WEBVTT 00:00:00.090 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.690 continue to offer high quality educational resources for free. 00:00:10.690 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.250 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.250 --> 00:00:18.220 at ocw.mit.edu. 00:00:22.190 --> 00:00:26.990 ROBERT FIELD: Last lecture, I did a fairly standard treatment 00:00:26.990 --> 00:00:29.750 of the harmonic oscillator, which is not 00:00:29.750 --> 00:00:31.430 supposed to make you excited, but just 00:00:31.430 --> 00:00:33.320 to see that you can do this. 00:00:33.320 --> 00:00:38.000 And the path to the solution was to define 00:00:38.000 --> 00:00:41.630 dimensionless position coordinate, 00:00:41.630 --> 00:00:45.680 and then you get a dimensionless Schrodinger equation. 00:00:45.680 --> 00:00:51.210 And the solution of that involves two steps. 00:00:51.210 --> 00:00:57.950 One is to insist that the solutions to the Schrodinger 00:00:57.950 --> 00:01:02.150 equation have an exponentially damped form. 00:01:02.150 --> 00:01:03.710 And then the Schrodinger equation 00:01:03.710 --> 00:01:07.441 is transformed into a new equation called the Hermite 00:01:07.441 --> 00:01:07.940 equation. 00:01:10.499 --> 00:01:13.040 You shouldn't get all excited about that-- the mathematicians 00:01:13.040 --> 00:01:14.650 take care of it. 00:01:14.650 --> 00:01:20.420 And so the solutions to this Hermite differential equation 00:01:20.420 --> 00:01:25.170 gives you a set of orthogonal and normalized wave functions. 00:01:25.170 --> 00:01:29.090 They give you the energy levels expressed as a quantum number 00:01:29.090 --> 00:01:31.940 plus 1/2, times constant. 00:01:31.940 --> 00:01:37.920 The energy levels with the quantum number v 00:01:37.920 --> 00:01:42.320 are even or odd in v, and they're even or odd in psi. 00:01:42.320 --> 00:01:46.600 Even v corresponds to even psi. 00:01:46.600 --> 00:01:50.000 V is the number of internal nodes, 00:01:50.000 --> 00:01:53.570 and there are all sorts of things you can do to say, 00:01:53.570 --> 00:01:57.080 well, I expect if I know how to do certain things 00:01:57.080 --> 00:01:58.772 in classical mechanics, they're are 00:01:58.772 --> 00:02:00.980 going to come out pretty much the same way in quantum 00:02:00.980 --> 00:02:02.490 mechanics. 00:02:02.490 --> 00:02:04.940 So that's the standard structure, 00:02:04.940 --> 00:02:10.130 but more importantly, I want you to have in your head 00:02:10.130 --> 00:02:14.570 the pictures of the wave functions, the idea that there 00:02:14.570 --> 00:02:19.476 is a zero point energy, and that there's a reason for that-- 00:02:19.476 --> 00:02:22.520 that the wave functions have tails extending out 00:02:22.520 --> 00:02:26.550 into the non-classical, or the classically forbidden region. 00:02:26.550 --> 00:02:30.380 And this turns out to be the beginning of tunneling. 00:02:30.380 --> 00:02:36.680 You'll be looking at tunneling more specifically. 00:02:36.680 --> 00:02:40.650 I also want you to know how the spacing of nodes is, 00:02:40.650 --> 00:02:45.950 and that involves generalization of the bright idea 00:02:45.950 --> 00:02:47.960 that the wavelength is h over p. 00:02:47.960 --> 00:02:51.260 But if the potential is not constant, 00:02:51.260 --> 00:02:53.720 then p is a function of x. 00:02:53.720 --> 00:02:58.020 This is not the quantum mechanical operator, 00:02:58.020 --> 00:03:00.950 this is a function which provides you 00:03:00.950 --> 00:03:03.770 with a lot of intuition. 00:03:03.770 --> 00:03:08.110 And then if you know where the node spacings are, and you 00:03:08.110 --> 00:03:12.260 know the shape of the envelope, you 00:03:12.260 --> 00:03:14.150 have basically everything you need 00:03:14.150 --> 00:03:18.470 to have a classical sense of what's going on. 00:03:18.470 --> 00:03:23.090 And then-- I guess it's supposed to be hidden-- 00:03:23.090 --> 00:03:27.410 I did a little bit of semiclassical theory, 00:03:27.410 --> 00:03:30.650 and I showed that if you integrate from the left turning 00:03:30.650 --> 00:03:32.690 point to the right turning point at a given 00:03:32.690 --> 00:03:37.230 energy of this momentum function, 00:03:37.230 --> 00:03:42.010 you get h over 2 times the number of nodes. 00:03:42.010 --> 00:03:44.380 And this is the semiclassical quantization-- 00:03:44.380 --> 00:03:47.380 it's incredibly important, and it's 00:03:47.380 --> 00:03:51.070 useful either as an exact or approximate result 00:03:51.070 --> 00:03:53.710 for all one-dimensional problems. 00:03:53.710 --> 00:03:57.160 And so it tells you how to begin. 00:03:57.160 --> 00:03:59.500 Now, before I start talking about what 00:03:59.500 --> 00:04:05.800 we're going to do today, I want to stress where we're going. 00:04:05.800 --> 00:04:09.525 So we're going to be looking at some exactly solved problems. 00:04:24.300 --> 00:04:29.140 And so we have a particle in a box, the harmonic oscillator, 00:04:29.140 --> 00:04:31.980 the hydrogen atoms-- you have to count them-- 00:04:31.980 --> 00:04:32.910 and the rigid rotor. 00:04:35.980 --> 00:04:40.660 Now, all of these problems have an infinite number 00:04:40.660 --> 00:04:46.180 eigenfunctions, an infinite number of energy levels, 00:04:46.180 --> 00:04:52.020 and that's intimidating, but it's true. 00:04:52.020 --> 00:04:55.370 Now, these infinite number of functions 00:04:55.370 --> 00:04:59.980 are explicit functions of the quantum number. 