigenvalues are already real, 183 00:12:52,980 --> 00:13:05,210 so it goes out as lambda 2 v2 v1, 184 00:13:05,210 --> 00:13:07,640 because the operator is Hermitian. 185 00:13:07,640 --> 00:13:10,440 So at this moment, you have these two equations. 186 00:13:10,440 --> 00:13:13,800 You bring, say, this to the right-hand side, 187 00:13:13,800 --> 00:13:23,670 and you get lambda 1 minus lambda 2 v1 v2 is equal to 0. 188 00:13:23,670 --> 00:13:26,780 And since the eigenvalues are supposed to be different, 189 00:13:26,780 --> 00:13:33,810 you conclude that v1 inner product with v2 is 0. 190 00:13:33,810 --> 00:13:35,405 So that's the end of the proof. 191 00:13:46,240 --> 00:13:48,160 And those are the two properties that 192 00:13:48,160 --> 00:13:56,660 are very quickly proven with rather little effort. 193 00:13:56,660 --> 00:14:02,240 So where do we go from now? 194 00:14:02,240 --> 00:14:05,750 Well there's one more class of operators 195 00:14:05,750 --> 00:14:08,320 that are crucial in the physics. 196 00:14:08,320 --> 00:14:13,280 They are perhaps as important as the Hermitian operators, 197 00:14:13,280 --> 00:14:15,070 if not more. 198 00:14:15,070 --> 00:14:20,350 They are some operators that are called unitary operators, 199 00:14:20,350 --> 00:14:22,580 and the way I will introduce them 200 00:14:22,580 --> 00:14:35,130 is as follows-- so I will say-- it's an economical way 201 00:14:35,130 --> 00:14:39,523 to introduce them-- so we'll talk about unitary operators. 202 00:14:45,700 --> 00:14:54,210 If S is unitary, and mathematicians 203 00:14:54,210 --> 00:15:07,180 call it anisometry-- if you find that S acting on any vector-- 204 00:15:07,180 --> 00:15:12,940 if you take the norm, it's equal to the norm 205 00:15:12,940 --> 00:15:17,820 of the vector for all u in the vector space. 206 00:15:24,140 --> 00:15:32,600 So let's follow this, and make a couple of comments. 207 00:15:32,600 --> 00:15:38,740 An example-- a trivial example-- this operator lambda 208 00:15:38,740 --> 00:15:39,560 times the identity. 209 00:15:42,960 --> 00:15:45,770 Lambda times the identity acts on vectors. 210 00:15:45,770 --> 00:15:47,810 What does it do, lambda times identity? 211 00:15:47,810 --> 00:15:50,040 The identity does nothing on the vector, 212 00:15:50,040 --> 00:15:52,800 and lambda stretches it. 213 00:15:52,800 --> 00:16:00,170 So lambda, in order not to change the length of any vector 214 00:16:00,170 --> 00:16:02,240 should be kind of 1. 215 00:16:02,240 --> 00:16:09,100 Well, in fact, it suffices-- it's unitary-- 216 00:16:09,100 --> 00:16:14,430 if the absolute value of lambda is equal to 1. 217 00:16:14,430 --> 00:16:19,400 Because then lambda is a phase, and it just rotates the vector. 218 00:16:19,400 --> 00:16:24,110 Or in other words, you know that the norm of av 219 00:16:24,110 --> 00:16:28,630 is equal to the absolute value of a times 220 00:16:28,630 --> 00:16:31,550 the norm of v, where this is a number. 221 00:16:34,920 --> 00:16:37,480 And remember these two norms are different. 222 00:16:37,480 --> 00:16:39,210 This is the norm of a vector. 223 00:16:39,210 --> 00:16:44,210 This is the normal of a complex number. 224 00:16:44,210 --> 00:16:51,090 And therefore, if you take lambda i u-- 225 00:16:51,090 --> 00:16:58,260 norm-- is lambda u is equal absolute value of lambda u, 226 00:16:58,260 --> 00:17:02,400 and absolute value of lambda is equal to 1 is the answer. 227 00:17:02,400 --> 00:17:09,599 So that's a simple unitary operator, but an important one. 228 00:17:09,599 --> 00:17:15,180 Another observation-- what are the vectors 229 00:17:15,180 --> 00:17:17,670 annihilated by this operator u? 230 00:17:24,150 --> 00:17:27,540 Zero-- it's the only vector, because any other vector that's 231 00:17:27,540 --> 00:17:30,570 nonzero has some length, so it's not killed. 232 00:17:30,570 --> 00:17:33,510 So it kills only zero. 233 00:17:33,510 --> 00:17:42,790 So the null space of S is equal to the 0 vector. 234 00:17:42,790 --> 00:17:48,480 So this operator has no kernel, nothing nontrivial 235 00:17:48,480 --> 00:17:49,820 is put to zero. 236 00:17:49,820 --> 00:17:52,740 It's an invertible operator. 237 00:17:52,740 --> 00:17:53,870 So s is invertible. 238 00:18:00,970 --> 00:18:05,680 So that's a few things that you get very cheaply. 239 00:18:05,680 --> 00:18:09,980 Now from this equation, S u equals 240 00:18:09,980 --> 00:18:12,270 u-- if you square that equation, you 241 00:18:12,270 --> 00:18:21,060 would have S u S u is equal to u u. 242 00:18:21,060 --> 00:18:22,700 Maybe I should probably call it v. 243 00:18:22,700 --> 00:18:26,890 I don't know why I called it u, but let's stick to u. 244 00:18:26,890 --> 00:18:31,980 Now, remember that we can move operators 245 00:18:31,980 --> 00:18:33,690 from one side to the other. 246 00:18:33,690 --> 00:18:36,970 So I'll move this one to that side. 247 00:18:36,970 --> 00:18:40,580 If you move an S here, you would put an S dagger. 248 00:18:40,580 --> 00:18:46,290 But since the dagger of an S dagger is S, 249 00:18:46,290 --> 00:18:50,860 you can move also the S to that side as S dagger. 250 00:18:50,860 --> 00:18:56,920 So u S dagger, S u-- you see that. 251 00:18:56,920 --> 00:18:58,940 If you want to move this one, you 252 00:18:58,940 --> 00:19:03,130 can move it by putting another dagger, and you get that one. 253 00:19:03,130 --> 00:19:07,910 And this is u u, and therefore you 254 00:19:07,910 --> 00:19:16,810 get u S dagger, S minus the identity acting on u 255 00:19:16,810 --> 00:19:19,210 is equal to 0 for all u. 256 00:19:22,210 --> 00:19:26,500 So for every vector, this is true, because this is true. 257 00:19:26,500 --> 00:19:29,200 We just squared it. 258 00:19:29,200 --> 00:19:32,100 And now you have our favorite theorem, 259 00:19:32,100 --> 00:19:35,020 that says if this is true in a complex vector space, 260 00:19:35,020 --> 00:19:37,830 this is 0, and therefore, you've shown 261 00:19:37,830 --> 00:19:41,035 that S dagger S is equal to 1. 262 00:19:44,680 --> 00:19:49,050 So that's another property of unitary operators. 263 00:19:49,050 --> 00:19:52,830 In fact that's the way it's many times defined. 264 00:19:52,830 --> 00:19:55,360 Unitary operators sometimes are said 265 00:19:55,360 --> 00:20:00,590 to be operators whose inverse is S dagger. 266 00:20:00,590 --> 00:20:02,710 I will not go into the subtleties 267 00:20:02,710 --> 00:20:07,270 of what steps in all these things I'm saying 268 00:20:07,270 --> 00:20:11,260 are true or not true for infinite dimensional 269 00:20:11,260 --> 00:20:15,340 operators-- infinite dimensional vector spaces. 270 00:20:15,340 --> 00:20:19,070 So I will assume, and it will be true in our examples, 271 00:20:19,070 --> 00:20:22,170 that if S dagger is an inverse from the left, 272 00:20:22,170 --> 00:20:23,880 it's also an inverse from the right. 273 00:20:23,880 --> 00:20:30,420 And perhaps everything is true for infinite dimensional vector 274 00:20:30,420 --> 00:20:35,240 spaces, but I'm not 100% positive. 275 00:20:35,240 --> 00:20:50,580 So S dagger is the inverse of S. And that's 276 00:20:50,580 --> 00:20:54,920 a pretty important thing. 277 00:20:54,920 --> 00:21:04,160 So one last comment on unitary operators has to do with basis. 278 00:21:04,160 --> 00:21:13,980 So suppose you have an orthonormal basis, e1 up to en. 279 00:21:17,880 --> 00:21:20,015 Now you can define another basis. 280 00:21:28,280 --> 00:21:33,550 fi equal-- I'll change to a letter 281 00:21:33,550 --> 00:21:39,910 U-- U e i where U is unitary, so it's 282 00:21:39,910 --> 00:21:45,720 like the S. In fact, most books in physics 283 00:21:45,720 --> 00:21:47,450 call it U for unitary. 284 00:21:47,450 --> 00:21:50,210 So maybe I should have changed that letter in there, too, 285 00:21:50,210 --> 00:21:51,340 as well. 286 00:21:51,340 --> 00:21:55,190 So suppose you change basis. 287 00:21:55,190 --> 00:22:02,580 You put-- oh, there was something else 288 00:22:02,580 --> 00:22:05,930 I wanted to say before. 