1 00:00:01,960 --> 00:00:03,760 PROFESSOR: So here it is. 2 00:00:03,760 --> 00:00:10,140 Suppose you have a serious expansion in lambda. 3 00:00:10,140 --> 00:00:13,250 So this is the state-- when lambda is equal to 0, 4 00:00:13,250 --> 00:00:15,320 this should be the state. 5 00:00:15,320 --> 00:00:17,540 But when lambda is different from 0, 6 00:00:17,540 --> 00:00:18,770 that will not be the state. 7 00:00:18,770 --> 00:00:20,090 We'll have lambda correction. 8 00:00:20,090 --> 00:00:25,230 So this is the first-order correction to this state. 9 00:00:25,230 --> 00:00:27,410 So that's why I put the 1. 10 00:00:27,410 --> 00:00:30,410 And you should think of it first order-- oh! 11 00:00:30,410 --> 00:00:33,110 --because it comes with a lambda. 12 00:00:33,110 --> 00:00:35,990 This is the second-order correction to the state, 13 00:00:35,990 --> 00:00:38,040 because it comes with a lambda squared. 14 00:00:38,040 --> 00:00:39,890 And the same thing here. 15 00:00:39,890 --> 00:00:45,230 So the superscript is telling you what order in lambda 16 00:00:45,230 --> 00:00:46,850 you are working-- 17 00:00:46,850 --> 00:00:48,180 to what accuracy. 18 00:00:48,180 --> 00:00:50,750 So what is the most urgent thing to find 19 00:00:50,750 --> 00:00:52,670 the first order of corrections? 20 00:00:52,670 --> 00:00:55,720 If you find them, and we still have time [LAUGH] 21 00:00:55,720 --> 00:01:00,010 for the second order, you go more and more. 22 00:01:00,010 --> 00:01:02,250 OK, let's continue. 23 00:01:02,250 --> 00:01:03,715 Let's solve some of this. 24 00:01:06,670 --> 00:01:10,450 So our next task is to solve this problem. 25 00:01:10,450 --> 00:01:12,870 And here we go. 26 00:01:15,720 --> 00:01:18,060 Let's solve that. 27 00:01:18,060 --> 00:01:19,570 So what am I going to do? 28 00:01:19,570 --> 00:01:25,840 I'm going to just write this equation slightly differently. 29 00:01:25,840 --> 00:01:31,770 I'll write it as h of lambda, which is-- 30 00:01:31,770 --> 00:01:37,870 OK, I'll write it differently. h0 plus lambda delta h-- 31 00:01:37,870 --> 00:01:39,930 that's h of lambda-- 32 00:01:39,930 --> 00:01:49,320 minus En of lambda on the state n of lambda is equal to 0. 33 00:01:49,320 --> 00:01:52,140 That's your Schrodinger equation, 34 00:01:52,140 --> 00:01:54,690 the time-independent Schrodinger equation, 35 00:01:54,690 --> 00:01:57,810 we're trying to solve. 36 00:01:57,810 --> 00:02:01,080 And now I'm going to just write it out, 37 00:02:01,080 --> 00:02:03,010 so that you can see what we get. 38 00:02:03,010 --> 00:02:05,670 So it's going to take a little bit of writing. 39 00:02:05,670 --> 00:02:11,370 Let me collect the terms that have no lambda. 40 00:02:11,370 --> 00:02:12,300 It's h0. 41 00:02:12,300 --> 00:02:20,040 This has a lambda, but En begins with En0 0 that has no lambda. 42 00:02:20,040 --> 00:02:23,685 So, from this parentheses, this is a term without a lambda. 43 00:02:27,020 --> 00:02:30,710 It came from here, En0-- 44 00:02:30,710 --> 00:02:32,150 it's here. 45 00:02:32,150 --> 00:02:37,340 Now let's look at the terms with a lambda. 46 00:02:37,340 --> 00:02:41,750 So I want to see how I'm writing. 