00:04:59.980 --> 00:05:04.160 And so we have an infinite number, 00:05:04.160 --> 00:05:07.610 but in order to describe systems, 00:05:07.610 --> 00:05:10.320 we're going to be calculating integrals. 00:05:10.320 --> 00:05:12.480 We're going to be calculating a lot of integrals 00:05:12.480 --> 00:05:15.910 between these infinite number of functions. 00:05:15.910 --> 00:05:19.320 So we have an infinity squared of integrals. 00:05:22.120 --> 00:05:25.300 Well, that shouldn't scare you because what 00:05:25.300 --> 00:05:28.960 I'm going to show you is that all of the integrals that we 00:05:28.960 --> 00:05:33.430 are going to encounter are explicit functions 00:05:33.430 --> 00:05:35.690 of the quantum numbers, and they have 00:05:35.690 --> 00:05:38.090 relatively selection rules. 00:05:38.090 --> 00:05:42.100 In other words, which integrals are non-zero 00:05:42.100 --> 00:05:43.780 based on the difference in quantum 00:05:43.780 --> 00:05:48.290 numbers between the left-hand side and the right-hand side? 00:05:48.290 --> 00:05:53.320 So we're collecting these things in order 00:05:53.320 --> 00:05:56.320 to calculate a whole bunch of stuff. 00:05:56.320 --> 00:05:59.410 Now, I told you that this is a course for use 00:05:59.410 --> 00:06:03.440 rather than philosophy or history. 00:06:03.440 --> 00:06:07.210 And so if you encounter any quantum mechanical problem, 00:06:07.210 --> 00:06:11.020 you'd like to be able to describe 00:06:11.020 --> 00:06:13.180 what you could observe with it. 00:06:13.180 --> 00:06:17.860 And so if you're armed with the infinite number of energy 00:06:17.860 --> 00:06:23.350 levels and eigen solutions for our problem, 00:06:23.350 --> 00:06:26.990 you can calculate any property. 00:06:26.990 --> 00:06:30.350 So you define some property you're interested in-- 00:06:30.350 --> 00:06:32.870 there is a quantum mechanical operator that 00:06:32.870 --> 00:06:35.290 corresponds to that property. 00:06:35.290 --> 00:06:40.010 And in order to be able to describe observations 00:06:40.010 --> 00:06:43.010 of that property, you need to calculate 00:06:43.010 --> 00:06:47.280 integrals of that operator. 00:06:47.280 --> 00:06:49.060 Well, la dee dah. 00:06:49.060 --> 00:06:51.860 That should be intimidating, but it's not 00:06:51.860 --> 00:06:54.830 because almost all of these integrals 00:06:54.830 --> 00:06:59.850 can be expressed as a simple constant times 00:06:59.850 --> 00:07:02.340 a function of the quantum numbers 00:07:02.340 --> 00:07:04.680 or the difference of quantum numbers, 00:07:04.680 --> 00:07:08.360 and that's a fantastic thing. 00:07:08.360 --> 00:07:15.290 So we have any operator-- 00:07:15.290 --> 00:07:20.600 suppose the Hamiltonian is an exactly solved problem 00:07:20.600 --> 00:07:24.530 plus something else, which we'll call h1. 00:07:24.530 --> 00:07:27.860 And this is a complexity in the-- 00:07:27.860 --> 00:07:30.330 or it's the reality in the problem. 00:07:30.330 --> 00:07:35.450 And in order to deal with this, again, you're 00:07:35.450 --> 00:07:40.410 going to need to calculate integrals of this operator. 00:07:40.410 --> 00:07:43.520 And the last thing that's really going to be exciting 00:07:43.520 --> 00:07:48.287 is once we look at the time dependent Schrodinger equation, 00:07:48.287 --> 00:07:49.620 we're going to get wave packets. 00:07:55.720 --> 00:08:01.020 And these are functions of position and time, 00:08:01.020 --> 00:08:04.350 and these wave packets are classical-like, localized 00:08:04.350 --> 00:08:08.610 objects that move following the Newton's equations 00:08:08.610 --> 00:08:11.640 of motion with the center of the wave packet. 00:08:11.640 --> 00:08:13.110 And again, there are a whole bunch 00:08:13.110 --> 00:08:15.210 of integrals you're going to need in order 00:08:15.210 --> 00:08:17.050 to do these things. 00:08:17.050 --> 00:08:21.090 And so right now, we're starting with the best problem 00:08:21.090 --> 00:08:24.300 for these integrals, because a harmonic oscillator 00:08:24.300 --> 00:08:26.880 has some special properties. 00:08:26.880 --> 00:08:31.170 And the lecture notes are incredibly tedious, 00:08:31.170 --> 00:08:32.520 and they're mostly proofs. 00:08:32.520 --> 00:08:37.289 And I'm going to try to go fast over the tedious stuff, 00:08:37.289 --> 00:08:42.390 and give you the important ideas, 00:08:42.390 --> 00:08:44.970 but since there is some important logic, 00:08:44.970 --> 00:08:46.680 you should really look at these notes. 00:08:49.590 --> 00:08:53.070 So what we're going to be doing today 00:08:53.070 --> 00:08:59.900 is we start with the coordinate momentum operators, 00:08:59.900 --> 00:09:07.560 we're going to get these operators 00:09:07.560 --> 00:09:11.400 in dimensionless form, and then we're 00:09:11.400 --> 00:09:14.165 going to get these a and a-dagger guys. 00:09:18.460 --> 00:09:20.880 So this step is reminiscent of what 00:09:20.880 --> 00:09:23.920 I did at the beginning of the previous lecture, 00:09:23.920 --> 00:09:28.380 and then this is magic because this magic enables 00:09:28.380 --> 00:09:32.610 you to just look at integrals and say, 00:09:32.610 --> 00:09:35.760 I know that integral is zero, or I know that area is not zero. 00:09:35.760 --> 00:09:37.800 And with a little bit more effort-- 00:09:37.800 --> 00:09:40.260 maybe something that you'd put on the back of a postage 00:09:40.260 --> 00:09:41.250 stamp-- 00:09:41.250 --> 00:09:45.150 you can evaluate that integral, not by knowing integral tables, 00:09:45.150 --> 00:09:47.920 but by knowing the properties of these simple little a 00:09:47.920 --> 00:09:49.080 and a-dagger. 