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 OpenCourseWare 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 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,850 at ocw.mit.edu. 8 00:00:24,570 --> 00:00:26,820 ROBERT FIELD: So today we're going 9 00:00:26,820 --> 00:00:29,190 to go from the classical mechanical treatment 10 00:00:29,190 --> 00:00:33,360 of the harmonic oscillator to a quantum mechanical treatment. 11 00:00:33,360 --> 00:00:37,520 And I warn you that I intentionally 12 00:00:37,520 --> 00:00:41,710 am going to make this look bad, because 13 00:00:41,710 --> 00:00:48,140 the semi-classical approach at the end of this lecture 14 00:00:48,140 --> 00:00:50,480 will make it all really simple. 15 00:00:50,480 --> 00:00:52,670 And then on Monday I'll introduce 16 00:00:52,670 --> 00:00:54,920 creation and annihilation operators, 17 00:00:54,920 --> 00:00:57,050 which makes the harmonic oscillator simpler 18 00:00:57,050 --> 00:00:59,530 than the particle in a box. 19 00:00:59,530 --> 00:01:02,470 You can't believe that, but all right. 20 00:01:02,470 --> 00:01:10,930 So last time we treated the harmonic oscillator classically 21 00:01:10,930 --> 00:01:16,150 and so we derive the equation of motion from forces 22 00:01:16,150 --> 00:01:19,910 equal to mass times acceleration, and we solved it. 23 00:01:19,910 --> 00:01:26,140 And we saw that we have this quantity omega, which initially 24 00:01:26,140 --> 00:01:28,090 I just introduced as a constant, which 25 00:01:28,090 --> 00:01:35,700 was a way of combining the force constant and the mass. 26 00:01:35,700 --> 00:01:40,320 And then I showed that the period of oscillation 27 00:01:40,320 --> 00:01:45,105 is 1 over the frequency, which is 2 pi over omega. 28 00:01:47,980 --> 00:01:50,140 Now one of the things that people 29 00:01:50,140 --> 00:01:54,160 have trouble remembering under exam pressure 30 00:01:54,160 --> 00:01:55,585 is turning points. 31 00:02:01,620 --> 00:02:05,490 And this comes when the energy is 32 00:02:05,490 --> 00:02:08,900 equal to the potential at a turning point. 33 00:02:15,490 --> 00:02:18,140 And since the potential is 1/2 kx 34 00:02:18,140 --> 00:02:23,540 squared we get the equation for the turning point 35 00:02:23,540 --> 00:02:28,520 at a given energy, which is equal to plus or minus 36 00:02:28,520 --> 00:02:30,260 the square root of 2e/k. 37 00:02:35,420 --> 00:02:37,940 Now when you're drawing pictures there 38 00:02:37,940 --> 00:02:40,542 are certain things that anchor the pictures, 39 00:02:40,542 --> 00:02:41,375 like turning points. 40 00:02:44,000 --> 00:02:49,120 And now at the end of the previous lecture 41 00:02:49,120 --> 00:02:54,170 I calculated the classical mechanical average, 42 00:02:54,170 --> 00:02:57,010 and we use we use this kind of notation 43 00:02:57,010 --> 00:02:58,090 in classical mechanics. 44 00:02:58,090 --> 00:03:00,380 Sometimes we use this notation, too, 45 00:03:00,380 --> 00:03:03,640 but this is what we mean by the average value in quantum 46 00:03:03,640 --> 00:03:04,850 mechanics. 47 00:03:04,850 --> 00:03:12,040 And we found that this was the total energy divided by 2. 48 00:03:12,040 --> 00:03:19,817 And the average momentum is the energy divided by 2. 49 00:03:19,817 --> 00:03:21,400 And that's the basis for some insight, 50 00:03:21,400 --> 00:03:24,370 because as a harmonic oscillator moves 51 00:03:24,370 --> 00:03:28,270 it throws energy back and forth between kinetic energy 52 00:03:28,270 --> 00:03:30,310 and potential energy. 53 00:03:30,310 --> 00:03:32,230 And then one of my favorite things, 54 00:03:32,230 --> 00:03:36,340 and one of my favorite tortures on short answers, 55 00:03:36,340 --> 00:03:41,800 is the average values of-- 56 00:03:47,460 --> 00:03:53,220 And does anybody want to remind the class the easy way, 57 00:03:53,220 --> 00:03:57,300 or the two easy ways to know what this is? 58 00:03:57,300 --> 00:04:00,190 And the person who answered the question last time 59 00:04:00,190 --> 00:04:02,040 is disqualified. 60 00:04:02,040 --> 00:04:02,966 Yes. 61 00:04:02,966 --> 00:04:04,590 AUDIENCE: There is the symmetry method. 62 00:04:04,590 --> 00:04:05,020 ROBERT FIELD: What? 63 00:04:05,020 --> 00:04:06,350 AUDIENCE: There's symmetry. 64 00:04:06,350 --> 00:04:10,650 So if the system's symmetrical that the average value will 65 00:04:10,650 --> 00:04:12,880 actually be that symmetry point. 66 00:04:12,880 --> 00:04:14,100 ROBERT FIELD: Yes. 67 00:04:14,100 --> 00:04:17,470 So and the other is that the harmonic oscillator 68 00:04:17,470 --> 00:04:22,390 isn't moving, and so there is no way that the coordinate-- 69 00:04:22,390 --> 00:04:28,720 that the average value of either the coordinate or the momentum 70 00:04:28,720 --> 00:04:30,730 could be different from 0. 71 00:04:30,730 --> 00:04:33,280 However you do it you want to be doing it 72 00:04:33,280 --> 00:04:35,260 in seconds, not minutes. 73 00:04:35,260 --> 00:04:37,990 And certainly not by calculating integral. 74 00:04:37,990 --> 00:04:43,180 And now x squared and p squared, they're easy too. 75 00:04:43,180 --> 00:04:50,560 Especially if you know t and v. And so we then use those 76 00:04:50,560 --> 00:04:55,820 to calculate the variance. 77 00:04:55,820 --> 00:04:58,990 And that's defined as the average value 78 00:04:58,990 --> 00:05:03,880 of the square minus the square of the average square root. 79 00:05:07,100 --> 00:05:11,690 And what we find is that the variance 80 00:05:11,690 --> 00:05:19,470 of x times the variance of p is equal to e over omega. 81 00:05:19,470 --> 00:05:26,160 So as you go up in energy this joint uncertainty increases. 82 00:05:26,160 --> 00:05:32,040 And we'll find that that also is true for quantum mechanics. 83 00:05:32,040 --> 00:05:35,550 So this is sort of the kind of questions 84 00:05:35,550 --> 00:05:38,940 you want to be asking in quantum mechanics. 85 00:05:38,940 --> 00:05:41,100 And you want to be able to be guided 86 00:05:41,100 --> 00:05:43,340 by what you know from classical mechanics, 87 00:05:43,340 --> 00:05:46,480 and you want to be able to do it fast. 88 00:05:46,480 --> 00:05:49,240 Today's menu is what I would call-- 89 00:05:59,630 --> 00:06:03,110 This lecture is gratuitous complexity. 90 00:06:03,110 --> 00:06:05,210 Does everybody know what gratuitous means? 91 00:06:09,380 --> 00:06:12,200 This is one of my favorite Bobisms. 92 00:06:15,410 --> 00:06:19,340 And you'll hear other Bobisms during the course 93 00:06:19,340 --> 00:06:21,320 of this course. 