WEBVTT 00:00:00.090 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.820 Commons license. 00:00:03.820 --> 00:00:06.030 Your support will help MIT OpenCourseWare 00:00:06.030 --> 00:00:10.120 continue to offer high quality educational resources for free. 00:00:10.120 --> 00:00:12.660 To make a donation or to view additional materials 00:00:12.660 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:17.850 at ocw.mit.edu. 00:00:24.570 --> 00:00:26.820 ROBERT FIELD: So today we're going 00:00:26.820 --> 00:00:29.190 to go from the classical mechanical treatment 00:00:29.190 --> 00:00:33.360 of the harmonic oscillator to a quantum mechanical treatment. 00:00:33.360 --> 00:00:37.520 And I warn you that I intentionally 00:00:37.520 --> 00:00:41.710 am going to make this look bad, because 00:00:41.710 --> 00:00:48.140 the semi-classical approach at the end of this lecture 00:00:48.140 --> 00:00:50.480 will make it all really simple. 00:00:50.480 --> 00:00:52.670 And then on Monday I'll introduce 00:00:52.670 --> 00:00:54.920 creation and annihilation operators, 00:00:54.920 --> 00:00:57.050 which makes the harmonic oscillator simpler 00:00:57.050 --> 00:00:59.530 than the particle in a box. 00:00:59.530 --> 00:01:02.470 You can't believe that, but all right. 00:01:02.470 --> 00:01:10.930 So last time we treated the harmonic oscillator classically 00:01:10.930 --> 00:01:16.150 and so we derive the equation of motion from forces 00:01:16.150 --> 00:01:19.910 equal to mass times acceleration, and we solved it. 00:01:19.910 --> 00:01:26.140 And we saw that we have this quantity omega, which initially 00:01:26.140 --> 00:01:28.090 I just introduced as a constant, which 00:01:28.090 --> 00:01:35.700 was a way of combining the force constant and the mass. 00:01:35.700 --> 00:01:40.320 And then I showed that the period of oscillation 00:01:40.320 --> 00:01:45.105 is 1 over the frequency, which is 2 pi over omega. 00:01:47.980 --> 00:01:50.140 Now one of the things that people 00:01:50.140 --> 00:01:54.160 have trouble remembering under exam pressure 00:01:54.160 --> 00:01:55.585 is turning points. 00:02:01.620 --> 00:02:05.490 And this comes when the energy is 00:02:05.490 --> 00:02:08.900 equal to the potential at a turning point. 00:02:15.490 --> 00:02:18.140 And since the potential is 1/2 kx 00:02:18.140 --> 00:02:23.540 squared we get the equation for the turning point 00:02:23.540 --> 00:02:28.520 at a given energy, which is equal to plus or minus 00:02:28.520 --> 00:02:30.260 the square root of 2e/k. 00:02:35.420 --> 00:02:37.940 Now when you're drawing pictures there 00:02:37.940 --> 00:02:40.542 are certain things that anchor the pictures, 00:02:40.542 --> 00:02:41.375 like turning points. 00:02:44.000 --> 00:02:49.120 And now at the end of the previous lecture 00:02:49.120 --> 00:02:54.170 I calculated the classical mechanical average, 00:02:54.170 --> 00:02:57.010 and we use we use this kind of notation 00:02:57.010 --> 00:02:58.090 in classical mechanics. 00:02:58.090 --> 00:03:00.380 Sometimes we use this notation, too, 00:03:00.380 --> 00:03:03.640 but this is what we mean by the average value in quantum 00:03:03.640 --> 00:03:04.850 mechanics. 00:03:04.850 --> 00:03:12.040 And we found that this was the total energy divided by 2. 00:03:12.040 --> 00:03:19.817 And the average momentum is the energy divided by 2. 00:03:19.817 --> 00:03:21.400 And that's the basis for some insight, 00:03:21.400 --> 00:03:24.370 because as a harmonic oscillator moves 00:03:24.370 --> 00:03:28.270 it throws energy back and forth between kinetic energy 00:03:28.270 --> 00:03:30.310 and potential energy. 00:03:30.310 --> 00:03:32.230 And then one of my favorite things, 00:03:32.230 --> 00:03:36.340 and one of my favorite tortures on short answers, 00:03:36.340 --> 00:03:41.800 is the average values of-- 00:03:47.460 --> 00:03:53.220 And does anybody want to remind the class the easy way, 00:03:53.220 --> 00:03:57.300 or the two easy ways to know what this is? 00:03:57.300 --> 00:04:00.190 And the person who answered the question last time 00:04:00.190 --> 00:04:02.040 is disqualified. 00:04:02.040 --> 00:04:02.966 Yes. 00:04:02.966 --> 00:04:04.590 AUDIENCE: There is the symmetry method. 00:04:04.590 --> 00:04:05.020 ROBERT FIELD: What? 00:04:05.020 --> 00:04:06.350 AUDIENCE: There's symmetry. 00:04:06.350 --> 00:04:10.650 So if the system's symmetrical that the average value will 00:04:10.650 --> 00:04:12.880 actually be that symmetry point. 00:04:12.880 --> 00:04:14.100 ROBERT FIELD: Yes. 00:04:14.100 --> 00:04:17.470 So and the other is that the harmonic oscillator 00:04:17.470 --> 00:04:22.390 isn't moving, and so there is no way that the coordinate-- 00:04:22.390 --> 00:04:28.720 that the average value of either the coordinate or the momentum 00:04:28.720 --> 00:04:30.730 could be different from 0. 00:04:30.730 --> 00:04:33.280 However you do it you want to be doing it 00:04:33.280 --> 00:04:35.260 in seconds, not minutes. 00:04:35.260 --> 00:04:37.990 And certainly not by calculating integral. 00:04:37.990 --> 00:04:43.180 And now x squared and p squared, they're easy too. 00:04:43.180 --> 00:04:50.560 Especially if you know t and v. And so we then use those 00:04:50.560 --> 00:04:55.820 to calculate the variance. 00:04:55.820 --> 00:04:58.990 And that's defined as the average value 00:04:58.990 --> 00:05:03.880 of the square minus the square of the average square root. 00:05:07.100 --> 00:05:11.690 And what we find is that the variance 00:05:11.690 --> 00:05:19.470 of x times the variance of p is equal to e over omega. 00:05:19.470 --> 00:05:26.160 So as you go up in energy this joint uncertainty increases. 00:05:26.160 --> 00:05:32.040 And we'll find that that also is true for quantum mechanics. 00:05:32.040 --> 00:05:35.550 So this is sort of the kind of questions 00:05:35.550 --> 00:05:38.940 you want to be asking in quantum mechanics. 00:05:38.940 --> 00:05:41.100 And you want to be able to be guided 00:05:41.100 --> 00:05:43.340 by what you know from classical mechanics, 00:05:43.340 --> 00:05:46.480 and you want to be able to do it fast. 00:05:46.480 --> 00:05:49.240 Today's menu is what I would call-- 00:05:59.630 --> 00:06:03.110 This lecture is gratuitous complexity. 