1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT Open Courseware 4 00:00:06,030 --> 00:00:10,120 continue to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,660 To make a donation or to view additional materials 6 00:00:12,660 --> 00:00:15,150 from hundreds of MIT courses, visit 7 00:00:15,150 --> 00:00:17,150 MITOpenCourseware@ocw.mit.edu. 8 00:00:22,210 --> 00:00:25,550 ROBERT FIELD: So as I said at the beginning of the course, 9 00:00:25,550 --> 00:00:30,010 this is quantum mechanics for use, not admiration, and not 10 00:00:30,010 --> 00:00:31,780 historical. 11 00:00:31,780 --> 00:00:34,480 You're going to leave this course knowing 12 00:00:34,480 --> 00:00:40,840 how to solve a very large number of quantum mechanical problems. 13 00:00:40,840 --> 00:00:42,880 Or if not to solve it, to get insight 14 00:00:42,880 --> 00:00:45,120 into how you would solve it. 15 00:00:45,120 --> 00:00:53,600 And so I presented a couple of exactly solved problems, 16 00:00:53,600 --> 00:00:56,940 and that's not because they're historically important. 17 00:00:56,940 --> 00:00:59,190 It's because you're going to use them, 18 00:00:59,190 --> 00:01:03,030 and you're going to embed the results of those exactly solved 19 00:01:03,030 --> 00:01:09,750 problems in an approximate approach to almost any problem. 20 00:01:09,750 --> 00:01:12,180 And the vast majority of problems 21 00:01:12,180 --> 00:01:16,340 that you would face use the harmonic oscillator 22 00:01:16,340 --> 00:01:19,700 as the core of your understanding 23 00:01:19,700 --> 00:01:24,260 because almost all potentials have a minimum, 24 00:01:24,260 --> 00:01:25,880 and so that means the first derivative 25 00:01:25,880 --> 00:01:27,530 is zero at the minimum. 26 00:01:27,530 --> 00:01:30,710 So you don't care about the first derivative. 27 00:01:30,710 --> 00:01:32,330 You care about where the minimum is. 28 00:01:32,330 --> 00:01:37,970 And the second derivative is the dominant thing, 29 00:01:37,970 --> 00:01:40,770 and that's the harmonic aspect. 30 00:01:40,770 --> 00:01:43,780 And so using a harmonic oscillator basis set, 31 00:01:43,780 --> 00:01:48,750 you're going to be able to attack every problem. 32 00:01:48,750 --> 00:01:51,330 And another thing about the harmonic oscillator 33 00:01:51,330 --> 00:01:54,345 is that it uses these As and A daggers, which are magic. 34 00:01:57,490 --> 00:02:02,110 And what happens is you forget, totally, 35 00:02:02,110 --> 00:02:04,120 about the wave functions. 36 00:02:04,120 --> 00:02:06,580 The wave functions are there if you 37 00:02:06,580 --> 00:02:09,039 want to calculate some kind of probability amplitude 38 00:02:09,039 --> 00:02:12,040 distribution, but you never look at them 39 00:02:12,040 --> 00:02:16,790 when you're solving the problem, and that's a fantastic thing. 40 00:02:16,790 --> 00:02:19,950 Now, there is one more exactly solved problem 41 00:02:19,950 --> 00:02:22,580 that I want to talk about today, which is not usually 42 00:02:22,580 --> 00:02:26,120 included in the list of exactly solved problems 43 00:02:26,120 --> 00:02:28,520 because it's kind of special. 44 00:02:28,520 --> 00:02:30,850 The two-level problem. 45 00:02:30,850 --> 00:02:32,730 The two-level problem is exactly solved 46 00:02:32,730 --> 00:02:34,830 because there are only two levels, 47 00:02:34,830 --> 00:02:37,877 and the solution of that problem involves 48 00:02:37,877 --> 00:02:38,835 the quadratic equation. 49 00:02:42,780 --> 00:02:45,330 So there is an exact solution. 50 00:02:45,330 --> 00:02:50,610 And this is used a lot in introducing new techniques 51 00:02:50,610 --> 00:02:53,070 in quantum mechanics, and so you're 52 00:02:53,070 --> 00:02:56,760 going to see the two-level problem again and again 53 00:02:56,760 --> 00:02:59,670 as opening the door to being able to deal 54 00:02:59,670 --> 00:03:01,980 with much more complicated problems, 55 00:03:01,980 --> 00:03:04,320 and I'm going to try to refer to that a lot. 56 00:03:04,320 --> 00:03:10,230 OK, so we're about to go from the Schrodinger picture 57 00:03:10,230 --> 00:03:13,290 to matrix mechanics, and I'd like 58 00:03:13,290 --> 00:03:16,746 to have some comments about what are the elements 59 00:03:16,746 --> 00:03:17,870 of the Schrodinger picture. 60 00:03:20,670 --> 00:03:22,410 And there's no wrong answer here, 61 00:03:22,410 --> 00:03:24,165 but I'm looking for certain things. 62 00:03:26,920 --> 00:03:28,320 Anybody want to tell me? 63 00:03:28,320 --> 00:03:29,840 Yes? 64 00:03:29,840 --> 00:03:33,530 AUDIENCE: Is it based off wave equation or the Schrodinger 65 00:03:33,530 --> 00:03:36,589 equation, which is very similar to the wave equation? 66 00:03:36,589 --> 00:03:37,380 ROBERT FIELD: Yeah. 67 00:03:41,320 --> 00:03:44,300 So we have a differential equation, 68 00:03:44,300 --> 00:03:48,151 and the solutions are wave functions. 69 00:03:48,151 --> 00:03:48,650 More? 70 00:03:57,488 --> 00:03:58,480 AUDIENCE: [INAUDIBLE] 71 00:03:58,480 --> 00:04:02,170 ROBERT FIELD: Well, mathematics is challenging to some people. 72 00:04:02,170 --> 00:04:03,920 Some people really love it. 73 00:04:03,920 --> 00:04:07,030 But yes, it's much more mathematical 74 00:04:07,030 --> 00:04:11,410 because you're faced with solving differential equations, 75 00:04:11,410 --> 00:04:14,650 coupled differential equations, and you're 76 00:04:14,650 --> 00:04:18,160 challenged with calculating a lot of intervals, 77 00:04:18,160 --> 00:04:21,070 and sometimes the intervals are not over simple functions. 78 00:04:21,070 --> 00:04:21,940 They're complicated. 79 00:04:27,110 --> 00:04:30,800 But the main thing is you have this thing which you could 80 00:04:30,800 --> 00:04:35,230 never observe, but is somehow the core of everything 81 00:04:35,230 --> 00:04:38,410 that you can know, and that really-- 82 00:04:38,410 --> 00:04:41,301 as I get older, it bothers me more and more. 83 00:04:41,301 --> 00:04:43,050 It's not that I'm going to invent some way 84 00:04:43,050 --> 00:04:45,720 to do quantum mechanics totally without a wave function. 85 00:04:49,410 --> 00:04:51,850 The most important thing, I think, 86 00:04:51,850 --> 00:04:55,050 is that when you work in the Schrodinger picture, 87 00:04:55,050 --> 00:05:03,970 you get one wave function and one eigenvalue at a time. 88 00:05:03,970 --> 00:05:07,770 And so yes, you can solve these problems, 89 00:05:07,770 --> 00:05:09,830 but no, you don't get insight. 