1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,250 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,250 --> 00:00:18,220 at ocw.mit.edu. 8 00:00:22,190 --> 00:00:26,990 ROBERT FIELD: Last lecture, I did a fairly standard treatment 9 00:00:26,990 --> 00:00:29,750 of the harmonic oscillator, which is not 10 00:00:29,750 --> 00:00:31,430 supposed to make you excited, but just 11 00:00:31,430 --> 00:00:33,320 to see that you can do this. 12 00:00:33,320 --> 00:00:38,000 And the path to the solution was to define 13 00:00:38,000 --> 00:00:41,630 dimensionless position coordinate, 14 00:00:41,630 --> 00:00:45,680 and then you get a dimensionless Schrodinger equation. 15 00:00:45,680 --> 00:00:51,210 And the solution of that involves two steps. 16 00:00:51,210 --> 00:00:57,950 One is to insist that the solutions to the Schrodinger 17 00:00:57,950 --> 00:01:02,150 equation have an exponentially damped form. 18 00:01:02,150 --> 00:01:03,710 And then the Schrodinger equation 19 00:01:03,710 --> 00:01:07,441 is transformed into a new equation called the Hermite 20 00:01:07,441 --> 00:01:07,940 equation. 21 00:01:10,499 --> 00:01:13,040 You shouldn't get all excited about that-- the mathematicians 22 00:01:13,040 --> 00:01:14,650 take care of it. 23 00:01:14,650 --> 00:01:20,420 And so the solutions to this Hermite differential equation 24 00:01:20,420 --> 00:01:25,170 gives you a set of orthogonal and normalized wave functions. 25 00:01:25,170 --> 00:01:29,090 They give you the energy levels expressed as a quantum number 26 00:01:29,090 --> 00:01:31,940 plus 1/2, times constant. 27 00:01:31,940 --> 00:01:37,920 The energy levels with the quantum number v 28 00:01:37,920 --> 00:01:42,320 are even or odd in v, and they're even or odd in psi. 29 00:01:42,320 --> 00:01:46,600 Even v corresponds to even psi. 30 00:01:46,600 --> 00:01:50,000 V is the number of internal nodes, 31 00:01:50,000 --> 00:01:53,570 and there are all sorts of things you can do to say, 32 00:01:53,570 --> 00:01:57,080 well, I expect if I know how to do certain things 33 00:01:57,080 --> 00:01:58,772 in classical mechanics, they're are 34 00:01:58,772 --> 00:02:00,980 going to come out pretty much the same way in quantum 35 00:02:00,980 --> 00:02:02,490 mechanics. 36 00:02:02,490 --> 00:02:04,940 So that's the standard structure, 37 00:02:04,940 --> 00:02:10,130 but more importantly, I want you to have in your head 38 00:02:10,130 --> 00:02:14,570 the pictures of the wave functions, the idea that there 39 00:02:14,570 --> 00:02:19,476 is a zero point energy, and that there's a reason for that-- 40 00:02:19,476 --> 00:02:22,520 that the wave functions have tails extending out 41 00:02:22,520 --> 00:02:26,550 into the non-classical, or the classically forbidden region. 42 00:02:26,550 --> 00:02:30,380 And this turns out to be the beginning of tunneling. 43 00:02:30,380 --> 00:02:36,680 You'll be looking at tunneling more specifically. 44 00:02:36,680 --> 00:02:40,650 I also want you to know how the spacing of nodes is, 45 00:02:40,650 --> 00:02:45,950 and that involves generalization of the bright idea 46 00:02:45,950 --> 00:02:47,960 that the wavelength is h over p. 47 00:02:47,960 --> 00:02:51,260 But if the potential is not constant, 48 00:02:51,260 --> 00:02:53,720 then p is a function of x. 49 00:02:53,720 --> 00:02:58,020 This is not the quantum mechanical operator, 50 00:02:58,020 --> 00:03:00,950 this is a function which provides you 51 00:03:00,950 --> 00:03:03,770 with a lot of intuition. 52 00:03:03,770 --> 00:03:08,110 And then if you know where the node spacings are, and you 53 00:03:08,110 --> 00:03:12,260 know the shape of the envelope, you 54 00:03:12,260 --> 00:03:14,150 have basically everything you need 55 00:03:14,150 --> 00:03:18,470 to have a classical sense of what's going on. 56 00:03:18,470 --> 00:03:23,090 And then-- I guess it's supposed to be hidden-- 57 00:03:23,090 --> 00:03:27,410 I did a little bit of semiclassical theory, 58 00:03:27,410 --> 00:03:30,650 and I showed that if you integrate from the left turning 59 00:03:30,650 --> 00:03:32,690 point to the right turning point at a given 60 00:03:32,690 --> 00:03:37,230 energy of this momentum function, 61 00:03:37,230 --> 00:03:42,010 you get h over 2 times the number of nodes. 62 00:03:42,010 --> 00:03:44,380 And this is the semiclassical quantization-- 63 00:03:44,380 --> 00:03:47,380 it's incredibly important, and it's 64 00:03:47,380 --> 00:03:51,070 useful either as an exact or approximate result 65 00:03:51,070 --> 00:03:53,710 for all one-dimensional problems. 66 00:03:53,710 --> 00:03:57,160 And so it tells you how to begin. 67 00:03:57,160 --> 00:03:59,500 Now, before I start talking about what 68 00:03:59,500 --> 00:04:05,800 we're going to do today, I want to stress where we're going. 69 00:04:05,800 --> 00:04:09,525 So we're going to be looking at some exactly solved problems. 70 00:04:24,300 --> 00:04:29,140 And so we have a particle in a box, the harmonic oscillator, 71 00:04:29,140 --> 00:04:31,980 the hydrogen atoms-- you have to count them-- 72 00:04:31,980 --> 00:04:32,910 and the rigid rotor. 73 00:04:35,980 --> 00:04:40,660 Now, all of these problems have an infinite number 74 00:04:40,660 --> 00:04:46,180 eigenfunctions, an infinite number of energy levels, 75 00:04:46,180 --> 00:04:52,020 and that's intimidating, but it's true. 76 00:04:52,020 --> 00:04:55,370 Now, these infinite number of functions 77 00:04:55,370 --> 00:04:59,980 are explicit functions of the quantum number. 78 00:04:59,980 --> 00:05:04,160 And so we have an infinite number, 79 00:05:04,160 --> 00:05:07,610 but in order to describe systems, 80 00:05:07,610 --> 00:05:10,320 we're going to be calculating integrals. 