00:06:03.110 --> 00:06:05.210 Does everybody know what gratuitous means? 00:06:09.380 --> 00:06:12.200 This is one of my favorite Bobisms. 00:06:15.410 --> 00:06:19.340 And you'll hear other Bobisms during the course 00:06:19.340 --> 00:06:21.320 of this course. 00:06:24.920 --> 00:06:27.550 What I want you to be able to do for a lot 00:06:27.550 --> 00:06:32.620 of mechanical problems is to know the answer, 00:06:32.620 --> 00:06:36.760 or know what things look like without doing a calculation. 00:06:36.760 --> 00:06:39.670 In particular, not solving a differential equation 00:06:39.670 --> 00:06:41.800 or evaluating in the integrals. 00:06:41.800 --> 00:06:47.050 You want to be able to draw these pictures instantly. 00:06:47.050 --> 00:06:53.860 Now in the modern age everyone has a cell phone, 00:06:53.860 --> 00:06:55.750 and one could have a program in there 00:06:55.750 --> 00:06:59.720 to calculate what anything you wanted for harmonic oscillator. 00:06:59.720 --> 00:07:03.700 But chances are you won't be prepared for that. 00:07:03.700 --> 00:07:08.260 And if you want to have insights into how do various things 00:07:08.260 --> 00:07:13.180 you want to know about harmonic oscillators come about? 00:07:13.180 --> 00:07:15.640 You need the pictures, as opposed 00:07:15.640 --> 00:07:17.710 to the computer program. 00:07:17.710 --> 00:07:23.650 Now the pictures involve an advance investment of energy. 00:07:23.650 --> 00:07:25.174 You want to understand every detail 00:07:25.174 --> 00:07:26.215 of these little pictures. 00:07:31.254 --> 00:07:32.920 I'm going to write this runner equation, 00:07:32.920 --> 00:07:35.770 I'm going to clean it up to get rid of units 00:07:35.770 --> 00:07:38.080 which makes it universal. 00:07:38.080 --> 00:07:41.690 So it becomes a dimensionless equation. 00:07:41.690 --> 00:07:46.180 And the unit removal, or the thing that 00:07:46.180 --> 00:07:50.740 takes you from a specific problem where there's 00:07:50.740 --> 00:07:55.960 a particular force constant and a particular reduced mass, 00:07:55.960 --> 00:07:57.640 and makes it into a general problem. 00:07:57.640 --> 00:08:03.500 There is one or two constants that combine those things. 00:08:03.500 --> 00:08:06.700 And you've taken them out at the end of a calculation. 00:08:06.700 --> 00:08:11.350 If you need to have real units you can put those back in. 00:08:11.350 --> 00:08:12.820 And that's a very wonderful thing, 00:08:12.820 --> 00:08:16.630 and that enables us to draw pictures 00:08:16.630 --> 00:08:21.180 without thinking about, what is the problem? 00:08:21.180 --> 00:08:24.600 And then the solution of this differential equation-- 00:08:24.600 --> 00:08:28.260 which is actually quite an awful differential equation at least 00:08:28.260 --> 00:08:30.900 for people who are not mathematicians-- 00:08:30.900 --> 00:08:33.750 and the solution can be expressed 00:08:33.750 --> 00:08:36.090 as the product of a Gaussian function 00:08:36.090 --> 00:08:39.990 which goes to 0 at plus and minus infinity, so it makes 00:08:39.990 --> 00:08:42.390 the function well behaved. 00:08:42.390 --> 00:08:46.200 Times something that produces nodes, a polynomial. 00:08:46.200 --> 00:08:48.960 Does anyone want to give me a definition of a polynomial? 00:08:56.720 --> 00:08:58.430 Silence. 00:08:58.430 --> 00:09:01.495 OK, your turn. 00:09:01.495 --> 00:09:06.740 AUDIENCE: It's the linear combination of some numbers 00:09:06.740 --> 00:09:08.417 taken to different power 00:09:08.417 --> 00:09:09.250 ROBERT FIELD: Right. 00:09:09.250 --> 00:09:12.185 A sum of integer powers of a variable. 00:09:14.700 --> 00:09:20.610 And when we take a derivative of a polynomial 00:09:20.610 --> 00:09:23.610 we reduce the order of the polynomial. 00:09:23.610 --> 00:09:26.820 A little bit of thought, if you have a first order 00:09:26.820 --> 00:09:29.260 polynomial there'll be one node. 00:09:29.260 --> 00:09:31.380 If there is a second order there'll be two nodes. 00:09:33.920 --> 00:09:37.130 And nodes are very important. 00:09:37.130 --> 00:09:40.870 And so when we're going to be dealing with cartoons 00:09:40.870 --> 00:09:44.650 of the wave function, and then using semi-classical ideas 00:09:44.650 --> 00:09:49.450 to actually semi-calculate things that you'd want to know, 00:09:49.450 --> 00:09:53.740 the nodes are really important. 00:09:53.740 --> 00:09:57.760 And what's going to happen Monday 00:09:57.760 --> 00:10:00.980 is we'll throw away all this garbage 00:10:00.980 --> 00:10:05.120 and we will replace everything by these creation 00:10:05.120 --> 00:10:07.370 and annihilation operators. 00:10:07.370 --> 00:10:12.360 Which do have really simple properties, 00:10:12.360 --> 00:10:16.190 which you can use to do astonishingly 00:10:16.190 --> 00:10:19.190 complicated things without breaking a sweat. 00:10:25.910 --> 00:10:30.380 And the final exam is in this room on the first day of exam 00:10:30.380 --> 00:10:33.290 period, at least it's on a Monday, 00:10:33.290 --> 00:10:34.864 and it's in the afternoon. 00:10:47.120 --> 00:10:51.980 In the non-lecture part of the notes 00:10:51.980 --> 00:10:58.060 I replaced the mass for one mass on a spring 00:10:58.060 --> 00:11:06.680 by the reduced mass, which is m1 m2 over m1 plus m2 for two 00:11:06.680 --> 00:11:09.180 masses connected by a spring. 00:11:09.180 --> 00:11:16.640 And I go back and forth between using mu and m, and that's OK. 00:11:16.640 --> 00:11:22.450 All right so in the notes the differential equations 00:11:22.450 --> 00:11:24.160 in the first few pages are expressed 00:11:24.160 --> 00:11:26.210 as partial differential equations. 00:11:26.210 --> 00:11:28.780 They're total differential equations. 00:11:28.780 --> 00:11:31.960 That'll get changed. 00:11:31.960 --> 00:11:43.610 The Hamiltonian is t plus v, and in the usual form t 00:11:43.610 --> 00:11:45.920 is p squared over 2 mu. 00:11:45.920 --> 00:11:54.460 And so we get minus h-bar squared over 2 mu par-- 00:11:54.460 --> 00:11:56.345 not partial, I'm so used to writing partials 00:11:56.345 --> 00:11:59.360 that I can't stop. 00:11:59.360 --> 00:12:08.390 Second derivative with respect to x, plus 1/2 kx squared. 00:12:08.390 --> 00:12:12.080 So that's the Hamiltonian. 