289 00:22:05,930 --> 00:22:09,170 Thanks to this equation, consider now 290 00:22:09,170 --> 00:22:14,830 the following thing-- S U Sv. 291 00:22:19,530 --> 00:22:24,040 SUSv-- you can move this S, for example, 292 00:22:24,040 --> 00:22:30,370 to the other side-- S dagger S U v, 293 00:22:30,370 --> 00:22:34,160 and S dagger S is equal to 1, and it's Uv. 294 00:22:34,160 --> 00:22:36,770 So this is a pretty nice property. 295 00:22:36,770 --> 00:22:41,030 We started from the fact that it preserved 296 00:22:41,030 --> 00:22:45,200 the norm of a single vector, of all vectors, 297 00:22:45,200 --> 00:22:49,670 and now you see that in fact, it preserved the inner product. 298 00:22:49,670 --> 00:22:53,160 So if you have two vectors, to compare their inner product, 299 00:22:53,160 --> 00:22:57,700 compute them after action with U or before action with U, 300 00:22:57,700 --> 00:23:00,620 and it doesn't make a difference. 301 00:23:00,620 --> 00:23:05,670 So suppose you define a second basis here. 302 00:23:05,670 --> 00:23:07,630 You have one orthonormal basis. 303 00:23:07,630 --> 00:23:10,980 You define basis vectors like this. 304 00:23:10,980 --> 00:23:21,660 Then the claim is that the f1 up to fn is orthonormal. 305 00:23:21,660 --> 00:23:26,480 And for that you simply do the following-- you just 306 00:23:26,480 --> 00:23:36,840 check f i f j is equal to U e i, U e j. 307 00:23:36,840 --> 00:23:40,540 By this property, you can delete both U's, rules, 308 00:23:40,540 --> 00:23:45,550 and therefore this is e i, e j. 309 00:23:45,550 --> 00:23:48,800 And that's delta i j. 310 00:23:48,800 --> 00:23:50,920 So the new basis is orthonormal. 311 00:23:56,400 --> 00:23:59,480 If you play with these things, it's 312 00:23:59,480 --> 00:24:06,460 easy to get some extra curious fact here. 313 00:24:06,460 --> 00:24:15,150 Let's think of the matrix representation of the operator 314 00:24:15,150 --> 00:24:24,310 U. Well, we know how these things 315 00:24:24,310 --> 00:24:28,770 are, and let's think of this in the basis e basis. 316 00:24:28,770 --> 00:24:37,480 So U k i is equal to ek U e i. 317 00:24:40,800 --> 00:24:46,300 That's the definition of U in the basis 318 00:24:46,300 --> 00:24:58,160 e-- the matrix elements of U. You can try to figure out 319 00:24:58,160 --> 00:25:02,320 what is Uki in the f basis. 320 00:25:02,320 --> 00:25:06,106 How does operator U look in the f basis? 321 00:25:08,700 --> 00:25:12,320 Well, let's just do it without thinking. 322 00:25:12,320 --> 00:25:18,720 So in the f basis, I would put fk U fi. 323 00:25:21,850 --> 00:25:30,750 Well, but fk is U ek, so I'll put Uek Ufi. 324 00:25:34,080 --> 00:25:44,000 Now we can delete both U's, and it's ek fi. 325 00:25:44,000 --> 00:25:53,930 And I can remember what fi was, which is ek U ei. 326 00:25:53,930 --> 00:25:57,700 And it's just the same as the one we had there. 327 00:25:57,700 --> 00:26:00,790 So the operator, unitary operator, 328 00:26:00,790 --> 00:26:02,456 looks the same in both bases. 329 00:26:06,030 --> 00:26:09,690 That might seem strange or a coincidence, but it's not. 330 00:26:09,690 --> 00:26:12,200 So I leave it to you to think about, 331 00:26:12,200 --> 00:26:17,570 and visualize why did that happen. 332 00:26:17,570 --> 00:26:20,740 What's the reason? 333 00:26:20,740 --> 00:26:30,000 So the bracket notation-- we've been using it here and there-- 334 00:26:30,000 --> 00:26:34,650 and I will ask you to please read the notes. 335 00:26:34,650 --> 00:26:37,410 The notes will be posted this afternoon, 336 00:26:37,410 --> 00:26:41,180 and they will have-- not maybe all we've done today, 337 00:26:41,180 --> 00:26:44,360 but they will have some of what we'll do today, 338 00:26:44,360 --> 00:26:47,190 and all of what we've been doing. 339 00:26:47,190 --> 00:26:48,900 And the way it's done-- it's first 340 00:26:48,900 --> 00:26:51,800 done in this sort of inner product language, 341 00:26:51,800 --> 00:26:55,790 and then things are done in the bracket language. 342 00:26:55,790 --> 00:26:58,210 And it's a little repetitious, and I'm 343 00:26:58,210 --> 00:27:00,630 trying to take out some things here and there, 344 00:27:00,630 --> 00:27:02,250 so it's less repetitious. 345 00:27:02,250 --> 00:27:06,260 But at this moment it's probably worth reading it, 346 00:27:06,260 --> 00:27:07,600 and reading it again. 347 00:27:07,600 --> 00:27:08,748 Yes. 348 00:27:08,748 --> 00:27:13,230 AUDIENCE: [INAUDIBLE] if you have two orthonormal bases, 349 00:27:13,230 --> 00:27:17,370 is the transformation between them necessarily unitary? 350 00:27:17,370 --> 00:27:18,597 PROFESSOR: Yes, yes. 351 00:27:23,170 --> 00:27:23,850 All right. 352 00:27:23,850 --> 00:27:28,590 So as I was saying we're going to go into the Dirac notation 353 00:27:28,590 --> 00:27:29,090 again. 354 00:27:29,090 --> 00:27:33,160 And here's an example of a place where everybody, I think, 355 00:27:33,160 --> 00:27:35,940 tends to use Dirac notation. 356 00:27:35,940 --> 00:27:40,060 And the reason is a little curious, 357 00:27:40,060 --> 00:27:44,553 and you will appreciate it quite fast. 358 00:27:47,510 --> 00:27:50,050 So this will be the case of where 359 00:27:50,050 --> 00:27:57,273 we return to x and p operators, on a non-denumerable basis. 360 00:28:08,470 --> 00:28:12,830 So we're going to try to do x and p. 361 00:28:12,830 --> 00:28:15,730 now this is the classic of Dirac notation. 362 00:28:15,730 --> 00:28:17,890 It's probably-- as I said-- the place 363 00:28:17,890 --> 00:28:21,370 where everybody likes to use Dirac notation. 364 00:28:21,370 --> 00:28:24,750 And the reason it's efficient is because it prevents you 365 00:28:24,750 --> 00:28:27,850 from confusing two things. 366 00:28:27,850 --> 00:28:32,340 So I've written in the notes, and we 367 00:28:32,340 --> 00:28:36,150 have all these v's that belong to the vector space. 368 00:28:36,150 --> 00:28:38,580 And then we put this, and we still 369 00:28:38,580 --> 00:28:41,150 say it belongs to the vector space. 370 00:28:41,150 --> 00:28:46,310 And this is just a decoration that doesn't do much. 371 00:28:46,310 --> 00:28:48,050 And we can play with this. 372 00:28:48,050 --> 00:28:50,660 Now, in the non-denumerable basis, 373 00:28:50,660 --> 00:28:53,490 the catch-- and the possible confusion-- 374 00:28:53,490 --> 00:29:02,030 is that the label is not quite a vector in the vector space. 375 00:29:02,030 --> 00:29:06,490 So that is the reason why the notation is helpful, 376 00:29:06,490 --> 00:29:11,300 because it helps you distinguish two things that you 377 00:29:11,300 --> 00:29:13,150 could confuse. 378 00:29:13,150 --> 00:29:14,580 So here we go. 379 00:29:14,580 --> 00:29:19,080 We're going to talk about coordinate x, 380 00:29:19,080 --> 00:29:24,260 and the x operator, and the states. 381 00:29:24,260 --> 00:29:26,500 Well, this is a state space. 382 00:29:26,500 --> 00:29:30,360 So what kind of states do we have here? 383 00:29:30,360 --> 00:29:32,625 Well, we've talked about wave functions, 384 00:29:32,625 --> 00:29:34,250 and we could give the value of the wave 385 00:29:34,250 --> 00:29:36,430 function of different places. 386 00:29:36,430 --> 00:29:40,560 We're going to go for a more intrinsic definition. 387 00:29:40,560 --> 00:29:44,525 We're going to try to introduce position states. 388 00:29:50,970 --> 00:29:56,060 And position states will be called this-- x. 389 00:30:03,650 --> 00:30:07,060 Now, what is the meaning of this position state? 390 00:30:07,060 --> 00:30:15,490 We should think of this intuitively as a particle at x. 391 00:30:18,230 --> 00:30:21,760 Now here's how you can go wrong with this thing, 392 00:30:21,760 --> 00:30:25,860 if you stop thinking for a second. 393 00:30:32,660 --> 00:30:37,330 What is, then, ax? 394 00:30:37,330 --> 00:30:48,250 Is it ax, a being a number. 395 00:30:48,250 --> 00:30:51,140 Is it the same thing? 396 00:30:51,140 --> 00:30:53,630 No, not at all. 397 00:30:53,630 --> 00:30:58,930 This is a particle at the coordinate ax, 398 00:30:58,930 --> 00:31:04,530 and this is a particle at x with some different amplitude-- very 399 00:31:04,530 --> 00:31:05,260 different. 400 00:31:05,260 --> 00:31:08,500 So this is not true-- typical mistake. 