94 00:06:24,920 --> 00:06:27,550 What I want you to be able to do for a lot 95 00:06:27,550 --> 00:06:32,620 of mechanical problems is to know the answer, 96 00:06:32,620 --> 00:06:36,760 or know what things look like without doing a calculation. 97 00:06:36,760 --> 00:06:39,670 In particular, not solving a differential equation 98 00:06:39,670 --> 00:06:41,800 or evaluating in the integrals. 99 00:06:41,800 --> 00:06:47,050 You want to be able to draw these pictures instantly. 100 00:06:47,050 --> 00:06:53,860 Now in the modern age everyone has a cell phone, 101 00:06:53,860 --> 00:06:55,750 and one could have a program in there 102 00:06:55,750 --> 00:06:59,720 to calculate what anything you wanted for harmonic oscillator. 103 00:06:59,720 --> 00:07:03,700 But chances are you won't be prepared for that. 104 00:07:03,700 --> 00:07:08,260 And if you want to have insights into how do various things 105 00:07:08,260 --> 00:07:13,180 you want to know about harmonic oscillators come about? 106 00:07:13,180 --> 00:07:15,640 You need the pictures, as opposed 107 00:07:15,640 --> 00:07:17,710 to the computer program. 108 00:07:17,710 --> 00:07:23,650 Now the pictures involve an advance investment of energy. 109 00:07:23,650 --> 00:07:25,174 You want to understand every detail 110 00:07:25,174 --> 00:07:26,215 of these little pictures. 111 00:07:31,254 --> 00:07:32,920 I'm going to write this runner equation, 112 00:07:32,920 --> 00:07:35,770 I'm going to clean it up to get rid of units 113 00:07:35,770 --> 00:07:38,080 which makes it universal. 114 00:07:38,080 --> 00:07:41,690 So it becomes a dimensionless equation. 115 00:07:41,690 --> 00:07:46,180 And the unit removal, or the thing that 116 00:07:46,180 --> 00:07:50,740 takes you from a specific problem where there's 117 00:07:50,740 --> 00:07:55,960 a particular force constant and a particular reduced mass, 118 00:07:55,960 --> 00:07:57,640 and makes it into a general problem. 119 00:07:57,640 --> 00:08:03,500 There is one or two constants that combine those things. 120 00:08:03,500 --> 00:08:06,700 And you've taken them out at the end of a calculation. 121 00:08:06,700 --> 00:08:11,350 If you need to have real units you can put those back in. 122 00:08:11,350 --> 00:08:12,820 And that's a very wonderful thing, 123 00:08:12,820 --> 00:08:16,630 and that enables us to draw pictures 124 00:08:16,630 --> 00:08:21,180 without thinking about, what is the problem? 125 00:08:21,180 --> 00:08:24,600 And then the solution of this differential equation-- 126 00:08:24,600 --> 00:08:28,260 which is actually quite an awful differential equation at least 127 00:08:28,260 --> 00:08:30,900 for people who are not mathematicians-- 128 00:08:30,900 --> 00:08:33,750 and the solution can be expressed 129 00:08:33,750 --> 00:08:36,090 as the product of a Gaussian function 130 00:08:36,090 --> 00:08:39,990 which goes to 0 at plus and minus infinity, so it makes 131 00:08:39,990 --> 00:08:42,390 the function well behaved. 132 00:08:42,390 --> 00:08:46,200 Times something that produces nodes, a polynomial. 133 00:08:46,200 --> 00:08:48,960 Does anyone want to give me a definition of a polynomial? 134 00:08:56,720 --> 00:08:58,430 Silence. 135 00:08:58,430 --> 00:09:01,495 OK, your turn. 136 00:09:01,495 --> 00:09:06,740 AUDIENCE: It's the linear combination of some numbers 137 00:09:06,740 --> 00:09:08,417 taken to different power 138 00:09:08,417 --> 00:09:09,250 ROBERT FIELD: Right. 139 00:09:09,250 --> 00:09:12,185 A sum of integer powers of a variable. 140 00:09:14,700 --> 00:09:20,610 And when we take a derivative of a polynomial 141 00:09:20,610 --> 00:09:23,610 we reduce the order of the polynomial. 142 00:09:23,610 --> 00:09:26,820 A little bit of thought, if you have a first order 143 00:09:26,820 --> 00:09:29,260 polynomial there'll be one node. 144 00:09:29,260 --> 00:09:31,380 If there is a second order there'll be two nodes. 145 00:09:33,920 --> 00:09:37,130 And nodes are very important. 146 00:09:37,130 --> 00:09:40,870 And so when we're going to be dealing with cartoons 147 00:09:40,870 --> 00:09:44,650 of the wave function, and then using semi-classical ideas 148 00:09:44,650 --> 00:09:49,450 to actually semi-calculate things that you'd want to know, 149 00:09:49,450 --> 00:09:53,740 the nodes are really important. 150 00:09:53,740 --> 00:09:57,760 And what's going to happen Monday 151 00:09:57,760 --> 00:10:00,980 is we'll throw away all this garbage 152 00:10:00,980 --> 00:10:05,120 and we will replace everything by these creation 153 00:10:05,120 --> 00:10:07,370 and annihilation operators. 154 00:10:07,370 --> 00:10:12,360 Which do have really simple properties, 155 00:10:12,360 --> 00:10:16,190 which you can use to do astonishingly 156 00:10:16,190 --> 00:10:19,190 complicated things without breaking a sweat. 157 00:10:25,910 --> 00:10:30,380 And the final exam is in this room on the first day of exam 158 00:10:30,380 --> 00:10:33,290 period, at least it's on a Monday, 159 00:10:33,290 --> 00:10:34,864 and it's in the afternoon. 160 00:10:47,120 --> 00:10:51,980 In the non-lecture part of the notes 161 00:10:51,980 --> 00:10:58,060 I replaced the mass for one mass on a spring 162 00:10:58,060 --> 00:11:06,680 by the reduced mass, which is m1 m2 over m1 plus m2 for two 163 00:11:06,680 --> 00:11:09,180 masses connected by a spring. 164 00:11:09,180 --> 00:11:16,640 And I go back and forth between using mu and m, and that's OK. 165 00:11:16,640 --> 00:11:22,450 All right so in the notes the differential equations 166 00:11:22,450 --> 00:11:24,160 in the first few pages are expressed 167 00:11:24,160 --> 00:11:26,210 as partial differential equations. 168 00:11:26,210 --> 00:11:28,780 They're total differential equations. 169 00:11:28,780 --> 00:11:31,960 That'll get changed. 170 00:11:31,960 --> 00:11:43,610 The Hamiltonian is t plus v, and in the usual form t 171 00:11:43,610 --> 00:11:45,920 is p squared over 2 mu. 172 00:11:45,920 --> 00:11:54,460 And so we get minus h-bar squared over 2 mu par-- 173 00:11:54,460 --> 00:11:56,345 not partial, I'm so used to writing partials 174 00:11:56,345 --> 00:11:59,360 that I can't stop. 175 00:11:59,360 --> 00:12:08,390 Second derivative with respect to x, plus 1/2 kx squared. 176 00:12:08,390 --> 00:12:12,080 So that's the Hamiltonian. 177 00:12:12,080 --> 00:12:16,310 Now that looks kind of innocent, but it isn't. 178 00:12:16,310 --> 00:12:18,350 And so the first thing we want to do 179 00:12:18,350 --> 00:12:22,260 is get rid of the dimensionality, the units. 