90 00:05:12,340 --> 00:05:15,040 You don't see the overall structure. 91 00:05:15,040 --> 00:05:17,260 You just see well, if you did this experiment, 92 00:05:17,260 --> 00:05:19,480 this is what you would observe. 93 00:05:19,480 --> 00:05:21,730 That's perfectly wonderful. 94 00:05:21,730 --> 00:05:25,420 And now, some anticipation. 95 00:05:25,420 --> 00:05:28,210 What's so special about the Heisenberg picture? 96 00:05:28,210 --> 00:05:30,130 Or what is the Heisenberg picture? 97 00:05:30,130 --> 00:05:33,430 Now, I haven't lectured on it so if you read my lecture notes, 98 00:05:33,430 --> 00:05:34,930 you know the answers to all of this, 99 00:05:34,930 --> 00:05:39,570 but can we speculate about what goes here? 100 00:05:39,570 --> 00:05:40,810 Yes? 101 00:05:40,810 --> 00:05:43,406 AUDIENCE: [INAUDIBLE] 102 00:05:55,760 --> 00:05:59,190 ROBERT FIELD: OK, so every operator 103 00:05:59,190 --> 00:06:06,150 can be represented by a matrix, and the elements of that matrix 104 00:06:06,150 --> 00:06:10,290 are calculated usually automatically. 105 00:06:10,290 --> 00:06:15,930 Often, they're an infinite number of basis states, 106 00:06:15,930 --> 00:06:18,130 so you sort of refer to the Schrodinger picture, 107 00:06:18,130 --> 00:06:23,030 but anyway, these matrices are arrays of numbers, 108 00:06:23,030 --> 00:06:27,450 and they can be infinite arrays of numbers, 109 00:06:27,450 --> 00:06:30,660 but you don't write an infinite array of numbers down. 110 00:06:30,660 --> 00:06:34,470 You write a few numbers and you recognize what the pattern is, 111 00:06:34,470 --> 00:06:37,410 and usually, you have some function of the quantum 112 00:06:37,410 --> 00:06:43,620 numbers, and that generates all the elements in this matrix. 113 00:06:43,620 --> 00:06:50,160 And the solutions are going to be certain eigenvectors, 114 00:06:50,160 --> 00:06:53,330 and instead of differential equations, it's linear algebra. 115 00:06:57,350 --> 00:07:01,490 Now, for differential equations, you'll learn a lot of tricks. 116 00:07:01,490 --> 00:07:04,710 For integrals, you learn a lot of tricks. 117 00:07:04,710 --> 00:07:07,230 For linear algebra, there aren't any tricks. 118 00:07:07,230 --> 00:07:08,190 It's all right there. 119 00:07:10,740 --> 00:07:16,150 And so it's a much more transparent-- 120 00:07:16,150 --> 00:07:19,270 now, it might be numerically demanding because you're 121 00:07:19,270 --> 00:07:22,480 dealing with very large arrays of numbers, 122 00:07:22,480 --> 00:07:24,400 and you're trying to get something from it, 123 00:07:24,400 --> 00:07:28,270 but the linear algebra is simple. 124 00:07:28,270 --> 00:07:34,310 OK, and we have these things called matrix elements. 125 00:07:34,310 --> 00:07:38,880 I can use this word now because the numbers in a matrix 126 00:07:38,880 --> 00:07:40,110 are called matrix elements. 127 00:07:40,110 --> 00:07:43,800 They're integrals, but they're integrals 128 00:07:43,800 --> 00:07:45,360 that are, more or less, given to you. 129 00:07:49,537 --> 00:07:50,995 And then we have infinite matrices. 130 00:07:57,070 --> 00:08:01,450 And in the Schrodinger picture, we might have infinite basis 131 00:08:01,450 --> 00:08:05,200 sets, but we don't really think too much 132 00:08:05,200 --> 00:08:07,960 about the problem of infinities because we're looking 133 00:08:07,960 --> 00:08:09,580 at things one at a time. 134 00:08:09,580 --> 00:08:14,280 Here, the operators are implicitly infinite, 135 00:08:14,280 --> 00:08:17,900 and we have to do something about that 136 00:08:17,900 --> 00:08:22,370 because no computer can find the eigenvalues and eigenvectors 137 00:08:22,370 --> 00:08:24,470 of an infinite matrix. 138 00:08:24,470 --> 00:08:26,780 So somehow, you have to truncate it, 139 00:08:26,780 --> 00:08:29,180 and you have to use some approximations, 140 00:08:29,180 --> 00:08:31,820 and the framework for that is perturbation theory. 141 00:08:37,690 --> 00:08:39,350 Now, I love perturbation theory. 142 00:08:39,350 --> 00:08:41,750 I've made my career out of perturbation theory 143 00:08:41,750 --> 00:08:45,120 and you're going to see a lot of it, but not today. 144 00:08:51,230 --> 00:08:54,170 The next exam is going to have a lot of perturbation theory. 145 00:08:57,480 --> 00:09:01,080 OK, so let's review where we've been. 146 00:09:01,080 --> 00:09:13,220 With a two-level problem, we have two energy levels, 147 00:09:13,220 --> 00:09:19,580 and there is some interaction between them, 148 00:09:19,580 --> 00:09:26,190 and we get E plus, psi plus, and E minus, psi minus. 149 00:09:35,280 --> 00:09:43,620 So we have these integrals, H2,2, H1,2, 150 00:09:43,620 --> 00:09:46,590 and because it's a Hermitian matrix, 151 00:09:46,590 --> 00:09:48,570 it's equal to H2,1 star. 152 00:09:51,390 --> 00:09:55,790 That's the definition of a Hermitian matrix, 153 00:09:55,790 --> 00:10:01,520 and that means if the H1,2 element is imaginary, 154 00:10:01,520 --> 00:10:04,750 the H2,1 matrix element is imaginary, 155 00:10:04,750 --> 00:10:08,369 but with the opposite sign. 156 00:10:08,369 --> 00:10:09,910 Now, almost all of the problems we're 157 00:10:09,910 --> 00:10:17,230 going to face are expressed in terms of real matrix elements, 158 00:10:17,230 --> 00:10:22,090 and so all you're really doing to go from the 1,2 to the 2,1 159 00:10:22,090 --> 00:10:24,940 is just reversing the order of the index. 160 00:10:24,940 --> 00:10:28,120 Nothing else is happening, and that has also 161 00:10:28,120 --> 00:10:30,940 some great simplicity in the solutions 162 00:10:30,940 --> 00:10:35,140 we use to solve the two-level problems. 163 00:10:35,140 --> 00:10:40,160 OK, the solutions to the two-level problem 164 00:10:40,160 --> 00:10:44,113 are based on some simplifications. 165 00:10:47,220 --> 00:10:50,430 So what we do is we make the two by two matrix 166 00:10:50,430 --> 00:10:53,610 symmetric by subtracting out the average energy. 