401 00:31:08,500 --> 00:31:14,770 This is not minus x. 402 00:31:14,770 --> 00:31:16,450 That's totally different. 403 00:31:16,450 --> 00:31:19,595 So there's no such thing as this, either. 404 00:31:25,760 --> 00:31:29,310 It doesn't mean anything. 405 00:31:29,310 --> 00:31:34,990 And the reason is that these things are not our vectors. 406 00:31:34,990 --> 00:31:40,670 Our vector is this whole thing that says a particle at x. 407 00:31:40,670 --> 00:31:45,250 Maybe to make a clearer impression, 408 00:31:45,250 --> 00:31:47,310 imagine you're in three dimensions, 409 00:31:47,310 --> 00:31:50,210 and you have an x vector. 410 00:31:50,210 --> 00:31:53,965 So then you have to ket this. 411 00:31:56,690 --> 00:32:01,370 This is the ket particle at x. 412 00:32:01,370 --> 00:32:02,930 x is now a vector. 413 00:32:02,930 --> 00:32:05,320 It's a three-dimensional vector. 414 00:32:05,320 --> 00:32:08,270 This is a vector, but it's a vector 415 00:32:08,270 --> 00:32:11,910 in an infinite dimensional space, 416 00:32:11,910 --> 00:32:15,480 because the particle can be anywhere. 417 00:32:15,480 --> 00:32:18,990 So this is a vector in quantum mechanics. 418 00:32:18,990 --> 00:32:21,090 This is a complex vector space. 419 00:32:21,090 --> 00:32:25,880 This is a real vector space, and it's the label here. 420 00:32:25,880 --> 00:32:33,730 So again, minus x is not minus x vector. 421 00:32:33,730 --> 00:32:35,575 It's not the vector. 422 00:32:35,575 --> 00:32:41,060 The addition of the bra has moved you from vectors 423 00:32:41,060 --> 00:32:43,720 that you're familiar with, to states 424 00:32:43,720 --> 00:32:45,830 that are a little more abstract. 425 00:32:49,270 --> 00:32:52,990 So the reason this notation is quite good is because this is 426 00:32:52,990 --> 00:32:56,490 the number, but this i--- or this is a coordinate, 427 00:32:56,490 --> 00:33:00,090 and this is a vector already. 428 00:33:00,090 --> 00:33:02,572 So these are going to be our basis states, 429 00:33:02,572 --> 00:33:03,780 and they are non-denumerable. 430 00:33:09,790 --> 00:33:13,260 And here you can have that all x must 431 00:33:13,260 --> 00:33:16,390 belong to the real numbers, because we have particles 432 00:33:16,390 --> 00:33:22,270 in a line, while this thing can be changed by real numbers. 433 00:33:22,270 --> 00:33:24,982 The states can be multiplied by complex numbers, 434 00:33:24,982 --> 00:33:26,565 because we're doing quantum mechanics. 435 00:33:29,210 --> 00:33:33,260 So if you want to define a vector space-- now, 436 00:33:33,260 --> 00:33:34,990 this is all infinite dimension. 437 00:33:34,990 --> 00:33:36,730 It's a little worse in this sense 438 00:33:36,730 --> 00:33:39,150 the basis is non-denumerable. 439 00:33:39,150 --> 00:33:43,800 If I use this basis, I cannot make a list of all the basis 440 00:33:43,800 --> 00:33:46,540 vectors. 441 00:33:46,540 --> 00:33:49,930 So for an inner product, we will take the following-- 442 00:33:49,930 --> 00:33:58,220 we will take x with y to be delta of x minus y. 443 00:33:58,220 --> 00:34:00,090 That will be our inner product. 444 00:34:00,090 --> 00:34:06,550 And it has all the properties of the inner product 445 00:34:06,550 --> 00:34:09,250 that we may want. 446 00:34:09,250 --> 00:34:13,449 And what else? 447 00:34:13,449 --> 00:34:18,370 Well at this moment, we can try to-- this 448 00:34:18,370 --> 00:34:20,800 is physically sensible, let me say, 449 00:34:20,800 --> 00:34:23,830 because if you have a particle at one point 450 00:34:23,830 --> 00:34:26,159 and a particle at another point, the amplitude 451 00:34:26,159 --> 00:34:31,440 that this particle at one point is at this other point is 0. 452 00:34:31,440 --> 00:34:33,710 And these states are not normalizable. 453 00:34:33,710 --> 00:34:35,880 They correspond to a particle at the point, 454 00:34:35,880 --> 00:34:38,710 so once you try to normalize them, you get infinity, 455 00:34:38,710 --> 00:34:41,000 and you can't do much. 456 00:34:41,000 --> 00:34:45,179 But what you can do here is state more of the properties, 457 00:34:45,179 --> 00:34:46,860 and learn how to manipulate this. 458 00:34:46,860 --> 00:34:57,590 So remember we had one was the sum of all e i e i. 459 00:34:57,590 --> 00:35:00,890 The unit operator was that. 460 00:35:00,890 --> 00:35:05,490 Well, let's try to write a similar one. 461 00:35:05,490 --> 00:35:09,505 The unit operator will be the sum over all x's. 462 00:35:12,940 --> 00:35:16,640 And you could say, well, looks reasonable, 463 00:35:16,640 --> 00:35:20,210 but maybe there's a 1/2 in here, or some factor. 464 00:35:20,210 --> 00:35:22,260 Well, no factor is needed. 465 00:35:22,260 --> 00:35:30,240 You can check that-- that you've defined this thing properly. 466 00:35:30,240 --> 00:35:34,650 So let me do it. 467 00:35:34,650 --> 00:35:39,010 So act to on this so-called resolution 468 00:35:39,010 --> 00:35:48,060 of the identity with the vector y, so 1 on y is equal to y. 469 00:35:48,060 --> 00:35:51,777 And now let's add on the right xxy. 470 00:35:55,320 --> 00:35:58,670 This is delta of x minus y. 471 00:35:58,670 --> 00:36:02,300 And then when you integrate, you get y. 472 00:36:02,300 --> 00:36:04,750 So we're fine. 473 00:36:09,610 --> 00:36:13,120 So this looks a little too abstract, 474 00:36:13,120 --> 00:36:17,520 but it's not the abstract if you now introduce wave functions. 475 00:36:17,520 --> 00:36:19,830 So let's do wave functions. 476 00:36:19,830 --> 00:36:26,930 So you have a particle, a state of the particle psi. 477 00:36:26,930 --> 00:36:30,790 Time would be random, so I will put just this psi like that 478 00:36:30,790 --> 00:36:34,690 without the bottom line. 479 00:36:34,690 --> 00:36:37,350 And let's look at it. 480 00:36:37,350 --> 00:36:39,343 Oh, I want to say one more thing. 481 00:36:44,070 --> 00:36:52,500 The x operator acts on the x states to give x x. 482 00:36:52,500 --> 00:36:56,400 So these are eigenstates of the x operator. 483 00:36:56,400 --> 00:36:58,400 We declare them to be eigenstates 484 00:36:58,400 --> 00:37:00,680 of the x operator with eigenvalue x. 485 00:37:00,680 --> 00:37:04,330 That's their physical interpretation. 486 00:37:04,330 --> 00:37:08,430 I probably should have said before. 487 00:37:08,430 --> 00:37:15,030 Now, if we have a psi as a state or a vector, how do 488 00:37:15,030 --> 00:37:16,530 we get the wave function? 489 00:37:16,530 --> 00:37:19,860 Well, in this language the wave function, 490 00:37:19,860 --> 00:37:23,260 which we call psi of x, is defined 491 00:37:23,260 --> 00:37:27,365 to be the overlap of x with psi. 492 00:37:31,660 --> 00:37:37,810 And that makes sense, because this overlap 493 00:37:37,810 --> 00:37:42,420 is a function of this label here, where the particle is. 494 00:37:42,420 --> 00:37:46,310 And therefore, the result is a complex number 495 00:37:46,310 --> 00:37:49,200 that is dependent on x. 496 00:37:49,200 --> 00:37:51,590 So this belongs to the complex numbers, 497 00:37:51,590 --> 00:37:55,480 because inner products can have complex numbers. 498 00:37:55,480 --> 00:37:58,040 Now, I didn't put any complex number here, 499 00:37:58,040 --> 00:38:01,200 but when you form states, you can superpose states 500 00:38:01,200 --> 00:38:02,610 with complex numbers. 501 00:38:02,610 --> 00:38:07,060 So this psi of x will come out this way. 502 00:38:07,060 --> 00:38:11,110 And now that you are armed with that, 503 00:38:11,110 --> 00:38:14,440 you can even think of this in a nicer way. 504 00:38:14,440 --> 00:38:19,290 The state psi is equal to 1 times psi. 505 00:38:19,290 --> 00:38:22,740 And then use the rest of this formula, 506 00:38:22,740 --> 00:38:30,940 so this is integral-- dx x x psi. 507 00:38:30,940 --> 00:38:34,460 And again, the bracket notation is quite nice, 508 00:38:34,460 --> 00:38:37,360 because the bra already meets the ket. 509 00:38:37,360 --> 00:38:43,875 This is a number, and this is dx x psi of x. 510 00:38:46,780 --> 00:38:49,760 This equation has a nice interpretation. 