180 00:12:22,260 --> 00:12:25,670 So this is xi and it's defined as square root 181 00:12:25,670 --> 00:12:27,880 of alpha times x. 182 00:12:27,880 --> 00:12:36,290 Where alpha is defined as k mu square root over h-bar. 183 00:12:40,120 --> 00:12:43,870 Now it would be a perfectly reasonable exam question 184 00:12:43,870 --> 00:12:49,210 for you to prove that if I take this combination 185 00:12:49,210 --> 00:12:53,140 of physical quantities this will have dimension of 1 186 00:12:53,140 --> 00:12:53,660 over length. 187 00:12:57,980 --> 00:13:01,220 That makes xi a dimensionless quantity. 188 00:13:05,670 --> 00:13:09,020 I'm not even going to bother going through the derivation. 189 00:13:09,020 --> 00:13:16,920 The Hamiltonian becomes h-bar of omega times 2, 190 00:13:16,920 --> 00:13:27,110 minus second derivative with respect to xi plus xi squared. 191 00:13:31,800 --> 00:13:42,080 So this is now dimensionless. 192 00:13:42,080 --> 00:13:43,380 This has units. 193 00:13:43,380 --> 00:13:48,060 We divide by h-bar omega to make now everything dimensionless. 194 00:13:48,060 --> 00:13:52,380 And we get a differential equation 195 00:13:52,380 --> 00:13:57,900 that has the form minus the second derivative with respect 196 00:13:57,900 --> 00:14:07,230 xi plus xi squared, minus 2e over h-bar omega, 197 00:14:07,230 --> 00:14:11,700 times the wave function, expressive function of xi, 198 00:14:11,700 --> 00:14:14,760 not x. 199 00:14:14,760 --> 00:14:16,980 This is the differential equation we want to solve, 200 00:14:16,980 --> 00:14:21,291 and we don't do that in 5.61. 201 00:14:21,291 --> 00:14:23,540 You're never going to be asked to solve a differential 202 00:14:23,540 --> 00:14:26,090 equation like this. 203 00:14:26,090 --> 00:14:27,830 But you're certainly going to be asked 204 00:14:27,830 --> 00:14:30,960 to understand what the solution looks like, 205 00:14:30,960 --> 00:14:34,030 and perhaps that it is in fact a solution. 206 00:14:34,030 --> 00:14:38,450 But that's still pretty high value stuff, 207 00:14:38,450 --> 00:14:41,510 though you wouldn't really have to do that. 208 00:14:41,510 --> 00:14:43,820 This is the simplest way of writing the differential 209 00:14:43,820 --> 00:14:46,182 equation and it's dimensionless. 210 00:14:48,900 --> 00:14:55,530 The standard way of dealing with many differential equations 211 00:14:55,530 --> 00:14:59,640 is to say, OK, we have some function 212 00:14:59,640 --> 00:15:07,500 and it's going to be written as an exponential, a Gaussian, 213 00:15:07,500 --> 00:15:09,600 times some new function. 214 00:15:12,300 --> 00:15:16,170 And for quantum mechanics this is perfectly reasonable 215 00:15:16,170 --> 00:15:20,430 because we have a function in a well 216 00:15:20,430 --> 00:15:25,070 and the wave functions have to go to 0 at plus minus infinity. 217 00:15:25,070 --> 00:15:27,985 And this thing goes to zero at plus and minus infinity pretty 218 00:15:27,985 --> 00:15:28,485 strongly. 219 00:15:31,030 --> 00:15:34,440 So it's a good way of building in some 220 00:15:34,440 --> 00:15:38,570 of the expected behavior of the solution. 221 00:15:38,570 --> 00:15:40,410 And that's perfectly legal, and it just 222 00:15:40,410 --> 00:15:43,110 then defines what is the difference equation remaining 223 00:15:43,110 --> 00:15:45,494 for this? 224 00:15:45,494 --> 00:15:46,910 And it turns out, well we're going 225 00:15:46,910 --> 00:15:52,120 to get the Hermite equation. 226 00:15:52,120 --> 00:15:56,140 And this will be a Hermite polynomial, the solutions. 227 00:16:07,050 --> 00:16:15,050 Now one way of dealing with this is to simply say, well, 228 00:16:15,050 --> 00:16:17,290 we know the solution of this differential equation 229 00:16:17,290 --> 00:16:20,770 if this term weren't there. 230 00:16:20,770 --> 00:16:25,330 Because this is now the equation for a Gaussian. 231 00:16:25,330 --> 00:16:31,390 So building in a Gaussian as a factor in the solution, 232 00:16:31,390 --> 00:16:33,010 it's a perfectly reasonable thing. 233 00:16:33,010 --> 00:16:35,400 And then we have to say, what happens now when 234 00:16:35,400 --> 00:16:38,200 we put this term back in? 235 00:16:38,200 --> 00:16:43,460 And when we do we get this thing. 236 00:16:43,460 --> 00:16:52,470 Second derivative with respect to this polynomial-- 237 00:16:52,470 --> 00:16:58,650 I mean of this polynomial is equal to minus 2 xi times 238 00:16:58,650 --> 00:17:04,970 the hn d xi plus 2n hn. 239 00:17:08,560 --> 00:17:13,089 This is a famous differential equation, the Hermite equation, 240 00:17:13,089 --> 00:17:17,030 which is of no interest to us. 241 00:17:17,030 --> 00:17:19,400 And it generates the Hermite polynomials. 242 00:17:19,400 --> 00:17:22,339 These things are the Hermite polynomials. 243 00:17:26,560 --> 00:17:29,380 And they are treated in some kind of sacred manner 244 00:17:29,380 --> 00:17:31,990 in most of the textbooks, and I think 245 00:17:31,990 --> 00:17:36,100 that's really an offense because, well, we're 246 00:17:36,100 --> 00:17:39,950 not interested in mathematical functions. 247 00:17:39,950 --> 00:17:42,430 We're interested in insight and this is just 248 00:17:42,430 --> 00:17:43,690 putting up another barrier. 249 00:17:48,510 --> 00:17:54,510 Now with this equation, you can derive two things called 250 00:17:54,510 --> 00:18:04,720 recursion relations, and one of them 251 00:18:04,720 --> 00:18:10,150 is the derivative of this polynomial with respect to xi 252 00:18:10,150 --> 00:18:17,050 is equal to 2n times hn minus 1 of xi. 253 00:18:17,050 --> 00:18:20,630 Now that's not a surprise, because this is a polynomial. 254 00:18:20,630 --> 00:18:24,920 If you take a derivative of the variable 255 00:18:24,920 --> 00:18:29,720 you're going to reduce the power of each term by 1. 256 00:18:29,720 --> 00:18:33,610 Now it just happens to be lucky that when we reduce it to 1 257 00:18:33,610 --> 00:18:37,480 you don't get a sum of many different lower order 258 00:18:37,480 --> 00:18:40,870 polynomials, you just get 1. 259 00:18:40,870 --> 00:18:44,050 And there is another one, another recursive relation, 260 00:18:44,050 --> 00:18:47,890 where it tells you if you want to increase 261 00:18:47,890 --> 00:18:56,200 the order you can do this, you can multiply hn by xi. 262 00:18:56,200 --> 00:18:59,860 And that's obviously-- it is going to increase the order, 263 00:18:59,860 --> 00:19:01,300 but it might not do it cleanly. 264 00:19:01,300 --> 00:19:02,480 And it doesn't. 265 00:19:02,480 --> 00:19:03,190 And so we get-- 266 00:19:06,710 --> 00:19:14,840 We have a relationship between these three 267 00:19:14,840 --> 00:19:15,950 different polynomials. 