81 00:05:10,320 --> 00:05:12,480 We're going to be calculating a lot of integrals 82 00:05:12,480 --> 00:05:15,910 between these infinite number of functions. 83 00:05:15,910 --> 00:05:19,320 So we have an infinity squared of integrals. 84 00:05:22,120 --> 00:05:25,300 Well, that shouldn't scare you because what 85 00:05:25,300 --> 00:05:28,960 I'm going to show you is that all of the integrals that we 86 00:05:28,960 --> 00:05:33,430 are going to encounter are explicit functions 87 00:05:33,430 --> 00:05:35,690 of the quantum numbers, and they have 88 00:05:35,690 --> 00:05:38,090 relatively selection rules. 89 00:05:38,090 --> 00:05:42,100 In other words, which integrals are non-zero 90 00:05:42,100 --> 00:05:43,780 based on the difference in quantum 91 00:05:43,780 --> 00:05:48,290 numbers between the left-hand side and the right-hand side? 92 00:05:48,290 --> 00:05:53,320 So we're collecting these things in order 93 00:05:53,320 --> 00:05:56,320 to calculate a whole bunch of stuff. 94 00:05:56,320 --> 00:05:59,410 Now, I told you that this is a course for use 95 00:05:59,410 --> 00:06:03,440 rather than philosophy or history. 96 00:06:03,440 --> 00:06:07,210 And so if you encounter any quantum mechanical problem, 97 00:06:07,210 --> 00:06:11,020 you'd like to be able to describe 98 00:06:11,020 --> 00:06:13,180 what you could observe with it. 99 00:06:13,180 --> 00:06:17,860 And so if you're armed with the infinite number of energy 100 00:06:17,860 --> 00:06:23,350 levels and eigen solutions for our problem, 101 00:06:23,350 --> 00:06:26,990 you can calculate any property. 102 00:06:26,990 --> 00:06:30,350 So you define some property you're interested in-- 103 00:06:30,350 --> 00:06:32,870 there is a quantum mechanical operator that 104 00:06:32,870 --> 00:06:35,290 corresponds to that property. 105 00:06:35,290 --> 00:06:40,010 And in order to be able to describe observations 106 00:06:40,010 --> 00:06:43,010 of that property, you need to calculate 107 00:06:43,010 --> 00:06:47,280 integrals of that operator. 108 00:06:47,280 --> 00:06:49,060 Well, la dee dah. 109 00:06:49,060 --> 00:06:51,860 That should be intimidating, but it's not 110 00:06:51,860 --> 00:06:54,830 because almost all of these integrals 111 00:06:54,830 --> 00:06:59,850 can be expressed as a simple constant times 112 00:06:59,850 --> 00:07:02,340 a function of the quantum numbers 113 00:07:02,340 --> 00:07:04,680 or the difference of quantum numbers, 114 00:07:04,680 --> 00:07:08,360 and that's a fantastic thing. 115 00:07:08,360 --> 00:07:15,290 So we have any operator-- 116 00:07:15,290 --> 00:07:20,600 suppose the Hamiltonian is an exactly solved problem 117 00:07:20,600 --> 00:07:24,530 plus something else, which we'll call h1. 118 00:07:24,530 --> 00:07:27,860 And this is a complexity in the-- 119 00:07:27,860 --> 00:07:30,330 or it's the reality in the problem. 120 00:07:30,330 --> 00:07:35,450 And in order to deal with this, again, you're 121 00:07:35,450 --> 00:07:40,410 going to need to calculate integrals of this operator. 122 00:07:40,410 --> 00:07:43,520 And the last thing that's really going to be exciting 123 00:07:43,520 --> 00:07:48,287 is once we look at the time dependent Schrodinger equation, 124 00:07:48,287 --> 00:07:49,620 we're going to get wave packets. 125 00:07:55,720 --> 00:08:01,020 And these are functions of position and time, 126 00:08:01,020 --> 00:08:04,350 and these wave packets are classical-like, localized 127 00:08:04,350 --> 00:08:08,610 objects that move following the Newton's equations 128 00:08:08,610 --> 00:08:11,640 of motion with the center of the wave packet. 129 00:08:11,640 --> 00:08:13,110 And again, there are a whole bunch 130 00:08:13,110 --> 00:08:15,210 of integrals you're going to need in order 131 00:08:15,210 --> 00:08:17,050 to do these things. 132 00:08:17,050 --> 00:08:21,090 And so right now, we're starting with the best problem 133 00:08:21,090 --> 00:08:24,300 for these integrals, because a harmonic oscillator 134 00:08:24,300 --> 00:08:26,880 has some special properties. 135 00:08:26,880 --> 00:08:31,170 And the lecture notes are incredibly tedious, 136 00:08:31,170 --> 00:08:32,520 and they're mostly proofs. 137 00:08:32,520 --> 00:08:37,289 And I'm going to try to go fast over the tedious stuff, 138 00:08:37,289 --> 00:08:42,390 and give you the important ideas, 139 00:08:42,390 --> 00:08:44,970 but since there is some important logic, 140 00:08:44,970 --> 00:08:46,680 you should really look at these notes. 141 00:08:49,590 --> 00:08:53,070 So what we're going to be doing today 142 00:08:53,070 --> 00:08:59,900 is we start with the coordinate momentum operators, 143 00:08:59,900 --> 00:09:07,560 we're going to get these operators 144 00:09:07,560 --> 00:09:11,400 in dimensionless form, and then we're 145 00:09:11,400 --> 00:09:14,165 going to get these a and a-dagger guys. 146 00:09:18,460 --> 00:09:20,880 So this step is reminiscent of what 147 00:09:20,880 --> 00:09:23,920 I did at the beginning of the previous lecture, 148 00:09:23,920 --> 00:09:28,380 and then this is magic because this magic enables 149 00:09:28,380 --> 00:09:32,610 you to just look at integrals and say, 150 00:09:32,610 --> 00:09:35,760 I know that integral is zero, or I know that area is not zero. 151 00:09:35,760 --> 00:09:37,800 And with a little bit more effort-- 152 00:09:37,800 --> 00:09:40,260 maybe something that you'd put on the back of a postage 153 00:09:40,260 --> 00:09:41,250 stamp-- 154 00:09:41,250 --> 00:09:45,150 you can evaluate that integral, not by knowing integral tables, 155 00:09:45,150 --> 00:09:47,920 but by knowing the properties of these simple little a 156 00:09:47,920 --> 00:09:49,080 and a-dagger. 157 00:09:49,080 --> 00:09:51,420 And that's a fantastic thing. 