511 00:38:49,760 --> 00:38:55,980 It says that the state is a superposition of the basis 512 00:38:55,980 --> 00:39:00,130 states, the position states, and the component 513 00:39:00,130 --> 00:39:03,690 of your original state along the basis state 514 00:39:03,690 --> 00:39:08,170 x is precisely the value of the wave function at x. 515 00:39:08,170 --> 00:39:10,270 So the wave function at x is giving you 516 00:39:10,270 --> 00:39:14,370 the weight of the state x as it enters into the sum. 517 00:39:17,310 --> 00:39:23,620 So one can compute more things. 518 00:39:23,620 --> 00:39:27,750 You will get practice in this type of computations. 519 00:39:27,750 --> 00:39:31,730 There are just a limited type of variations that you can do, 520 00:39:31,730 --> 00:39:37,120 so it's not that complicated. 521 00:39:37,120 --> 00:39:41,180 Basically, you can introduce resolutions of the identity 522 00:39:41,180 --> 00:39:42,710 wherever you need them. 523 00:39:42,710 --> 00:39:47,050 And if you introduce too many, you waste time, 524 00:39:47,050 --> 00:39:51,220 but you typically get the answer anyway. 525 00:39:51,220 --> 00:39:54,970 So it's not too serious. 526 00:39:54,970 --> 00:39:56,810 So suppose you want to understand 527 00:39:56,810 --> 00:39:59,213 what is the inner product of two states. 528 00:40:02,130 --> 00:40:05,000 Put the resolution of the identity in between. 529 00:40:05,000 --> 00:40:14,300 So put phi, and then put the integral dx x x psi. 530 00:40:14,300 --> 00:40:23,380 Well, the integral goes out, and you get phi x x psi. 531 00:40:23,380 --> 00:40:29,740 And remember, if x psi is psi of x, 532 00:40:29,740 --> 00:40:36,450 phi x is the complex conjugate, so it's phi star of x. 533 00:40:36,450 --> 00:40:39,240 And you knew that. 534 00:40:39,240 --> 00:40:41,100 If you have two wave functions, and you 535 00:40:41,100 --> 00:40:45,700 want to compute the overlap, you integrate the complex conjugate 536 00:40:45,700 --> 00:40:49,700 of one against the other. 537 00:40:49,700 --> 00:40:54,235 So this notation is doing all what you want from this. 538 00:40:58,040 --> 00:41:01,560 You want to compute a matrix element of x. 539 00:41:06,920 --> 00:41:12,920 Well, put another resolution of the identity here. 540 00:41:12,920 --> 00:41:19,740 So this would be integral dx phi-- the x hat is here. 541 00:41:19,740 --> 00:41:23,960 And then you put x x psi. 542 00:41:26,820 --> 00:41:30,670 The x hat on x is x. 543 00:41:30,670 --> 00:41:33,610 That's what this operator does, so you 544 00:41:33,610 --> 00:41:46,830 get integral dx of-- I'll put x phi x x psi, which 545 00:41:46,830 --> 00:41:50,160 is what you expect it to be-- integral 546 00:41:50,160 --> 00:41:55,260 of x phi star of x, psi of x. 547 00:42:00,990 --> 00:42:06,930 Now we can do exactly the same thing with momentum states. 548 00:42:06,930 --> 00:42:09,790 So I don't want to bore you, so I just 549 00:42:09,790 --> 00:42:15,520 list the properties-- basis states are 550 00:42:15,520 --> 00:42:19,830 momenta where the momenta is real. 551 00:42:19,830 --> 00:42:26,520 p prime p is equal delta of p minus p prime. 552 00:42:26,520 --> 00:42:33,900 One is the integral dp of p p. 553 00:42:33,900 --> 00:42:38,725 And p hat p is equal to p p. 554 00:42:44,170 --> 00:42:47,350 So these are the momentum bases. 555 00:42:47,350 --> 00:42:49,530 They're exactly analogous. 556 00:42:49,530 --> 00:42:53,600 So all what we've done for x is true. 557 00:42:53,600 --> 00:42:57,440 The completeness and normalization 558 00:42:57,440 --> 00:42:59,930 work well together, like we checked there, 559 00:42:59,930 --> 00:43:02,610 and everything is true. 560 00:43:02,610 --> 00:43:07,830 The only thing that you need to make this more interesting 561 00:43:07,830 --> 00:43:12,860 is a relation between the x basis and the p basis. 562 00:43:12,860 --> 00:43:15,440 And that's where physics comes in. 563 00:43:15,440 --> 00:43:20,250 Anybody can define these two, but then a physical assumption 564 00:43:20,250 --> 00:43:27,560 as to what you really mean by momentum is necessary. 565 00:43:27,560 --> 00:43:31,070 And what we've said is that the wave function 566 00:43:31,070 --> 00:43:36,091 of a particle with momentum p is e to the i 567 00:43:36,091 --> 00:43:42,370 px over h bar over square root of 2 pi 568 00:43:42,370 --> 00:43:49,280 h-- convenient normalization, but that was it. 569 00:43:49,280 --> 00:43:52,090 That was our physical interpretation 570 00:43:52,090 --> 00:43:55,010 of the wave function of a particle with some momentum. 571 00:43:55,010 --> 00:43:58,665 And therefore, if this is a wave function, that's xp. 572 00:44:01,340 --> 00:44:06,650 A state of momentum p has this wave function. 573 00:44:06,650 --> 00:44:08,400 So we write this. 574 00:44:16,180 --> 00:44:21,130 OK, there are tricks you can do, and please read the notes. 575 00:44:21,130 --> 00:44:24,270 But let's do a little computation. 576 00:44:24,270 --> 00:44:29,930 Suppose you want to compute what is p on psi. 577 00:44:29,930 --> 00:44:32,350 You could say, well, I don't know 578 00:44:32,350 --> 00:44:34,770 why would I want to do something with that? 579 00:44:34,770 --> 00:44:36,370 Looks simple enough. 580 00:44:36,370 --> 00:44:38,000 Well, it's simple enough, but you 581 00:44:38,000 --> 00:44:42,840 could say I want to see that in terms of wave functions, 582 00:44:42,840 --> 00:44:44,660 coordinate space wave functions. 583 00:44:44,660 --> 00:44:47,530 Well, if you want to see them in terms of coordinate space wave 584 00:44:47,530 --> 00:44:51,330 functions, you have to introduce a complete set of states. 585 00:44:51,330 --> 00:44:58,430 So introduce p x x psi. 586 00:44:58,430 --> 00:45:00,240 Then you have this wave function, 587 00:45:00,240 --> 00:45:03,380 and oh, this is sort of known, because it's 588 00:45:03,380 --> 00:45:10,566 the complex conjugate of this, so it's integral dx 589 00:45:10,566 --> 00:45:19,300 px over h bar, square root of 2 pi h bar times psi of x. 590 00:45:19,300 --> 00:45:22,140 And this was the Fourier transform-- 591 00:45:22,140 --> 00:45:26,260 what we call the Fourier transform of the wave function. 592 00:45:26,260 --> 00:45:33,840 So we can call it psi tilde of p, just to distinguish it, 593 00:45:33,840 --> 00:45:37,320 because we called psi with x, psi of x. 594 00:45:37,320 --> 00:45:39,190 So if I didn't put a tilde, you might 595 00:45:39,190 --> 00:45:41,450 think it's the same functional form, 596 00:45:41,450 --> 00:45:45,400 but it's the momentum space wave function. 597 00:45:45,400 --> 00:45:51,180 So here is the wave function in the p basis. 598 00:45:51,180 --> 00:45:53,690 It's the Fourier transform of the wave 599 00:45:53,690 --> 00:45:57,220 function in the x basis. 600 00:45:57,220 --> 00:46:01,020 One last computation, and then we change subjects again. 601 00:46:05,710 --> 00:46:10,360 It's the classic computation that you 602 00:46:10,360 --> 00:46:15,530 have now a mixed situation, in which you have the momentum 603 00:46:15,530 --> 00:46:20,430 operator states and the coordinate bra. 604 00:46:20,430 --> 00:46:34,753 So what is the following expression-- X p hat psi? 605 00:46:44,696 --> 00:46:45,195 OK. 606 00:46:49,900 --> 00:46:51,750 What is your temptation? 607 00:46:51,750 --> 00:46:57,170 Your temptation is to say, look, this is like the momentum 608 00:46:57,170 --> 00:47:02,490 operator acting on the wave function in the x basis. 609 00:47:02,490 --> 00:47:09,790 It can only be h bar over i d dx of psi of x. 610 00:47:09,790 --> 00:47:13,020 That's probably what it means. 611 00:47:13,020 --> 00:47:15,730 But the notation is clear enough, 612 00:47:15,730 --> 00:47:19,120 so we can check if that is exactly what it is. 613 00:47:19,120 --> 00:47:22,360 We can manipulate things already. 614 00:47:22,360 --> 00:47:24,250 So let's do it. 615 00:47:24,250 --> 00:47:28,900 So for that, I first have to try to get rid of this operator. 616 00:47:28,900 --> 00:47:32,840 Now the only way I know how to get rid of this operator p 617 00:47:32,840 --> 00:47:36,090 is because it has eigenstates. 