268 00:19:19,740 --> 00:19:23,490 Now it turns out that these two equations are 269 00:19:23,490 --> 00:19:27,630 going to reappear, or at least their progeny 270 00:19:27,630 --> 00:19:32,700 will reappear, on Monday in terms of raising and lowering 271 00:19:32,700 --> 00:19:34,770 operators. 272 00:19:34,770 --> 00:19:38,800 And what you intuit about what happens if you multiply 273 00:19:38,800 --> 00:19:42,790 polynomial by the variable? 274 00:19:42,790 --> 00:19:44,940 Or what happens if you take its derivative? 275 00:19:44,940 --> 00:19:47,006 And it is very simple and beautiful, 276 00:19:47,006 --> 00:19:49,380 but I don't think this is very beautiful for our purposes 277 00:19:49,380 --> 00:19:52,500 as chemists. 278 00:19:52,500 --> 00:19:57,300 And one of the things that these recursive relationships do, 279 00:19:57,300 --> 00:20:01,470 which also hints at what's to come on Monday, 280 00:20:01,470 --> 00:20:05,925 is that we can calculate integrals like this. 281 00:20:11,242 --> 00:20:12,200 That's not what I want. 282 00:20:17,880 --> 00:20:20,460 This is a quantum number, it's an integer. 283 00:20:20,460 --> 00:20:28,236 It's v, not nu, and multiply it by x to the n, 284 00:20:28,236 --> 00:20:34,010 p to the m, psi that should be complex 285 00:20:34,010 --> 00:20:41,050 conjugated, v plus l, dx. 286 00:20:44,630 --> 00:20:47,480 It turns out for almost everything 287 00:20:47,480 --> 00:20:50,400 we want to do with harmonic oscillators 288 00:20:50,400 --> 00:20:54,210 we're going to want to know a lot of integrals like this. 289 00:20:58,570 --> 00:21:01,770 And one of the things we like is when 290 00:21:01,770 --> 00:21:03,919 an integral is promised to be 0 so we don't ever 291 00:21:03,919 --> 00:21:04,710 have to look at it. 292 00:21:07,370 --> 00:21:09,640 And so there are selection rules. 293 00:21:09,640 --> 00:21:13,720 And the selection rules for this kind of integral 294 00:21:13,720 --> 00:21:22,950 is l is equal to m plus n, m plus n minus 2, 295 00:21:22,950 --> 00:21:27,470 down to minus m plus n. 296 00:21:27,470 --> 00:21:33,260 So the only possible non-zero integrals of this form 297 00:21:33,260 --> 00:21:37,280 are for the change in quantum number 298 00:21:37,280 --> 00:21:44,550 by this l, which goes from m plus n down to minus m 299 00:21:44,550 --> 00:21:47,780 plus n in steps of two. 300 00:21:47,780 --> 00:21:49,910 The two shouldn't be too surprising, 301 00:21:49,910 --> 00:21:52,850 because there is symmetry and we have 302 00:21:52,850 --> 00:21:55,640 odd functions for odd quantum numbers 303 00:21:55,640 --> 00:21:58,340 and even functions for even quantum numbers. 304 00:21:58,340 --> 00:22:00,680 And so something like this is going 305 00:22:00,680 --> 00:22:02,390 to have a definite symmetry and it's 306 00:22:02,390 --> 00:22:06,360 going to change things within a symmetry, 307 00:22:06,360 --> 00:22:11,400 and so it's going to change the selection rule in steps two. 308 00:22:11,400 --> 00:22:13,400 Now you don't know what selection rules are for, 309 00:22:13,400 --> 00:22:16,220 or why you should get excited about these sorts of things, 310 00:22:16,220 --> 00:22:20,180 but it's really nice to know that almost all the integrals 311 00:22:20,180 --> 00:22:23,990 you are ever going to face for a particular problem are zero. 312 00:22:23,990 --> 00:22:27,170 And you can focus on a small number of non-zero ones, 313 00:22:27,170 --> 00:22:29,120 and it just turns out that the non-zero ones 314 00:22:29,120 --> 00:22:30,360 have really simple value. 315 00:22:35,950 --> 00:22:45,440 There also exists what's called a generating function, which 316 00:22:45,440 --> 00:22:51,850 is the Rodriguez formula. 317 00:22:51,850 --> 00:23:00,740 And that is the hn of xi is equal to minus 1 to the n, 318 00:23:00,740 --> 00:23:11,830 e to the psi squared, the derivative with respect to xi, 319 00:23:11,830 --> 00:23:13,992 e to minus xi squared. 320 00:23:16,500 --> 00:23:19,397 We have one that has a positive exponent, 321 00:23:19,397 --> 00:23:21,480 and one has a negative exponent, and we have this. 322 00:23:21,480 --> 00:23:25,080 So we could calculate any Hermite polynomial 323 00:23:25,080 --> 00:23:30,160 using this formula, which you will never do. 324 00:23:30,160 --> 00:23:36,870 But it's treated with great fanfare in textbooks. 325 00:23:36,870 --> 00:23:43,640 Now the solution to the harmonic oscillator wave function 326 00:23:43,640 --> 00:23:49,980 in real units, as opposed to dimensionless quantities, is-- 327 00:23:49,980 --> 00:23:54,880 and I'm just writing this down because I never would ever 328 00:23:54,880 --> 00:23:59,900 think about it this way, but I have to at least provide you 329 00:23:59,900 --> 00:24:01,910 with guidance-- 330 00:24:01,910 --> 00:24:07,580 so we have a factor 2 to the v, again this is v. 331 00:24:07,580 --> 00:24:15,320 Now the reason I'm emphasizing this is that in all texts 332 00:24:15,320 --> 00:24:19,590 v quantum numbers are italicized. 333 00:24:19,590 --> 00:24:24,170 And if you've thought about it for a minute, an italic v-- 334 00:24:24,170 --> 00:24:27,250 for mortals-- looks like a nu. 335 00:24:27,250 --> 00:24:28,270 It isn't quite. 336 00:24:28,270 --> 00:24:29,770 I don't know what the difference is, 337 00:24:29,770 --> 00:24:33,010 but if you have them side by side they are different. 338 00:24:33,010 --> 00:24:37,120 And so a large number of people who should know better 339 00:24:37,120 --> 00:24:41,540 refer to the vibrational quantum number as nu, which 340 00:24:41,540 --> 00:24:44,660 marks that person as, well, I won't 341 00:24:44,660 --> 00:24:48,350 say but it's not complimentary! 342 00:24:48,350 --> 00:24:56,880 We have this factor to the square root, 343 00:24:56,880 --> 00:25:00,180 and that's a normalization-- oh we got another part of it. 344 00:25:00,180 --> 00:25:06,390 Alpha over pi to the 1/4 power. 345 00:25:06,390 --> 00:25:08,240 So that's normalization. 346 00:25:08,240 --> 00:25:09,980 Then we have the Hermite polynomial. 347 00:25:13,560 --> 00:25:16,020 And you notice I've got xi back in here, 348 00:25:16,020 --> 00:25:17,460 which is really a shame. 349 00:25:17,460 --> 00:25:18,150 And we have-- 350 00:25:21,290 --> 00:25:25,520 So this is the general solution, we 351 00:25:25,520 --> 00:25:30,320 have the exponentially damped function, we have polynomials. 