158 00:09:51,420 --> 00:09:53,790 And it's so fantastic that this is 159 00:09:53,790 --> 00:09:59,200 one of the reasons why almost all problems in quantum 160 00:09:59,200 --> 00:10:03,310 mechanics start with a harmonic oscillator approximation, 161 00:10:03,310 --> 00:10:07,030 because there is so much you can do with this a and a-dagger 162 00:10:07,030 --> 00:10:08,770 formalism. 163 00:10:08,770 --> 00:10:11,530 Now, at the beginning I also told you 164 00:10:11,530 --> 00:10:16,030 that in quantum mechanics, the important thing that 165 00:10:16,030 --> 00:10:18,981 contains everything we're allowed to know about a system 166 00:10:18,981 --> 00:10:19,855 is the wave function. 167 00:10:22,850 --> 00:10:27,310 But I also told you we can never measure the wave function. 168 00:10:27,310 --> 00:10:31,760 We can never experimentally determine it, 169 00:10:31,760 --> 00:10:35,690 and so we need to be able to calculate what this wave 170 00:10:35,690 --> 00:10:40,580 function does as far as what we can observe, 171 00:10:40,580 --> 00:10:43,670 and these a's and a-daggers are really important in being 172 00:10:43,670 --> 00:10:44,360 able to do that. 173 00:10:48,090 --> 00:10:51,300 So I'm going to start with covering 174 00:10:51,300 --> 00:10:56,040 what I did in the notes, but I'm going to jump to final results 175 00:10:56,040 --> 00:10:57,300 at some point-- 176 00:10:57,300 --> 00:10:59,250 governed by the clock. 177 00:10:59,250 --> 00:11:03,870 And so the first thing we're going to do is these. 178 00:11:03,870 --> 00:11:10,920 And so what we do is we define the relationship 179 00:11:10,920 --> 00:11:15,720 between the ordinary position coordinate. 180 00:11:15,720 --> 00:11:25,050 And this little twiddle means it's dimensionless, 181 00:11:25,050 --> 00:11:29,260 and so we can write the inverse of that. 182 00:11:29,260 --> 00:11:32,120 And that's the one we are going to want to-- 183 00:11:32,120 --> 00:11:33,730 well, actually, we go both ways. 184 00:11:42,230 --> 00:11:45,500 And we do the same thing for the momentum-- 185 00:11:45,500 --> 00:11:52,880 p is equal to h-bar mu omega square 186 00:11:52,880 --> 00:12:00,030 root, p twiddle and the inverse which I don't need to write. 187 00:12:00,030 --> 00:12:04,190 And finally, we get the Hamiltonian, 188 00:12:04,190 --> 00:12:10,760 which is p squared over 2 mu plus 1/2 kx squared. 189 00:12:14,080 --> 00:12:17,390 And we'll put that into these new units. 190 00:12:17,390 --> 00:12:27,630 So we have h-bar mu omega over 2 mu, p twiddle squared 191 00:12:27,630 --> 00:12:29,430 plus k over 2. 192 00:12:33,160 --> 00:12:38,130 This is all very tedious, but the payoff is very soon. 193 00:12:40,980 --> 00:12:50,640 k over 2 times hr mu omega, x twiddle squared. 194 00:12:55,200 --> 00:12:56,900 Oh, isn't that interesting? 195 00:12:56,900 --> 00:13:05,630 We can combine-- we can absorb a k over mu in omega, 196 00:13:05,630 --> 00:13:08,360 and so we get, actually, a big simplification. 197 00:13:08,360 --> 00:13:14,140 We get h bar omega over 2 times p twiddle 198 00:13:14,140 --> 00:13:17,545 squared plus x twiddle squared. 199 00:13:20,110 --> 00:13:21,505 Well, that looks simpler. 200 00:13:26,770 --> 00:13:30,760 And so the next thing we do is-- it looks like a simple problem 201 00:13:30,760 --> 00:13:35,069 from algebra, let's factor this. 202 00:13:35,069 --> 00:13:36,610 Now, it's a little tricky because you 203 00:13:36,610 --> 00:13:40,630 know you can factor something in real terms 204 00:13:40,630 --> 00:13:42,500 if this is a minus sign. 205 00:13:42,500 --> 00:13:45,400 But we are allowed to talk about complex quantities, 206 00:13:45,400 --> 00:13:48,400 so we can factor that. 207 00:13:48,400 --> 00:14:00,030 And so this term, p twiddle squared plus x twiddle squared 208 00:14:00,030 --> 00:14:16,280 is equal to ip twiddle, plus x twiddle times minus ip twiddle, 209 00:14:16,280 --> 00:14:17,610 plus x. 210 00:14:22,390 --> 00:14:27,990 And you can work that out-- that ip times minus ip is p squared, 211 00:14:27,990 --> 00:14:30,810 and x times x is x squared. 212 00:14:30,810 --> 00:14:34,800 And then we have these cross-terms, ip times 213 00:14:34,800 --> 00:14:37,410 x and x times minus ip. 214 00:14:37,410 --> 00:14:41,102 Whoops, they don't commute. 215 00:14:41,102 --> 00:14:42,530 If this were algebra-- 216 00:14:42,530 --> 00:14:45,640 well, they would go away, but they don't. 217 00:14:45,640 --> 00:14:52,010 And so what you end up getting is 218 00:14:52,010 --> 00:15:06,160 p twiddle squared plus x twiddle squared, plus i times p-- 219 00:15:15,360 --> 00:15:17,580 I'm going to stop writing the twiddles. 220 00:15:17,580 --> 00:15:18,870 So we have this. 221 00:15:24,820 --> 00:15:28,330 I want to make sure that I haven't sabotaged myself-- 222 00:15:28,330 --> 00:15:31,550 that's going to be-- yeah, that's right. 223 00:15:31,550 --> 00:15:36,941 So we have something here that isn't zero. 224 00:15:36,941 --> 00:15:40,340 And it looks like i times the commutator of p twiddle 225 00:15:40,340 --> 00:15:42,746 with x twiddle. 226 00:15:42,746 --> 00:15:44,550 But we can work that out because we 227 00:15:44,550 --> 00:15:51,440 know the commutator of p ordinary with x ordinary. 228 00:15:51,440 --> 00:15:54,190 And so I did that. 229 00:15:54,190 --> 00:16:07,690 And so we have this commutator, p twiddle, x twiddle. 230 00:16:07,690 --> 00:16:10,495 After some algebra, we get plus 1. 231 00:16:15,187 --> 00:16:17,950 A number-- pure number-- 232 00:16:17,950 --> 00:16:18,940 no? 233 00:16:18,940 --> 00:16:23,270 I want you to check my algebra. 234 00:16:23,270 --> 00:16:28,000 So you just substitute in what this 235 00:16:28,000 --> 00:16:31,960 is in terms of ordinary p and the ordinary x. 236 00:16:31,960 --> 00:16:36,640 Use a commutator for ordinary xp, which is ih-bar, 237 00:16:36,640 --> 00:16:40,160 and magically, you get plus 1. 