618 00:47:36,090 --> 00:47:40,890 So it suggests very strongly that we should introduce 619 00:47:40,890 --> 00:47:43,430 momentum states, complete them. 620 00:47:43,430 --> 00:47:53,080 So I'll put v p x p hat p p psi. 621 00:48:00,660 --> 00:48:03,770 And now I can evaluate the little-- 622 00:48:03,770 --> 00:48:09,050 because p hat and p is little p, or p without the hat. 623 00:48:09,050 --> 00:48:17,570 So this is p xp p psi. 624 00:48:22,470 --> 00:48:28,990 Now you can look at that, and think 625 00:48:28,990 --> 00:48:32,800 carefully what should you do. 626 00:48:32,800 --> 00:48:35,250 And there's one thing that you can 627 00:48:35,250 --> 00:48:40,200 do is look at the equation on top. 628 00:48:40,200 --> 00:48:44,100 And this is a way to avoid working very hard. 629 00:48:44,100 --> 00:48:48,250 So look at the equation on top-- x p is equal to that. 630 00:48:48,250 --> 00:48:53,240 How do I get a p to multiply this? 631 00:48:53,240 --> 00:48:57,390 I can get a p to multiply this xp 632 00:48:57,390 --> 00:49:06,530 by doing h bar over i d dx of x p. 633 00:49:11,930 --> 00:49:17,790 Because if I see it there, I see that differentiating by d dx 634 00:49:17,790 --> 00:49:21,160 brings down an ip over h bar. 635 00:49:21,160 --> 00:49:24,760 So if I multiply by h bar over i, I get that. 636 00:49:24,760 --> 00:49:26,055 So let's do this. 637 00:49:36,670 --> 00:49:43,730 Now I claim we can take the h over i d 638 00:49:43,730 --> 00:49:47,690 dx out of this integral. 639 00:49:47,690 --> 00:49:51,380 And the reason is that first, it's not an x integral. 640 00:49:51,380 --> 00:49:55,440 It's a p integral, and nothing else except this factor 641 00:49:55,440 --> 00:49:56,630 depends on x. 642 00:49:56,630 --> 00:49:59,170 So I take it out and I want to bring it back, 643 00:49:59,170 --> 00:50:01,970 it will only act on this, because this is not 644 00:50:01,970 --> 00:50:03,920 x dependent. 645 00:50:03,920 --> 00:50:10,930 So you should think of psi, psi doesn't have an x dependence. 646 00:50:10,930 --> 00:50:15,050 Psi is a state, and here is p-- doesn't have an x dependence? 647 00:50:15,050 --> 00:50:18,280 You say no, it does, it looks here. 648 00:50:18,280 --> 00:50:22,060 No, but it doesn't have it, because it's been integrated. 649 00:50:22,060 --> 00:50:24,030 It really doesn't have x dependence. 650 00:50:24,030 --> 00:50:27,140 So we can take this out. 651 00:50:27,140 --> 00:50:31,490 We'll have h over i d dx. 652 00:50:31,490 --> 00:50:37,140 And now we have vp x p p psi. 653 00:50:40,750 --> 00:50:44,850 And now by completeness, this is just 1. 654 00:50:44,850 --> 00:50:47,070 So this becomes x psi. 655 00:50:47,070 --> 00:50:55,750 So h bar over i d dx of x psi, which is what we claimed it 656 00:50:55,750 --> 00:50:56,340 would be. 657 00:50:59,480 --> 00:51:06,083 So this is rigorous-- a rigorous derivation. 658 00:51:08,880 --> 00:51:11,260 There's no guessing. 659 00:51:11,260 --> 00:51:13,670 We've introduced complete states until you 660 00:51:13,670 --> 00:51:17,490 can see how things act. 661 00:51:17,490 --> 00:51:19,860 But the moral is here that you shouldn't 662 00:51:19,860 --> 00:51:22,730 have to go through this more than once in your life, 663 00:51:22,730 --> 00:51:23,750 or practice it. 664 00:51:23,750 --> 00:51:27,270 But once you see something like that, you think. 665 00:51:27,270 --> 00:51:29,780 You're using x representation, and you're 666 00:51:29,780 --> 00:51:32,040 talking about the operator p. 667 00:51:32,040 --> 00:51:35,140 It cannot be anything like that. 668 00:51:35,140 --> 00:51:39,650 If you want to practice something different, 669 00:51:39,650 --> 00:51:49,860 show that the analogue p x hat psi is equal i h bar d dp 670 00:51:49,860 --> 00:51:52,920 of psi tilde. 671 00:51:52,920 --> 00:51:55,670 So it's the opposite relation. 672 00:51:59,140 --> 00:52:00,550 All right. 673 00:52:00,550 --> 00:52:01,050 Questions? 674 00:52:04,240 --> 00:52:05,212 Yes. 675 00:52:05,212 --> 00:52:09,256 AUDIENCE: So how's one supposed to-- so what it appears 676 00:52:09,256 --> 00:52:11,755 is happening is you're basically taking some state like psi, 677 00:52:11,755 --> 00:52:15,486 and you're basically writing in terms of some basis. 678 00:52:15,486 --> 00:52:18,430 And then you're basically using the [INAUDIBLE] coordinates 679 00:52:18,430 --> 00:52:19,470 of this thing. 680 00:52:19,470 --> 00:52:23,780 But the question is, what does this basis actually look like? 681 00:52:23,780 --> 00:52:27,100 Like, what do these vectors-- because if you put them 682 00:52:27,100 --> 00:52:29,400 in their own coordinates, they're just infinite. 683 00:52:29,400 --> 00:52:30,430 PROFESSOR: Yup. 684 00:52:30,430 --> 00:52:32,290 AUDIENCE: They're not even delta-- I mean-- 685 00:52:32,290 --> 00:52:33,790 PROFESSOR: They are delta functions. 686 00:52:33,790 --> 00:52:34,980 AUDIENCE: [INAUDIBLE] 687 00:52:34,980 --> 00:52:36,980 PROFESSOR: These vectors are delta functions 688 00:52:36,980 --> 00:52:43,720 because if you have a state that has this as the position 689 00:52:43,720 --> 00:52:45,550 state of a particle, you find the wave 690 00:52:45,550 --> 00:52:47,580 function by doing x on it. 691 00:52:47,580 --> 00:52:50,120 That's our definition of a wave function. 692 00:52:50,120 --> 00:52:52,670 And its infinite. 693 00:52:52,670 --> 00:52:59,150 So there's is not too much one can say about this. 694 00:52:59,150 --> 00:53:02,830 If people want to work more mathematically, 695 00:53:02,830 --> 00:53:05,155 the more comfortable way, what you do 696 00:53:05,155 --> 00:53:07,950 is, instead of taking infinite things, 697 00:53:07,950 --> 00:53:10,540 you put everything on a big circle. 698 00:53:10,540 --> 00:53:13,220 And then you have a Fourier series 699 00:53:13,220 --> 00:53:18,400 and they transform as sums, and everything goes into sums. 700 00:53:18,400 --> 00:53:20,210 But there's no real need. 701 00:53:20,210 --> 00:53:22,510 These operations are safe. 702 00:53:22,510 --> 00:53:27,920 And we managed to do them, and we're OK with them. 703 00:53:27,920 --> 00:53:30,150 Other questions? 704 00:53:30,150 --> 00:53:31,334 Yes. 705 00:53:31,334 --> 00:53:32,209 AUDIENCE: [INAUDIBLE] 706 00:53:38,926 --> 00:53:39,925 PROFESSOR: Probably not. 707 00:53:39,925 --> 00:53:45,000 You know, infinite bases are delicate. 708 00:53:45,000 --> 00:53:48,510 Hilbert spaces are infinite dimensional vector spaces, 709 00:53:48,510 --> 00:53:53,670 and they-- not every infinite dimensional space 710 00:53:53,670 --> 00:53:55,550 is a Hilbert space. 711 00:53:55,550 --> 00:53:57,930 The most important thing of a Hilbert space 712 00:53:57,930 --> 00:54:00,840 is this norm, this inner product. 713 00:54:00,840 --> 00:54:05,980 But the other important thing is some convergence facts 714 00:54:05,980 --> 00:54:09,170 about sequences of vectors that converge 715 00:54:09,170 --> 00:54:11,220 to points that are on the space. 716 00:54:11,220 --> 00:54:12,900 So it's delicate. 717 00:54:12,900 --> 00:54:15,880 Infinite dimensional spaces can be pretty bad. 718 00:54:15,880 --> 00:54:19,550 A Banach space is not a Hilbert space. 719 00:54:19,550 --> 00:54:20,190 It's more com-- 720 00:54:20,190 --> 00:54:21,800 AUDIENCE: [INAUDIBLE] 721 00:54:21,800 --> 00:54:24,640 PROFESSOR: Only for Hilbert spaces, 722 00:54:24,640 --> 00:54:31,505 and basically, this problem of a particle in a line, 723 00:54:31,505 --> 00:54:35,630 or a particle in three space is sufficiently well known 724 00:54:35,630 --> 00:54:40,410 that we're totally comfortable with this somewhat 725 00:54:40,410 --> 00:54:42,100 singular operation. 726 00:54:42,100 --> 00:54:46,110 So the operator x or the operator p 727 00:54:46,110 --> 00:54:48,960 may not be what mathematicians like them 728 00:54:48,960 --> 00:54:51,950 to be-- bounded operators in Hilbert spaces. 729 00:54:51,950 --> 00:54:56,260 But we know how not to make mistakes with them. 730 00:54:56,260 --> 00:55:00,230 And if you have a very subtle problem, one day 731 00:55:00,230 --> 00:55:01,950 you probably have to be more careful. 732 00:55:01,950 --> 00:55:04,740 But for the problems we're interested in now, we don't. 733 00:55:07,300 --> 00:55:15,270 So our last topic today is uncertainties and uncertainty 734 00:55:15,270 --> 00:55:17,040 relations. 