352 00:25:30,320 --> 00:25:31,820 These are all the actors that we're 353 00:25:31,820 --> 00:25:32,944 going to have to deal with. 354 00:25:36,310 --> 00:25:40,050 And I promise you, you will never 355 00:25:40,050 --> 00:25:43,710 use this unless you want to program a computer 356 00:25:43,710 --> 00:25:46,230 to calculate the wave function for God knows what reason. 357 00:25:49,840 --> 00:25:56,840 The quantum numbers we are 0, 1, 2. 358 00:25:56,840 --> 00:26:00,630 And for a harmonic oscillator, which goes to infinity, 359 00:26:00,630 --> 00:26:01,950 v goes to infinity, too. 360 00:26:01,950 --> 00:26:05,100 There's an infinite number of eigenfunctions 361 00:26:05,100 --> 00:26:08,990 of the quantum mechanical Hamiltonian-- 362 00:26:08,990 --> 00:26:12,940 of quantum mechanical harmonic oscillator. 363 00:26:12,940 --> 00:26:21,200 We have-- and the functions are normalized, 364 00:26:21,200 --> 00:26:26,690 we have psi plus and minus infinity goes to 0, 365 00:26:26,690 --> 00:26:31,370 we have psi v of 0. 366 00:26:31,370 --> 00:26:32,300 So this for all the-- 367 00:26:32,300 --> 00:26:33,400 I put a v there. 368 00:26:36,400 --> 00:26:50,050 Psi v is 0 for odd v, derivative of psi with respect to x, at x 369 00:26:50,050 --> 00:26:58,900 equals 0 is 0 per even v. So we have symmetric functions. 370 00:27:03,040 --> 00:27:11,140 And we have the energy levels is equal to h-bar omega, v 371 00:27:11,140 --> 00:27:13,810 plus 1/2. 372 00:27:13,810 --> 00:27:16,960 Now this is h over 2 pi, and this 373 00:27:16,960 --> 00:27:20,680 is nu times 2 pi, the frequency times 2 pi. 374 00:27:20,680 --> 00:27:23,050 So it could also be h nu. 375 00:27:23,050 --> 00:27:27,700 I have trouble remembering when there is a 2 pi involved. 376 00:27:34,280 --> 00:27:37,360 And we have this wonderful thing. 377 00:27:47,620 --> 00:27:51,750 It's as if v prime is equal to v this integral is 1, 378 00:27:51,750 --> 00:27:53,520 it's normalized. 379 00:27:53,520 --> 00:27:57,500 And a v prime is not equal to v, it's zero. 380 00:27:57,500 --> 00:28:00,630 And that stems from a theorem I mentioned before, 381 00:28:00,630 --> 00:28:10,000 is if you have two eigenvalues of the same Hamiltonian, 382 00:28:10,000 --> 00:28:12,300 eigenfunctions of the same Hamiltonian, 383 00:28:12,300 --> 00:28:15,710 and they belong to different eigenvalues 384 00:28:15,710 --> 00:28:19,520 their overlap integral is 0. 385 00:28:19,520 --> 00:28:21,890 We like zeros. 386 00:28:21,890 --> 00:28:25,490 We like normalization because the integral is just 1, 387 00:28:25,490 --> 00:28:26,930 it goes away. 388 00:28:26,930 --> 00:28:31,470 Or the integral is 0, the whole thing goes away. 389 00:28:31,470 --> 00:28:34,330 So that's really good. 390 00:28:34,330 --> 00:28:39,990 So we call this set of v's is orthonormal. 391 00:28:39,990 --> 00:28:41,760 Orthogonal and normalized. 392 00:28:41,760 --> 00:28:44,250 And the orthonormal terminology is used a lot. 393 00:28:44,250 --> 00:28:47,580 And in almost all quantum mechanical problems 394 00:28:47,580 --> 00:28:52,800 we like using an orthonormal set of functions 395 00:28:52,800 --> 00:28:54,348 to solve everything. 396 00:28:57,340 --> 00:29:00,300 Sometimes we have to do a little work to establish that, 397 00:29:00,300 --> 00:29:02,460 and I'll show much later in the course 398 00:29:02,460 --> 00:29:06,080 how when you have functions that are not orthogonal, 399 00:29:06,080 --> 00:29:08,280 and not normalized, you can create 400 00:29:08,280 --> 00:29:11,320 a set of functions which are. 401 00:29:11,320 --> 00:29:13,070 And this is something that a computer will 402 00:29:13,070 --> 00:29:14,380 do without breaking a sweat. 403 00:29:20,150 --> 00:29:27,190 Now we're back to my favorite topic, semi-classical. 404 00:29:32,100 --> 00:29:35,790 Because it's really easy to understand. 405 00:29:35,790 --> 00:29:38,220 Not just to understand the harmonic oscillator, 406 00:29:38,220 --> 00:29:40,050 but to use it in many problems. 407 00:29:43,560 --> 00:29:48,590 So in classic mechanics the kinetic energy 408 00:29:48,590 --> 00:29:58,540 is e minus v of x, or p squared over 2 mu. 409 00:29:58,540 --> 00:30:01,770 And so we can derive an equation for p 410 00:30:01,770 --> 00:30:09,330 of x classical mechanically, which is 2 mu e 411 00:30:09,330 --> 00:30:13,598 minus v of x square root. 412 00:30:16,670 --> 00:30:20,000 This is an extremely useful function. 413 00:30:20,000 --> 00:30:22,720 It's not an operator. 414 00:30:22,720 --> 00:30:24,240 It's a thing that we're going to use 415 00:30:24,240 --> 00:30:29,410 to make sense of everything, but it's not an operator. 416 00:30:29,410 --> 00:30:31,380 And so this is classic mechanics, and then 417 00:30:31,380 --> 00:30:36,900 in quantum mechanics, we know that Mr. de Broglie told us 418 00:30:36,900 --> 00:30:41,730 that the wavelength is equal to h over p. 419 00:30:41,730 --> 00:30:46,350 And we can generalize and say, well, maybe the wavelength 420 00:30:46,350 --> 00:30:49,390 is a function of x for potential, 421 00:30:49,390 --> 00:30:50,640 which is not constant. 422 00:30:57,710 --> 00:31:02,740 And even though this is not an operator in quantum mechanics 423 00:31:02,740 --> 00:31:04,740 this is true. 424 00:31:04,740 --> 00:31:09,870 That you can say the distance between consecutive nodes 425 00:31:09,870 --> 00:31:12,560 is lambda over 2. 426 00:31:12,560 --> 00:31:15,650 We can use this node relationship 427 00:31:15,650 --> 00:31:16,670 to great advantage. 428 00:31:21,830 --> 00:31:25,940 For the pair of nodes closest to x 429 00:31:25,940 --> 00:31:29,600 we can use this to calculate the distance between them. 430 00:31:32,710 --> 00:31:34,180 Very valuable. 431 00:31:34,180 --> 00:31:36,760 Because I also want to mention something. 432 00:31:36,760 --> 00:31:41,140 If you have an integrand which is rapidly oscillating, 433 00:31:41,140 --> 00:31:43,270 or if you have two rapidly oscillating functions 434 00:31:43,270 --> 00:31:46,120 and you're multiplying them together, 435 00:31:46,120 --> 00:31:55,040 that integral will accumulate to its final value at the position 436 00:31:55,040 --> 00:31:57,710 where the two oscillating functions are oscillating 437 00:31:57,710 --> 00:31:59,390 at the same frequency. 438 00:31:59,390 --> 00:32:01,460 That's the stationary phase point. 439 00:32:01,460 --> 00:32:03,980 And this is also a wonderful thing, 440 00:32:03,980 --> 00:32:06,560 because if you can figure out where 441 00:32:06,560 --> 00:32:08,540 the things you're multiplying together 442 00:32:08,540 --> 00:32:10,920 are oscillating at the same frequency, 443 00:32:10,920 --> 00:32:14,640 your integral becomes a number. 