00:09:49.080 --> 00:09:51.420 And that's a fantastic thing. 00:09:51.420 --> 00:09:53.790 And it's so fantastic that this is 00:09:53.790 --> 00:09:59.200 one of the reasons why almost all problems in quantum 00:09:59.200 --> 00:10:03.310 mechanics start with a harmonic oscillator approximation, 00:10:03.310 --> 00:10:07.030 because there is so much you can do with this a and a-dagger 00:10:07.030 --> 00:10:08.770 formalism. 00:10:08.770 --> 00:10:11.530 Now, at the beginning I also told you 00:10:11.530 --> 00:10:16.030 that in quantum mechanics, the important thing that 00:10:16.030 --> 00:10:18.981 contains everything we're allowed to know about a system 00:10:18.981 --> 00:10:19.855 is the wave function. 00:10:22.850 --> 00:10:27.310 But I also told you we can never measure the wave function. 00:10:27.310 --> 00:10:31.760 We can never experimentally determine it, 00:10:31.760 --> 00:10:35.690 and so we need to be able to calculate what this wave 00:10:35.690 --> 00:10:40.580 function does as far as what we can observe, 00:10:40.580 --> 00:10:43.670 and these a's and a-daggers are really important in being 00:10:43.670 --> 00:10:44.360 able to do that. 00:10:48.090 --> 00:10:51.300 So I'm going to start with covering 00:10:51.300 --> 00:10:56.040 what I did in the notes, but I'm going to jump to final results 00:10:56.040 --> 00:10:57.300 at some point-- 00:10:57.300 --> 00:10:59.250 governed by the clock. 00:10:59.250 --> 00:11:03.870 And so the first thing we're going to do is these. 00:11:03.870 --> 00:11:10.920 And so what we do is we define the relationship 00:11:10.920 --> 00:11:15.720 between the ordinary position coordinate. 00:11:15.720 --> 00:11:25.050 And this little twiddle means it's dimensionless, 00:11:25.050 --> 00:11:29.260 and so we can write the inverse of that. 00:11:29.260 --> 00:11:32.120 And that's the one we are going to want to-- 00:11:32.120 --> 00:11:33.730 well, actually, we go both ways. 00:11:42.230 --> 00:11:45.500 And we do the same thing for the momentum-- 00:11:45.500 --> 00:11:52.880 p is equal to h-bar mu omega square 00:11:52.880 --> 00:12:00.030 root, p twiddle and the inverse which I don't need to write. 00:12:00.030 --> 00:12:04.190 And finally, we get the Hamiltonian, 00:12:04.190 --> 00:12:10.760 which is p squared over 2 mu plus 1/2 kx squared. 00:12:14.080 --> 00:12:17.390 And we'll put that into these new units. 00:12:17.390 --> 00:12:27.630 So we have h-bar mu omega over 2 mu, p twiddle squared 00:12:27.630 --> 00:12:29.430 plus k over 2. 00:12:33.160 --> 00:12:38.130 This is all very tedious, but the payoff is very soon. 00:12:40.980 --> 00:12:50.640 k over 2 times hr mu omega, x twiddle squared. 00:12:55.200 --> 00:12:56.900 Oh, isn't that interesting? 00:12:56.900 --> 00:13:05.630 We can combine-- we can absorb a k over mu in omega, 00:13:05.630 --> 00:13:08.360 and so we get, actually, a big simplification. 00:13:08.360 --> 00:13:14.140 We get h bar omega over 2 times p twiddle 00:13:14.140 --> 00:13:17.545 squared plus x twiddle squared. 00:13:20.110 --> 00:13:21.505 Well, that looks simpler. 00:13:26.770 --> 00:13:30.760 And so the next thing we do is-- it looks like a simple problem 00:13:30.760 --> 00:13:35.069 from algebra, let's factor this. 00:13:35.069 --> 00:13:36.610 Now, it's a little tricky because you 00:13:36.610 --> 00:13:40.630 know you can factor something in real terms 00:13:40.630 --> 00:13:42.500 if this is a minus sign. 00:13:42.500 --> 00:13:45.400 But we are allowed to talk about complex quantities, 00:13:45.400 --> 00:13:48.400 so we can factor that. 00:13:48.400 --> 00:14:00.030 And so this term, p twiddle squared plus x twiddle squared 00:14:00.030 --> 00:14:16.280 is equal to ip twiddle, plus x twiddle times minus ip twiddle, 00:14:16.280 --> 00:14:17.610 plus x. 00:14:22.390 --> 00:14:27.990 And you can work that out-- that ip times minus ip is p squared, 00:14:27.990 --> 00:14:30.810 and x times x is x squared. 00:14:30.810 --> 00:14:34.800 And then we have these cross-terms, ip times 00:14:34.800 --> 00:14:37.410 x and x times minus ip. 00:14:37.410 --> 00:14:41.102 Whoops, they don't commute. 00:14:41.102 --> 00:14:42.530 If this were algebra-- 00:14:42.530 --> 00:14:45.640 well, they would go away, but they don't. 00:14:45.640 --> 00:14:52.010 And so what you end up getting is 00:14:52.010 --> 00:15:06.160 p twiddle squared plus x twiddle squared, plus i times p-- 00:15:15.360 --> 00:15:17.580 I'm going to stop writing the twiddles. 00:15:17.580 --> 00:15:18.870 So we have this. 00:15:24.820 --> 00:15:28.330 I want to make sure that I haven't sabotaged myself-- 00:15:28.330 --> 00:15:31.550 that's going to be-- yeah, that's right. 00:15:31.550 --> 00:15:36.941 So we have something here that isn't zero. 00:15:36.941 --> 00:15:40.340 And it looks like i times the commutator of p twiddle 00:15:40.340 --> 00:15:42.746 with x twiddle. 00:15:42.746 --> 00:15:44.550 But we can work that out because we 00:15:44.550 --> 00:15:51.440 know the commutator of p ordinary with x ordinary. 00:15:51.440 --> 00:15:54.190 And so I did that. 00:15:54.190 --> 00:16:07.690 And so we have this commutator, p twiddle, x twiddle. 00:16:07.690 --> 00:16:10.495 After some algebra, we get plus 1. 00:16:15.187 --> 00:16:17.950 A number-- pure number-- 00:16:17.950 --> 00:16:18.940 no? 00:16:18.940 --> 00:16:23.270 I want you to check my algebra. 