238 00:16:40,160 --> 00:16:50,585 So this very strange and boring derivation says, OK-- 239 00:16:55,900 --> 00:17:02,830 well, let's now give these two things names. 240 00:17:02,830 --> 00:17:09,069 This guy, we're going to call as the square root of 2 times a, 241 00:17:09,069 --> 00:17:11,585 and this one is going to be the square root 242 00:17:11,585 --> 00:17:13,869 of 2 times a-dagger. 243 00:17:13,869 --> 00:17:31,890 So H is going to be h-bar omega over 2, 244 00:17:31,890 --> 00:17:40,280 times square root of 2a-hat times 245 00:17:40,280 --> 00:17:47,496 the square root of 2a-dagger-hat minus 1. 246 00:17:47,496 --> 00:17:49,080 Remember, when we factored it, we 247 00:17:49,080 --> 00:17:51,150 got this extra term which was 1. 248 00:17:51,150 --> 00:17:54,240 And in order to make it correct, we have to subtract it away. 249 00:17:58,940 --> 00:18:12,130 And so this becomes h-bar omega, a-dagger-hat minus 1/2. 250 00:18:12,130 --> 00:18:13,750 Well, isn't that nice? 251 00:18:13,750 --> 00:18:17,620 Now, we have the Hamiltonian expressed as a constant, which 252 00:18:17,620 --> 00:18:20,500 we know is important because it's related to the energy 253 00:18:20,500 --> 00:18:23,940 levels, and times these two little things, 254 00:18:23,940 --> 00:18:29,040 which turn out to be the gift from God. 255 00:18:29,040 --> 00:18:31,380 It's an incredible thing, what these do. 256 00:18:35,740 --> 00:18:38,710 So we have gone through some algebra, 257 00:18:38,710 --> 00:18:43,450 and we know the relationship between a, and the x and p 258 00:18:43,450 --> 00:18:46,660 twiddles, and similarly for a-dagger. 259 00:18:46,660 --> 00:18:49,340 And we can go in the other direction, 260 00:18:49,340 --> 00:18:51,610 and we know the commutator, and now we're 261 00:18:51,610 --> 00:18:53,940 going to start doing some really great stuff. 262 00:18:59,860 --> 00:19:02,700 Well, one thing we're going to want to know about is a-hat. 263 00:19:02,700 --> 00:19:09,640 a-hat, a-dagger, that commutator-hat. 264 00:19:09,640 --> 00:19:12,340 And that turns out to be-- 265 00:19:15,820 --> 00:19:20,960 well, I already derived it-- it turns out to be plus 1. 266 00:19:20,960 --> 00:19:23,910 And as a result, we can say things like this-- a a-dagger-- 267 00:19:56,290 --> 00:20:03,040 So using this trick, we can show we can always replace something 268 00:20:03,040 --> 00:20:10,780 like a, a-dagger by a-dagger a plus this commutator, which 269 00:20:10,780 --> 00:20:11,280 is 1. 270 00:20:15,580 --> 00:20:19,070 And so we have this really neat way 271 00:20:19,070 --> 00:20:25,330 of reversing the order of the a's and a-daggers. 272 00:20:25,330 --> 00:20:28,300 So with this, we're going to soon discover 273 00:20:28,300 --> 00:20:33,980 the a operating on the eigenfunction gives square root 274 00:20:33,980 --> 00:20:37,845 of v times psi v minus 1. 275 00:20:37,845 --> 00:20:42,430 And a-dagger operating on this wave function 276 00:20:42,430 --> 00:20:48,550 gives v plus 1 square root of psi v plus 1. 277 00:20:48,550 --> 00:20:55,010 Which is the reason these things are valuable, 278 00:20:55,010 --> 00:21:01,980 because if you have any eigenfunction, 279 00:21:01,980 --> 00:21:04,780 you can get all the others. 280 00:21:04,780 --> 00:21:07,350 So suppose you have the lowest eigenfunction-- 281 00:21:07,350 --> 00:21:10,310 you apply a-dagger on it as many times 282 00:21:10,310 --> 00:21:12,126 as you need to get to, say vth function. 283 00:21:15,070 --> 00:21:17,790 So you don't actually-- 284 00:21:17,790 --> 00:21:20,640 you're not going to be evaluating integrals, you're 285 00:21:20,640 --> 00:21:24,060 going to be counting a's and a-daggers, 286 00:21:24,060 --> 00:21:26,410 and permuting them around, and getting 1's, and stuff 287 00:21:26,410 --> 00:21:26,910 like that. 288 00:21:26,910 --> 00:21:27,760 Yes? 289 00:21:27,760 --> 00:21:29,551 AUDIENCE: In this line with the commutator, 290 00:21:29,551 --> 00:21:30,904 you didn't move the dagger. 291 00:21:30,904 --> 00:21:32,070 ROBERT FIELD: I didn't what? 292 00:21:32,070 --> 00:21:34,785 AUDIENCE: For a, a times the square root of a a-dagger plus 293 00:21:34,785 --> 00:21:38,945 a-dagger a, it should be a a-dagger. 294 00:21:38,945 --> 00:21:42,470 And on the right-hand side, you need to move the dagger. 295 00:21:47,870 --> 00:21:52,110 ROBERT FIELD: OK, so this is to switch the order, 296 00:21:52,110 --> 00:21:53,610 and I've done that. 297 00:21:53,610 --> 00:21:55,035 And that then is-- 298 00:22:00,150 --> 00:22:03,030 no, I think-- wait a second. 299 00:22:03,030 --> 00:22:04,400 So we have a a-dagger-- 300 00:22:04,400 --> 00:22:10,070 so that's a a-dagger minus a-dagger a, and that's-- 301 00:22:10,070 --> 00:22:12,020 oh, yeah. 302 00:22:12,020 --> 00:22:13,940 Thank you. 303 00:22:13,940 --> 00:22:17,990 It's very, very easy to get lost, and once you're lost, 304 00:22:17,990 --> 00:22:19,910 you can never be found because you've 305 00:22:19,910 --> 00:22:23,300 made a mistake that's built into your logic, 306 00:22:23,300 --> 00:22:25,330 and you're never going to see it. 307 00:22:25,330 --> 00:22:27,870 You see it took me a couple of minutes to even accept-- 308 00:22:27,870 --> 00:22:31,640 that the insight from my TA who's sitting there calmly 309 00:22:31,640 --> 00:22:34,300 thinking, and I'm trying to do several things in addition 310 00:22:34,300 --> 00:22:35,810 to the thinking. 