47 00:02:41,750 --> 00:02:44,750 I want to write it with a minus sign. 48 00:02:44,750 --> 00:02:47,000 So, with a lambda, we have minus-- 49 00:02:50,990 --> 00:03:03,290 from here, we have a term En1 minus delta h. 50 00:03:03,290 --> 00:03:06,420 That is all the terms with a lambda. 51 00:03:06,420 --> 00:03:09,455 So I should put the lambda, as well, here. 52 00:03:09,455 --> 00:03:11,690 Probably I want to put it in front. 53 00:03:11,690 --> 00:03:15,920 Minus lambda [INAUDIBLE]. 54 00:03:15,920 --> 00:03:21,020 En1, from there, and the lambda delta h, 55 00:03:21,020 --> 00:03:22,385 with a double minus sign. 56 00:03:25,940 --> 00:03:28,580 Then it goes simple, now. 57 00:03:28,580 --> 00:03:30,800 I've taken into account these two terms. 58 00:03:30,800 --> 00:03:32,210 All the rest come from here. 59 00:03:32,210 --> 00:03:38,680 So you have a minus lambda squared En2, 60 00:03:38,680 --> 00:03:45,560 and, at some point, a minus lambda to the k Enk. 61 00:03:45,560 --> 00:03:47,990 And then it goes on. 62 00:03:47,990 --> 00:03:52,610 And then we write it like a big bracket, here. 63 00:03:52,610 --> 00:03:54,870 That's the parentheses. 64 00:03:54,870 --> 00:03:57,500 And now the state. 65 00:03:57,500 --> 00:04:11,140 You have n0 lambda n1 plus lambda squared n2 plus 66 00:04:11,140 --> 00:04:16,910 lambda to the k, the k-th correction to the state. 67 00:04:16,910 --> 00:04:19,880 And it goes on forever. 68 00:04:19,880 --> 00:04:21,410 And it is here. 69 00:04:21,410 --> 00:04:25,012 And all that is equal to 0. 70 00:04:25,012 --> 00:04:30,360 [LAUGH] Looks daunting, but it's not. 71 00:04:34,370 --> 00:04:35,730 What should we do? 72 00:04:35,730 --> 00:04:40,910 Well, here is, again, lambda helpful for you. 73 00:04:40,910 --> 00:04:44,250 Lambda is a parameter. 74 00:04:44,250 --> 00:04:48,090 The left-hand side is a polynomial on lambda. 75 00:04:48,090 --> 00:04:51,660 It should vanish for all values of lambda, 76 00:04:51,660 --> 00:04:53,460 because the Schrodinger equation should 77 00:04:53,460 --> 00:04:55,710 hold for all values of lambda. 78 00:04:55,710 --> 00:05:00,660 When a polynomial vanishes for all values of lambda, 79 00:05:00,660 --> 00:05:04,170 the argument of the polynomial, all the coefficients 80 00:05:04,170 --> 00:05:06,330 must vanish, of the polynomial. 81 00:05:06,330 --> 00:05:13,800 Therefore, we must look at what is 0-th order in lambda, here, 82 00:05:13,800 --> 00:05:15,190 and see what we get. 83 00:05:15,190 --> 00:05:23,490 Well, 0-th order in lambda, we get this equation, h0 minus En0 84 00:05:23,490 --> 00:05:30,340 [? on ?] n0 equals 0. 85 00:05:30,340 --> 00:05:32,790 That's 0-th order in lambda, and that's 86 00:05:32,790 --> 00:05:36,495 an equation that is not new. 87 00:05:36,495 --> 00:05:38,010 [LAUGH] You knew it! 88 00:05:38,010 --> 00:05:42,300 That's a statement that n0 was an eigenstate 89 00:05:42,300 --> 00:05:44,250 of the original Hamiltonian. 90 00:05:44,250 --> 00:05:45,120 So it's good. 91 00:05:45,120 --> 00:05:48,030 You know, the 0-th order things had to work, 92 00:05:48,030 --> 00:05:53,430 because we said, to 0-th order you have the known Hamiltonian. 93 00:05:53,430 --> 00:05:57,045 Let's look at the term with order lambda. 94 00:06:02,600 --> 00:06:09,140 Lambda can get from this term in the Hamiltonian acting on n1. 