00:12:12.080 --> 00:12:16.310 Now that looks kind of innocent, but it isn't. 00:12:16.310 --> 00:12:18.350 And so the first thing we want to do 00:12:18.350 --> 00:12:22.260 is get rid of the dimensionality, the units. 00:12:22.260 --> 00:12:25.670 So this is xi and it's defined as square root 00:12:25.670 --> 00:12:27.880 of alpha times x. 00:12:27.880 --> 00:12:36.290 Where alpha is defined as k mu square root over h-bar. 00:12:40.120 --> 00:12:43.870 Now it would be a perfectly reasonable exam question 00:12:43.870 --> 00:12:49.210 for you to prove that if I take this combination 00:12:49.210 --> 00:12:53.140 of physical quantities this will have dimension of 1 00:12:53.140 --> 00:12:53.660 over length. 00:12:57.980 --> 00:13:01.220 That makes xi a dimensionless quantity. 00:13:05.670 --> 00:13:09.020 I'm not even going to bother going through the derivation. 00:13:09.020 --> 00:13:16.920 The Hamiltonian becomes h-bar of omega times 2, 00:13:16.920 --> 00:13:27.110 minus second derivative with respect to xi plus xi squared. 00:13:31.800 --> 00:13:42.080 So this is now dimensionless. 00:13:42.080 --> 00:13:43.380 This has units. 00:13:43.380 --> 00:13:48.060 We divide by h-bar omega to make now everything dimensionless. 00:13:48.060 --> 00:13:52.380 And we get a differential equation 00:13:52.380 --> 00:13:57.900 that has the form minus the second derivative with respect 00:13:57.900 --> 00:14:07.230 xi plus xi squared, minus 2e over h-bar omega, 00:14:07.230 --> 00:14:11.700 times the wave function, expressive function of xi, 00:14:11.700 --> 00:14:14.760 not x. 00:14:14.760 --> 00:14:16.980 This is the differential equation we want to solve, 00:14:16.980 --> 00:14:21.291 and we don't do that in 5.61. 00:14:21.291 --> 00:14:23.540 You're never going to be asked to solve a differential 00:14:23.540 --> 00:14:26.090 equation like this. 00:14:26.090 --> 00:14:27.830 But you're certainly going to be asked 00:14:27.830 --> 00:14:30.960 to understand what the solution looks like, 00:14:30.960 --> 00:14:34.030 and perhaps that it is in fact a solution. 00:14:34.030 --> 00:14:38.450 But that's still pretty high value stuff, 00:14:38.450 --> 00:14:41.510 though you wouldn't really have to do that. 00:14:41.510 --> 00:14:43.820 This is the simplest way of writing the differential 00:14:43.820 --> 00:14:46.182 equation and it's dimensionless. 00:14:48.900 --> 00:14:55.530 The standard way of dealing with many differential equations 00:14:55.530 --> 00:14:59.640 is to say, OK, we have some function 00:14:59.640 --> 00:15:07.500 and it's going to be written as an exponential, a Gaussian, 00:15:07.500 --> 00:15:09.600 times some new function. 00:15:12.300 --> 00:15:16.170 And for quantum mechanics this is perfectly reasonable 00:15:16.170 --> 00:15:20.430 because we have a function in a well 00:15:20.430 --> 00:15:25.070 and the wave functions have to go to 0 at plus minus infinity. 00:15:25.070 --> 00:15:27.985 And this thing goes to zero at plus and minus infinity pretty 00:15:27.985 --> 00:15:28.485 strongly. 00:15:31.030 --> 00:15:34.440 So it's a good way of building in some 00:15:34.440 --> 00:15:38.570 of the expected behavior of the solution. 00:15:38.570 --> 00:15:40.410 And that's perfectly legal, and it just 00:15:40.410 --> 00:15:43.110 then defines what is the difference equation remaining 00:15:43.110 --> 00:15:45.494 for this? 00:15:45.494 --> 00:15:46.910 And it turns out, well we're going 00:15:46.910 --> 00:15:52.120 to get the Hermite equation. 00:15:52.120 --> 00:15:56.140 And this will be a Hermite polynomial, the solutions. 00:16:07.050 --> 00:16:15.050 Now one way of dealing with this is to simply say, well, 00:16:15.050 --> 00:16:17.290 we know the solution of this differential equation 00:16:17.290 --> 00:16:20.770 if this term weren't there. 00:16:20.770 --> 00:16:25.330 Because this is now the equation for a Gaussian. 00:16:25.330 --> 00:16:31.390 So building in a Gaussian as a factor in the solution, 00:16:31.390 --> 00:16:33.010 it's a perfectly reasonable thing. 00:16:33.010 --> 00:16:35.400 And then we have to say, what happens now when 00:16:35.400 --> 00:16:38.200 we put this term back in? 00:16:38.200 --> 00:16:43.460 And when we do we get this thing. 00:16:43.460 --> 00:16:52.470 Second derivative with respect to this polynomial-- 00:16:52.470 --> 00:16:58.650 I mean of this polynomial is equal to minus 2 xi times 00:16:58.650 --> 00:17:04.970 the hn d xi plus 2n hn. 00:17:08.560 --> 00:17:13.089 This is a famous differential equation, the Hermite equation, 00:17:13.089 --> 00:17:17.030 which is of no interest to us. 00:17:17.030 --> 00:17:19.400 And it generates the Hermite polynomials. 00:17:19.400 --> 00:17:22.339 These things are the Hermite polynomials. 00:17:26.560 --> 00:17:29.380 And they are treated in some kind of sacred manner 00:17:29.380 --> 00:17:31.990 in most of the textbooks, and I think 00:17:31.990 --> 00:17:36.100 that's really an offense because, well, we're 00:17:36.100 --> 00:17:39.950 not interested in mathematical functions. 00:17:39.950 --> 00:17:42.430 We're interested in insight and this is just 00:17:42.430 --> 00:17:43.690 putting up another barrier. 00:17:48.510 --> 00:17:54.510 Now with this equation, you can derive two things called 00:17:54.510 --> 00:18:04.720 recursion relations, and one of them 00:18:04.720 --> 00:18:10.150 is the derivative of this polynomial with respect to xi 00:18:10.150 --> 00:18:17.050 is equal to 2n times hn minus 1 of xi. 00:18:17.050 --> 00:18:20.630 Now that's not a surprise, because this is a polynomial. 00:18:20.630 --> 00:18:24.920 If you take a derivative of the variable 00:18:24.920 --> 00:18:29.720 you're going to reduce the power of each term by 1. 00:18:29.720 --> 00:18:33.610 Now it just happens to be lucky that when we reduce it to 1 00:18:33.610 --> 00:18:37.480 you don't get a sum of many different lower order 00:18:37.480 --> 00:18:40.870 polynomials, you just get 1. 00:18:40.870 --> 00:18:44.050 And there is another one, another recursive relation, 00:18:44.050 --> 00:18:47.890 where it tells you if you want to increase 00:18:47.890 --> 00:18:56.200 the order you can do this, you can multiply hn by xi. 00:18:56.200 --> 00:18:59.860 And that's obviously-- it is going to increase the order, 00:18:59.860 --> 00:19:01.300 but it might not do it cleanly. 