167 00:10:56,180 --> 00:11:01,850 We define then, two numbers, which are H1,1 minus H2,2 over 168 00:11:01,850 --> 00:11:03,670 two and this. 169 00:11:03,670 --> 00:11:11,230 So we have E bar and delta, and we have the interaction, 170 00:11:11,230 --> 00:11:22,160 which is H1,2, and we call it v. And we have to be a little 171 00:11:22,160 --> 00:11:28,190 careful because v could be complex or imaginary, 172 00:11:28,190 --> 00:11:34,340 but for now, we're just going to treat it always as real. 173 00:11:34,340 --> 00:11:40,140 OK, so for this two-level problem, 174 00:11:40,140 --> 00:11:50,880 the energy levels are given by E bar plus or minus delta squared 175 00:11:50,880 --> 00:11:55,810 plus v-squared square root. 176 00:11:55,810 --> 00:12:00,280 Or E bar plus or minus the square root 177 00:12:00,280 --> 00:12:03,840 of x, where that's x. 178 00:12:06,820 --> 00:12:12,280 And the eigenfunctions are expressed in terms 179 00:12:12,280 --> 00:12:13,960 of these coefficients-- 180 00:12:13,960 --> 00:12:21,730 one plus or minus is equal to 1/2 one plus or minus 181 00:12:21,730 --> 00:12:27,790 delta over x to the 1/2 square root. 182 00:12:27,790 --> 00:12:30,250 Square root is outside the bracket. 183 00:12:30,250 --> 00:12:38,940 And c2 plus or minus looks almost the same, 184 00:12:38,940 --> 00:12:47,240 except we have 1 minus or plus delta over square root of x. 185 00:12:50,480 --> 00:12:52,460 So in this reduced picture, there's 186 00:12:52,460 --> 00:12:55,400 not too much to remember. 187 00:12:55,400 --> 00:13:05,220 The difference between the eigenfunctions for the higher 188 00:13:05,220 --> 00:13:09,780 energy and the lower energy eigenstates 189 00:13:09,780 --> 00:13:12,120 differ only by these signs. 190 00:13:12,120 --> 00:13:17,330 OK, now, this is a lot. 191 00:13:17,330 --> 00:13:20,610 I derived it in lecture last time, 192 00:13:20,610 --> 00:13:23,540 and the algebra is horrible, and I'm not 193 00:13:23,540 --> 00:13:26,880 good at presenting algebra, but it's all in the notes. 194 00:13:26,880 --> 00:13:32,510 But if you take these formulas and you try them on for size, 195 00:13:32,510 --> 00:13:36,140 you will be able to verify that these 196 00:13:36,140 --> 00:13:42,770 give normalized and orthogonal eigenfunctions, which 197 00:13:42,770 --> 00:13:45,700 I recommend, mostly. 198 00:13:45,700 --> 00:13:50,360 OK, we take this expression for the eigenvalues, 199 00:13:50,360 --> 00:13:53,660 and let's just take one of the eigenfunctions 200 00:13:53,660 --> 00:13:58,850 and find out whether it belongs to the correct eigenvalue. 201 00:14:02,120 --> 00:14:06,850 If it doesn't, you've made an algebraic mistake. 202 00:14:06,850 --> 00:14:08,120 It's a very useful thing. 203 00:14:08,120 --> 00:14:13,030 OK, so now we're going to take the results 204 00:14:13,030 --> 00:14:16,450 for this two-level problem solved algebraically, 205 00:14:16,450 --> 00:14:21,610 and at the core of that was the quadratic energy formula. 206 00:14:21,610 --> 00:14:26,650 The quadratic formula for the equation 207 00:14:26,650 --> 00:14:29,448 that determines the energy. 208 00:14:29,448 --> 00:14:32,420 The quadratic formula is always applicable. 209 00:14:32,420 --> 00:14:33,020 It's exact. 210 00:14:35,720 --> 00:14:39,000 So what we end up getting is analytical expressions. 211 00:14:39,000 --> 00:14:44,000 It doesn't matter what the value of the two critical quantities, 212 00:14:44,000 --> 00:14:46,570 delta and x, are. 213 00:14:46,570 --> 00:14:47,720 You have solutions. 214 00:14:54,770 --> 00:14:57,280 So now we're going to take all this stuff 215 00:14:57,280 --> 00:14:59,590 and we're going to start over. 216 00:14:59,590 --> 00:15:03,520 We're going to rearrange it so that we have a different way 217 00:15:03,520 --> 00:15:04,660 of approaching the problem. 218 00:15:08,370 --> 00:15:12,790 So we're going to start talking about matrices 219 00:15:12,790 --> 00:15:14,410 rather than operators. 220 00:15:14,410 --> 00:15:22,380 And so I represent a matrix by a boldface-- 221 00:15:22,380 --> 00:15:28,050 this symbol is how you ask a computer to make it-- 222 00:15:28,050 --> 00:15:32,760 yes, it's how I represent a matrix, instead of a hat, 223 00:15:32,760 --> 00:15:35,590 which is the way you represent an operator. 224 00:15:35,590 --> 00:15:38,790 But now we're going to say every operator is 225 00:15:38,790 --> 00:15:44,080 represented by a matrix rather than a differential operator. 226 00:15:44,080 --> 00:15:54,580 And so this Hamiltonian is E bar, zero, E bar plus delta, v, 227 00:15:54,580 --> 00:16:00,240 v star, minus delta. 228 00:16:00,240 --> 00:16:03,000 And v is usually real. 229 00:16:03,000 --> 00:16:05,480 And another way we can say this is 230 00:16:05,480 --> 00:16:14,350 it's E bar times this fancy 1 with an under bar plus H bar. 231 00:16:14,350 --> 00:16:20,530 So this is a symmetric Hamiltonian matrix. 232 00:16:20,530 --> 00:16:23,450 This is the unit matrix. 233 00:16:23,450 --> 00:16:31,190 And now, since we're going to be doing matrix multiplications, 234 00:16:31,190 --> 00:16:33,620 let me just give you some mnemonics. 235 00:16:33,620 --> 00:16:39,380 So if we have a square matrix multiplying a square matrix, 236 00:16:39,380 --> 00:16:51,919 what we do is we multiply this row by that column, 237 00:16:51,919 --> 00:16:54,335 and we get one number, and you fill out the square matrix. 238 00:16:57,830 --> 00:17:00,680 And with a little practice, this will be permanently 239 00:17:00,680 --> 00:17:03,340 ingrained in your head. 240 00:17:03,340 --> 00:17:08,900 We also can have a matrix multiplying a vector. 241 00:17:08,900 --> 00:17:16,890 And so a matrix multiplying a vector gives a vector, 242 00:17:16,890 --> 00:17:19,690 and this product gives a number here. 243 00:17:24,690 --> 00:17:27,660 And you've probably all seen these sorts of things 244 00:17:27,660 --> 00:17:31,250 or could grasp them very quickly, 245 00:17:31,250 --> 00:17:34,240 but it's useful just to not get confused. 246 00:17:34,240 --> 00:17:40,920 We can also do something like this, 247 00:17:40,920 --> 00:17:46,640 and again, we use the usual vector and we get a vector. 248 00:17:46,640 --> 00:17:48,195 I'm sorry, we get a column. 249 00:17:53,480 --> 00:18:00,560 And this is a difficult symbol to make on a computer, 250 00:18:00,560 --> 00:18:05,560 but you get this first element here like that. 251 00:18:05,560 --> 00:18:12,280 And of course, you can do this times that, 252 00:18:12,280 --> 00:18:14,800 and you get a number. 253 00:18:14,800 --> 00:18:17,120 And you can also do it the other way around. 254 00:18:17,120 --> 00:18:18,670 You can do this times that-- 255 00:18:21,990 --> 00:18:23,730 I'm sorry, don't do that. 256 00:18:28,224 --> 00:18:29,390 And you get a square matrix. 257 00:18:33,150 --> 00:18:35,720 So those are the things that you have to practice, 258 00:18:35,720 --> 00:18:40,790 and it becomes second nature very quickly, 259 00:18:40,790 --> 00:18:45,410 and it's a lot easier than doing differential equations, 260 00:18:45,410 --> 00:18:48,421 or matrices, or integrals. 