735 00:55:17,040 --> 00:55:21,095 I probably won't get through all of it, but we'll get started. 736 00:55:27,340 --> 00:55:29,750 And so we'll have uncertainties. 737 00:55:38,800 --> 00:55:43,850 And we will talk about operators, 738 00:55:43,850 --> 00:55:46,760 and Hermitian operators. 739 00:55:46,760 --> 00:55:49,560 So here is the question, basically-- 740 00:55:49,560 --> 00:55:53,550 if you have a state, we know the result 741 00:55:53,550 --> 00:55:57,380 of a measurement of an observable 742 00:55:57,380 --> 00:56:02,090 is the eigenvalue of a Hermitian operator. 743 00:56:02,090 --> 00:56:06,730 Now, if the state is an eigenstate of the Hermitian 744 00:56:06,730 --> 00:56:09,100 operator, you measure the observable, 745 00:56:09,100 --> 00:56:12,860 and out comes eigenvalue. 746 00:56:12,860 --> 00:56:17,290 And there's no uncertainty in the measured observable, 747 00:56:17,290 --> 00:56:20,000 because the measured observable is an eigenvalue 748 00:56:20,000 --> 00:56:23,420 and its state is an eigenstate. 749 00:56:23,420 --> 00:56:26,480 The problem arises when the state that you're 750 00:56:26,480 --> 00:56:30,600 trying to measure this property is not 751 00:56:30,600 --> 00:56:34,240 an eigenstate of the observable. 752 00:56:34,240 --> 00:56:36,510 So you know that the interpretation of quantum 753 00:56:36,510 --> 00:56:40,040 mechanics is a probabilistic distribution. 754 00:56:40,040 --> 00:56:43,610 You sometimes get one thing, sometimes get another thing, 755 00:56:43,610 --> 00:56:46,280 depending on the amplitudes of the states 756 00:56:46,280 --> 00:56:49,500 to be in those particular eigenstates. 757 00:56:49,500 --> 00:56:51,450 But there's an uncertainty. 758 00:56:51,450 --> 00:56:54,420 At this time, you don't know what 759 00:56:54,420 --> 00:56:57,690 the measured value will be. 760 00:56:57,690 --> 00:57:01,620 So we'll define the uncertainty associated to a Hermitian 761 00:57:01,620 --> 00:57:05,855 operator, and we want to define this uncertainty. 762 00:57:05,855 --> 00:57:08,000 So A will be a Hermitian operator. 763 00:57:16,890 --> 00:57:20,400 And you were talking about the uncertainty. 764 00:57:20,400 --> 00:57:22,812 Now the uncertainty of that operator-- 765 00:57:22,812 --> 00:57:24,770 the first thing that you should remember is you 766 00:57:24,770 --> 00:57:26,870 can't talk about the uncertainty of the operator 767 00:57:26,870 --> 00:57:29,500 unless you give me a state. 768 00:57:29,500 --> 00:57:32,370 So all the formulas we're going to write for uncertainties 769 00:57:32,370 --> 00:57:36,790 are uncertainties of operators in some state. 770 00:57:36,790 --> 00:57:39,635 So let's call the state psi. 771 00:57:47,250 --> 00:57:50,510 And time will not be relevant, so maybe I 772 00:57:50,510 --> 00:57:56,890 should delete the-- well, I'll leave that bar there, 773 00:57:56,890 --> 00:57:58,470 just in case. 774 00:57:58,470 --> 00:58:04,300 So we're going to try to define uncertainty. 775 00:58:04,300 --> 00:58:08,970 But before we do that, let's try to define 776 00:58:08,970 --> 00:58:11,350 another thing-- the expectation value. 777 00:58:11,350 --> 00:58:14,240 Well, the expectation value-- you know it. 778 00:58:14,240 --> 00:58:17,740 The expectation value of A, and you could put a psi here 779 00:58:17,740 --> 00:58:24,600 if you wish, to remind you that it depends on the state-- is, 780 00:58:24,600 --> 00:58:29,080 well, psi A psi. 781 00:58:29,080 --> 00:58:31,520 That's what we call expectation value. 782 00:58:31,520 --> 00:58:36,710 In the inner product notation would be psi A psi. 783 00:58:41,520 --> 00:58:48,570 And one thing you know-- that this thing is real, 784 00:58:48,570 --> 00:58:55,490 because the expectation values of Hermitian operators is real. 785 00:58:55,490 --> 00:58:58,280 That's something we reviewed at the beginning of the lecture 786 00:58:58,280 --> 00:59:01,080 today. 787 00:59:01,080 --> 00:59:06,070 So now comes the question, what can I 788 00:59:06,070 --> 00:59:10,240 do to define an uncertainty of an operator? 789 00:59:10,240 --> 00:59:15,120 And an uncertainty-- now we've said already something. 790 00:59:15,120 --> 00:59:18,970 I wish to define an uncertainty that is such 791 00:59:18,970 --> 00:59:25,310 that the uncertainty is 0 if the state is an eigenstate, 792 00:59:25,310 --> 00:59:27,720 and the uncertainty is different from 0 793 00:59:27,720 --> 00:59:29,910 if it's not an eigenstate. 794 00:59:29,910 --> 00:59:33,850 In fact, I wish that the uncertainty is 0 if 795 00:59:33,850 --> 00:59:38,130 and only if the state is an eigenstate. 796 00:59:38,130 --> 00:59:41,700 So actually, we can achieve that. 797 00:59:41,700 --> 00:59:50,140 And in some sense, I think, the most intuitive definition 798 00:59:50,140 --> 00:59:55,670 is the one that I will show here. 799 00:59:55,670 --> 01:00:00,080 It's that we define the uncertainty, delta A, 800 01:00:00,080 --> 01:00:01,950 and I'll put the psi here. 801 01:00:01,950 --> 01:00:10,060 So this is called the uncertainty of A 802 01:00:10,060 --> 01:00:11,285 in the state psi. 803 01:00:17,330 --> 01:00:19,390 So we'll define it a simple way. 804 01:00:19,390 --> 01:00:20,720 What else do we want? 805 01:00:20,720 --> 01:00:25,010 We said this should be 0 if and only 806 01:00:25,010 --> 01:00:28,490 if the state is an eigenstate. 807 01:00:28,490 --> 01:00:32,900 Second, I want this thing to be a real number-- 808 01:00:32,900 --> 01:00:35,790 in fact, a positive number. 809 01:00:35,790 --> 01:00:40,990 What function do we know in quantum mechanics 810 01:00:40,990 --> 01:00:43,000 that can do that magic? 811 01:00:43,000 --> 01:00:44,940 Well, it's the norm. 812 01:00:44,940 --> 01:00:48,610 The norm function is always real and positive. 813 01:00:48,610 --> 01:00:52,860 So this-- we'll try to set it equal to a norm. 814 01:00:52,860 --> 01:01:00,870 So it's the norm of the state A minus the expectation 815 01:01:00,870 --> 01:01:08,240 value of A times 1 acting on psi. 816 01:01:08,240 --> 01:01:12,170 This will be our definition of the uncertainty. 817 01:01:16,010 --> 01:01:18,233 So it's the norm of this vector. 818 01:01:23,280 --> 01:01:25,700 Now let's look at this. 819 01:01:25,700 --> 01:01:32,730 Suppose the norm uncertainty is 0. 820 01:01:36,230 --> 01:01:40,180 And if the uncertainty is 0, this vector must be 0. 821 01:01:40,180 --> 01:01:49,960 So A minus expectation value of A on psi is 0. 822 01:01:49,960 --> 01:02:00,330 Or A psi is equal to expectation value of A on psi. 823 01:02:00,330 --> 01:02:02,090 The 1 doesn't do much. 824 01:02:02,090 --> 01:02:04,460 Many people don't write the 1. 825 01:02:04,460 --> 01:02:08,030 I could get tired and stop writing it. 826 01:02:08,030 --> 01:02:13,630 You should-- probably it's good manners to write the i, 827 01:02:13,630 --> 01:02:16,910 but it's not all that necessary. 828 01:02:16,910 --> 01:02:18,500 You don't get that confused. 829 01:02:18,500 --> 01:02:21,970 If there's an operator and a number here, 830 01:02:21,970 --> 01:02:24,310 it must be an identity matrix. 831 01:02:24,310 --> 01:02:28,880 So the uncertainty is 0, the vector is 0, then this is true. 832 01:02:28,880 --> 01:02:32,200 Now, you say, well, this equation looks kind of funny, 833 01:02:32,200 --> 01:02:37,270 but it says that psi is an eigenstate of A, 834 01:02:37,270 --> 01:02:40,230 because this is a number. 835 01:02:40,230 --> 01:02:42,300 It looks a little funny, because we're 836 01:02:42,300 --> 01:02:46,580 accustomed to A psi lambda psi, but this is a number. 837 01:02:46,580 --> 01:02:49,410 And in fact, let me show you one thing. 838 01:02:49,410 --> 01:02:57,320 If you have A psi equal lambda psi-- oh, I 839 01:02:57,320 --> 01:03:00,426 should say here that psi is normalized. 840 01:03:04,220 --> 01:03:08,010 If psi would not be normalized, you change the normalization. 841 01:03:08,010 --> 01:03:10,765 You change the uncertainty. 842 01:03:10,765 --> 01:03:11,890 So it should be normalized. 843 01:03:14,530 --> 01:03:19,980 And look at this-- if you have a psi equal lambda psi, 844 01:03:19,980 --> 01:03:27,130 do the inner product with psi. 845 01:03:27,130 --> 01:03:33,510 Psi comma A psi would be equal to lambda, 846 01:03:33,510 --> 01:03:36,810 because psi inner product with psi is 1. 