00:19:01.300 --> 00:19:02.480 And it doesn't. 00:19:02.480 --> 00:19:03.190 And so we get-- 00:19:06.710 --> 00:19:14.840 We have a relationship between these three 00:19:14.840 --> 00:19:15.950 different polynomials. 00:19:19.740 --> 00:19:23.490 Now it turns out that these two equations are 00:19:23.490 --> 00:19:27.630 going to reappear, or at least their progeny 00:19:27.630 --> 00:19:32.700 will reappear, on Monday in terms of raising and lowering 00:19:32.700 --> 00:19:34.770 operators. 00:19:34.770 --> 00:19:38.800 And what you intuit about what happens if you multiply 00:19:38.800 --> 00:19:42.790 polynomial by the variable? 00:19:42.790 --> 00:19:44.940 Or what happens if you take its derivative? 00:19:44.940 --> 00:19:47.006 And it is very simple and beautiful, 00:19:47.006 --> 00:19:49.380 but I don't think this is very beautiful for our purposes 00:19:49.380 --> 00:19:52.500 as chemists. 00:19:52.500 --> 00:19:57.300 And one of the things that these recursive relationships do, 00:19:57.300 --> 00:20:01.470 which also hints at what's to come on Monday, 00:20:01.470 --> 00:20:05.925 is that we can calculate integrals like this. 00:20:11.242 --> 00:20:12.200 That's not what I want. 00:20:17.880 --> 00:20:20.460 This is a quantum number, it's an integer. 00:20:20.460 --> 00:20:28.236 It's v, not nu, and multiply it by x to the n, 00:20:28.236 --> 00:20:34.010 p to the m, psi that should be complex 00:20:34.010 --> 00:20:41.050 conjugated, v plus l, dx. 00:20:44.630 --> 00:20:47.480 It turns out for almost everything 00:20:47.480 --> 00:20:50.400 we want to do with harmonic oscillators 00:20:50.400 --> 00:20:54.210 we're going to want to know a lot of integrals like this. 00:20:58.570 --> 00:21:01.770 And one of the things we like is when 00:21:01.770 --> 00:21:03.919 an integral is promised to be 0 so we don't ever 00:21:03.919 --> 00:21:04.710 have to look at it. 00:21:07.370 --> 00:21:09.640 And so there are selection rules. 00:21:09.640 --> 00:21:13.720 And the selection rules for this kind of integral 00:21:13.720 --> 00:21:22.950 is l is equal to m plus n, m plus n minus 2, 00:21:22.950 --> 00:21:27.470 down to minus m plus n. 00:21:27.470 --> 00:21:33.260 So the only possible non-zero integrals of this form 00:21:33.260 --> 00:21:37.280 are for the change in quantum number 00:21:37.280 --> 00:21:44.550 by this l, which goes from m plus n down to minus m 00:21:44.550 --> 00:21:47.780 plus n in steps of two. 00:21:47.780 --> 00:21:49.910 The two shouldn't be too surprising, 00:21:49.910 --> 00:21:52.850 because there is symmetry and we have 00:21:52.850 --> 00:21:55.640 odd functions for odd quantum numbers 00:21:55.640 --> 00:21:58.340 and even functions for even quantum numbers. 00:21:58.340 --> 00:22:00.680 And so something like this is going 00:22:00.680 --> 00:22:02.390 to have a definite symmetry and it's 00:22:02.390 --> 00:22:06.360 going to change things within a symmetry, 00:22:06.360 --> 00:22:11.400 and so it's going to change the selection rule in steps two. 00:22:11.400 --> 00:22:13.400 Now you don't know what selection rules are for, 00:22:13.400 --> 00:22:16.220 or why you should get excited about these sorts of things, 00:22:16.220 --> 00:22:20.180 but it's really nice to know that almost all the integrals 00:22:20.180 --> 00:22:23.990 you are ever going to face for a particular problem are zero. 00:22:23.990 --> 00:22:27.170 And you can focus on a small number of non-zero ones, 00:22:27.170 --> 00:22:29.120 and it just turns out that the non-zero ones 00:22:29.120 --> 00:22:30.360 have really simple value. 00:22:35.950 --> 00:22:45.440 There also exists what's called a generating function, which 00:22:45.440 --> 00:22:51.850 is the Rodriguez formula. 00:22:51.850 --> 00:23:00.740 And that is the hn of xi is equal to minus 1 to the n, 00:23:00.740 --> 00:23:11.830 e to the psi squared, the derivative with respect to xi, 00:23:11.830 --> 00:23:13.992 e to minus xi squared. 00:23:16.500 --> 00:23:19.397 We have one that has a positive exponent, 00:23:19.397 --> 00:23:21.480 and one has a negative exponent, and we have this. 00:23:21.480 --> 00:23:25.080 So we could calculate any Hermite polynomial 00:23:25.080 --> 00:23:30.160 using this formula, which you will never do. 00:23:30.160 --> 00:23:36.870 But it's treated with great fanfare in textbooks. 00:23:36.870 --> 00:23:43.640 Now the solution to the harmonic oscillator wave function 00:23:43.640 --> 00:23:49.980 in real units, as opposed to dimensionless quantities, is-- 00:23:49.980 --> 00:23:54.880 and I'm just writing this down because I never would ever 00:23:54.880 --> 00:23:59.900 think about it this way, but I have to at least provide you 00:23:59.900 --> 00:24:01.910 with guidance-- 00:24:01.910 --> 00:24:07.580 so we have a factor 2 to the v, again this is v. 00:24:07.580 --> 00:24:15.320 Now the reason I'm emphasizing this is that in all texts 00:24:15.320 --> 00:24:19.590 v quantum numbers are italicized. 00:24:19.590 --> 00:24:24.170 And if you've thought about it for a minute, an italic v-- 00:24:24.170 --> 00:24:27.250 for mortals-- looks like a nu. 00:24:27.250 --> 00:24:28.270 It isn't quite. 00:24:28.270 --> 00:24:29.770 I don't know what the difference is, 00:24:29.770 --> 00:24:33.010 but if you have them side by side they are different. 00:24:33.010 --> 00:24:37.120 And so a large number of people who should know better 00:24:37.120 --> 00:24:41.540 refer to the vibrational quantum number as nu, which 00:24:41.540 --> 00:24:44.660 marks that person as, well, I won't 00:24:44.660 --> 00:24:48.350 say but it's not complimentary! 00:24:48.350 --> 00:24:56.880 We have this factor to the square root, 00:24:56.880 --> 00:25:00.180 and that's a normalization-- oh we got another part of it. 00:25:00.180 --> 00:25:06.390 Alpha over pi to the 1/4 power. 00:25:06.390 --> 00:25:08.240 So that's normalization. 00:25:08.240 --> 00:25:09.980 Then we have the Hermite polynomial. 00:25:13.560 --> 00:25:16.020 And you notice I've got xi back in here, 00:25:16.020 --> 00:25:17.460 which is really a shame. 