95 00:06:09,140 --> 00:06:12,790 That's order lambda, so let's write it here. 96 00:06:12,790 --> 00:06:17,190 h0 minus En0 on n1. 97 00:06:21,260 --> 00:06:25,880 And the other term comes from a lambda in the first factor 98 00:06:25,880 --> 00:06:27,470 and no lambda in the second. 99 00:06:27,470 --> 00:06:32,210 So it's this term, this acting on that state. 100 00:06:32,210 --> 00:06:34,760 Look-- there's a lambda, there's a minus sign, 101 00:06:34,760 --> 00:06:37,250 so you can put it on the right-hand side. 102 00:06:37,250 --> 00:06:45,850 And we get En1 minus delta h acting on n0. 103 00:06:51,880 --> 00:06:55,975 Let's be a little daring and try to get the lambda to the k. 104 00:07:03,120 --> 00:07:08,600 So h0 minus En0. 105 00:07:08,600 --> 00:07:11,280 And I want to see what are the terms that 106 00:07:11,280 --> 00:07:14,760 have lambda to the k, power k. 107 00:07:18,050 --> 00:07:23,660 Well, H0 minus En0 acting on this one has lambda to the k. 108 00:07:23,660 --> 00:07:27,530 So you have n to the k, here, nk-- 109 00:07:27,530 --> 00:07:28,610 not "to the k." 110 00:07:34,110 --> 00:07:41,100 And then, to get lambda to the k, I could have a lambda here, 111 00:07:41,100 --> 00:07:46,020 and the term that is before this, lambda to the k minus 1, 112 00:07:46,020 --> 00:07:48,180 nk minus 1. 113 00:07:48,180 --> 00:07:52,930 And it goes with a minus sign to the right-hand side. 114 00:07:52,930 --> 00:08:03,060 So you would have En1 minus delta h on n k minus 1. 115 00:08:07,730 --> 00:08:21,410 And then you'll have En2 on Enk minus 2. 116 00:08:21,410 --> 00:08:24,600 And it will go all the way until you'll 117 00:08:24,600 --> 00:08:34,929 have Enk acting on n0, the original state. 118 00:08:38,159 --> 00:08:40,873 So let me box this, and, uh-- 119 00:08:46,392 --> 00:08:48,530 those are the equations that we get. 120 00:08:54,920 --> 00:08:57,800 And we have to solve them. 121 00:08:57,800 --> 00:08:59,150 And we can solve them. 122 00:08:59,150 --> 00:09:02,920 That's the nice thing about this. 123 00:09:02,920 --> 00:09:06,250 Well, this one, we argued, it's simple enough. 124 00:09:06,250 --> 00:09:08,650 We don't have to do much about it. 125 00:09:11,500 --> 00:09:15,100 Then we have to solve for n1. 126 00:09:15,100 --> 00:09:19,390 Oh, but the second equation actually has two unknowns. 127 00:09:19,390 --> 00:09:23,560 We don't know the state n1, the first correction, 128 00:09:23,560 --> 00:09:27,510 and we don't know the energy correction. 129 00:09:27,510 --> 00:09:31,420 But that's kind of the useful thing 130 00:09:31,420 --> 00:09:32,860 that is Schrodinger equation. 131 00:09:32,860 --> 00:09:35,060 You don't know the energies, [LAUGH] 132 00:09:35,060 --> 00:09:36,790 and you don't know the eigenstate. 133 00:09:36,790 --> 00:09:39,650 So you couldn't expect this. 134 00:09:39,650 --> 00:09:41,410 It's kind of interesting. 135 00:09:41,410 --> 00:09:48,220 If you have solved for n1 and En1 and n2 and En2, 136 00:09:48,220 --> 00:09:53,320 up to some point, the next state involves 137 00:09:53,320 --> 00:09:59,140 nk, the energy of the state nk, and all the things 138 00:09:59,140 --> 00:10:03,010 that you already know-- the lower energies, and the lower 139 00:10:03,010 --> 00:10:04,120 states. 