00:16:23.270 --> 00:16:28.000 So you just substitute in what this 00:16:28.000 --> 00:16:31.960 is in terms of ordinary p and the ordinary x. 00:16:31.960 --> 00:16:36.640 Use a commutator for ordinary xp, which is ih-bar, 00:16:36.640 --> 00:16:40.160 and magically, you get plus 1. 00:16:40.160 --> 00:16:50.585 So this very strange and boring derivation says, OK-- 00:16:55.900 --> 00:17:02.830 well, let's now give these two things names. 00:17:02.830 --> 00:17:09.069 This guy, we're going to call as the square root of 2 times a, 00:17:09.069 --> 00:17:11.585 and this one is going to be the square root 00:17:11.585 --> 00:17:13.869 of 2 times a-dagger. 00:17:13.869 --> 00:17:31.890 So H is going to be h-bar omega over 2, 00:17:31.890 --> 00:17:40.280 times square root of 2a-hat times 00:17:40.280 --> 00:17:47.496 the square root of 2a-dagger-hat minus 1. 00:17:47.496 --> 00:17:49.080 Remember, when we factored it, we 00:17:49.080 --> 00:17:51.150 got this extra term which was 1. 00:17:51.150 --> 00:17:54.240 And in order to make it correct, we have to subtract it away. 00:17:58.940 --> 00:18:12.130 And so this becomes h-bar omega, a-dagger-hat minus 1/2. 00:18:12.130 --> 00:18:13.750 Well, isn't that nice? 00:18:13.750 --> 00:18:17.620 Now, we have the Hamiltonian expressed as a constant, which 00:18:17.620 --> 00:18:20.500 we know is important because it's related to the energy 00:18:20.500 --> 00:18:23.940 levels, and times these two little things, 00:18:23.940 --> 00:18:29.040 which turn out to be the gift from God. 00:18:29.040 --> 00:18:31.380 It's an incredible thing, what these do. 00:18:35.740 --> 00:18:38.710 So we have gone through some algebra, 00:18:38.710 --> 00:18:43.450 and we know the relationship between a, and the x and p 00:18:43.450 --> 00:18:46.660 twiddles, and similarly for a-dagger. 00:18:46.660 --> 00:18:49.340 And we can go in the other direction, 00:18:49.340 --> 00:18:51.610 and we know the commutator, and now we're 00:18:51.610 --> 00:18:53.940 going to start doing some really great stuff. 00:18:59.860 --> 00:19:02.700 Well, one thing we're going to want to know about is a-hat. 00:19:02.700 --> 00:19:09.640 a-hat, a-dagger, that commutator-hat. 00:19:09.640 --> 00:19:12.340 And that turns out to be-- 00:19:15.820 --> 00:19:20.960 well, I already derived it-- it turns out to be plus 1. 00:19:20.960 --> 00:19:23.910 And as a result, we can say things like this-- a a-dagger-- 00:19:56.290 --> 00:20:03.040 So using this trick, we can show we can always replace something 00:20:03.040 --> 00:20:10.780 like a, a-dagger by a-dagger a plus this commutator, which 00:20:10.780 --> 00:20:11.280 is 1. 00:20:15.580 --> 00:20:19.070 And so we have this really neat way 00:20:19.070 --> 00:20:25.330 of reversing the order of the a's and a-daggers. 00:20:25.330 --> 00:20:28.300 So with this, we're going to soon discover 00:20:28.300 --> 00:20:33.980 the a operating on the eigenfunction gives square root 00:20:33.980 --> 00:20:37.845 of v times psi v minus 1. 00:20:37.845 --> 00:20:42.430 And a-dagger operating on this wave function 00:20:42.430 --> 00:20:48.550 gives v plus 1 square root of psi v plus 1. 00:20:48.550 --> 00:20:55.010 Which is the reason these things are valuable, 00:20:55.010 --> 00:21:01.980 because if you have any eigenfunction, 00:21:01.980 --> 00:21:04.780 you can get all the others. 00:21:04.780 --> 00:21:07.350 So suppose you have the lowest eigenfunction-- 00:21:07.350 --> 00:21:10.310 you apply a-dagger on it as many times 00:21:10.310 --> 00:21:12.126 as you need to get to, say vth function. 00:21:15.070 --> 00:21:17.790 So you don't actually-- 00:21:17.790 --> 00:21:20.640 you're not going to be evaluating integrals, you're 00:21:20.640 --> 00:21:24.060 going to be counting a's and a-daggers, 00:21:24.060 --> 00:21:26.410 and permuting them around, and getting 1's, and stuff 00:21:26.410 --> 00:21:26.910 like that. 00:21:26.910 --> 00:21:27.760 Yes? 00:21:27.760 --> 00:21:29.551 AUDIENCE: In this line with the commutator, 00:21:29.551 --> 00:21:30.904 you didn't move the dagger. 00:21:30.904 --> 00:21:32.070 ROBERT FIELD: I didn't what? 00:21:32.070 --> 00:21:34.785 AUDIENCE: For a, a times the square root of a a-dagger plus 00:21:34.785 --> 00:21:38.945 a-dagger a, it should be a a-dagger. 00:21:38.945 --> 00:21:42.470 And on the right-hand side, you need to move the dagger. 00:21:47.870 --> 00:21:52.110 ROBERT FIELD: OK, so this is to switch the order, 00:21:52.110 --> 00:21:53.610 and I've done that. 00:21:53.610 --> 00:21:55.035 And that then is-- 00:22:00.150 --> 00:22:03.030 no, I think-- wait a second. 00:22:03.030 --> 00:22:04.400 So we have a a-dagger-- 00:22:04.400 --> 00:22:10.070 so that's a a-dagger minus a-dagger a, and that's-- 00:22:10.070 --> 00:22:12.020 oh, yeah. 00:22:12.020 --> 00:22:13.940 Thank you. 00:22:13.940 --> 00:22:17.990 It's very, very easy to get lost, and once you're lost, 00:22:17.990 --> 00:22:19.910 you can never be found because you've 00:22:19.910 --> 00:22:23.300 made a mistake that's built into your logic, 00:22:23.300 --> 00:22:25.330 and you're never going to see it. 00:22:25.330 --> 00:22:27.870 You see it took me a couple of minutes to even accept-- 00:22:27.870 --> 00:22:31.640 that the insight from my TA who's sitting there calmly 00:22:31.640 --> 00:22:34.300 thinking, and I'm trying to do several things in addition 00:22:34.300 --> 00:22:35.810 to the thinking. 