261 00:18:51,010 --> 00:18:56,380 OK, now, we use this superscript. 262 00:18:59,570 --> 00:19:01,810 This means transpose. 263 00:19:01,810 --> 00:19:05,530 This means complex conjugate and transpose. 264 00:19:09,866 --> 00:19:15,260 The theory deals with this, but when 265 00:19:15,260 --> 00:19:19,970 the Hamiltonian or the matrix you're interested in is real, 266 00:19:19,970 --> 00:19:23,150 the transformation that diagonalizes it 267 00:19:23,150 --> 00:19:24,600 is always just given. 268 00:19:24,600 --> 00:19:26,660 You need this to transpose. 269 00:19:26,660 --> 00:19:28,940 So these two symbols look similar 270 00:19:28,940 --> 00:19:31,670 and you won't have any trouble with that. 271 00:19:31,670 --> 00:19:37,920 And now, we have this kind of symbol. 272 00:19:37,920 --> 00:19:41,900 So this is a normalization. 273 00:19:41,900 --> 00:19:44,480 That should be equal to one, and it 274 00:19:44,480 --> 00:19:48,350 is because one times one is one plus zero times zero. 275 00:19:52,280 --> 00:19:57,600 And we can also look at this, and that's zero 276 00:19:57,600 --> 00:20:03,150 because zero times one is zero and one times zero is zero. 277 00:20:03,150 --> 00:20:05,807 So this is normalization, this is orthogonality. 278 00:20:09,150 --> 00:20:14,260 OK, we're playing with numbers, and we don't really 279 00:20:14,260 --> 00:20:18,580 look at the size, even though the numbers all 280 00:20:18,580 --> 00:20:23,290 are obtained by doing an integral over the wave 281 00:20:23,290 --> 00:20:28,520 functions and the operator, but that's something 282 00:20:28,520 --> 00:20:31,949 that you sort of do in first grade of quantum mechanics, 283 00:20:31,949 --> 00:20:33,740 and you forget how you did it, and you just 284 00:20:33,740 --> 00:20:35,870 know that they're there to be played with. 285 00:20:38,770 --> 00:20:46,660 OK, now, a unitary operator is one 286 00:20:46,660 --> 00:20:52,220 where the conjugate transpose-- 287 00:20:52,220 --> 00:20:59,280 the unitary matrix-- is equal to the inverse, which means 288 00:20:59,280 --> 00:21:06,790 t minus one times t equals one. 289 00:21:06,790 --> 00:21:10,780 So it's nice to be able to derive the inverse of a matrix. 290 00:21:10,780 --> 00:21:13,370 And for certain kinds of matrices, 291 00:21:13,370 --> 00:21:15,580 this is really easy because all you do 292 00:21:15,580 --> 00:21:19,080 is tip along the main diagonal. 293 00:21:19,080 --> 00:21:22,800 There are other matrices where you have to do a lot of work, 294 00:21:22,800 --> 00:21:25,150 but whenever you're dealing with matrices, 295 00:21:25,150 --> 00:21:26,670 you're not doing the work. 296 00:21:26,670 --> 00:21:28,470 The computer is doing the work. 297 00:21:28,470 --> 00:21:32,280 You teach the computer how to do matrix operations, 298 00:21:32,280 --> 00:21:36,580 and even if it's a hard one, the computer says OK, here it is. 299 00:21:40,140 --> 00:21:54,380 OK, so if you have a real symmetric matrix, 300 00:21:54,380 --> 00:22:03,100 then you can say OK, T transpose matrix T 301 00:22:03,100 --> 00:22:12,240 gives a1, a n, zero, zero. 302 00:22:12,240 --> 00:22:17,990 So you can diagonalize a real symmetric matrix 303 00:22:17,990 --> 00:22:20,630 by this kind of a transformation. 304 00:22:20,630 --> 00:22:23,250 That's called an orthogonal transformation. 305 00:22:23,250 --> 00:22:26,930 And if it's not real, then you use the conjugate 306 00:22:26,930 --> 00:22:31,250 transpose, use the dagger, and you still 307 00:22:31,250 --> 00:22:34,980 get the diagonalization. 308 00:22:34,980 --> 00:22:37,320 Now, in most of the books that you'll ever 309 00:22:37,320 --> 00:22:40,110 look at about unitary transformations, 310 00:22:40,110 --> 00:22:42,060 they actually are giving you what's called 311 00:22:42,060 --> 00:22:45,550 the orthogonal transformation, and it's 312 00:22:45,550 --> 00:22:52,040 what works for a real matrix, and I'm going to do that too. 313 00:22:52,040 --> 00:22:55,400 So when we have something like this, 314 00:22:55,400 --> 00:23:00,440 we say that this transformation diagonalizes A, or H, 315 00:23:00,440 --> 00:23:01,810 or whatever. 316 00:23:01,810 --> 00:23:05,716 And the word "diagonalizes" is really important 317 00:23:05,716 --> 00:23:07,340 because that's what you want because it 318 00:23:07,340 --> 00:23:09,620 presents all the eigenvalues. 319 00:23:09,620 --> 00:23:11,900 Remember, one of the things about quantum mechanics 320 00:23:11,900 --> 00:23:14,300 is you have an operator. 321 00:23:14,300 --> 00:23:16,330 You're going to observe something 322 00:23:16,330 --> 00:23:18,010 connected with an operator. 323 00:23:18,010 --> 00:23:21,860 The only things you can get are the eigenvalues. 324 00:23:21,860 --> 00:23:24,154 So here they are, all of them. 325 00:23:24,154 --> 00:23:25,070 That's kind of useful. 326 00:23:28,090 --> 00:23:37,030 OK, so in the Heisenberg picture, 327 00:23:37,030 --> 00:23:41,020 the key equation is the Hamiltonian 328 00:23:41,020 --> 00:23:44,260 as a matrix, some vector-- 329 00:23:47,920 --> 00:23:52,240 OK, this is the analog of the Schrodinger equation, 330 00:23:52,240 --> 00:23:55,180 but it's the Heisenberg equation. 331 00:23:55,180 --> 00:23:58,159 And mostly, it's just notation, and you 332 00:23:58,159 --> 00:23:59,200 have to get used to that. 333 00:24:07,360 --> 00:24:13,060 We want to find the vectors that are eigenvectors 334 00:24:13,060 --> 00:24:16,672 of this Hamiltonian with eigenvalue E. 335 00:24:16,672 --> 00:24:18,130 It's just like this the Schrodinger 336 00:24:18,130 --> 00:24:22,270 equation, except it's now looking for eigenvectors rather 337 00:24:22,270 --> 00:24:26,290 than eigenfunctions and eigenenergies. 338 00:24:26,290 --> 00:24:29,180 Now, in order to solve this problem, 339 00:24:29,180 --> 00:24:33,820 we exploit this kind of transformation, 340 00:24:33,820 --> 00:24:45,074 and we insert T T dagger between H and c, and that's just one. 341 00:24:45,074 --> 00:24:47,240 So we don't even have to worry about the other side. 342 00:24:50,860 --> 00:24:52,510 We're just playing with matrices, 343 00:24:52,510 --> 00:24:57,610 but they look like functions or variables, and everything is-- 344 00:24:57,610 --> 00:25:01,030 it's really neat how beautiful linear algebra 345 00:25:01,030 --> 00:25:05,830 is because you are now dealing with an infinite number 346 00:25:05,830 --> 00:25:07,300 of equations at once. 347 00:25:07,300 --> 00:25:10,150 You're dealing with these objects 348 00:25:10,150 --> 00:25:13,450 and you're using your insight from algebra 349 00:25:13,450 --> 00:25:18,940 as much as possible in order to figure out what's going on. 350 00:25:18,940 --> 00:25:21,070 Really beautiful. 