311 00:22:35,810 --> 00:22:39,730 So we can do things like this. 312 00:22:39,730 --> 00:22:42,080 Suppose we have psi-- 313 00:22:42,080 --> 00:22:45,530 I can't use this notation yet. 314 00:22:45,530 --> 00:22:50,660 So suppose we have psi-star v, and we 315 00:22:50,660 --> 00:22:58,990 have a-dagger, a-dagger, a-dagger, psi, v prime, dx. 316 00:23:04,490 --> 00:23:06,140 These are raising operators, so this 317 00:23:06,140 --> 00:23:11,120 is going to take v prime to v prime plus 3. 318 00:23:11,120 --> 00:23:15,440 That's the only integral that's not 0. 319 00:23:15,440 --> 00:23:29,554 And we get v prime plus 1, v prime plus 2, v prime plus 3, 320 00:23:29,554 --> 00:23:31,760 square what? 321 00:23:31,760 --> 00:23:33,280 It has the constants. 322 00:23:36,800 --> 00:23:38,960 And this would be v prime plus 3. 323 00:23:42,760 --> 00:23:45,430 So instead of evaluating an integral, 324 00:23:45,430 --> 00:23:47,730 looking at what the x's and p's are, 325 00:23:47,730 --> 00:23:49,840 we just have a little game we play. 326 00:23:55,700 --> 00:24:05,530 So now, we have to prove some of the things I've said. 327 00:24:05,530 --> 00:24:15,890 So we have h, and we're going to operate on a-dagger psi v. 328 00:24:15,890 --> 00:24:20,430 So what does the Hamiltonian do to this thing? 329 00:24:20,430 --> 00:24:22,250 So what we're going to want to do 330 00:24:22,250 --> 00:24:28,880 is to show that this thing is an eigenvalue-- 331 00:24:28,880 --> 00:24:32,060 eigenfunction of v plus 1, and that's 332 00:24:32,060 --> 00:24:33,180 what we are going to get. 333 00:24:33,180 --> 00:24:35,640 So let's just go through this. 334 00:24:35,640 --> 00:24:47,560 So we have h-bar omega, a-dagger a plus 1/2, times psi v. 335 00:24:47,560 --> 00:24:49,935 So what I did is-- 336 00:24:53,566 --> 00:24:59,830 where did I-- yeah, I showed that the Hamiltonian-- 337 00:24:59,830 --> 00:25:01,230 or did I not do that yet? 338 00:25:06,100 --> 00:25:07,500 Oh, yeah-- I did it right here. 339 00:25:07,500 --> 00:25:13,770 The Hamiltonian is h-bar omega, a a-dagger minus 1/2, 340 00:25:13,770 --> 00:25:21,030 or if we reverse these, it's equal to h-bar omega, a-dagger, 341 00:25:21,030 --> 00:25:25,410 a plus 1/2. 342 00:25:25,410 --> 00:25:35,000 So we can use either one, so I'm using that one-- 343 00:25:44,900 --> 00:25:48,140 except I wanted an a-dagger here, 344 00:25:48,140 --> 00:25:52,520 because we want to show what the Hamiltonian does to this. 345 00:25:52,520 --> 00:26:00,180 Now, we can pull in a-dagger out to the right, because this-- 346 00:26:06,942 --> 00:26:10,130 if it's 1/2 times a-dagger, well, that doesn't matter. 347 00:26:10,130 --> 00:26:12,410 This a-dagger, a, a-dagger-- well, 348 00:26:12,410 --> 00:26:14,840 we can pull this a-dagger out. 349 00:26:14,840 --> 00:26:25,854 So we have h-bar omega, a-dagger is equal to a a-dagger plus 350 00:26:25,854 --> 00:26:26,354 1/2-- 351 00:26:30,242 --> 00:26:31,214 so iv. 352 00:26:35,590 --> 00:26:38,740 Now, we use our magic commutator trick 353 00:26:38,740 --> 00:26:52,152 to replace this by a-dagger a plus 1. 354 00:26:59,580 --> 00:27:05,050 So now, we have h-bar omega, a-dagger, 355 00:27:05,050 --> 00:27:12,760 and we have a-dagger a plus three halves is iv. 356 00:27:22,650 --> 00:27:34,640 Well, this is Ev plus h-bar omega. 357 00:27:34,640 --> 00:27:39,260 We've increased the number that started here. 358 00:27:39,260 --> 00:27:42,800 Here is Ev-- that was Ev plus 1. 359 00:27:46,130 --> 00:27:48,740 And so now, we have no operators in here, 360 00:27:48,740 --> 00:27:51,910 and we can stick the a-dagger back here. 361 00:27:51,910 --> 00:28:10,080 And so we have h-bar omega, Ev plus 1, a-dagger 362 00:28:10,080 --> 00:28:11,840 Well, what do we have here? 363 00:28:11,840 --> 00:28:15,800 We have an operator, we have this function, 364 00:28:15,800 --> 00:28:20,650 we have some constant times the same function. 365 00:28:20,650 --> 00:28:25,600 So what we've shown is that this thing 366 00:28:25,600 --> 00:28:28,650 is an eigenfunction of the Hamiltonian 367 00:28:28,650 --> 00:28:33,720 that belongs to the eigenvalue Ev plus 1. 368 00:28:33,720 --> 00:28:36,690 We've increased the energy by 1. 369 00:28:42,390 --> 00:28:43,260 So what we have-- 370 00:29:01,420 --> 00:29:05,590 so we can show that we apply a-dagger 371 00:29:05,590 --> 00:29:12,700 to any function-- we increase its energy, 372 00:29:12,700 --> 00:29:15,820 and we can do this forever. 373 00:29:15,820 --> 00:29:21,820 We could also do a similar thing if we apply a to psi v. We 374 00:29:21,820 --> 00:29:27,550 can go down, but at some point, we run out of steam 375 00:29:27,550 --> 00:29:33,740 because we've gone to the lowest energy, and if we go lower, 376 00:29:33,740 --> 00:29:35,290 we get 0. 377 00:29:35,290 --> 00:29:43,810 So a operating on psi min gives 0. 378 00:29:43,810 --> 00:29:50,050 So we have this stack of energy levels and wave functions, 379 00:29:50,050 --> 00:29:53,290 and we have the same stack being repeated as we go down, 380 00:29:53,290 --> 00:29:55,240 but this one has an end. 381 00:30:08,880 --> 00:30:17,000 We bring back what a is, and so a psi min is 0-- 382 00:30:17,000 --> 00:30:18,390 that's the equation. 383 00:30:18,390 --> 00:30:32,060 We bring in what a is, and it's ip twiddle x plus x twiddle. 384 00:30:39,980 --> 00:30:42,040 So we do some algebra, and what we end up 385 00:30:42,040 --> 00:30:46,060 with is a differential equation, psi min. 386 00:30:46,060 --> 00:30:52,630 dxx twiddle is equal to-- 387 00:30:52,630 --> 00:30:54,400 again, a little bit more algebra-- 388 00:30:54,400 --> 00:31:00,790 minus mu omega over h-bar times psi min. 389 00:31:04,320 --> 00:31:09,880 So what function gives-- 390 00:31:09,880 --> 00:31:13,360 there's an x in here too. 391 00:31:13,360 --> 00:31:16,940 So what function has a derivative, 392 00:31:16,940 --> 00:31:19,690 which is the function you had started 393 00:31:19,690 --> 00:31:22,330 with, times the variable, times a constant? 