444 00:32:14,640 --> 00:32:17,170 No work ever. 445 00:32:17,170 --> 00:32:18,800 And that's a useful thing. 446 00:32:18,800 --> 00:32:22,200 OK, so the stationary phase method 447 00:32:22,200 --> 00:32:26,040 enables you to use this in a really fantastic way. 448 00:32:26,040 --> 00:32:29,690 And it's a little bit like Feynman's path integral idea, 449 00:32:29,690 --> 00:32:36,660 that you can calculate a complicated thing by evaluating 450 00:32:36,660 --> 00:32:40,410 an integral over a convenient path 451 00:32:40,410 --> 00:32:42,750 as opposed to integrating overall space, 452 00:32:42,750 --> 00:32:45,750 because everything that you care about 453 00:32:45,750 --> 00:32:47,790 comes from a stationary phase. 454 00:32:47,790 --> 00:32:50,400 Quantum mechanics is full of oscillations, 455 00:32:50,400 --> 00:32:53,220 classical mechanics doesn't have oscillations, 456 00:32:53,220 --> 00:32:56,190 and the two meet at the stationary phase point. 457 00:32:58,760 --> 00:33:02,120 Now we're going to use these ideas to calculate 458 00:33:02,120 --> 00:33:07,760 useful stuff for quantum mechanical vibrational wave 459 00:33:07,760 --> 00:33:08,570 functions. 460 00:33:08,570 --> 00:33:20,460 The shapes of psi of x gets exponentially damped, 461 00:33:20,460 --> 00:33:31,930 but it extends into the classically forbidden 462 00:33:31,930 --> 00:33:34,120 e less than v of x regions. 463 00:33:38,220 --> 00:33:42,250 The wave function, if we have potential 464 00:33:42,250 --> 00:33:45,650 and we have a wave function, that wave function 465 00:33:45,650 --> 00:33:48,620 is going to not go to 0 at the edge 466 00:33:48,620 --> 00:33:50,930 but it's going to have a tail. 467 00:33:50,930 --> 00:33:53,620 And that tail goes to 0 at infinity. 468 00:33:53,620 --> 00:33:58,400 And so there is some amplitude where the particle isn't 469 00:33:58,400 --> 00:34:00,400 allowed to be, classically. 470 00:34:00,400 --> 00:34:04,760 And that's where tunneling comes in. 471 00:34:04,760 --> 00:34:07,730 But the important thing, the important insight 472 00:34:07,730 --> 00:34:12,260 is that there are no nodes in the classically forbidden 473 00:34:12,260 --> 00:34:13,280 region. 474 00:34:13,280 --> 00:34:17,449 There is only exponential decay towards 0, 475 00:34:17,449 --> 00:34:22,290 and if you've chosen the wrong value of the energy, 476 00:34:22,290 --> 00:34:26,870 in other words a place where there is no eigenfunction, 477 00:34:26,870 --> 00:34:29,780 the wave function in the classically forbidden region 478 00:34:29,780 --> 00:34:31,730 will usually go to infinity. 479 00:34:31,730 --> 00:34:34,020 Either over here or over here, and says, 480 00:34:34,020 --> 00:34:37,969 well, it's clearly not a good function. 481 00:34:37,969 --> 00:34:39,370 But there are no 0 crossings. 482 00:34:44,340 --> 00:34:52,170 It's oscillating in e greater than v 483 00:34:52,170 --> 00:34:53,840 of x, the classically allowed region. 484 00:34:58,240 --> 00:35:05,340 The number of nodes is v. So we can 485 00:35:05,340 --> 00:35:07,860 have a v equals 0 function that just goes up and goes down, 486 00:35:07,860 --> 00:35:09,220 no internal nodes. 487 00:35:09,220 --> 00:35:12,060 v equals one, it crosses zero right in the middle. 488 00:35:15,980 --> 00:35:17,510 And we have the even oddness. 489 00:35:17,510 --> 00:35:26,116 Even v, even function. 490 00:35:26,116 --> 00:35:30,110 Odd v, odd function. 491 00:35:30,110 --> 00:35:35,390 For an even function you have a relative maximum at x equals 0, 492 00:35:35,390 --> 00:35:37,670 and for an odd function you have a 0. 493 00:35:37,670 --> 00:35:40,175 And the opposite further derivatives. 494 00:36:05,360 --> 00:36:08,150 The outer lobes, the ones on the ends 495 00:36:08,150 --> 00:36:16,300 just before the particle encounters the classical wall, 496 00:36:16,300 --> 00:36:18,560 you get the maximum amplitude. 497 00:36:21,260 --> 00:36:31,110 And so you can draw cartoons which look sort of like that. 498 00:36:31,110 --> 00:36:35,600 Most of the valuable stuff is at the other turning point. 499 00:36:35,600 --> 00:36:38,400 And there's oscillations in between, 500 00:36:38,400 --> 00:36:41,730 but often you really care about these two outer lobes. 501 00:36:44,880 --> 00:36:47,200 That's a pretty good simplification. 502 00:36:47,200 --> 00:36:53,630 Now there's a nice picture in McQuarrie on page 226 503 00:36:53,630 --> 00:36:57,390 which shows, especially for psi squared, 504 00:36:57,390 --> 00:36:59,870 that the nodes are pretty big. 505 00:36:59,870 --> 00:37:03,860 But they're not as big for relatively low quantum numbers 506 00:37:03,860 --> 00:37:07,110 as I've implied. 507 00:37:07,110 --> 00:37:08,640 But at really high quantum numbers 508 00:37:08,640 --> 00:37:12,930 we have a thing called the correspondence principle. 509 00:37:12,930 --> 00:37:18,450 And the corresponding principle says that quantum mechanics 510 00:37:18,450 --> 00:37:20,929 will do what classical mechanics does 511 00:37:20,929 --> 00:37:22,470 in the limit of high quantum numbers. 512 00:37:22,470 --> 00:37:24,136 And in the limit of high quantum numbers 513 00:37:24,136 --> 00:37:28,470 essentially all the amplitude is at the turning points. 514 00:37:28,470 --> 00:37:31,280 And in classic mechanics the particle 515 00:37:31,280 --> 00:37:35,570 is moving fast in the middle, and stops and turns around, 516 00:37:35,570 --> 00:37:37,340 and essentially all of the amplitude 517 00:37:37,340 --> 00:37:40,060 is at the turning point. 518 00:37:40,060 --> 00:37:41,640 So this is nice. 519 00:37:44,830 --> 00:37:48,610 Now we're getting into Bobism territory, 520 00:37:48,610 --> 00:37:50,110 because I'm really going to show you 521 00:37:50,110 --> 00:37:57,610 how to calculate whatever you need using 522 00:37:57,610 --> 00:37:58,780 these semi-classical ideas. 523 00:38:04,340 --> 00:38:09,420 We have the probability envelope. 524 00:38:14,220 --> 00:38:22,056 Psi star of x psi of x, and we're 525 00:38:22,056 --> 00:38:24,180 going to have both of these having the same quantum 526 00:38:24,180 --> 00:38:26,550 number, dx. 527 00:38:26,550 --> 00:38:32,490 So this is the probability of finding the particle near x 528 00:38:32,490 --> 00:38:36,030 in a region with dx. 529 00:38:44,330 --> 00:38:54,490 And this is the same thing as dx over v classical. 530 00:38:54,490 --> 00:38:58,480 It's not the same thing, there is a constant here, sorry. 