847 01:03:36,810 --> 01:03:38,310 But what is this? 848 01:03:38,310 --> 01:03:41,730 This is the expectation value of A. 849 01:03:41,730 --> 01:03:46,010 So actually, given our definition, 850 01:03:46,010 --> 01:03:50,750 the eigenvalue of some operator on this state 851 01:03:50,750 --> 01:03:53,720 is the expectation value of the operator in the state. 852 01:03:53,720 --> 01:03:58,910 So back to the argument-- if the uncertainty is 0, 853 01:03:58,910 --> 01:04:01,000 the state is an eigenstate. 854 01:04:01,000 --> 01:04:06,130 And the eigenvalue happens to be the expectation value-- that 855 01:04:06,130 --> 01:04:07,480 is, if the uncertainty is 0. 856 01:04:07,480 --> 01:04:12,220 On the other hand, if you are in an eigenstate, you're here. 857 01:04:12,220 --> 01:04:15,180 Then lambda is A, and this equation 858 01:04:15,180 --> 01:04:18,470 shows that this vector is 0, and therefore you get 0. 859 01:04:18,470 --> 01:04:25,860 So you've shown that this norm or this uncertainty is 0, if 860 01:04:25,860 --> 01:04:29,820 and only if the state is an eigenstate. 861 01:04:29,820 --> 01:04:32,550 And that's a very powerful statement. 862 01:04:32,550 --> 01:04:36,130 The statement that's always known by everybody 863 01:04:36,130 --> 01:04:40,480 is that if you have an eigenstate-- yes-- 864 01:04:40,480 --> 01:04:41,450 no uncertainty. 865 01:04:41,450 --> 01:04:45,700 But if there's no uncertainty, you must have an eigenstate. 866 01:04:45,700 --> 01:04:48,360 That's the second part, and uses the fact 867 01:04:48,360 --> 01:04:53,420 that the only vector with 0 norm is the zero vector-- a thing 868 01:04:53,420 --> 01:04:56,490 that we use over and over again. 869 01:04:56,490 --> 01:05:00,930 So let me make a couple more comments 870 01:05:00,930 --> 01:05:02,430 on how you compute this. 871 01:05:05,020 --> 01:05:09,525 So that's the uncertainty so far. 872 01:05:15,620 --> 01:05:17,970 So the uncertainty vanishes in that case. 873 01:05:17,970 --> 01:05:24,420 Now, we can square this equation to find a formula that 874 01:05:24,420 --> 01:05:30,130 is perhaps more familiar-- not necessarily more useful, 875 01:05:30,130 --> 01:05:31,440 but also good. 876 01:05:31,440 --> 01:05:34,500 For computations, it's pretty good-- delta A 877 01:05:34,500 --> 01:05:38,180 of psi, which is real-- we square it. 878 01:05:38,180 --> 01:05:42,100 Well, the norm square is the inner product 879 01:05:42,100 --> 01:05:58,560 of this A minus A psi A minus A psi. 880 01:05:58,560 --> 01:06:02,090 Norm squared is the inner product of these two vectors. 881 01:06:02,090 --> 01:06:04,800 Now, the thing that we like to do 882 01:06:04,800 --> 01:06:09,420 is to move this factor to that side. 883 01:06:09,420 --> 01:06:13,650 How do you move a factor on the first input to the other input? 884 01:06:13,650 --> 01:06:16,600 You take the adjoint. 885 01:06:16,600 --> 01:06:18,660 So I should move it with an adjoint. 886 01:06:18,660 --> 01:06:20,370 So what do I get? 887 01:06:20,370 --> 01:06:31,210 Psi, and then I get the adjoint and this factor again. 888 01:06:39,010 --> 01:06:44,320 Now, I should put a dagger here, but let me not put it, 889 01:06:44,320 --> 01:06:47,990 because A is Hermitian. 890 01:06:47,990 --> 01:06:53,800 And moreover, expectation value of A is real. 891 01:06:53,800 --> 01:06:57,140 Remember-- so no need for the dagger, 892 01:06:57,140 --> 01:06:58,920 so you can put the dagger, and then 893 01:06:58,920 --> 01:07:02,290 explain that this is Hermitian and this is real-- 894 01:07:02,290 --> 01:07:05,910 or just not put it. 895 01:07:05,910 --> 01:07:08,700 And now look at this. 896 01:07:08,700 --> 01:07:10,580 This is a typical calculation. 897 01:07:10,580 --> 01:07:12,255 You'll do it many, many times. 898 01:07:14,780 --> 01:07:18,440 You just spread out the things. 899 01:07:18,440 --> 01:07:20,800 So let me just do it once. 900 01:07:20,800 --> 01:07:27,360 Here you get A squared minus A expectation value of A minus 901 01:07:27,360 --> 01:07:32,960 expectation value of A A plus expectation 902 01:07:32,960 --> 01:07:36,135 value of A squared psi. 903 01:07:41,030 --> 01:07:44,850 So I multiplied everything, but you shouldn't be all 904 01:07:44,850 --> 01:07:49,820 that-- I should put a 1 here, probably-- 905 01:07:49,820 --> 01:07:51,450 shouldn't worry about this much. 906 01:07:51,450 --> 01:07:54,850 This is just a number and an A, a number and an A. 907 01:07:54,850 --> 01:07:56,010 The order doesn't matter. 908 01:07:56,010 --> 01:07:59,120 These two terms are really the same. 909 01:07:59,120 --> 01:08:04,520 Well, let me go slowly on this once. 910 01:08:04,520 --> 01:08:06,020 What is the first term? 911 01:08:06,020 --> 01:08:09,770 It's psi A squared psi, so it's the expectation value 912 01:08:09,770 --> 01:08:12,870 of A squared. 913 01:08:12,870 --> 01:08:15,260 Now, what is this term? 914 01:08:15,260 --> 01:08:18,960 Well, you have a number here, which is real. 915 01:08:18,960 --> 01:08:23,220 It goes out of whatever you're doing, and you have psi A psi. 916 01:08:23,220 --> 01:08:26,090 So this is expectation value of A. 917 01:08:26,090 --> 01:08:28,590 And from the leftover psi A psi, you 918 01:08:28,590 --> 01:08:36,640 get another expectation value of A. So this is A A. 919 01:08:36,640 --> 01:08:39,430 Here the same thing-- the number goes out, 920 01:08:39,430 --> 01:08:42,220 and you're left with a psi A psi, which 921 01:08:42,220 --> 01:08:49,010 is another expectation value of A, so you get minus A A. 922 01:08:49,010 --> 01:08:53,430 And you have a plus expectation value of A squared. 923 01:08:53,430 --> 01:08:56,890 And I don't need the i anymore, because the expectation values 924 01:08:56,890 --> 01:08:58,490 have been taken. 925 01:08:58,490 --> 01:09:00,689 And this always happens. 926 01:09:00,689 --> 01:09:03,920 It's a minus here, a minus here, and a plus 927 01:09:03,920 --> 01:09:06,964 here, so there's just one minus at the end of the day. 928 01:09:16,100 --> 01:09:21,529 One minus at the end of the day, and a familiar, or famous 929 01:09:21,529 --> 01:09:31,779 formula comes out that delta of A on psi squared 930 01:09:31,779 --> 01:09:36,250 is equal to the expectation value of A squared minus 931 01:09:36,250 --> 01:09:41,420 expectation value of A squared. 932 01:09:41,420 --> 01:09:44,430 Which shows something quite powerful. 933 01:09:44,430 --> 01:09:48,120 This has connections, of course, with statistical mechanics 934 01:09:48,120 --> 01:09:49,193 and standard deviations. 935 01:09:52,920 --> 01:09:56,390 It's a probabilistic interpretation of this formula, 936 01:09:56,390 --> 01:10:00,940 but one fact that this has allowed us to prove 937 01:10:00,940 --> 01:10:04,050 is that the expectation value of A squared 938 01:10:04,050 --> 01:10:06,550 is always greater or equal than that, 939 01:10:06,550 --> 01:10:10,380 because this number is positive, because it 940 01:10:10,380 --> 01:10:13,670 is the square of a real positive number. 941 01:10:13,670 --> 01:10:17,610 So that's a slightly non-trivial thing, 942 01:10:17,610 --> 01:10:19,256 and it's good to know it. 943 01:10:23,290 --> 01:10:26,380 And this formula, of course, is very well known. 944 01:10:29,020 --> 01:10:36,030 Now, I'm going to leave a funny geometrical interpretation 945 01:10:36,030 --> 01:10:37,020 of the uncertainty. 946 01:10:37,020 --> 01:10:40,110 Maybe you will find it illuminating, 947 01:10:40,110 --> 01:10:42,580 in some ways turning into pictures 948 01:10:42,580 --> 01:10:45,460 all these calculations we've done. 949 01:10:45,460 --> 01:10:48,460 I think it actually adds value to it, 950 01:10:48,460 --> 01:10:52,680 and I don't think it's very well known, 951 01:10:52,680 --> 01:10:57,380 or it's kind of funny, because it must not be very well known. 952 01:10:57,380 --> 01:11:01,630 But maybe people don't find it that suggestive. 953 01:11:01,630 --> 01:11:04,020 I kind of find it suggestive. 954 01:11:04,020 --> 01:11:09,550 So here's what I want to say geometrically. 