394 00:31:27,370 --> 00:31:31,660 And so the answer to that is that psi min 395 00:31:31,660 --> 00:31:33,680 has to have the form-- 396 00:31:33,680 --> 00:31:39,700 some normalization factor times e to the minus m omega-- 397 00:31:39,700 --> 00:31:40,760 or mu omega-- sorry-- 398 00:31:43,600 --> 00:31:49,680 over 2 h-bar x squared-- 399 00:31:49,680 --> 00:31:50,420 a Gaussian. 400 00:31:52,950 --> 00:31:54,990 Well, it had to be a Gaussian, right? 401 00:31:54,990 --> 00:31:57,990 We know when we did the algebra that we're 402 00:31:57,990 --> 00:32:01,890 going to get some function times a Gaussian. 403 00:32:01,890 --> 00:32:05,910 But for the lowest function, the Hermite polynomial is 1, 404 00:32:05,910 --> 00:32:08,550 and all there is is the Gaussian. 405 00:32:08,550 --> 00:32:15,430 And so we found the lowest level, and we can normalize it. 406 00:32:21,880 --> 00:32:23,320 So let's start over here. 407 00:32:29,640 --> 00:32:35,030 So what we have found is psi min of x 408 00:32:35,030 --> 00:32:47,260 is equal to mu omega over pi h-bar to the 1/4 power, e 409 00:32:47,260 --> 00:32:55,797 to the minus mu omega over 2 h-bar squared. 410 00:32:55,797 --> 00:32:56,630 Well, that's useful. 411 00:32:56,630 --> 00:33:00,050 We knew that, but this time we got it out 412 00:33:00,050 --> 00:33:01,530 of a completely different path. 413 00:33:05,900 --> 00:33:16,120 And now, we can get all higher v by 414 00:33:16,120 --> 00:33:21,140 a-dagger, a-dagger, et cetera. 415 00:33:21,140 --> 00:33:26,170 So remember, we don't care anything 416 00:33:26,170 --> 00:33:29,890 about what the function is, we just 417 00:33:29,890 --> 00:33:34,100 know that we can bring it in and get rid of it at will, 418 00:33:34,100 --> 00:33:37,880 because what we want is the values of integrals involving 419 00:33:37,880 --> 00:33:41,280 that function and some operator. 420 00:33:41,280 --> 00:33:44,280 So yeah, we can have all of those functions, 421 00:33:44,280 --> 00:33:46,820 and this is a way of generating all of the functions. 422 00:33:46,820 --> 00:33:56,880 And so if we wanted psi v, we would do a-dagger to the vth 423 00:33:56,880 --> 00:34:04,850 power divided by v-dagger-- 424 00:34:04,850 --> 00:34:12,300 v-- what do you call this with an exclamation point? 425 00:34:12,300 --> 00:34:13,419 Factorial-- ha! 426 00:34:18,909 --> 00:34:24,719 So we apply this operator that raises us to whatever level 427 00:34:24,719 --> 00:34:29,080 we want starting from this Gaussian at the bottom, 428 00:34:29,080 --> 00:34:32,710 and we have this normalization factor which cancels out 429 00:34:32,710 --> 00:34:35,800 the fact the stuff that you get by applying av. 430 00:34:47,120 --> 00:34:49,300 Now, there is some more logic in my notes, 431 00:34:49,300 --> 00:34:51,340 and I don't want to do that, but what 432 00:34:51,340 --> 00:34:57,080 we'd like to be able to show is that a-dagger on psi v 433 00:34:57,080 --> 00:35:04,850 gives some constant, and that this constant has 434 00:35:04,850 --> 00:35:07,460 some value-- we're going to evaluate what it is. 435 00:35:07,460 --> 00:35:16,280 And similarly, a psi v gives dv, and some constant v minus that. 436 00:35:16,280 --> 00:35:20,030 We can derive those things, and I'm not going 437 00:35:20,030 --> 00:35:23,120 to waste time deriving them-- 438 00:35:23,120 --> 00:35:25,150 I'm going to just give you the values. 439 00:35:25,150 --> 00:35:32,105 But we already know that cv is square of v plus 1/2-- 440 00:35:32,105 --> 00:35:34,355 e plus 1 and dv-- 441 00:35:38,170 --> 00:35:40,210 and you can see the derivation in my notes. 442 00:35:40,210 --> 00:35:43,950 I don't think going through them is going to be instructive. 443 00:35:43,950 --> 00:35:49,650 And that's just going to be v and 1/2. 444 00:35:49,650 --> 00:35:55,780 So now, we have something that's wonderful, 445 00:35:55,780 --> 00:36:01,270 because everything you need to know about getting numbers 446 00:36:01,270 --> 00:36:05,890 concerning harmonic oscillator is obtained from these five 447 00:36:05,890 --> 00:36:07,300 equations. 448 00:36:07,300 --> 00:36:18,810 a-dagger on psi v is v plus 1 square root psi v plus 1. 449 00:36:18,810 --> 00:36:24,260 a on psi v is v square root psi, v minus 1. 450 00:36:24,260 --> 00:36:28,450 I've said this before, but these are the most useful things 451 00:36:28,450 --> 00:36:29,650 you'll ever encounter. 452 00:36:29,650 --> 00:36:32,150 We have this thing called the number operator, 453 00:36:32,150 --> 00:36:35,890 and that number operator is a-dagger-hat, 454 00:36:35,890 --> 00:36:43,140 and the number operator operating on psi v 455 00:36:43,140 --> 00:36:49,410 gives v psi v. And so that's a kind of benign operator 456 00:36:49,410 --> 00:36:56,310 that can suck up all sorts of factors of a-dagger a, 457 00:36:56,310 --> 00:37:00,720 because it just gives a useful thing. 458 00:37:00,720 --> 00:37:07,350 And then we have a, a-dagger, and this is 1. 459 00:37:07,350 --> 00:37:08,940 Well, you sort of know it's going 460 00:37:08,940 --> 00:37:13,980 to be 1, because a a-dagger gives an increase-- it gives v 461 00:37:13,980 --> 00:37:22,260 plus 1, and a gives v minus 1. 462 00:37:22,260 --> 00:37:26,735 So it's plus 1, not minus 1-- you know that it's hardwired. 463 00:37:31,264 --> 00:37:31,930 Well, I did it-- 464 00:37:31,930 --> 00:37:36,400 I got to the point where it starts to get interesting. 465 00:37:36,400 --> 00:37:43,120 So we're going to be using this notation, a-dagger and a, 466 00:37:43,120 --> 00:37:45,930 for all sorts of stuff. 467 00:37:45,930 --> 00:37:57,070 And one sort of thing is transition intensities 468 00:37:57,070 --> 00:37:58,250 and selection rules. 