00:25:17.460 --> 00:25:18.150 And we have-- 00:25:21.290 --> 00:25:25.520 So this is the general solution, we 00:25:25.520 --> 00:25:30.320 have the exponentially damped function, we have polynomials. 00:25:30.320 --> 00:25:31.820 These are all the actors that we're 00:25:31.820 --> 00:25:32.944 going to have to deal with. 00:25:36.310 --> 00:25:40.050 And I promise you, you will never 00:25:40.050 --> 00:25:43.710 use this unless you want to program a computer 00:25:43.710 --> 00:25:46.230 to calculate the wave function for God knows what reason. 00:25:49.840 --> 00:25:56.840 The quantum numbers we are 0, 1, 2. 00:25:56.840 --> 00:26:00.630 And for a harmonic oscillator, which goes to infinity, 00:26:00.630 --> 00:26:01.950 v goes to infinity, too. 00:26:01.950 --> 00:26:05.100 There's an infinite number of eigenfunctions 00:26:05.100 --> 00:26:08.990 of the quantum mechanical Hamiltonian-- 00:26:08.990 --> 00:26:12.940 of quantum mechanical harmonic oscillator. 00:26:12.940 --> 00:26:21.200 We have-- and the functions are normalized, 00:26:21.200 --> 00:26:26.690 we have psi plus and minus infinity goes to 0, 00:26:26.690 --> 00:26:31.370 we have psi v of 0. 00:26:31.370 --> 00:26:32.300 So this for all the-- 00:26:32.300 --> 00:26:33.400 I put a v there. 00:26:36.400 --> 00:26:50.050 Psi v is 0 for odd v, derivative of psi with respect to x, at x 00:26:50.050 --> 00:26:58.900 equals 0 is 0 per even v. So we have symmetric functions. 00:27:03.040 --> 00:27:11.140 And we have the energy levels is equal to h-bar omega, v 00:27:11.140 --> 00:27:13.810 plus 1/2. 00:27:13.810 --> 00:27:16.960 Now this is h over 2 pi, and this 00:27:16.960 --> 00:27:20.680 is nu times 2 pi, the frequency times 2 pi. 00:27:20.680 --> 00:27:23.050 So it could also be h nu. 00:27:23.050 --> 00:27:27.700 I have trouble remembering when there is a 2 pi involved. 00:27:34.280 --> 00:27:37.360 And we have this wonderful thing. 00:27:47.620 --> 00:27:51.750 It's as if v prime is equal to v this integral is 1, 00:27:51.750 --> 00:27:53.520 it's normalized. 00:27:53.520 --> 00:27:57.500 And a v prime is not equal to v, it's zero. 00:27:57.500 --> 00:28:00.630 And that stems from a theorem I mentioned before, 00:28:00.630 --> 00:28:10.000 is if you have two eigenvalues of the same Hamiltonian, 00:28:10.000 --> 00:28:12.300 eigenfunctions of the same Hamiltonian, 00:28:12.300 --> 00:28:15.710 and they belong to different eigenvalues 00:28:15.710 --> 00:28:19.520 their overlap integral is 0. 00:28:19.520 --> 00:28:21.890 We like zeros. 00:28:21.890 --> 00:28:25.490 We like normalization because the integral is just 1, 00:28:25.490 --> 00:28:26.930 it goes away. 00:28:26.930 --> 00:28:31.470 Or the integral is 0, the whole thing goes away. 00:28:31.470 --> 00:28:34.330 So that's really good. 00:28:34.330 --> 00:28:39.990 So we call this set of v's is orthonormal. 00:28:39.990 --> 00:28:41.760 Orthogonal and normalized. 00:28:41.760 --> 00:28:44.250 And the orthonormal terminology is used a lot. 00:28:44.250 --> 00:28:47.580 And in almost all quantum mechanical problems 00:28:47.580 --> 00:28:52.800 we like using an orthonormal set of functions 00:28:52.800 --> 00:28:54.348 to solve everything. 00:28:57.340 --> 00:29:00.300 Sometimes we have to do a little work to establish that, 00:29:00.300 --> 00:29:02.460 and I'll show much later in the course 00:29:02.460 --> 00:29:06.080 how when you have functions that are not orthogonal, 00:29:06.080 --> 00:29:08.280 and not normalized, you can create 00:29:08.280 --> 00:29:11.320 a set of functions which are. 00:29:11.320 --> 00:29:13.070 And this is something that a computer will 00:29:13.070 --> 00:29:14.380 do without breaking a sweat. 00:29:20.150 --> 00:29:27.190 Now we're back to my favorite topic, semi-classical. 00:29:32.100 --> 00:29:35.790 Because it's really easy to understand. 00:29:35.790 --> 00:29:38.220 Not just to understand the harmonic oscillator, 00:29:38.220 --> 00:29:40.050 but to use it in many problems. 00:29:43.560 --> 00:29:48.590 So in classic mechanics the kinetic energy 00:29:48.590 --> 00:29:58.540 is e minus v of x, or p squared over 2 mu. 00:29:58.540 --> 00:30:01.770 And so we can derive an equation for p 00:30:01.770 --> 00:30:09.330 of x classical mechanically, which is 2 mu e 00:30:09.330 --> 00:30:13.598 minus v of x square root. 00:30:16.670 --> 00:30:20.000 This is an extremely useful function. 00:30:20.000 --> 00:30:22.720 It's not an operator. 00:30:22.720 --> 00:30:24.240 It's a thing that we're going to use 00:30:24.240 --> 00:30:29.410 to make sense of everything, but it's not an operator. 00:30:29.410 --> 00:30:31.380 And so this is classic mechanics, and then 00:30:31.380 --> 00:30:36.900 in quantum mechanics, we know that Mr. de Broglie told us 00:30:36.900 --> 00:30:41.730 that the wavelength is equal to h over p. 00:30:41.730 --> 00:30:46.350 And we can generalize and say, well, maybe the wavelength 00:30:46.350 --> 00:30:49.390 is a function of x for potential, 00:30:49.390 --> 00:30:50.640 which is not constant. 00:30:57.710 --> 00:31:02.740 And even though this is not an operator in quantum mechanics 00:31:02.740 --> 00:31:04.740 this is true. 00:31:04.740 --> 00:31:09.870 That you can say the distance between consecutive nodes 00:31:09.870 --> 00:31:12.560 is lambda over 2. 00:31:12.560 --> 00:31:15.650 We can use this node relationship 00:31:15.650 --> 00:31:16.670 to great advantage. 00:31:21.830 --> 00:31:25.940 For the pair of nodes closest to x 00:31:25.940 --> 00:31:29.600 we can use this to calculate the distance between them. 00:31:32.710 --> 00:31:34.180 Very valuable. 00:31:34.180 --> 00:31:36.760 Because I also want to mention something. 00:31:36.760 --> 00:31:41.140 If you have an integrand which is rapidly oscillating, 00:31:41.140 --> 00:31:43.270 or if you have two rapidly oscillating functions 00:31:43.270 --> 00:31:46.120 and you're multiplying them together, 00:31:46.120 --> 00:31:55.040 that integral will accumulate to its final value at the position 00:31:55.040 --> 00:31:57.710 where the two oscillating functions are oscillating 00:31:57.710 --> 00:31:59.390 at the same frequency. 00:31:59.390 --> 00:32:01.460 That's the stationary phase point. 