351 00:25:21,070 --> 00:25:25,640 OK, and so now we must multiply this equation on the left by T 352 00:25:25,640 --> 00:25:26,140 dagger. 353 00:25:36,060 --> 00:25:39,490 OK, I'm dropping the under-bars now. 354 00:25:39,490 --> 00:25:45,310 OK, so now we say OK, here we have H twiddle, 355 00:25:45,310 --> 00:25:50,830 and here we have c twiddle, and here we have c twiddle. 356 00:25:50,830 --> 00:25:59,350 So this is now a different eigenvector equation, 357 00:25:59,350 --> 00:26:04,450 but we insist that this guy, H twiddle, 358 00:26:04,450 --> 00:26:11,921 looks like E1, E2, En, 0,0. 359 00:26:11,921 --> 00:26:16,000 A diagonal matrix, where all the eigenvalues 360 00:26:16,000 --> 00:26:19,310 are along the diagonal. 361 00:26:19,310 --> 00:26:25,820 And so this is what we want, and lo and behold, 362 00:26:25,820 --> 00:26:29,060 this is what we need in order to say, 363 00:26:29,060 --> 00:26:35,980 well, yeah, this thing has to be the eigenvector of this for one 364 00:26:35,980 --> 00:26:40,360 of the eigenvalues because this is an eigenvalue equation 365 00:26:40,360 --> 00:26:42,500 or an eigenvector equation. 366 00:26:42,500 --> 00:26:45,760 So if we can diagonalize the Hamiltonian, 367 00:26:45,760 --> 00:26:49,090 the transformation that diagonalizes the Hamiltonian 368 00:26:49,090 --> 00:26:55,630 gives you the linear combination of basis vectors 369 00:26:55,630 --> 00:27:00,070 that is the eigenvector, and we'll 370 00:27:00,070 --> 00:27:02,830 talk about this some more. 371 00:27:02,830 --> 00:27:06,370 So for the two-level problem, we want 372 00:27:06,370 --> 00:27:13,510 to find E plus, 0, 0, E minus. 373 00:27:13,510 --> 00:27:21,770 And usually, E plus is the higher energy eigenvalue 374 00:27:21,770 --> 00:27:23,480 than E minus. 375 00:27:23,480 --> 00:27:29,450 Always, when you do this stuff, you get eigenvalues 376 00:27:29,450 --> 00:27:32,390 and you get eigenvectors. 377 00:27:32,390 --> 00:27:35,210 And frequently, when you do the algebra as opposed 378 00:27:35,210 --> 00:27:36,980 to the computer during the algebra, 379 00:27:36,980 --> 00:27:39,590 you don't know that a particular eigenvector 380 00:27:39,590 --> 00:27:43,160 belongs to which eigenvalues. 381 00:27:43,160 --> 00:27:45,220 So it's useful to have a couple of things 382 00:27:45,220 --> 00:27:47,830 that you normally insist on. 383 00:27:47,830 --> 00:27:51,280 And so I like to label these things 384 00:27:51,280 --> 00:27:53,380 by plus and minus, corresponding to which 385 00:27:53,380 --> 00:27:55,540 is higher energy and which is lower. 386 00:27:55,540 --> 00:27:57,730 You could also say, well, the plus is 387 00:27:57,730 --> 00:28:01,530 going to correspond to a plus linear combination somewhere, 388 00:28:01,530 --> 00:28:03,960 but that's really dangerous. 389 00:28:09,080 --> 00:28:11,870 So now, let's just play a little bit. 390 00:28:22,120 --> 00:28:30,220 So we have simplified H magically so far 391 00:28:30,220 --> 00:28:32,890 to diagonal form. 392 00:28:32,890 --> 00:28:35,110 So H C twiddle-- 393 00:28:39,020 --> 00:28:49,510 I'm sorry, yes H C twiddle is going to be E c twiddle. 394 00:28:54,500 --> 00:29:10,790 So H c plus is going to be E plus 0, 0, E minus, one, 395 00:29:10,790 --> 00:29:19,950 zero because that gives us E plus times one 396 00:29:19,950 --> 00:29:21,840 and zero times one. 397 00:29:25,070 --> 00:29:26,440 So multiply these two things. 398 00:29:26,440 --> 00:29:29,740 We have a column vector, and that's the same thing as E 399 00:29:29,740 --> 00:29:31,780 plus times one, zero. 400 00:29:39,060 --> 00:29:43,920 And we do the same sort of thing to-- 401 00:29:43,920 --> 00:29:46,770 instead of using c plus, we use c minus. 402 00:29:46,770 --> 00:29:50,250 That's a zero, one, and that will give us E minus times 403 00:29:50,250 --> 00:29:52,390 zero, one. 404 00:29:52,390 --> 00:29:55,840 This is all just playing with notation 405 00:29:55,840 --> 00:29:58,190 and we're about to start doing some work. 406 00:29:58,190 --> 00:30:12,350 OK, so T dagger times c is supposedly equal to c plus. 407 00:30:12,350 --> 00:30:16,310 OK, and so well, we can write this formula 408 00:30:16,310 --> 00:30:22,240 in a schematic way, and so we have T dagger. 409 00:30:29,480 --> 00:30:32,540 Now, I always remember this because there 410 00:30:32,540 --> 00:30:36,260 used to be something analogous to Coke and Pepsi called Royal 411 00:30:36,260 --> 00:30:41,000 Crown Cola, and for Royal Crown, that just reminds me 412 00:30:41,000 --> 00:30:43,910 that row first, column second. 413 00:30:43,910 --> 00:30:48,230 I don't believe that any of you have ever heard of Royal Crown, 414 00:30:48,230 --> 00:30:52,321 but you could think of some other mnemonic. 415 00:30:52,321 --> 00:30:55,100 Now, it's really important to keep the rows and the columns 416 00:30:55,100 --> 00:30:57,710 straight. 417 00:30:57,710 --> 00:31:02,240 So we have 1, 1, and T dagger. 418 00:31:02,240 --> 00:31:03,800 Now, what goes here? 419 00:31:03,800 --> 00:31:08,280 This is in the first row, second column. 420 00:31:08,280 --> 00:31:11,270 So what do I put here? 421 00:31:11,270 --> 00:31:12,690 Yeah. 422 00:31:12,690 --> 00:31:13,800 You could even say it. 423 00:31:16,530 --> 00:31:20,760 OK, and here we have T dagger 2,1, 424 00:31:20,760 --> 00:31:29,940 and here we have T dagger 2,2. 425 00:31:29,940 --> 00:31:45,160 Now, if we multiply one, zero, because that's 426 00:31:45,160 --> 00:32:02,450 what we're supposed to do here, we'll get T 1,1 plus T 1,2 427 00:32:02,450 --> 00:32:11,850 times zero, and then T 2,1 plus T 2,2 times zero. 428 00:32:11,850 --> 00:32:19,870 OK, and that's simply T 1,1 dagger times one, 429 00:32:19,870 --> 00:32:26,760 zero plus T 2,1 dagger times zero, one. 430 00:32:26,760 --> 00:32:30,730 So this thing gives the linear combination of the basis 431 00:32:30,730 --> 00:32:34,770 vectors that is equal to a particular eigenvector. 432 00:32:46,930 --> 00:32:52,690 So that means if we can find T, we can get T dagger, 433 00:32:52,690 --> 00:33:00,520 and we can get E plus and E minus, and c plus and c minus. 