140 00:10:04,120 --> 00:10:06,250 So you can solve this recursively, 141 00:10:06,250 --> 00:10:08,050 one equation at a time. 142 00:10:08,050 --> 00:10:10,210 Depending how much work you want to do, 143 00:10:10,210 --> 00:10:12,580 you go more and more equations. 144 00:10:12,580 --> 00:10:17,350 We'll typically go the first and the second 145 00:10:17,350 --> 00:10:20,320 and sometimes make some remarks about these things. 146 00:10:23,440 --> 00:10:28,040 There's one important simplifying assumption 147 00:10:28,040 --> 00:10:33,070 we can make that helps us a lot. 148 00:10:33,070 --> 00:10:48,240 I can claim you can choose n1 and all the higher ones, n2, 149 00:10:48,240 --> 00:10:55,890 to be orthogonal to n0. 150 00:10:55,890 --> 00:10:58,510 Think a little about this. 151 00:10:58,510 --> 00:11:01,000 What does that tell us? 152 00:11:01,000 --> 00:11:07,770 It says, oh, this vector should have no component along n0. 153 00:11:07,770 --> 00:11:12,990 And these vectors should have no component along n0. 154 00:11:12,990 --> 00:11:15,870 The intuitive reason why this is the case 155 00:11:15,870 --> 00:11:18,990 that you can choose that and simplifies your life 156 00:11:18,990 --> 00:11:23,610 is that, if it had some component along n0, 157 00:11:23,610 --> 00:11:27,210 you could just sort of move it here, 158 00:11:27,210 --> 00:11:32,130 and now you would have n0 plus a function of lambda times n0, 159 00:11:32,130 --> 00:11:36,040 and you can divide this by this function 160 00:11:36,040 --> 00:11:41,100 and rescale the state back, to have an n0 here. 161 00:11:41,100 --> 00:11:43,830 The normalization of this state-- 162 00:11:43,830 --> 00:11:46,140 originally, we have them normalized, 163 00:11:46,140 --> 00:11:50,820 but it would make our life extremely more complicated 164 00:11:50,820 --> 00:11:54,120 if we tried to do this perturbation series 165 00:11:54,120 --> 00:11:56,280 and keep the normalization. 166 00:11:56,280 --> 00:11:58,800 The states are not going to be normalized, 167 00:11:58,800 --> 00:12:01,540 but you know that's not the problem. 168 00:12:01,540 --> 00:12:05,580 If they are not normalized but are normalizable, 169 00:12:05,580 --> 00:12:08,280 you can always work with them. 170 00:12:08,280 --> 00:12:10,540 So we won't normalize them. 171 00:12:10,540 --> 00:12:14,970 But the idea is that any piece that is proportional to and n0, 172 00:12:14,970 --> 00:12:17,100 you could reabsorb it. 173 00:12:17,100 --> 00:12:19,450 Now, that's vague. 174 00:12:19,450 --> 00:12:21,360 If you didn't understand that argument, 175 00:12:21,360 --> 00:12:24,930 I commend you, because it's a vague argument. 176 00:12:24,930 --> 00:12:28,440 So let me do a more precise argument. 177 00:12:28,440 --> 00:12:34,440 Suppose, for example, you're solving this equation, 178 00:12:34,440 --> 00:12:37,050 and you solve n0. 179 00:12:37,050 --> 00:12:40,410 And suppose you've solved now for n1. 180 00:12:40,410 --> 00:12:41,940 And you got your n1. 181 00:12:41,940 --> 00:12:45,390 You're done, you solve the second equation, 182 00:12:45,390 --> 00:12:48,150 you're perfectly happy, but somebody says, 183 00:12:48,150 --> 00:12:53,070 you know, it has some component along n0. 