00:32:01.460 --> 00:32:03.980 And this is also a wonderful thing, 00:32:03.980 --> 00:32:06.560 because if you can figure out where 00:32:06.560 --> 00:32:08.540 the things you're multiplying together 00:32:08.540 --> 00:32:10.920 are oscillating at the same frequency, 00:32:10.920 --> 00:32:14.640 your integral becomes a number. 00:32:14.640 --> 00:32:17.170 No work ever. 00:32:17.170 --> 00:32:18.800 And that's a useful thing. 00:32:18.800 --> 00:32:22.200 OK, so the stationary phase method 00:32:22.200 --> 00:32:26.040 enables you to use this in a really fantastic way. 00:32:26.040 --> 00:32:29.690 And it's a little bit like Feynman's path integral idea, 00:32:29.690 --> 00:32:36.660 that you can calculate a complicated thing by evaluating 00:32:36.660 --> 00:32:40.410 an integral over a convenient path 00:32:40.410 --> 00:32:42.750 as opposed to integrating overall space, 00:32:42.750 --> 00:32:45.750 because everything that you care about 00:32:45.750 --> 00:32:47.790 comes from a stationary phase. 00:32:47.790 --> 00:32:50.400 Quantum mechanics is full of oscillations, 00:32:50.400 --> 00:32:53.220 classical mechanics doesn't have oscillations, 00:32:53.220 --> 00:32:56.190 and the two meet at the stationary phase point. 00:32:58.760 --> 00:33:02.120 Now we're going to use these ideas to calculate 00:33:02.120 --> 00:33:07.760 useful stuff for quantum mechanical vibrational wave 00:33:07.760 --> 00:33:08.570 functions. 00:33:08.570 --> 00:33:20.460 The shapes of psi of x gets exponentially damped, 00:33:20.460 --> 00:33:31.930 but it extends into the classically forbidden 00:33:31.930 --> 00:33:34.120 e less than v of x regions. 00:33:38.220 --> 00:33:42.250 The wave function, if we have potential 00:33:42.250 --> 00:33:45.650 and we have a wave function, that wave function 00:33:45.650 --> 00:33:48.620 is going to not go to 0 at the edge 00:33:48.620 --> 00:33:50.930 but it's going to have a tail. 00:33:50.930 --> 00:33:53.620 And that tail goes to 0 at infinity. 00:33:53.620 --> 00:33:58.400 And so there is some amplitude where the particle isn't 00:33:58.400 --> 00:34:00.400 allowed to be, classically. 00:34:00.400 --> 00:34:04.760 And that's where tunneling comes in. 00:34:04.760 --> 00:34:07.730 But the important thing, the important insight 00:34:07.730 --> 00:34:12.260 is that there are no nodes in the classically forbidden 00:34:12.260 --> 00:34:13.280 region. 00:34:13.280 --> 00:34:17.449 There is only exponential decay towards 0, 00:34:17.449 --> 00:34:22.290 and if you've chosen the wrong value of the energy, 00:34:22.290 --> 00:34:26.870 in other words a place where there is no eigenfunction, 00:34:26.870 --> 00:34:29.780 the wave function in the classically forbidden region 00:34:29.780 --> 00:34:31.730 will usually go to infinity. 00:34:31.730 --> 00:34:34.020 Either over here or over here, and says, 00:34:34.020 --> 00:34:37.969 well, it's clearly not a good function. 00:34:37.969 --> 00:34:39.370 But there are no 0 crossings. 00:34:44.340 --> 00:34:52.170 It's oscillating in e greater than v 00:34:52.170 --> 00:34:53.840 of x, the classically allowed region. 00:34:58.240 --> 00:35:05.340 The number of nodes is v. So we can 00:35:05.340 --> 00:35:07.860 have a v equals 0 function that just goes up and goes down, 00:35:07.860 --> 00:35:09.220 no internal nodes. 00:35:09.220 --> 00:35:12.060 v equals one, it crosses zero right in the middle. 00:35:15.980 --> 00:35:17.510 And we have the even oddness. 00:35:17.510 --> 00:35:26.116 Even v, even function. 00:35:26.116 --> 00:35:30.110 Odd v, odd function. 00:35:30.110 --> 00:35:35.390 For an even function you have a relative maximum at x equals 0, 00:35:35.390 --> 00:35:37.670 and for an odd function you have a 0. 00:35:37.670 --> 00:35:40.175 And the opposite further derivatives. 00:36:05.360 --> 00:36:08.150 The outer lobes, the ones on the ends 00:36:08.150 --> 00:36:16.300 just before the particle encounters the classical wall, 00:36:16.300 --> 00:36:18.560 you get the maximum amplitude. 00:36:21.260 --> 00:36:31.110 And so you can draw cartoons which look sort of like that. 00:36:31.110 --> 00:36:35.600 Most of the valuable stuff is at the other turning point. 00:36:35.600 --> 00:36:38.400 And there's oscillations in between, 00:36:38.400 --> 00:36:41.730 but often you really care about these two outer lobes. 00:36:44.880 --> 00:36:47.200 That's a pretty good simplification. 00:36:47.200 --> 00:36:53.630 Now there's a nice picture in McQuarrie on page 226 00:36:53.630 --> 00:36:57.390 which shows, especially for psi squared, 00:36:57.390 --> 00:36:59.870 that the nodes are pretty big. 00:36:59.870 --> 00:37:03.860 But they're not as big for relatively low quantum numbers 00:37:03.860 --> 00:37:07.110 as I've implied. 00:37:07.110 --> 00:37:08.640 But at really high quantum numbers 00:37:08.640 --> 00:37:12.930 we have a thing called the correspondence principle. 00:37:12.930 --> 00:37:18.450 And the corresponding principle says that quantum mechanics 00:37:18.450 --> 00:37:20.929 will do what classical mechanics does 00:37:20.929 --> 00:37:22.470 in the limit of high quantum numbers. 00:37:22.470 --> 00:37:24.136 And in the limit of high quantum numbers 00:37:24.136 --> 00:37:28.470 essentially all the amplitude is at the turning points. 00:37:28.470 --> 00:37:31.280 And in classic mechanics the particle 00:37:31.280 --> 00:37:35.570 is moving fast in the middle, and stops and turns around, 00:37:35.570 --> 00:37:37.340 and essentially all of the amplitude 00:37:37.340 --> 00:37:40.060 is at the turning point. 00:37:40.060 --> 00:37:41.640 So this is nice. 00:37:44.830 --> 00:37:48.610 Now we're getting into Bobism territory, 00:37:48.610 --> 00:37:50.110 because I'm really going to show you 00:37:50.110 --> 00:37:57.610 how to calculate whatever you need using 00:37:57.610 --> 00:37:58.780 these semi-classical ideas. 00:38:04.340 --> 00:38:09.420 We have the probability envelope. 00:38:14.220 --> 00:38:22.056 Psi star of x psi of x, and we're 00:38:22.056 --> 00:38:24.180 going to have both of these having the same quantum 00:38:24.180 --> 00:38:26.550 number, dx. 00:38:26.550 --> 00:38:32.490 So this is the probability of finding the particle near x 00:38:32.490 --> 00:38:36.030 in a region with dx. 00:38:44.330 --> 00:38:54.490 And this is the same thing as dx over v classical. 