434 00:33:00,520 --> 00:33:04,385 So we have completely solved the problem if we know what T is. 435 00:33:07,290 --> 00:33:11,640 Well, with a two-level problem, we 436 00:33:11,640 --> 00:33:14,970 know algebraically that there is such a T, 437 00:33:14,970 --> 00:33:19,770 and that it's analytically determined. 438 00:33:19,770 --> 00:33:22,770 There is another way of approaching this, and that 439 00:33:22,770 --> 00:33:31,050 is to say the general orthogonal transformation, 440 00:33:31,050 --> 00:33:33,570 which we will call a unitary transformation, 441 00:33:33,570 --> 00:33:36,440 but it's missing a little bit of stuff 442 00:33:36,440 --> 00:33:38,700 if it really wants to be unitary. 443 00:33:38,700 --> 00:33:40,080 I'm going to call it-- 444 00:33:40,080 --> 00:33:48,940 so T dagger is cos theta, sin theta, 445 00:33:48,940 --> 00:33:52,060 minus sin theta, cos theta. 446 00:33:55,500 --> 00:33:58,530 So this is a matrix which is determined by one thing, theta. 447 00:34:01,100 --> 00:34:04,750 We want to find what theta needs to be in order 448 00:34:04,750 --> 00:34:06,400 to diagonalize the matrix. 449 00:34:09,440 --> 00:34:14,440 Now, since we know we're talking about sines and cosines, 450 00:34:14,440 --> 00:34:18,010 and that there is one theta, we abbreviate this to c, s, 451 00:34:18,010 --> 00:34:23,639 minus s, c because the algebra is heinous. 452 00:34:23,639 --> 00:34:29,860 Not as bad as in the Schrodinger picture, but it's terrible, 453 00:34:29,860 --> 00:34:34,070 and so you want to compress the symbols as much as possible. 454 00:34:34,070 --> 00:34:42,620 OK, so we want T dagger HT because that's H twiddle. 455 00:34:42,620 --> 00:34:49,219 That's the thing we want, and we want T dagger HT. 456 00:34:49,219 --> 00:34:56,150 And now since we've expressed the T in this form, 457 00:34:56,150 --> 00:34:58,760 we can multiply this out. 458 00:34:58,760 --> 00:35:09,190 And so we have c, c, minus s, c, delta, v, v, delta. 459 00:35:09,190 --> 00:35:15,730 Delta, v, v, minus delta, c, minus s, s, c. 460 00:35:15,730 --> 00:35:18,175 So we have three two by two matrices to multiply. 461 00:35:20,680 --> 00:35:23,420 Now, that's not something you do in your head. 462 00:35:23,420 --> 00:35:25,480 You could do two. 463 00:35:25,480 --> 00:35:27,430 So you multiply these two, and then you 464 00:35:27,430 --> 00:35:32,810 multiply by that, and the result-- 465 00:35:32,810 --> 00:35:37,030 I would be here for hours doing this, 466 00:35:37,030 --> 00:35:42,280 and you wouldn't learn anything except that I'm a real klutz. 467 00:35:42,280 --> 00:35:44,710 I should write this on its own board 468 00:35:44,710 --> 00:35:46,260 because it's really important. 469 00:36:00,910 --> 00:36:04,690 So that matrix becomes a big matrix, 470 00:36:04,690 --> 00:36:15,640 c squared minus s squared times delta plus 2sc times V, 471 00:36:15,640 --> 00:36:26,530 and c squared minus s squared times V minus 2cs delta. 472 00:36:26,530 --> 00:36:31,650 And we have the same thing down here, c squared minus s squared 473 00:36:31,650 --> 00:36:36,430 times V minus 2cs times delta. 474 00:36:36,430 --> 00:36:42,280 And the last one is minus c squared minus s squared 475 00:36:42,280 --> 00:36:51,260 times delta minus 2cs times V. So this 476 00:36:51,260 --> 00:36:55,650 is what we get when we take the general form 477 00:36:55,650 --> 00:36:59,010 for the unitary transformation, and transform 478 00:36:59,010 --> 00:37:02,240 the Hamiltonian with it. 479 00:37:02,240 --> 00:37:04,530 And the first thing we do is we say, 480 00:37:04,530 --> 00:37:09,480 well, we want this to be zero. 481 00:37:09,480 --> 00:37:11,230 If this is zero, then this is zero, right? 482 00:37:14,980 --> 00:37:18,540 So this turns out to be an equation that 483 00:37:18,540 --> 00:37:21,380 tells us what theta has to be. 484 00:37:21,380 --> 00:37:24,020 OK, and we also know from trigonometry, 485 00:37:24,020 --> 00:37:28,480 c squared minus s squared is what? 486 00:37:39,890 --> 00:37:46,010 I'm actually jumping ahead, but it's just cosine two theta, 487 00:37:46,010 --> 00:37:54,010 and 2sc is sine two theta. 488 00:37:54,010 --> 00:37:57,570 So we're going to get a simplification based on this, 489 00:37:57,570 --> 00:38:00,480 but now let's just say we want this to be zero. 490 00:38:00,480 --> 00:38:05,550 So that means that c squared minus s squared times V has 491 00:38:05,550 --> 00:38:07,980 to be equal to 2cs times delta. 492 00:38:14,050 --> 00:38:15,610 Which way am I going? 493 00:38:15,610 --> 00:38:20,770 2cs over c squared minus s squared is V over delta. 494 00:38:27,060 --> 00:38:29,580 Well, that looks pretty good, especially 495 00:38:29,580 --> 00:38:32,550 because this is cosine-- 496 00:38:35,620 --> 00:38:42,930 this is sine two theta, and this is cosine two theta, which 497 00:38:42,930 --> 00:38:47,220 is tan theta is equal to this. 498 00:38:50,060 --> 00:38:52,370 So now we have a simple equation. 499 00:38:52,370 --> 00:38:56,750 We have the theta, and we have the V and a D-- 500 00:38:56,750 --> 00:38:58,500 a V over delta. 501 00:39:01,057 --> 00:39:01,890 I shouldn't do that. 502 00:39:07,030 --> 00:39:08,370 So now I can cover this again. 503 00:39:17,050 --> 00:39:20,500 So we can take this equation and solve it, 504 00:39:20,500 --> 00:39:35,050 and we have that theta is equal to 1/2 inverse tangent of V 505 00:39:35,050 --> 00:39:37,970 over delta. 506 00:39:37,970 --> 00:39:38,470 There it is. 507 00:39:38,470 --> 00:39:46,964 That's an analytic result. So for any value of V over delta, 508 00:39:46,964 --> 00:39:47,880 we know what theta is. 509 00:39:51,750 --> 00:39:54,060 That's not an iterative solution. 510 00:39:54,060 --> 00:39:58,590 That's complete analytical result, and that's fantastic, 511 00:39:58,590 --> 00:40:04,680 and it says, just like with the quadratic formula, which 512 00:40:04,680 --> 00:40:09,940 we used to look at the original Hamiltonian 513 00:40:09,940 --> 00:40:12,670 at the eigenvalues of the original-- 514 00:40:12,670 --> 00:40:17,020 well, yeah, it says no matter what, there is a solution, 515 00:40:17,020 --> 00:40:19,600 and you can express this solution 516 00:40:19,600 --> 00:40:21,320 as some combination of V and delta. 517 00:40:26,700 --> 00:40:32,440 OK, and so when you do that, you get that the energy levels are 518 00:40:32,440 --> 00:40:44,400 E bar plus or minus delta times cos two theta plus V times sin 519 00:40:44,400 --> 00:40:45,600 two theta. 