531 00:38:58,480 --> 00:39:02,380 This probability density that you want 532 00:39:02,380 --> 00:39:05,890 is basically 1 over the classical velocity. 533 00:39:05,890 --> 00:39:08,649 And I demonstrated that when I walked 534 00:39:08,649 --> 00:39:10,690 across the room, when I walked fast in the middle 535 00:39:10,690 --> 00:39:13,110 and slow the outside. 536 00:39:13,110 --> 00:39:16,380 And you get the probability, you get this constant, 537 00:39:16,380 --> 00:39:19,140 by saying, OK, how long did it take for me to go from one end 538 00:39:19,140 --> 00:39:20,880 to the other? 539 00:39:20,880 --> 00:39:22,890 And comparing that, how long it took 540 00:39:22,890 --> 00:39:27,270 for me to go in some differential position. 541 00:39:27,270 --> 00:39:32,450 You get this constant in a simple way. 542 00:39:37,450 --> 00:39:50,490 v classical is equal to p classical over the mass, 543 00:39:50,490 --> 00:39:51,190 over the mu. 544 00:39:53,900 --> 00:39:56,820 But we know the function for p classical. 545 00:39:56,820 --> 00:40:05,857 We have 1 over mu times 2 mu e, minus v of x. 546 00:40:10,150 --> 00:40:12,340 So we know the velocity everywhere, 547 00:40:12,340 --> 00:40:16,930 and there's nothing terribly hard about figuring that out. 548 00:40:19,854 --> 00:40:22,270 And now we want to know what this proportionality constant 549 00:40:22,270 --> 00:40:23,140 is. 550 00:40:23,140 --> 00:40:33,140 And so for that we say the time to go from x to x plus dx, 551 00:40:33,140 --> 00:40:40,957 over the time to go from x minus to x plus. 552 00:40:40,957 --> 00:40:42,540 Because what's happening, the particle 553 00:40:42,540 --> 00:40:44,180 is going back and forth inside this 554 00:40:44,180 --> 00:40:49,310 well and so this is the time it takes to go one pass, 555 00:40:49,310 --> 00:40:51,770 and this is the time it takes to go 556 00:40:51,770 --> 00:40:53,350 through the region of interest. 557 00:40:53,350 --> 00:40:55,564 And so this ratio is the probability. 558 00:41:22,410 --> 00:41:27,920 And so we have the probability moving from left turning point 559 00:41:27,920 --> 00:41:30,270 the right turning point, and we want 560 00:41:30,270 --> 00:41:33,210 to know the probability in that interval. 561 00:41:33,210 --> 00:41:45,450 And so that's just dx over v classical at x, over tau, 562 00:41:45,450 --> 00:41:46,530 over 2. 563 00:41:46,530 --> 00:41:50,970 Because tau is the period, and we have half of the period, 564 00:41:50,970 --> 00:41:54,860 and so it's all together. 565 00:41:54,860 --> 00:41:58,020 I'm going to skip a little step, because it's taking too long. 566 00:41:58,020 --> 00:42:12,890 Psi star psi dx is equal to k over 2 pi squared, 567 00:42:12,890 --> 00:42:19,590 e minus v of x square of dx. 568 00:42:19,590 --> 00:42:24,290 Now so if we know the potential, and we know the energy, 569 00:42:24,290 --> 00:42:26,760 and we know the force constant we can say, 570 00:42:26,760 --> 00:42:30,600 well, this is the probability and-- 571 00:42:33,890 --> 00:42:41,330 but this is the probability of the semi-classical 572 00:42:41,330 --> 00:42:46,254 representation of psi star psi dx at x. 573 00:42:48,960 --> 00:42:52,250 Now this is oscillating, and this is not. 574 00:42:57,000 --> 00:42:58,470 So what we really want to know-- 575 00:43:05,490 --> 00:43:17,000 here's the-- if we have psi star whoops, slow down. 576 00:43:21,870 --> 00:43:28,980 This is oscillating and what we've calculated before 577 00:43:28,980 --> 00:43:32,290 is something that looks sort of like that. 578 00:43:32,290 --> 00:43:35,410 And if we multiply by 2 we have a curve 579 00:43:35,410 --> 00:43:38,210 that goes to the maximum of all these oscillations. 580 00:43:44,130 --> 00:43:49,920 The envelope psi star psi has the form. 581 00:43:49,920 --> 00:43:52,770 We've multiplied by 2, and so we end up 582 00:43:52,770 --> 00:44:08,677 getting 2k over pi squared, e minus v of x square root. 583 00:44:08,677 --> 00:44:11,010 Now you might say, well these are complicated functions, 584 00:44:11,010 --> 00:44:12,840 why should I bother with them? 585 00:44:12,840 --> 00:44:16,690 But if you wanted anything starting 586 00:44:16,690 --> 00:44:19,480 from the correct solution to the harmonic oscillator, 587 00:44:19,480 --> 00:44:21,220 using the Hermite polynomials, there's 588 00:44:21,220 --> 00:44:22,555 a whole lot more overhead. 589 00:44:25,620 --> 00:44:27,930 Notice also this is a function of x, not xi. 590 00:44:38,440 --> 00:44:40,960 This is the overlap function, it's 591 00:44:40,960 --> 00:44:44,440 the curve that touches the maximum of all these things 592 00:44:44,440 --> 00:44:47,470 and it's very useful if you want to know the probability 593 00:44:47,470 --> 00:44:49,233 of finding the system anywhere. 594 00:44:55,980 --> 00:45:03,696 And we get the node spacing from the equation h over p of x. 595 00:45:03,696 --> 00:45:05,700 And now here comes something really nice. 596 00:45:08,970 --> 00:45:12,040 It's called the semi-classical quantization interval. 597 00:45:14,560 --> 00:45:18,500 If we have any one dimensional potential, 598 00:45:18,500 --> 00:45:21,510 and we're at some energy, we'd like 599 00:45:21,510 --> 00:45:27,360 to be able to know how many levels are at that energy, 600 00:45:27,360 --> 00:45:28,710 or below. 601 00:45:28,710 --> 00:45:31,240 Or where are the energy levels? 602 00:45:31,240 --> 00:45:33,870 And we get that from this really incredible thing. 603 00:45:40,860 --> 00:45:43,140 We want to know that-- 604 00:45:50,710 --> 00:45:53,785 this is the difference between nodes. 605 00:45:53,785 --> 00:45:56,160 Right? 606 00:45:56,160 --> 00:45:59,160 And now if we would like to know x 607 00:45:59,160 --> 00:46:14,050 minus to x plus and some energy, we can replace lambda of x by h 608 00:46:14,050 --> 00:46:15,010 over p of x. 609 00:46:15,010 --> 00:46:24,790 And so we get p of x at that energy over h dx. 610 00:46:24,790 --> 00:46:28,920 pdx, that's called an action integral. 611 00:46:28,920 --> 00:46:30,960 Now I have to tell a little story. 612 00:46:30,960 --> 00:46:34,850 When I was a senior at Amherst College 613 00:46:34,850 --> 00:46:44,810 we had a oral exam for whether my thesis was 614 00:46:44,810 --> 00:46:47,310 going to be accepted or not. 615 00:46:47,310 --> 00:46:55,020 And one of my examiners asked me, what is the unit of h? 616 00:46:55,020 --> 00:46:58,340 Well, it's energy times time. 617 00:46:58,340 --> 00:46:59,990 And he wouldn't stop. 618 00:46:59,990 --> 00:47:02,720 He said no, I want something else. 619 00:47:02,720 --> 00:47:04,040 It's called action. 