00:38:54.490 --> 00:38:58.480 It's not the same thing, there is a constant here, sorry. 00:38:58.480 --> 00:39:02.380 This probability density that you want 00:39:02.380 --> 00:39:05.890 is basically 1 over the classical velocity. 00:39:05.890 --> 00:39:08.649 And I demonstrated that when I walked 00:39:08.649 --> 00:39:10.690 across the room, when I walked fast in the middle 00:39:10.690 --> 00:39:13.110 and slow the outside. 00:39:13.110 --> 00:39:16.380 And you get the probability, you get this constant, 00:39:16.380 --> 00:39:19.140 by saying, OK, how long did it take for me to go from one end 00:39:19.140 --> 00:39:20.880 to the other? 00:39:20.880 --> 00:39:22.890 And comparing that, how long it took 00:39:22.890 --> 00:39:27.270 for me to go in some differential position. 00:39:27.270 --> 00:39:32.450 You get this constant in a simple way. 00:39:37.450 --> 00:39:50.490 v classical is equal to p classical over the mass, 00:39:50.490 --> 00:39:51.190 over the mu. 00:39:53.900 --> 00:39:56.820 But we know the function for p classical. 00:39:56.820 --> 00:40:05.857 We have 1 over mu times 2 mu e, minus v of x. 00:40:10.150 --> 00:40:12.340 So we know the velocity everywhere, 00:40:12.340 --> 00:40:16.930 and there's nothing terribly hard about figuring that out. 00:40:19.854 --> 00:40:22.270 And now we want to know what this proportionality constant 00:40:22.270 --> 00:40:23.140 is. 00:40:23.140 --> 00:40:33.140 And so for that we say the time to go from x to x plus dx, 00:40:33.140 --> 00:40:40.957 over the time to go from x minus to x plus. 00:40:40.957 --> 00:40:42.540 Because what's happening, the particle 00:40:42.540 --> 00:40:44.180 is going back and forth inside this 00:40:44.180 --> 00:40:49.310 well and so this is the time it takes to go one pass, 00:40:49.310 --> 00:40:51.770 and this is the time it takes to go 00:40:51.770 --> 00:40:53.350 through the region of interest. 00:40:53.350 --> 00:40:55.564 And so this ratio is the probability. 00:41:22.410 --> 00:41:27.920 And so we have the probability moving from left turning point 00:41:27.920 --> 00:41:30.270 the right turning point, and we want 00:41:30.270 --> 00:41:33.210 to know the probability in that interval. 00:41:33.210 --> 00:41:45.450 And so that's just dx over v classical at x, over tau, 00:41:45.450 --> 00:41:46.530 over 2. 00:41:46.530 --> 00:41:50.970 Because tau is the period, and we have half of the period, 00:41:50.970 --> 00:41:54.860 and so it's all together. 00:41:54.860 --> 00:41:58.020 I'm going to skip a little step, because it's taking too long. 00:41:58.020 --> 00:42:12.890 Psi star psi dx is equal to k over 2 pi squared, 00:42:12.890 --> 00:42:19.590 e minus v of x square of dx. 00:42:19.590 --> 00:42:24.290 Now so if we know the potential, and we know the energy, 00:42:24.290 --> 00:42:26.760 and we know the force constant we can say, 00:42:26.760 --> 00:42:30.600 well, this is the probability and-- 00:42:33.890 --> 00:42:41.330 but this is the probability of the semi-classical 00:42:41.330 --> 00:42:46.254 representation of psi star psi dx at x. 00:42:48.960 --> 00:42:52.250 Now this is oscillating, and this is not. 00:42:57.000 --> 00:42:58.470 So what we really want to know-- 00:43:05.490 --> 00:43:17.000 here's the-- if we have psi star whoops, slow down. 00:43:21.870 --> 00:43:28.980 This is oscillating and what we've calculated before 00:43:28.980 --> 00:43:32.290 is something that looks sort of like that. 00:43:32.290 --> 00:43:35.410 And if we multiply by 2 we have a curve 00:43:35.410 --> 00:43:38.210 that goes to the maximum of all these oscillations. 00:43:44.130 --> 00:43:49.920 The envelope psi star psi has the form. 00:43:49.920 --> 00:43:52.770 We've multiplied by 2, and so we end up 00:43:52.770 --> 00:44:08.677 getting 2k over pi squared, e minus v of x square root. 00:44:08.677 --> 00:44:11.010 Now you might say, well these are complicated functions, 00:44:11.010 --> 00:44:12.840 why should I bother with them? 00:44:12.840 --> 00:44:16.690 But if you wanted anything starting 00:44:16.690 --> 00:44:19.480 from the correct solution to the harmonic oscillator, 00:44:19.480 --> 00:44:21.220 using the Hermite polynomials, there's 00:44:21.220 --> 00:44:22.555 a whole lot more overhead. 00:44:25.620 --> 00:44:27.930 Notice also this is a function of x, not xi. 00:44:38.440 --> 00:44:40.960 This is the overlap function, it's 00:44:40.960 --> 00:44:44.440 the curve that touches the maximum of all these things 00:44:44.440 --> 00:44:47.470 and it's very useful if you want to know the probability 00:44:47.470 --> 00:44:49.233 of finding the system anywhere. 00:44:55.980 --> 00:45:03.696 And we get the node spacing from the equation h over p of x. 00:45:03.696 --> 00:45:05.700 And now here comes something really nice. 00:45:08.970 --> 00:45:12.040 It's called the semi-classical quantization interval. 00:45:14.560 --> 00:45:18.500 If we have any one dimensional potential, 00:45:18.500 --> 00:45:21.510 and we're at some energy, we'd like 00:45:21.510 --> 00:45:27.360 to be able to know how many levels are at that energy, 00:45:27.360 --> 00:45:28.710 or below. 00:45:28.710 --> 00:45:31.240 Or where are the energy levels? 00:45:31.240 --> 00:45:33.870 And we get that from this really incredible thing. 00:45:40.860 --> 00:45:43.140 We want to know that-- 00:45:50.710 --> 00:45:53.785 this is the difference between nodes. 00:45:53.785 --> 00:45:56.160 Right? 00:45:56.160 --> 00:45:59.160 And now if we would like to know x 00:45:59.160 --> 00:46:14.050 minus to x plus and some energy, we can replace lambda of x by h 00:46:14.050 --> 00:46:15.010 over p of x. 00:46:15.010 --> 00:46:24.790 And so we get p of x at that energy over h dx. 00:46:24.790 --> 00:46:28.920 pdx, that's called an action integral. 00:46:28.920 --> 00:46:30.960 Now I have to tell a little story. 00:46:30.960 --> 00:46:34.850 When I was a senior at Amherst College 00:46:34.850 --> 00:46:44.810 we had a oral exam for whether my thesis was 00:46:44.810 --> 00:46:47.310 going to be accepted or not. 00:46:47.310 --> 00:46:55.020 And one of my examiners asked me, what is the unit of h? 00:46:55.020 --> 00:46:58.340 Well, it's energy times time. 00:46:58.340 --> 00:46:59.990 And he wouldn't stop. 00:46:59.990 --> 00:47:02.720 He said no, I want something else. 00:47:02.720 --> 00:47:04.040 It's called action. 