469 00:38:03,350 --> 00:38:05,255 So you have a harmonic oscillator. 470 00:38:05,255 --> 00:38:09,160 A harmonic oscillator is, say a diatomic molecule 471 00:38:09,160 --> 00:38:11,410 which is heteronuclear. 472 00:38:11,410 --> 00:38:15,250 And so as the molecule vibrates, you 473 00:38:15,250 --> 00:38:18,792 have a dipole moment which is oscillating. 474 00:38:21,690 --> 00:38:25,420 And so any oscillating electric field 475 00:38:25,420 --> 00:38:27,940 will grab a hold of that dipole moment 476 00:38:27,940 --> 00:38:31,900 and stretch or compress it, especially 477 00:38:31,900 --> 00:38:37,390 if that field is in resonance with h-bar omega. 478 00:38:37,390 --> 00:38:41,675 And I've got some beautiful animation showing this, 479 00:38:41,675 --> 00:38:44,050 but we can't do that until we have time dependent quantum 480 00:38:44,050 --> 00:38:45,260 mechanics. 481 00:38:45,260 --> 00:38:48,550 So we have a time dependent radiation 482 00:38:48,550 --> 00:38:50,770 field, which is going to interact 483 00:38:50,770 --> 00:38:56,170 with the dipole associated with the vibrating molecule, 484 00:38:56,170 --> 00:38:59,060 and it's going to cause transitions. 485 00:38:59,060 --> 00:39:06,760 And so we can write the quantum mechanical operator 486 00:39:06,760 --> 00:39:08,860 that causes the transitions-- 487 00:39:08,860 --> 00:39:14,380 this is the electric dipole moment operator-- 488 00:39:14,380 --> 00:39:17,410 as a function of coordinate. 489 00:39:17,410 --> 00:39:21,670 And we can do a power series expansion of this, 490 00:39:21,670 --> 00:39:22,270 and we have-- 491 00:39:36,580 --> 00:39:40,240 so we have mu 0-- the constant term-- 492 00:39:40,240 --> 00:39:42,130 the first derivative of the dipole 493 00:39:42,130 --> 00:39:45,100 with respect to x, and the second derivative of the dipole 494 00:39:45,100 --> 00:39:46,930 with respect to x. 495 00:39:46,930 --> 00:39:50,840 And we have the x cofactor and the x squared cofactor. 496 00:39:50,840 --> 00:39:55,000 And so this guy doesn't have any x on it-- it's a constant. 497 00:39:55,000 --> 00:39:57,730 The only integrals involving-- 498 00:40:09,880 --> 00:40:18,070 the only integrals are delta v, v prime following the selection 499 00:40:18,070 --> 00:40:19,190 rule delta v v prime. 500 00:40:19,190 --> 00:40:24,440 So these integrals are 0 unless v and v prime are the same. 501 00:40:26,980 --> 00:40:30,220 And that says, well, an isolating field 502 00:40:30,220 --> 00:40:33,580 isn't going to do anything key to it, 503 00:40:33,580 --> 00:40:36,160 it's just going to leave it in the same vibration level. 504 00:40:36,160 --> 00:40:39,880 But it might have an electric Stark effect, 505 00:40:39,880 --> 00:40:41,320 but that's something else. 506 00:40:41,320 --> 00:40:45,940 So this term does nothing as far as vibration is concerned. 507 00:40:45,940 --> 00:40:50,395 This guy, which is a plus a-dagger, 508 00:40:50,395 --> 00:40:54,544 has a selection rule, delta v of plus and minus 1, 509 00:40:54,544 --> 00:40:58,255 and this guy has a selection rule, delta v 510 00:40:58,255 --> 00:41:00,220 of plus and minus 2 and 0. 511 00:41:03,600 --> 00:41:06,270 So if we're interested in the intensities 512 00:41:06,270 --> 00:41:08,910 of vibrational transitions, it says, 513 00:41:08,910 --> 00:41:12,420 well, this is the important term and it causes transitions, 514 00:41:12,420 --> 00:41:16,080 changing the vibrational quantum number by one, which 515 00:41:16,080 --> 00:41:18,540 is called the fundamental. 516 00:41:18,540 --> 00:41:22,530 This gives rise to overtones. 517 00:41:22,530 --> 00:41:27,330 So all of a sudden, we're in real problem land, where 518 00:41:27,330 --> 00:41:31,590 if we're looking at vibrational transitions in a molecule, 519 00:41:31,590 --> 00:41:34,740 that this enables us to calculate what's important, 520 00:41:34,740 --> 00:41:37,560 or to say these are the intensities I measure, 521 00:41:37,560 --> 00:41:41,850 and these are the first and second derivative 522 00:41:41,850 --> 00:41:45,330 of the dipole moment operator as a function of internuclear 523 00:41:45,330 --> 00:41:46,160 distance. 524 00:41:46,160 --> 00:41:46,860 Isn't that neat? 525 00:41:53,370 --> 00:41:59,460 I've gone so fast, I'm more or less at the end of my notes, 526 00:41:59,460 --> 00:42:01,120 but I can blather on for a while. 527 00:42:03,790 --> 00:42:09,420 So suppose you have some integral involving 528 00:42:09,420 --> 00:42:13,680 an operator and a vibrational wave function. 529 00:42:13,680 --> 00:42:21,810 So we have psi v star, some operator, psi v prime dx. 530 00:42:25,300 --> 00:42:28,630 And we'd like to know how to focus our energies. 531 00:42:28,630 --> 00:42:30,070 We're very busy people-- 532 00:42:30,070 --> 00:42:31,590 we don't want any value integrals 533 00:42:31,590 --> 00:42:36,280 that come out to be 0, we'd like to just know. 534 00:42:36,280 --> 00:42:45,100 So if this operator is some function of x or function of p, 535 00:42:45,100 --> 00:42:48,550 we'd have a power series expansion of the operator, 536 00:42:48,550 --> 00:42:51,580 and we then know what the selection rules are. 537 00:42:56,460 --> 00:42:58,600 So usually, you look at the operator 538 00:42:58,600 --> 00:43:05,040 and you find that it's a linear quadratic cubic function of x-- 539 00:43:05,040 --> 00:43:07,580 the leading term is usually linear. 540 00:43:07,580 --> 00:43:10,920 Bang-- you have a delta v of plus 1-- 541 00:43:10,920 --> 00:43:12,900 selection rule. 542 00:43:12,900 --> 00:43:16,770 Or if someone has bothered to actually convert 543 00:43:16,770 --> 00:43:18,250 the operator to some form-- 544 00:43:20,855 --> 00:43:23,666 oh, it's the operator-- 545 00:43:29,130 --> 00:43:34,772 this might have some form, a-dagger cubed times 546 00:43:34,772 --> 00:43:35,355 some constant. 