184 00:12:53,070 --> 00:12:55,490 What can you do? 185 00:12:55,490 --> 00:12:58,020 OK, you say, look-- 186 00:12:58,020 --> 00:13:09,090 if n1 solves this equation, n1 plus any number c times n0 187 00:13:09,090 --> 00:13:12,830 still is a solution, I claim. 188 00:13:12,830 --> 00:13:14,050 Why? 189 00:13:14,050 --> 00:13:17,950 Because n1, the state n1 that you're trying to find, only 190 00:13:17,950 --> 00:13:20,590 appears on the left-hand side. 191 00:13:20,590 --> 00:13:25,060 And n0 is killed by this combination 192 00:13:25,060 --> 00:13:27,340 in the first equation. 193 00:13:27,340 --> 00:13:32,530 So, if you have a solution n1, you can replace it by this one. 194 00:13:32,530 --> 00:13:39,530 And you can choose c to cancel whatever n0 you had in here. 195 00:13:39,530 --> 00:13:44,360 So you can always produce a state that is orthogonal to it. 196 00:13:44,360 --> 00:13:47,350 And it's easier to work with that. 197 00:13:47,350 --> 00:13:49,030 And this goes on forever. 198 00:13:49,030 --> 00:13:54,100 Suppose you've solved now n1 that has no piece along n0, n2, 199 00:13:54,100 --> 00:13:54,970 n3, n4-- 200 00:13:54,970 --> 00:13:56,770 all those-- and you go up to here, 201 00:13:56,770 --> 00:13:59,380 and nk has a piece along n0. 202 00:13:59,380 --> 00:14:03,505 You can still add the constant to nk times n0 203 00:14:03,505 --> 00:14:04,700 and make it work. 204 00:14:04,700 --> 00:14:06,860 So you can always do that. 205 00:14:06,860 --> 00:14:10,820 They're orthogonal to n0. 206 00:14:10,820 --> 00:14:18,680 So let me say one more thing that is very amazing. 207 00:14:18,680 --> 00:14:22,650 Let's look a little a first look at the equation lambda 208 00:14:22,650 --> 00:14:27,090 1 of order lambda 1. 209 00:14:27,090 --> 00:14:32,750 Or-- I'm sorry, these parentheses are not good. 210 00:14:32,750 --> 00:14:34,940 This is lambda to the 1. 211 00:14:34,940 --> 00:14:36,740 This is lambda to the k. 212 00:14:36,740 --> 00:14:38,000 The parentheses is bad. 213 00:14:40,580 --> 00:14:41,710 Lambda to the 0. 214 00:14:45,800 --> 00:14:47,250 Lambda 1. 215 00:14:47,250 --> 00:14:49,590 So this is our equation. 216 00:14:49,590 --> 00:15:07,455 We'll have h0 minus En0 n1 equal En1 minus delta h n0. 217 00:15:11,650 --> 00:15:13,025 I'm going to do one thing. 218 00:15:16,050 --> 00:15:22,460 I'm going to push a bra n0 on the left. 219 00:15:22,460 --> 00:15:24,500 Should I do it on that same equation? 220 00:15:24,500 --> 00:15:26,630 Let's save a little time. 221 00:15:26,630 --> 00:15:29,340 That equation, we already had it here. 222 00:15:29,340 --> 00:15:31,430 So let's put an n0 here. 223 00:15:34,620 --> 00:15:37,810 bra n0 here. 224 00:15:43,790 --> 00:15:44,860 OK. 225 00:15:44,860 --> 00:15:47,740 So here is the challenge. 226 00:15:47,740 --> 00:15:52,090 We've put a lot of notation on the blackboard. 227 00:15:52,090 --> 00:15:56,110 And maybe by now all the symbols are floating in your head 228 00:15:56,110 --> 00:16:00,040 and not making much sense. 229 00:16:00,040 --> 00:16:03,580 I want you to figure out what is the value 230 00:16:03,580 --> 00:16:06,360 of this left-hand side.