00:22:35.810 --> 00:22:39.730 So we can do things like this. 00:22:39.730 --> 00:22:42.080 Suppose we have psi-- 00:22:42.080 --> 00:22:45.530 I can't use this notation yet. 00:22:45.530 --> 00:22:50.660 So suppose we have psi-star v, and we 00:22:50.660 --> 00:22:58.990 have a-dagger, a-dagger, a-dagger, psi, v prime, dx. 00:23:04.490 --> 00:23:06.140 These are raising operators, so this 00:23:06.140 --> 00:23:11.120 is going to take v prime to v prime plus 3. 00:23:11.120 --> 00:23:15.440 That's the only integral that's not 0. 00:23:15.440 --> 00:23:29.554 And we get v prime plus 1, v prime plus 2, v prime plus 3, 00:23:29.554 --> 00:23:31.760 square what? 00:23:31.760 --> 00:23:33.280 It has the constants. 00:23:36.800 --> 00:23:38.960 And this would be v prime plus 3. 00:23:42.760 --> 00:23:45.430 So instead of evaluating an integral, 00:23:45.430 --> 00:23:47.730 looking at what the x's and p's are, 00:23:47.730 --> 00:23:49.840 we just have a little game we play. 00:23:55.700 --> 00:24:05.530 So now, we have to prove some of the things I've said. 00:24:05.530 --> 00:24:15.890 So we have h, and we're going to operate on a-dagger psi v. 00:24:15.890 --> 00:24:20.430 So what does the Hamiltonian do to this thing? 00:24:20.430 --> 00:24:22.250 So what we're going to want to do 00:24:22.250 --> 00:24:28.880 is to show that this thing is an eigenvalue-- 00:24:28.880 --> 00:24:32.060 eigenfunction of v plus 1, and that's 00:24:32.060 --> 00:24:33.180 what we are going to get. 00:24:33.180 --> 00:24:35.640 So let's just go through this. 00:24:35.640 --> 00:24:47.560 So we have h-bar omega, a-dagger a plus 1/2, times psi v. 00:24:47.560 --> 00:24:49.935 So what I did is-- 00:24:53.566 --> 00:24:59.830 where did I-- yeah, I showed that the Hamiltonian-- 00:24:59.830 --> 00:25:01.230 or did I not do that yet? 00:25:06.100 --> 00:25:07.500 Oh, yeah-- I did it right here. 00:25:07.500 --> 00:25:13.770 The Hamiltonian is h-bar omega, a a-dagger minus 1/2, 00:25:13.770 --> 00:25:21.030 or if we reverse these, it's equal to h-bar omega, a-dagger, 00:25:21.030 --> 00:25:25.410 a plus 1/2. 00:25:25.410 --> 00:25:35.000 So we can use either one, so I'm using that one-- 00:25:44.900 --> 00:25:48.140 except I wanted an a-dagger here, 00:25:48.140 --> 00:25:52.520 because we want to show what the Hamiltonian does to this. 00:25:52.520 --> 00:26:00.180 Now, we can pull in a-dagger out to the right, because this-- 00:26:06.942 --> 00:26:10.130 if it's 1/2 times a-dagger, well, that doesn't matter. 00:26:10.130 --> 00:26:12.410 This a-dagger, a, a-dagger-- well, 00:26:12.410 --> 00:26:14.840 we can pull this a-dagger out. 00:26:14.840 --> 00:26:25.854 So we have h-bar omega, a-dagger is equal to a a-dagger plus 00:26:25.854 --> 00:26:26.354 1/2-- 00:26:30.242 --> 00:26:31.214 so iv. 00:26:35.590 --> 00:26:38.740 Now, we use our magic commutator trick 00:26:38.740 --> 00:26:52.152 to replace this by a-dagger a plus 1. 00:26:59.580 --> 00:27:05.050 So now, we have h-bar omega, a-dagger, 00:27:05.050 --> 00:27:12.760 and we have a-dagger a plus three halves is iv. 00:27:22.650 --> 00:27:34.640 Well, this is Ev plus h-bar omega. 00:27:34.640 --> 00:27:39.260 We've increased the number that started here. 00:27:39.260 --> 00:27:42.800 Here is Ev-- that was Ev plus 1. 00:27:46.130 --> 00:27:48.740 And so now, we have no operators in here, 00:27:48.740 --> 00:27:51.910 and we can stick the a-dagger back here. 00:27:51.910 --> 00:28:10.080 And so we have h-bar omega, Ev plus 1, a-dagger 00:28:10.080 --> 00:28:11.840 Well, what do we have here? 00:28:11.840 --> 00:28:15.800 We have an operator, we have this function, 00:28:15.800 --> 00:28:20.650 we have some constant times the same function. 00:28:20.650 --> 00:28:25.600 So what we've shown is that this thing 00:28:25.600 --> 00:28:28.650 is an eigenfunction of the Hamiltonian 00:28:28.650 --> 00:28:33.720 that belongs to the eigenvalue Ev plus 1. 00:28:33.720 --> 00:28:36.690 We've increased the energy by 1. 00:28:42.390 --> 00:28:43.260 So what we have-- 00:29:01.420 --> 00:29:05.590 so we can show that we apply a-dagger 00:29:05.590 --> 00:29:12.700 to any function-- we increase its energy, 00:29:12.700 --> 00:29:15.820 and we can do this forever. 00:29:15.820 --> 00:29:21.820 We could also do a similar thing if we apply a to psi v. We 00:29:21.820 --> 00:29:27.550 can go down, but at some point, we run out of steam 00:29:27.550 --> 00:29:33.740 because we've gone to the lowest energy, and if we go lower, 00:29:33.740 --> 00:29:35.290 we get 0. 00:29:35.290 --> 00:29:43.810 So a operating on psi min gives 0. 00:29:43.810 --> 00:29:50.050 So we have this stack of energy levels and wave functions, 00:29:50.050 --> 00:29:53.290 and we have the same stack being repeated as we go down, 00:29:53.290 --> 00:29:55.240 but this one has an end. 00:30:08.880 --> 00:30:17.000 We bring back what a is, and so a psi min is 0-- 00:30:17.000 --> 00:30:18.390 that's the equation. 00:30:18.390 --> 00:30:32.060 We bring in what a is, and it's ip twiddle x plus x twiddle. 00:30:39.980 --> 00:30:42.040 So we do some algebra, and what we end up 00:30:42.040 --> 00:30:46.060 with is a differential equation, psi min. 00:30:46.060 --> 00:30:52.630 dxx twiddle is equal to-- 00:30:52.630 --> 00:30:54.400 again, a little bit more algebra-- 00:30:54.400 --> 00:31:00.790 minus mu omega over h-bar times psi min. 00:31:04.320 --> 00:31:09.880 So what function gives-- 00:31:09.880 --> 00:31:13.360 there's an x in here too. 00:31:13.360 --> 00:31:16.940 So what function has a derivative, 00:31:16.940 --> 00:31:19.690 which is the function you had started 00:31:19.690 --> 00:31:22.330 with, times the variable, times a constant? 