00:47:04.040 --> 00:47:07.400 Momentum times position. 00:47:07.400 --> 00:47:09.110 This is an action integral. 00:47:09.110 --> 00:47:11.150 And so anyway that's just a story. 00:47:11.150 --> 00:47:14.900 I spent a half an hour, and I was damn stubborn. 00:47:14.900 --> 00:47:19.520 I was not-- you know, it was energy times time. 00:47:19.520 --> 00:47:22.130 But that is much more insight here 00:47:22.130 --> 00:47:24.230 and that's maybe why I got so excited 00:47:24.230 --> 00:47:26.540 about this sort of an integral. 00:47:31.060 --> 00:47:36.250 If we want to know if we have an eigenvalue 00:47:36.250 --> 00:47:41.980 this integral has to be equal to h over 00:47:41.980 --> 00:47:47.180 2 times the number of nodes. 00:47:47.180 --> 00:47:49.440 Well it's pretty simple. 00:47:49.440 --> 00:47:55.690 So we can adjust e to satisfy this. 00:47:55.690 --> 00:48:00.100 Or if we wanted to know how many energy levels are 00:48:00.100 --> 00:48:02.470 at an energy below the energy we've 00:48:02.470 --> 00:48:05.870 chosen we evaluate this integral, 00:48:05.870 --> 00:48:10.420 and we get a number like 13.5. 00:48:10.420 --> 00:48:14.670 Well, it means there are 13 energy levels below that. 00:48:14.670 --> 00:48:19.400 Now often you want to know the density of states, 00:48:19.400 --> 00:48:23.120 the number of energy levels per unit energy, 00:48:23.120 --> 00:48:25.460 because that turns out to be the critical quantity 00:48:25.460 --> 00:48:28.190 in calculating many things you want to know. 00:48:28.190 --> 00:48:31.321 And you can get that from the semi-classical quantization. 00:48:35.480 --> 00:48:43.100 We're close to the end, I just want to say where we're going. 00:48:43.100 --> 00:48:47.750 We have classical pictures and I really, really 00:48:47.750 --> 00:48:50.900 want you to think about these classical pictures 00:48:50.900 --> 00:48:53.990 and use them rather than thinking, well, 00:48:53.990 --> 00:48:57.050 I'm going to have my cell phone program to evaluate 00:48:57.050 --> 00:48:59.619 all the necessary stuff. 00:48:59.619 --> 00:49:01.160 And there are certain things you want 00:49:01.160 --> 00:49:05.170 to remember about this semi-classical picture. 00:49:05.170 --> 00:49:10.420 And now we have the ability to calculate an infinite number 00:49:10.420 --> 00:49:13.270 of integrals involving harmonic oscillator 00:49:13.270 --> 00:49:17.230 functions in certain operators. 00:49:17.230 --> 00:49:18.980 Well, la-di-da. 00:49:18.980 --> 00:49:20.170 Why do we want them? 00:49:20.170 --> 00:49:22.000 Well one of the things we want is 00:49:22.000 --> 00:49:23.740 to be able to calculate the probability 00:49:23.740 --> 00:49:26.960 of a vibrational transition. 00:49:26.960 --> 00:49:29.060 That's called a transition element, 00:49:29.060 --> 00:49:30.680 and that's an easy thing to calculate. 00:49:34.000 --> 00:49:37.580 Another thing we want to do is to say, well, 00:49:37.580 --> 00:49:39.320 nature screwed up. 00:49:39.320 --> 00:49:43.790 This oscillator isn't harmonic, there's anharmonic term, 00:49:43.790 --> 00:49:46.080 and I would like to know what is the contribution 00:49:46.080 --> 00:49:51.450 of a constant times x cubed in the potential 00:49:51.450 --> 00:49:53.700 to the energy levels. 00:49:53.700 --> 00:49:55.710 And that's called perturbation theory. 00:49:55.710 --> 00:49:59.100 Or I want to have many harmonic oscillators 00:49:59.100 --> 00:50:01.800 in a polyatomic molecule, they talk to each other, 00:50:01.800 --> 00:50:06.360 and I want to calculate the interactions 00:50:06.360 --> 00:50:08.460 between these harmonic oscillators 00:50:08.460 --> 00:50:09.990 affect the energy level. 00:50:09.990 --> 00:50:13.470 Remember, when we have a separable Hamiltonian 00:50:13.470 --> 00:50:15.510 we can just write the energy levels as the sum 00:50:15.510 --> 00:50:16.670 of the individual Hamilton. 00:50:16.670 --> 00:50:19.080 And then there's coupling terms, and we 00:50:19.080 --> 00:50:22.950 deal with those but perturbation theory. 00:50:22.950 --> 00:50:26.680 There's all sorts of wonderful things we do. 00:50:26.680 --> 00:50:35.320 But we're going to consider these magical operators, 00:50:35.320 --> 00:50:39.430 a creation and annihilation operator, where a star-- 00:50:39.430 --> 00:50:49.010 a dagger operating on psi v gives 00:50:49.010 --> 00:50:54.570 the square root of v plus one, times psi v plus 1. 00:50:54.570 --> 00:50:58.960 And a dagger operating on v gives the square root of v-- 00:50:58.960 --> 00:51:00.220 let's write it-- 00:51:00.220 --> 00:51:04.540 I mean a non-dagger gives v square root times 00:51:04.540 --> 00:51:07.060 psi, v minus 1. 00:51:07.060 --> 00:51:13.210 And that x is equal to a plus a dagger times a constant. 00:51:13.210 --> 00:51:17.980 And so all of a sudden we can evaluate all integrals 00:51:17.980 --> 00:51:23.440 involving x, or powers of x, or momenta, or powers of momenta 00:51:23.440 --> 00:51:24.640 without thinking. 00:51:24.640 --> 00:51:28.420 Without ever looking at a function. 00:51:28.420 --> 00:51:32.550 And I guarantee you that this is embodied 00:51:32.550 --> 00:51:35.310 in the incredible amount of work done wherever 00:51:35.310 --> 00:51:39.510 we pretend almost every problem is a harmonic 00:51:39.510 --> 00:51:41.680 oscillator in disguise. 00:51:41.680 --> 00:51:44.710 Because these a's and a daggers enable 00:51:44.710 --> 00:51:47.500 you to generate everything without ever 00:51:47.500 --> 00:51:52.600 converting from x to xi, without ever looking at an integral. 00:51:52.600 --> 00:51:57.190 It's all just a manipulation of algebra. 00:51:57.190 --> 00:52:01.380 And it's not just convenient, but there's insight 00:52:01.380 --> 00:52:04.950 and so this is what I want to convey. 00:52:04.950 --> 00:52:08.370 That you will get tremendous insight. 00:52:08.370 --> 00:52:11.270 Maybe, maybe I sold you on semi-classical, 00:52:11.270 --> 00:52:16.340 and I don't apologize for that because that's very useful. 00:52:16.340 --> 00:52:20.630 But the next lecture, when you get the a's and a daggers, 00:52:20.630 --> 00:52:23.290 it'll just knock your socks off. 00:52:23.290 --> 00:52:25.430 OK, that's it.