955 01:11:09,550 --> 01:11:12,110 You have this vector space, and you have a vector psi. 956 01:11:17,770 --> 01:11:22,180 Then you come along, and you add with the operator 957 01:11:22,180 --> 01:11:27,890 A. Now the fact that this thing is not 958 01:11:27,890 --> 01:11:32,510 and eigenstate means that after you add with A, 959 01:11:32,510 --> 01:11:35,500 you don't keep in the same direction. 960 01:11:35,500 --> 01:11:37,180 You go in different directions. 961 01:11:37,180 --> 01:11:41,670 So here is A psi. 962 01:11:48,350 --> 01:11:50,630 So what can we say here? 963 01:11:50,630 --> 01:11:55,700 Well, actually here is this thing. 964 01:11:55,700 --> 01:12:00,030 Think of this vector space spanned by psi. 965 01:12:03,740 --> 01:12:06,850 Let's call it U psi. 966 01:12:06,850 --> 01:12:10,860 So it's that line there. 967 01:12:10,860 --> 01:12:16,680 You can project this in here, orthogonally. 968 01:12:23,930 --> 01:12:27,030 Here is the first claim-- the vector 969 01:12:27,030 --> 01:12:30,130 that you get up to here-- this vector-- is 970 01:12:30,130 --> 01:12:37,210 nothing else but expectation value of A times psi. 971 01:12:37,210 --> 01:12:40,140 And that makes sense, because it's a number times psi. 972 01:12:40,140 --> 01:12:44,120 But precisely the orthogonal projection is this. 973 01:12:44,120 --> 01:12:46,850 And here, you get an orthogonal vector. 974 01:12:46,850 --> 01:12:48,740 We'll call it psi perp. 975 01:12:52,830 --> 01:12:56,000 And the funny thing about this psi perp 976 01:12:56,000 --> 01:13:00,863 is that its length is precisely the uncertainty. 977 01:13:05,890 --> 01:13:12,220 So all this, but you could prove-- I'm going to do it. 978 01:13:12,220 --> 01:13:14,220 I'm going to show you all these things are true, 979 01:13:14,220 --> 01:13:16,550 but it gives you a bit of an insight. 980 01:13:16,550 --> 01:13:17,870 you have a vector. 981 01:13:17,870 --> 01:13:19,980 A moves you out. 982 01:13:19,980 --> 01:13:24,524 What is the uncertainty is this vertical projection-- 983 01:13:24,524 --> 01:13:25,940 vertical thing is the uncertainty. 984 01:13:25,940 --> 01:13:30,560 If you're down there, you get nothing. 985 01:13:30,560 --> 01:13:35,130 So how do we prove that? 986 01:13:35,130 --> 01:13:41,440 Well, let's construct a projector 987 01:13:41,440 --> 01:13:49,440 down to the space U psi, which is psi psi. 988 01:13:49,440 --> 01:13:56,880 This is a projector, just like any e1. 989 01:13:56,880 --> 01:14:00,660 e1 is a projection into the direction of 1. 990 01:14:00,660 --> 01:14:04,450 Well, take your first basis vector to be psi, 991 01:14:04,450 --> 01:14:07,310 and that's a projection to psi. 992 01:14:07,310 --> 01:14:11,950 So let's see what it-- so the projection to psi. 993 01:14:11,950 --> 01:14:15,600 So now let's see what it gives you 994 01:14:15,600 --> 01:14:24,390 when it acts on A psi-- this project acting on A psi 995 01:14:24,390 --> 01:14:31,120 is equal to psi psi A psi. 996 01:14:34,280 --> 01:14:36,590 And again, the usefulness of bracket notation 997 01:14:36,590 --> 01:14:38,400 is kind of nice here. 998 01:14:38,400 --> 01:14:39,310 So what is this? 999 01:14:39,310 --> 01:14:44,780 The expectation value of A. So indeed psi expectation value 1000 01:14:44,780 --> 01:14:49,870 of A is what you get when you project this down. 1001 01:15:00,882 --> 01:15:05,550 So then, the rest is sort of simple. 1002 01:15:05,550 --> 01:15:13,885 If you take psi, and subtract from psi-- well, 1003 01:15:13,885 --> 01:15:31,360 I'll subtract from psi, psi times expectation value of A. 1004 01:15:31,360 --> 01:15:34,270 I'm sorry, I was saying it wrong. 1005 01:15:34,270 --> 01:15:41,040 If you think the original vector-- A psi, 1006 01:15:41,040 --> 01:15:45,520 and subtract from it what we took out, 1007 01:15:45,520 --> 01:15:50,610 which is psi times expectation value of A, the projected 1008 01:15:50,610 --> 01:15:53,840 thing-- this is some vector. 1009 01:15:53,840 --> 01:16:01,880 But the main thing is that this vector is orthogonal to psi. 1010 01:16:01,880 --> 01:16:02,630 Why? 1011 01:16:02,630 --> 01:16:09,450 If you take a psi on the left, this is orthogonal to psi. 1012 01:16:09,450 --> 01:16:10,830 And how do you see it? 1013 01:16:10,830 --> 01:16:13,980 Put the psi from the left. 1014 01:16:13,980 --> 01:16:15,340 And what do you get here? 1015 01:16:15,340 --> 01:16:19,180 Psi A psi, which is expectation value of A, 1016 01:16:19,180 --> 01:16:23,840 psi psi, which is 1, and expectation value A is 0. 1017 01:16:23,840 --> 01:16:30,110 So this is a vector psi perp. 1018 01:16:30,110 --> 01:16:35,280 And this is, of course, A minus expectation value 1019 01:16:35,280 --> 01:16:40,270 of A acting on the state psi. 1020 01:16:40,270 --> 01:16:46,300 Well, precisely the norm of psi perp is the norm of this, 1021 01:16:46,300 --> 01:16:48,510 but that's what we defined to be the uncertainty. 1022 01:16:51,830 --> 01:17:04,840 So indeed, the norm of psi perp is delta A of psi. 1023 01:17:04,840 --> 01:17:09,230 So our ideas of projectors and orthogonal projectors 1024 01:17:09,230 --> 01:17:11,340 allow you to understand better what 1025 01:17:11,340 --> 01:17:14,010 is the uncertainty-- more pictorially. 1026 01:17:14,010 --> 01:17:18,080 You have pictures of vectors, and orthogonal projections, 1027 01:17:18,080 --> 01:17:20,550 and you want to make the uncertainty 0, 1028 01:17:20,550 --> 01:17:24,010 you have to push the A psi into psi. 1029 01:17:24,010 --> 01:17:28,160 You have to be an eigenstate, and you're there. 1030 01:17:28,160 --> 01:17:31,430 Now, the last thing of-- I'll use the last five minutes 1031 01:17:31,430 --> 01:17:37,965 to motivate the uncertainty, the famous uncertainty theorem. 1032 01:17:59,260 --> 01:18:01,470 And typically, the uncertainly theorem 1033 01:18:01,470 --> 01:18:09,460 is useful for A and B-- two Hermitian operators. 1034 01:18:13,220 --> 01:18:22,250 And it relates the uncertainty in A on the state psi 1035 01:18:22,250 --> 01:18:27,570 to the uncertainty in B of psi, saying 1036 01:18:27,570 --> 01:18:31,535 it must be greater than or equal than some number. 1037 01:18:38,300 --> 01:18:42,600 Now, if you look at that, and you 1038 01:18:42,600 --> 01:18:46,580 think of all the math we've been talking about, 1039 01:18:46,580 --> 01:18:49,870 you maybe know exactly how you're 1040 01:18:49,870 --> 01:18:51,860 supposed to prove the uncertainty theorem. 1041 01:18:57,300 --> 01:19:00,070 Well, what does this remind you of? 1042 01:19:02,830 --> 01:19:05,750 Cauchy-Schwarz-- Schwarz inequality, 1043 01:19:05,750 --> 01:19:08,350 I'm sorry-- not Cauchy-Schwarz. 1044 01:19:08,350 --> 01:19:08,850 Why? 1045 01:19:08,850 --> 01:19:16,870 Because for Schwarz inequality, you have norm of u, norm of v 1046 01:19:16,870 --> 01:19:21,600 is greater than or equal than the norm of the inner product 1047 01:19:21,600 --> 01:19:29,290 of u and v-- absolute value of the inner product of u and v. 1048 01:19:29,290 --> 01:19:31,660 Remember, in this thing, this is norm of a vector, 1049 01:19:31,660 --> 01:19:34,910 this is norm of a vector, and this is value of a scalar. 1050 01:19:34,910 --> 01:19:42,160 And our uncertainties are norms. 1051 01:19:42,160 --> 01:19:44,720 So it better be that. 1052 01:19:44,720 --> 01:19:46,640 That inequality is the only inequality 1053 01:19:46,640 --> 01:19:49,800 that can possibly give you the answer. 1054 01:19:49,800 --> 01:19:54,500 So how would you set this up? 1055 01:19:54,500 --> 01:20:04,990 You would say define-- as we'll say f equal A minus A acting 1056 01:20:04,990 --> 01:20:15,060 on psi, and g is equal to B minus B acting on psi. 1057 01:20:15,060 --> 01:20:25,730 And then f f, or f f is delta A squared. 1058 01:20:25,730 --> 01:20:32,806 f g g is delta B squared. 1059 01:20:37,730 --> 01:20:42,090 And you just need to compute the inner product of f g, 1060 01:20:42,090 --> 01:20:44,330 because you need the mixed one. 1061 01:20:44,330 --> 01:20:48,420 So if you want to have fun, try it. 1062 01:20:48,420 --> 01:20:50,380 We'll do it next time anyway. 1063 01:20:50,380 --> 01:20:53,600 All right that's it for today.