00:31:27.370 --> 00:31:31.660 And so the answer to that is that psi min 00:31:31.660 --> 00:31:33.680 has to have the form-- 00:31:33.680 --> 00:31:39.700 some normalization factor times e to the minus m omega-- 00:31:39.700 --> 00:31:40.760 or mu omega-- sorry-- 00:31:43.600 --> 00:31:49.680 over 2 h-bar x squared-- 00:31:49.680 --> 00:31:50.420 a Gaussian. 00:31:52.950 --> 00:31:54.990 Well, it had to be a Gaussian, right? 00:31:54.990 --> 00:31:57.990 We know when we did the algebra that we're 00:31:57.990 --> 00:32:01.890 going to get some function times a Gaussian. 00:32:01.890 --> 00:32:05.910 But for the lowest function, the Hermite polynomial is 1, 00:32:05.910 --> 00:32:08.550 and all there is is the Gaussian. 00:32:08.550 --> 00:32:15.430 And so we found the lowest level, and we can normalize it. 00:32:21.880 --> 00:32:23.320 So let's start over here. 00:32:29.640 --> 00:32:35.030 So what we have found is psi min of x 00:32:35.030 --> 00:32:47.260 is equal to mu omega over pi h-bar to the 1/4 power, e 00:32:47.260 --> 00:32:55.797 to the minus mu omega over 2 h-bar squared. 00:32:55.797 --> 00:32:56.630 Well, that's useful. 00:32:56.630 --> 00:33:00.050 We knew that, but this time we got it out 00:33:00.050 --> 00:33:01.530 of a completely different path. 00:33:05.900 --> 00:33:16.120 And now, we can get all higher v by 00:33:16.120 --> 00:33:21.140 a-dagger, a-dagger, et cetera. 00:33:21.140 --> 00:33:26.170 So remember, we don't care anything 00:33:26.170 --> 00:33:29.890 about what the function is, we just 00:33:29.890 --> 00:33:34.100 know that we can bring it in and get rid of it at will, 00:33:34.100 --> 00:33:37.880 because what we want is the values of integrals involving 00:33:37.880 --> 00:33:41.280 that function and some operator. 00:33:41.280 --> 00:33:44.280 So yeah, we can have all of those functions, 00:33:44.280 --> 00:33:46.820 and this is a way of generating all of the functions. 00:33:46.820 --> 00:33:56.880 And so if we wanted psi v, we would do a-dagger to the vth 00:33:56.880 --> 00:34:04.850 power divided by v-dagger-- 00:34:04.850 --> 00:34:12.300 v-- what do you call this with an exclamation point? 00:34:12.300 --> 00:34:13.419 Factorial-- ha! 00:34:18.909 --> 00:34:24.719 So we apply this operator that raises us to whatever level 00:34:24.719 --> 00:34:29.080 we want starting from this Gaussian at the bottom, 00:34:29.080 --> 00:34:32.710 and we have this normalization factor which cancels out 00:34:32.710 --> 00:34:35.800 the fact the stuff that you get by applying av. 00:34:47.120 --> 00:34:49.300 Now, there is some more logic in my notes, 00:34:49.300 --> 00:34:51.340 and I don't want to do that, but what 00:34:51.340 --> 00:34:57.080 we'd like to be able to show is that a-dagger on psi v 00:34:57.080 --> 00:35:04.850 gives some constant, and that this constant has 00:35:04.850 --> 00:35:07.460 some value-- we're going to evaluate what it is. 00:35:07.460 --> 00:35:16.280 And similarly, a psi v gives dv, and some constant v minus that. 00:35:16.280 --> 00:35:20.030 We can derive those things, and I'm not going 00:35:20.030 --> 00:35:23.120 to waste time deriving them-- 00:35:23.120 --> 00:35:25.150 I'm going to just give you the values. 00:35:25.150 --> 00:35:32.105 But we already know that cv is square of v plus 1/2-- 00:35:32.105 --> 00:35:34.355 e plus 1 and dv-- 00:35:38.170 --> 00:35:40.210 and you can see the derivation in my notes. 00:35:40.210 --> 00:35:43.950 I don't think going through them is going to be instructive. 00:35:43.950 --> 00:35:49.650 And that's just going to be v and 1/2. 00:35:49.650 --> 00:35:55.780 So now, we have something that's wonderful, 00:35:55.780 --> 00:36:01.270 because everything you need to know about getting numbers 00:36:01.270 --> 00:36:05.890 concerning harmonic oscillator is obtained from these five 00:36:05.890 --> 00:36:07.300 equations. 00:36:07.300 --> 00:36:18.810 a-dagger on psi v is v plus 1 square root psi v plus 1. 00:36:18.810 --> 00:36:24.260 a on psi v is v square root psi, v minus 1. 00:36:24.260 --> 00:36:28.450 I've said this before, but these are the most useful things 00:36:28.450 --> 00:36:29.650 you'll ever encounter. 00:36:29.650 --> 00:36:32.150 We have this thing called the number operator, 00:36:32.150 --> 00:36:35.890 and that number operator is a-dagger-hat, 00:36:35.890 --> 00:36:43.140 and the number operator operating on psi v 00:36:43.140 --> 00:36:49.410 gives v psi v. And so that's a kind of benign operator 00:36:49.410 --> 00:36:56.310 that can suck up all sorts of factors of a-dagger a, 00:36:56.310 --> 00:37:00.720 because it just gives a useful thing. 00:37:00.720 --> 00:37:07.350 And then we have a, a-dagger, and this is 1. 00:37:07.350 --> 00:37:08.940 Well, you sort of know it's going 00:37:08.940 --> 00:37:13.980 to be 1, because a a-dagger gives an increase-- it gives v 00:37:13.980 --> 00:37:22.260 plus 1, and a gives v minus 1. 00:37:22.260 --> 00:37:26.735 So it's plus 1, not minus 1-- you know that it's hardwired. 00:37:31.264 --> 00:37:31.930 Well, I did it-- 00:37:31.930 --> 00:37:36.400 I got to the point where it starts to get interesting. 00:37:36.400 --> 00:37:43.120 So we're going to be using this notation, a-dagger and a, 00:37:43.120 --> 00:37:45.930 for all sorts of stuff. 00:37:45.930 --> 00:37:57.070 And one sort of thing is transition intensities 00:37:57.070 --> 00:37:58.250 and selection rules. 