620 00:47:04,040 --> 00:47:07,400 Momentum times position. 621 00:47:07,400 --> 00:47:09,110 This is an action integral. 622 00:47:09,110 --> 00:47:11,150 And so anyway that's just a story. 623 00:47:11,150 --> 00:47:14,900 I spent a half an hour, and I was damn stubborn. 624 00:47:14,900 --> 00:47:19,520 I was not-- you know, it was energy times time. 625 00:47:19,520 --> 00:47:22,130 But that is much more insight here 626 00:47:22,130 --> 00:47:24,230 and that's maybe why I got so excited 627 00:47:24,230 --> 00:47:26,540 about this sort of an integral. 628 00:47:31,060 --> 00:47:36,250 If we want to know if we have an eigenvalue 629 00:47:36,250 --> 00:47:41,980 this integral has to be equal to h over 630 00:47:41,980 --> 00:47:47,180 2 times the number of nodes. 631 00:47:47,180 --> 00:47:49,440 Well it's pretty simple. 632 00:47:49,440 --> 00:47:55,690 So we can adjust e to satisfy this. 633 00:47:55,690 --> 00:48:00,100 Or if we wanted to know how many energy levels are 634 00:48:00,100 --> 00:48:02,470 at an energy below the energy we've 635 00:48:02,470 --> 00:48:05,870 chosen we evaluate this integral, 636 00:48:05,870 --> 00:48:10,420 and we get a number like 13.5. 637 00:48:10,420 --> 00:48:14,670 Well, it means there are 13 energy levels below that. 638 00:48:14,670 --> 00:48:19,400 Now often you want to know the density of states, 639 00:48:19,400 --> 00:48:23,120 the number of energy levels per unit energy, 640 00:48:23,120 --> 00:48:25,460 because that turns out to be the critical quantity 641 00:48:25,460 --> 00:48:28,190 in calculating many things you want to know. 642 00:48:28,190 --> 00:48:31,321 And you can get that from the semi-classical quantization. 643 00:48:35,480 --> 00:48:43,100 We're close to the end, I just want to say where we're going. 644 00:48:43,100 --> 00:48:47,750 We have classical pictures and I really, really 645 00:48:47,750 --> 00:48:50,900 want you to think about these classical pictures 646 00:48:50,900 --> 00:48:53,990 and use them rather than thinking, well, 647 00:48:53,990 --> 00:48:57,050 I'm going to have my cell phone program to evaluate 648 00:48:57,050 --> 00:48:59,619 all the necessary stuff. 649 00:48:59,619 --> 00:49:01,160 And there are certain things you want 650 00:49:01,160 --> 00:49:05,170 to remember about this semi-classical picture. 651 00:49:05,170 --> 00:49:10,420 And now we have the ability to calculate an infinite number 652 00:49:10,420 --> 00:49:13,270 of integrals involving harmonic oscillator 653 00:49:13,270 --> 00:49:17,230 functions in certain operators. 654 00:49:17,230 --> 00:49:18,980 Well, la-di-da. 655 00:49:18,980 --> 00:49:20,170 Why do we want them? 656 00:49:20,170 --> 00:49:22,000 Well one of the things we want is 657 00:49:22,000 --> 00:49:23,740 to be able to calculate the probability 658 00:49:23,740 --> 00:49:26,960 of a vibrational transition. 659 00:49:26,960 --> 00:49:29,060 That's called a transition element, 660 00:49:29,060 --> 00:49:30,680 and that's an easy thing to calculate. 661 00:49:34,000 --> 00:49:37,580 Another thing we want to do is to say, well, 662 00:49:37,580 --> 00:49:39,320 nature screwed up. 663 00:49:39,320 --> 00:49:43,790 This oscillator isn't harmonic, there's anharmonic term, 664 00:49:43,790 --> 00:49:46,080 and I would like to know what is the contribution 665 00:49:46,080 --> 00:49:51,450 of a constant times x cubed in the potential 666 00:49:51,450 --> 00:49:53,700 to the energy levels. 667 00:49:53,700 --> 00:49:55,710 And that's called perturbation theory. 668 00:49:55,710 --> 00:49:59,100 Or I want to have many harmonic oscillators 669 00:49:59,100 --> 00:50:01,800 in a polyatomic molecule, they talk to each other, 670 00:50:01,800 --> 00:50:06,360 and I want to calculate the interactions 671 00:50:06,360 --> 00:50:08,460 between these harmonic oscillators 672 00:50:08,460 --> 00:50:09,990 affect the energy level. 673 00:50:09,990 --> 00:50:13,470 Remember, when we have a separable Hamiltonian 674 00:50:13,470 --> 00:50:15,510 we can just write the energy levels as the sum 675 00:50:15,510 --> 00:50:16,670 of the individual Hamilton. 676 00:50:16,670 --> 00:50:19,080 And then there's coupling terms, and we 677 00:50:19,080 --> 00:50:22,950 deal with those but perturbation theory. 678 00:50:22,950 --> 00:50:26,680 There's all sorts of wonderful things we do. 679 00:50:26,680 --> 00:50:35,320 But we're going to consider these magical operators, 680 00:50:35,320 --> 00:50:39,430 a creation and annihilation operator, where a star-- 681 00:50:39,430 --> 00:50:49,010 a dagger operating on psi v gives 682 00:50:49,010 --> 00:50:54,570 the square root of v plus one, times psi v plus 1. 683 00:50:54,570 --> 00:50:58,960 And a dagger operating on v gives the square root of v-- 684 00:50:58,960 --> 00:51:00,220 let's write it-- 685 00:51:00,220 --> 00:51:04,540 I mean a non-dagger gives v square root times 686 00:51:04,540 --> 00:51:07,060 psi, v minus 1. 687 00:51:07,060 --> 00:51:13,210 And that x is equal to a plus a dagger times a constant. 688 00:51:13,210 --> 00:51:17,980 And so all of a sudden we can evaluate all integrals 689 00:51:17,980 --> 00:51:23,440 involving x, or powers of x, or momenta, or powers of momenta 690 00:51:23,440 --> 00:51:24,640 without thinking. 691 00:51:24,640 --> 00:51:28,420 Without ever looking at a function. 692 00:51:28,420 --> 00:51:32,550 And I guarantee you that this is embodied 693 00:51:32,550 --> 00:51:35,310 in the incredible amount of work done wherever 694 00:51:35,310 --> 00:51:39,510 we pretend almost every problem is a harmonic 695 00:51:39,510 --> 00:51:41,680 oscillator in disguise. 696 00:51:41,680 --> 00:51:44,710 Because these a's and a daggers enable 697 00:51:44,710 --> 00:51:47,500 you to generate everything without ever 698 00:51:47,500 --> 00:51:52,600 converting from x to xi, without ever looking at an integral. 699 00:51:52,600 --> 00:51:57,190 It's all just a manipulation of algebra. 700 00:51:57,190 --> 00:52:01,380 And it's not just convenient, but there's insight 701 00:52:01,380 --> 00:52:04,950 and so this is what I want to convey. 702 00:52:04,950 --> 00:52:08,370 That you will get tremendous insight. 703 00:52:08,370 --> 00:52:11,270 Maybe, maybe I sold you on semi-classical, 704 00:52:11,270 --> 00:52:16,340 and I don't apologize for that because that's very useful. 705 00:52:16,340 --> 00:52:20,630 But the next lecture, when you get the a's and a daggers, 706 00:52:20,630 --> 00:52:23,290 it'll just knock your socks off. 707 00:52:23,290 --> 00:52:25,430 OK, that's it.