520 00:40:49,240 --> 00:40:50,620 And there's no square root here. 521 00:40:50,620 --> 00:40:51,820 Why do you know that? 522 00:40:51,820 --> 00:40:53,350 Well, this is dimensionless. 523 00:40:53,350 --> 00:40:57,000 This is the units of energy, and so 524 00:40:57,000 --> 00:40:58,704 square roots keep coming popping up, 525 00:40:58,704 --> 00:41:00,120 but you don't want to put one here 526 00:41:00,120 --> 00:41:01,286 because that would be wrong. 527 00:41:03,780 --> 00:41:05,820 Even if you didn't do the derivation, 528 00:41:05,820 --> 00:41:07,320 if you saw a square root here, you'd 529 00:41:07,320 --> 00:41:11,460 know somebody is just writing down things from memory 530 00:41:11,460 --> 00:41:16,040 or copying badly, and making corrections. 531 00:41:16,040 --> 00:41:24,810 And that leads to E plus minus is equal to E bar plus or minus 532 00:41:24,810 --> 00:41:29,091 delta squared plus V squared, and there is a square root 533 00:41:29,091 --> 00:41:30,270 there. 534 00:41:30,270 --> 00:41:34,350 This is what we derived via the quadratic formula. 535 00:41:34,350 --> 00:41:38,130 We knew this, and it came out to be the same. 536 00:41:38,130 --> 00:41:40,890 Well, it better have. 537 00:41:40,890 --> 00:41:43,950 And we can also determine what T is. 538 00:41:46,890 --> 00:41:50,130 And I'm not going to write it, it's in the notes. 539 00:41:50,130 --> 00:41:54,390 It's a lot of symbols, but it's something-- 540 00:41:54,390 --> 00:42:00,830 it's so compressed that you can guess the form, 541 00:42:00,830 --> 00:42:04,600 and so you should look at that. 542 00:42:04,600 --> 00:42:09,850 We derived the eigenfunctions of the Hamiltonian, 543 00:42:09,850 --> 00:42:14,140 and they are exactly the same as what we get here. 544 00:42:14,140 --> 00:42:23,410 And remember that the column of T dagger or T transpose 545 00:42:23,410 --> 00:42:32,980 is eigenvector, and sometimes we want 546 00:42:32,980 --> 00:42:34,738 to know those eigenvectors. 547 00:42:42,230 --> 00:42:43,170 We're doing semi-OK. 548 00:43:02,080 --> 00:43:04,390 What happens if we go beyond the two-level problem? 549 00:43:08,070 --> 00:43:14,100 Well, you know from algebra that there is no general solution 550 00:43:14,100 --> 00:43:16,760 to a cubic equation. 551 00:43:16,760 --> 00:43:19,040 There are some limited range over which 552 00:43:19,040 --> 00:43:23,240 there is an analytic solution, but mostly, you 553 00:43:23,240 --> 00:43:26,960 don't use a cubic formula to solve the cubic equation. 554 00:43:26,960 --> 00:43:30,930 You do some kind of iteration. 555 00:43:30,930 --> 00:43:36,180 So for the number of levels greater than two, 556 00:43:36,180 --> 00:43:38,520 we know we're going to have a problem because just 557 00:43:38,520 --> 00:43:42,330 approaching it by transformation theory or linear algebra as 558 00:43:42,330 --> 00:43:48,130 opposed to the Schrodinger picture-- 559 00:43:48,130 --> 00:43:51,850 if you can't get a solution in one picture which 560 00:43:51,850 --> 00:43:56,290 requires solving an algebraic equation, 561 00:43:56,290 --> 00:43:59,410 you're not going to get it by playing 562 00:43:59,410 --> 00:44:01,210 with these unitary matrices. 563 00:44:07,290 --> 00:44:14,430 So we're going to be approaching a problem where 564 00:44:14,430 --> 00:44:20,550 we have to find the eigenvalues and the elements 565 00:44:20,550 --> 00:44:23,700 of the transformation matrix in some sort 566 00:44:23,700 --> 00:44:25,170 of computer-based way. 567 00:44:25,170 --> 00:44:26,700 We're not going to do it. 568 00:44:26,700 --> 00:44:29,850 It would be nuts, even for a three by three. 569 00:44:29,850 --> 00:44:34,110 Although, I will give you an exam problem which 570 00:44:34,110 --> 00:44:36,435 will be a three by three, and you're 571 00:44:36,435 --> 00:44:38,310 going to use perturbation theory to solve it. 572 00:44:41,030 --> 00:44:42,920 I haven't told you about perturbation theory. 573 00:44:42,920 --> 00:44:48,230 That's going to be next week, but we're leading up to it. 574 00:44:48,230 --> 00:44:54,470 OK, but now the point is we have the machinery in place. 575 00:44:54,470 --> 00:45:08,300 We have exactly solved problems, and we have the key parameters 576 00:45:08,300 --> 00:45:10,340 for exactly solved problems. 577 00:45:10,340 --> 00:45:11,360 So the structural-- 578 00:45:18,530 --> 00:45:21,230 So for the harmonic oscillator, we 579 00:45:21,230 --> 00:45:26,750 have the force constant and the reduced mass. 580 00:45:26,750 --> 00:45:36,590 For the particle in the box, we have the bottom of the box, v0, 581 00:45:36,590 --> 00:45:40,440 and we have the width of the box. 582 00:45:40,440 --> 00:45:45,030 For the rigid rotor, we're going to have 583 00:45:45,030 --> 00:45:52,780 the reduced mass and we're going to have the internuclear 584 00:45:52,780 --> 00:45:53,280 distance. 585 00:45:55,810 --> 00:46:00,510 There's things that determine all of the energy 586 00:46:00,510 --> 00:46:02,790 levels for exact solved problems, 587 00:46:02,790 --> 00:46:04,740 and they are basically the things that 588 00:46:04,740 --> 00:46:08,040 appear in the Hamiltonian, and we call 589 00:46:08,040 --> 00:46:11,500 them structural parameters. 590 00:46:11,500 --> 00:46:12,820 And we have energy levels. 591 00:46:18,410 --> 00:46:22,020 And often, these are some function of a quantum number. 592 00:46:25,170 --> 00:46:27,870 This is what we can observe. 593 00:46:27,870 --> 00:46:32,600 We observe the energy levels, and we represent them 594 00:46:32,600 --> 00:46:36,530 by some formula, and the coefficients 595 00:46:36,530 --> 00:46:42,307 of the quantum numbers relate to these things 596 00:46:42,307 --> 00:46:43,140 that we really what. 597 00:46:46,390 --> 00:46:49,910 So when you're not dealing with an exactly solved problem-- 598 00:46:49,910 --> 00:46:52,030 like instead of having a harmonic oscillator, 599 00:46:52,030 --> 00:46:55,030 you have a harmonic oscillator with something at the bottom, 600 00:46:55,030 --> 00:46:58,570 or you have a particle in a box with a slant bottom, 601 00:46:58,570 --> 00:47:02,200 or you have a rigid rotor where it's not rigid, 602 00:47:02,200 --> 00:47:06,020 you have additional terms in the Hamiltonian, 603 00:47:06,020 --> 00:47:08,540 and they are going to-- we are going to use perturbation 604 00:47:08,540 --> 00:47:12,710 theory to relate the numerical values of the things 605 00:47:12,710 --> 00:47:17,560 we want to know to the things we can observe, 606 00:47:17,560 --> 00:47:23,860 and perturbation theory gives us the formulas of the quantum 607 00:47:23,860 --> 00:47:30,220 numbers, and tells us the explicit relationships 608 00:47:30,220 --> 00:47:35,380 of the coefficients of each term in the quantum number 609 00:47:35,380 --> 00:47:38,640 expression to the things we want. 610 00:47:38,640 --> 00:47:41,880 This is how we learn everything. 