00:38:03.350 --> 00:38:05.255 So you have a harmonic oscillator. 00:38:05.255 --> 00:38:09.160 A harmonic oscillator is, say a diatomic molecule 00:38:09.160 --> 00:38:11.410 which is heteronuclear. 00:38:11.410 --> 00:38:15.250 And so as the molecule vibrates, you 00:38:15.250 --> 00:38:18.792 have a dipole moment which is oscillating. 00:38:21.690 --> 00:38:25.420 And so any oscillating electric field 00:38:25.420 --> 00:38:27.940 will grab a hold of that dipole moment 00:38:27.940 --> 00:38:31.900 and stretch or compress it, especially 00:38:31.900 --> 00:38:37.390 if that field is in resonance with h-bar omega. 00:38:37.390 --> 00:38:41.675 And I've got some beautiful animation showing this, 00:38:41.675 --> 00:38:44.050 but we can't do that until we have time dependent quantum 00:38:44.050 --> 00:38:45.260 mechanics. 00:38:45.260 --> 00:38:48.550 So we have a time dependent radiation 00:38:48.550 --> 00:38:50.770 field, which is going to interact 00:38:50.770 --> 00:38:56.170 with the dipole associated with the vibrating molecule, 00:38:56.170 --> 00:38:59.060 and it's going to cause transitions. 00:38:59.060 --> 00:39:06.760 And so we can write the quantum mechanical operator 00:39:06.760 --> 00:39:08.860 that causes the transitions-- 00:39:08.860 --> 00:39:14.380 this is the electric dipole moment operator-- 00:39:14.380 --> 00:39:17.410 as a function of coordinate. 00:39:17.410 --> 00:39:21.670 And we can do a power series expansion of this, 00:39:21.670 --> 00:39:22.270 and we have-- 00:39:36.580 --> 00:39:40.240 so we have mu 0-- the constant term-- 00:39:40.240 --> 00:39:42.130 the first derivative of the dipole 00:39:42.130 --> 00:39:45.100 with respect to x, and the second derivative of the dipole 00:39:45.100 --> 00:39:46.930 with respect to x. 00:39:46.930 --> 00:39:50.840 And we have the x cofactor and the x squared cofactor. 00:39:50.840 --> 00:39:55.000 And so this guy doesn't have any x on it-- it's a constant. 00:39:55.000 --> 00:39:57.730 The only integrals involving-- 00:40:09.880 --> 00:40:18.070 the only integrals are delta v, v prime following the selection 00:40:18.070 --> 00:40:19.190 rule delta v v prime. 00:40:19.190 --> 00:40:24.440 So these integrals are 0 unless v and v prime are the same. 00:40:26.980 --> 00:40:30.220 And that says, well, an isolating field 00:40:30.220 --> 00:40:33.580 isn't going to do anything key to it, 00:40:33.580 --> 00:40:36.160 it's just going to leave it in the same vibration level. 00:40:36.160 --> 00:40:39.880 But it might have an electric Stark effect, 00:40:39.880 --> 00:40:41.320 but that's something else. 00:40:41.320 --> 00:40:45.940 So this term does nothing as far as vibration is concerned. 00:40:45.940 --> 00:40:50.395 This guy, which is a plus a-dagger, 00:40:50.395 --> 00:40:54.544 has a selection rule, delta v of plus and minus 1, 00:40:54.544 --> 00:40:58.255 and this guy has a selection rule, delta v 00:40:58.255 --> 00:41:00.220 of plus and minus 2 and 0. 00:41:03.600 --> 00:41:06.270 So if we're interested in the intensities 00:41:06.270 --> 00:41:08.910 of vibrational transitions, it says, 00:41:08.910 --> 00:41:12.420 well, this is the important term and it causes transitions, 00:41:12.420 --> 00:41:16.080 changing the vibrational quantum number by one, which 00:41:16.080 --> 00:41:18.540 is called the fundamental. 00:41:18.540 --> 00:41:22.530 This gives rise to overtones. 00:41:22.530 --> 00:41:27.330 So all of a sudden, we're in real problem land, where 00:41:27.330 --> 00:41:31.590 if we're looking at vibrational transitions in a molecule, 00:41:31.590 --> 00:41:34.740 that this enables us to calculate what's important, 00:41:34.740 --> 00:41:37.560 or to say these are the intensities I measure, 00:41:37.560 --> 00:41:41.850 and these are the first and second derivative 00:41:41.850 --> 00:41:45.330 of the dipole moment operator as a function of internuclear 00:41:45.330 --> 00:41:46.160 distance. 00:41:46.160 --> 00:41:46.860 Isn't that neat? 00:41:53.370 --> 00:41:59.460 I've gone so fast, I'm more or less at the end of my notes, 00:41:59.460 --> 00:42:01.120 but I can blather on for a while. 00:42:03.790 --> 00:42:09.420 So suppose you have some integral involving 00:42:09.420 --> 00:42:13.680 an operator and a vibrational wave function. 00:42:13.680 --> 00:42:21.810 So we have psi v star, some operator, psi v prime dx. 00:42:25.300 --> 00:42:28.630 And we'd like to know how to focus our energies. 00:42:28.630 --> 00:42:30.070 We're very busy people-- 00:42:30.070 --> 00:42:31.590 we don't want any value integrals 00:42:31.590 --> 00:42:36.280 that come out to be 0, we'd like to just know. 00:42:36.280 --> 00:42:45.100 So if this operator is some function of x or function of p, 00:42:45.100 --> 00:42:48.550 we'd have a power series expansion of the operator, 00:42:48.550 --> 00:42:51.580 and we then know what the selection rules are. 00:42:56.460 --> 00:42:58.600 So usually, you look at the operator 00:42:58.600 --> 00:43:05.040 and you find that it's a linear quadratic cubic function of x-- 00:43:05.040 --> 00:43:07.580 the leading term is usually linear. 00:43:07.580 --> 00:43:10.920 Bang-- you have a delta v of plus 1-- 00:43:10.920 --> 00:43:12.900 selection rule. 00:43:12.900 --> 00:43:16.770 Or if someone has bothered to actually convert 00:43:16.770 --> 00:43:18.250 the operator to some form-- 00:43:20.855 --> 00:43:23.666 oh, it's the operator-- 00:43:29.130 --> 00:43:34.772 this might have some form, a-dagger cubed times 00:43:34.772 --> 00:43:35.355 some constant. 