547 00:43:40,100 --> 00:43:43,940 So if the operator looks like a-dagger cubed, 548 00:43:43,940 --> 00:43:49,740 we know that the selection rule is v to v plus 3, 549 00:43:49,740 --> 00:43:54,550 and we know that the matrix of the integral is v plus 1, 550 00:43:54,550 --> 00:43:59,550 times v plus 2, times v plus 3, square root of that, 551 00:43:59,550 --> 00:44:01,870 times the constant. 552 00:44:01,870 --> 00:44:04,740 So there is a huge number of problems 553 00:44:04,740 --> 00:44:08,860 that, instead of being pages and pages of algebra, 554 00:44:08,860 --> 00:44:14,520 are just reduced something that you can tell by inspection. 555 00:44:14,520 --> 00:44:19,520 So one of the tricks is we have an operator 556 00:44:19,520 --> 00:44:21,290 like x squared or x cubed-- 557 00:44:24,320 --> 00:44:27,560 what we want to do is write this in terms 558 00:44:27,560 --> 00:44:36,720 of a squared, a-dagger squared, and maybe 559 00:44:36,720 --> 00:44:40,920 some combination of a-dagger a. 560 00:44:40,920 --> 00:44:44,340 So we want to take the a-dagger a's with the a a-daggers 561 00:44:44,340 --> 00:44:47,550 and combine them using the commutation rule. 562 00:44:47,550 --> 00:44:51,720 And then we have expressed this in this maximally simple form, 563 00:44:51,720 --> 00:44:54,600 and then you just apply a squared, 564 00:44:54,600 --> 00:44:59,050 apply a-dagger squared, and you apply this, 565 00:44:59,050 --> 00:45:02,800 then you've got the value of the integral. 566 00:45:02,800 --> 00:45:04,930 So if you're a busy person and you 567 00:45:04,930 --> 00:45:08,640 want to actually calculate stuff, 568 00:45:08,640 --> 00:45:12,960 you want to know how to reduce operators-- 569 00:45:12,960 --> 00:45:17,010 usually expressed as some power of the coordinate 570 00:45:17,010 --> 00:45:18,240 or the momentum-- 571 00:45:18,240 --> 00:45:23,730 into a sum of terms involving these organized 572 00:45:23,730 --> 00:45:25,500 products of a's and a-daggers. 573 00:45:28,150 --> 00:45:32,980 And you're going to be absolutely shocked at how 574 00:45:32,980 --> 00:45:35,140 perturbation theory-- 575 00:45:35,140 --> 00:45:41,910 which leads to basically all of the formulas and spectroscopy-- 576 00:45:41,910 --> 00:45:46,230 it's an ugly theory, but it reduces everything to things 577 00:45:46,230 --> 00:45:50,280 that you can just write down at the speed of your pen 578 00:45:50,280 --> 00:45:54,040 or pencil, and that's a fantastic thing. 579 00:45:54,040 --> 00:45:58,140 So you can't do this with the particle in a box, 580 00:45:58,140 --> 00:46:00,480 you can't do this with a hydrogen atom, 581 00:46:00,480 --> 00:46:02,040 you can't do this with the rigid-- 582 00:46:02,040 --> 00:46:04,230 well, you can do some of this with a rigid rotor. 583 00:46:04,230 --> 00:46:07,920 But the harmonic oscillator is so ubiquitous 584 00:46:07,920 --> 00:46:09,930 because every one-dimensional problem 585 00:46:09,930 --> 00:46:12,647 is harmonic at the bottom. 586 00:46:12,647 --> 00:46:15,230 And so you can use it and then you can put in the corrections. 587 00:46:15,230 --> 00:46:21,530 But also because you want to describe dynamics, 588 00:46:21,530 --> 00:46:24,830 you almost always use the harmonic oscillator, 589 00:46:24,830 --> 00:46:27,440 because not only do you know the integrals, 590 00:46:27,440 --> 00:46:29,540 but you know there's only a few. 591 00:46:29,540 --> 00:46:31,640 Normally, you're going to be summing 592 00:46:31,640 --> 00:46:36,290 over an infinite number of quantum numbers, 593 00:46:36,290 --> 00:46:39,950 and that takes time, and it takes judgment to say, 594 00:46:39,950 --> 00:46:44,230 well, only certain of these are important. 595 00:46:44,230 --> 00:46:49,240 But for the harmonic oscillator, the sums are finite. 596 00:46:49,240 --> 00:46:50,710 All of these things are wonderful, 597 00:46:50,710 --> 00:46:54,880 and that's why whenever you look at a theory, 598 00:46:54,880 --> 00:46:58,390 you're going to discover that hidden in there 599 00:46:58,390 --> 00:47:01,550 is the harmonic oscillator approximation, 600 00:47:01,550 --> 00:47:07,290 because everything is doable in no effort. 601 00:47:07,290 --> 00:47:10,880 And sometimes when you look at a paper like that, 602 00:47:10,880 --> 00:47:13,280 it doesn't show you the intermediate steps, 603 00:47:13,280 --> 00:47:17,480 because everybody knows what a harmonic oscillator does. 604 00:47:17,480 --> 00:47:18,950 And there's also a lot of insight 605 00:47:18,950 --> 00:47:23,730 because something like this-- 606 00:47:23,730 --> 00:47:27,440 this is an odd symmetry term, this is an even symmetry term, 607 00:47:27,440 --> 00:47:30,370 and there are all sorts of things that have to do with, 608 00:47:30,370 --> 00:47:33,050 are you're conserving symmetry or changing symmetry? 609 00:47:33,050 --> 00:47:38,660 And sometimes, the issue is how does the molecule spontaneously 610 00:47:38,660 --> 00:47:40,610 change symmetry by doing something 611 00:47:40,610 --> 00:47:43,370 interacting with a field, or interacting 612 00:47:43,370 --> 00:47:47,280 with some feature of the potential surface. 613 00:47:47,280 --> 00:47:51,560 So this is a place where it's labor saving and insight 614 00:47:51,560 --> 00:47:55,520 generating, and it's really amazing. 615 00:47:55,520 --> 00:47:59,110 So maybe I've bored you with this, 616 00:47:59,110 --> 00:48:03,070 but this is the beginning of almost every theory 617 00:48:03,070 --> 00:48:07,670 that you encounter just because of the simplicity of the a's 618 00:48:07,670 --> 00:48:10,210 and a-daggers. 619 00:48:10,210 --> 00:48:11,470 OK, I'm done. 620 00:48:11,470 --> 00:48:13,054 Thank you.