611 00:47:41,880 --> 00:47:46,500 When we do spectroscopy, we measure these energy levels, 612 00:47:46,500 --> 00:47:52,290 and these energy levels encode all of the structure 613 00:47:52,290 --> 00:47:53,340 and all of the dynamics. 614 00:47:56,950 --> 00:47:58,840 It's really neat. 615 00:47:58,840 --> 00:48:01,630 OK, now, the last thing I want to-- do I have time? 616 00:48:01,630 --> 00:48:02,263 Yeah, maybe. 617 00:48:15,200 --> 00:48:19,150 Remember when we do time dependent quantum mechanics 618 00:48:19,150 --> 00:48:21,627 with a time independent Hamiltonian. 619 00:48:26,870 --> 00:48:32,090 We want to have psi of x and t. 620 00:48:35,450 --> 00:48:41,150 And usually, we're given psi of x, t equals zero. 621 00:48:41,150 --> 00:48:46,040 We're given the initial state, and that initial state, 622 00:48:46,040 --> 00:48:47,540 if this is an interesting problem, 623 00:48:47,540 --> 00:48:51,050 is not an eigenstate of the Hamiltonian. 624 00:48:51,050 --> 00:48:53,880 It's a linear combination of eigenstates 625 00:48:53,880 --> 00:48:57,860 of that Hamiltonian, and the kinds of flux 626 00:48:57,860 --> 00:49:01,070 we almost always use to test our insight, 627 00:49:01,070 --> 00:49:04,010 or actually, because they're feasible experimentally, 628 00:49:04,010 --> 00:49:09,440 is the initial state is one of the eigenstates of an exactly 629 00:49:09,440 --> 00:49:11,500 solved problem. 630 00:49:11,500 --> 00:49:18,110 It's some special combination of easy stuff. 631 00:49:18,110 --> 00:49:22,570 And we need to know how the coefficient-- 632 00:49:22,570 --> 00:49:29,892 this thing-- how that is expressed as j 633 00:49:29,892 --> 00:49:40,280 equals one to n of c j psi j. 634 00:49:40,280 --> 00:49:46,790 Because if we can express the t equals zero pluck 635 00:49:46,790 --> 00:49:49,220 as a linear combination of the eigenstates, 636 00:49:49,220 --> 00:49:58,160 then it's just a matter of mindless manipulation 637 00:49:58,160 --> 00:50:04,760 because we have c j, e to the minus i e j t over h bar times 638 00:50:04,760 --> 00:50:05,780 j. 639 00:50:05,780 --> 00:50:08,190 Bang, it's done. 640 00:50:08,190 --> 00:50:13,280 So what we want to know is how to go from a not-eigenstate 641 00:50:13,280 --> 00:50:15,530 to an eigenstate. 642 00:50:15,530 --> 00:50:17,540 And lo and behold, that's just the inverse 643 00:50:17,540 --> 00:50:20,960 of the transformation. 644 00:50:20,960 --> 00:50:25,050 So what we want to know is OK, since I don't have time 645 00:50:25,050 --> 00:50:28,210 to spell it out for you exactly, we 646 00:50:28,210 --> 00:50:32,590 have t dagger, which relates the zero order 647 00:50:32,590 --> 00:50:36,100 states to the eigenstates. 648 00:50:36,100 --> 00:50:39,940 We want to go in the opposite direction. 649 00:50:39,940 --> 00:50:41,810 We want the inverse transformation. 650 00:50:41,810 --> 00:50:47,890 So we want t, or we want to take instead of the columns of t 651 00:50:47,890 --> 00:50:51,680 dagger, we take the rows. 652 00:50:51,680 --> 00:50:57,390 And so if you have a machine or a brain-- 653 00:50:57,390 --> 00:50:59,000 and I'm not doubting this!-- 654 00:50:59,000 --> 00:51:03,770 that enables you to write down all of the elements in the t 655 00:51:03,770 --> 00:51:08,990 dagger matrix, you are armed to go both from zero order 656 00:51:08,990 --> 00:51:12,620 states to eigenstates, and from plucks 657 00:51:12,620 --> 00:51:15,780 to time evolving wave packets. 658 00:51:15,780 --> 00:51:18,030 It's really beautiful and simple. 659 00:51:18,030 --> 00:51:21,270 And normally, when it's presented, 660 00:51:21,270 --> 00:51:24,390 these are presented as separate project problems, 661 00:51:24,390 --> 00:51:27,960 and the whole point is you've got a unified picture that 662 00:51:27,960 --> 00:51:32,310 enables you to go get whatever you need in a simple way, 663 00:51:32,310 --> 00:51:36,240 as long as a computer is able to diagonalize 664 00:51:36,240 --> 00:51:37,950 your critical matrices. 665 00:51:42,477 --> 00:51:47,800 Well, I don't have time to talk about this in any detail, 666 00:51:47,800 --> 00:51:55,380 but if we look at the eigenfunctions or eigenvectors 667 00:51:55,380 --> 00:52:00,550 for the two-level problem, and we do power series expansions 668 00:52:00,550 --> 00:52:05,900 in theta, where theta is V over d-- 669 00:52:05,900 --> 00:52:09,620 theta is also called the mixing angle-- 670 00:52:09,620 --> 00:52:13,100 we discover that we have some formulas which 671 00:52:13,100 --> 00:52:17,570 says that the energy levels-- 672 00:52:17,570 --> 00:52:26,630 let's say the j-th energy level is equal to E bar plus-- 673 00:52:26,630 --> 00:52:30,490 now I'm doing this and I have to somehow grab something 674 00:52:30,490 --> 00:52:31,080 from in here. 675 00:52:36,980 --> 00:52:45,360 We have a sum of k not equal to j of V-- 676 00:52:45,360 --> 00:52:57,890 of H. 677 00:52:57,890 --> 00:52:59,840 This is the formula for the correction 678 00:52:59,840 --> 00:53:05,960 of the energy by a second order perturbation theory, 679 00:53:05,960 --> 00:53:09,650 and we can also write the formula for the corrected wave 680 00:53:09,650 --> 00:53:12,170 function. 681 00:53:12,170 --> 00:53:15,590 By doing a power series expansion 682 00:53:15,590 --> 00:53:21,500 in terms of powers of V/d or theta, 683 00:53:21,500 --> 00:53:26,540 we find what the structure has to be for the solutions 684 00:53:26,540 --> 00:53:29,150 to the general problem when n is not two. 685 00:53:31,860 --> 00:53:36,380 I'll develop the formal theory for non-degenerate perturbation 686 00:53:36,380 --> 00:53:40,880 theory in Monday's lecture, and that's really empowering. 687 00:53:40,880 --> 00:53:45,650 It's really ugly, but it gives you the answers 688 00:53:45,650 --> 00:53:47,960 to essentially every problem you will ever 689 00:53:47,960 --> 00:53:51,300 face in quantum mechanics. 690 00:53:51,300 --> 00:53:54,160 OK, have a nice weekend.