00:43:40.100 --> 00:43:43.940 So if the operator looks like a-dagger cubed, 00:43:43.940 --> 00:43:49.740 we know that the selection rule is v to v plus 3, 00:43:49.740 --> 00:43:54.550 and we know that the matrix of the integral is v plus 1, 00:43:54.550 --> 00:43:59.550 times v plus 2, times v plus 3, square root of that, 00:43:59.550 --> 00:44:01.870 times the constant. 00:44:01.870 --> 00:44:04.740 So there is a huge number of problems 00:44:04.740 --> 00:44:08.860 that, instead of being pages and pages of algebra, 00:44:08.860 --> 00:44:14.520 are just reduced something that you can tell by inspection. 00:44:14.520 --> 00:44:19.520 So one of the tricks is we have an operator 00:44:19.520 --> 00:44:21.290 like x squared or x cubed-- 00:44:24.320 --> 00:44:27.560 what we want to do is write this in terms 00:44:27.560 --> 00:44:36.720 of a squared, a-dagger squared, and maybe 00:44:36.720 --> 00:44:40.920 some combination of a-dagger a. 00:44:40.920 --> 00:44:44.340 So we want to take the a-dagger a's with the a a-daggers 00:44:44.340 --> 00:44:47.550 and combine them using the commutation rule. 00:44:47.550 --> 00:44:51.720 And then we have expressed this in this maximally simple form, 00:44:51.720 --> 00:44:54.600 and then you just apply a squared, 00:44:54.600 --> 00:44:59.050 apply a-dagger squared, and you apply this, 00:44:59.050 --> 00:45:02.800 then you've got the value of the integral. 00:45:02.800 --> 00:45:04.930 So if you're a busy person and you 00:45:04.930 --> 00:45:08.640 want to actually calculate stuff, 00:45:08.640 --> 00:45:12.960 you want to know how to reduce operators-- 00:45:12.960 --> 00:45:17.010 usually expressed as some power of the coordinate 00:45:17.010 --> 00:45:18.240 or the momentum-- 00:45:18.240 --> 00:45:23.730 into a sum of terms involving these organized 00:45:23.730 --> 00:45:25.500 products of a's and a-daggers. 00:45:28.150 --> 00:45:32.980 And you're going to be absolutely shocked at how 00:45:32.980 --> 00:45:35.140 perturbation theory-- 00:45:35.140 --> 00:45:41.910 which leads to basically all of the formulas and spectroscopy-- 00:45:41.910 --> 00:45:46.230 it's an ugly theory, but it reduces everything to things 00:45:46.230 --> 00:45:50.280 that you can just write down at the speed of your pen 00:45:50.280 --> 00:45:54.040 or pencil, and that's a fantastic thing. 00:45:54.040 --> 00:45:58.140 So you can't do this with the particle in a box, 00:45:58.140 --> 00:46:00.480 you can't do this with a hydrogen atom, 00:46:00.480 --> 00:46:02.040 you can't do this with the rigid-- 00:46:02.040 --> 00:46:04.230 well, you can do some of this with a rigid rotor. 00:46:04.230 --> 00:46:07.920 But the harmonic oscillator is so ubiquitous 00:46:07.920 --> 00:46:09.930 because every one-dimensional problem 00:46:09.930 --> 00:46:12.647 is harmonic at the bottom. 00:46:12.647 --> 00:46:15.230 And so you can use it and then you can put in the corrections. 00:46:15.230 --> 00:46:21.530 But also because you want to describe dynamics, 00:46:21.530 --> 00:46:24.830 you almost always use the harmonic oscillator, 00:46:24.830 --> 00:46:27.440 because not only do you know the integrals, 00:46:27.440 --> 00:46:29.540 but you know there's only a few. 00:46:29.540 --> 00:46:31.640 Normally, you're going to be summing 00:46:31.640 --> 00:46:36.290 over an infinite number of quantum numbers, 00:46:36.290 --> 00:46:39.950 and that takes time, and it takes judgment to say, 00:46:39.950 --> 00:46:44.230 well, only certain of these are important. 00:46:44.230 --> 00:46:49.240 But for the harmonic oscillator, the sums are finite. 00:46:49.240 --> 00:46:50.710 All of these things are wonderful, 00:46:50.710 --> 00:46:54.880 and that's why whenever you look at a theory, 00:46:54.880 --> 00:46:58.390 you're going to discover that hidden in there 00:46:58.390 --> 00:47:01.550 is the harmonic oscillator approximation, 00:47:01.550 --> 00:47:07.290 because everything is doable in no effort. 00:47:07.290 --> 00:47:10.880 And sometimes when you look at a paper like that, 00:47:10.880 --> 00:47:13.280 it doesn't show you the intermediate steps, 00:47:13.280 --> 00:47:17.480 because everybody knows what a harmonic oscillator does. 00:47:17.480 --> 00:47:18.950 And there's also a lot of insight 00:47:18.950 --> 00:47:23.730 because something like this-- 00:47:23.730 --> 00:47:27.440 this is an odd symmetry term, this is an even symmetry term, 00:47:27.440 --> 00:47:30.370 and there are all sorts of things that have to do with, 00:47:30.370 --> 00:47:33.050 are you're conserving symmetry or changing symmetry? 00:47:33.050 --> 00:47:38.660 And sometimes, the issue is how does the molecule spontaneously 00:47:38.660 --> 00:47:40.610 change symmetry by doing something 00:47:40.610 --> 00:47:43.370 interacting with a field, or interacting 00:47:43.370 --> 00:47:47.280 with some feature of the potential surface. 00:47:47.280 --> 00:47:51.560 So this is a place where it's labor saving and insight 00:47:51.560 --> 00:47:55.520 generating, and it's really amazing. 00:47:55.520 --> 00:47:59.110 So maybe I've bored you with this, 00:47:59.110 --> 00:48:03.070 but this is the beginning of almost every theory 00:48:03.070 --> 00:48:07.670 that you encounter just because of the simplicity of the a's 00:48:07.670 --> 00:48:10.210 and a-daggers. 00:48:10.210 --> 00:48:11.470 OK, I'm done. 00:48:11.470 --> 00:48:13.054 Thank you.