WEBVTT 00:00:00.500 --> 00:00:04.080 PROFESSOR: Today, we have to discuss harmonic perturbations. 00:00:04.080 --> 00:00:08.700 So we've done Fermi's golden rule for constant transitions. 00:00:08.700 --> 00:00:13.950 We saw transitions from a discrete state to a continuum. 00:00:13.950 --> 00:00:16.470 And by integrating over the continuum, 00:00:16.470 --> 00:00:20.460 we found a nice rule, Fermi's golden rule, 00:00:20.460 --> 00:00:25.510 that govern the transition rate for this process. 00:00:25.510 --> 00:00:28.020 So the only thing we have to do different 00:00:28.020 --> 00:00:31.620 now is consider the case that the perturbation is not 00:00:31.620 --> 00:00:35.230 just a step that gets up and stays there, 00:00:35.230 --> 00:00:37.830 but it has a frequency dependence. 00:00:37.830 --> 00:00:42.100 So that will bring a couple of novel features. 00:00:42.100 --> 00:00:44.520 But at the end of the day, as we will see, 00:00:44.520 --> 00:00:48.560 our Fermi's golden rule is going to look pretty similar 00:00:48.560 --> 00:00:51.930 to the original Fermi's golden rule. 00:00:51.930 --> 00:00:56.580 A nice application of Fermi's golden rule 00:00:56.580 --> 00:01:01.550 is the calculation of the ionization rate for hydrogen, 00:01:01.550 --> 00:01:04.010 in which you take a hydrogen atom, 00:01:04.010 --> 00:01:09.820 you put it in an electric field or send a light wave, 00:01:09.820 --> 00:01:13.350 and then suddenly the electron and the hydrogen atom 00:01:13.350 --> 00:01:15.870 from the ground state ionizes. 00:01:15.870 --> 00:01:19.530 And we can compute already-- we have the technology 00:01:19.530 --> 00:01:23.460 to compute the ionization rate. 00:01:23.460 --> 00:01:26.740 That's a pretty physical quantity. 00:01:26.740 --> 00:01:30.990 And that will be an example we'll develop today. 00:01:30.990 --> 00:01:34.050 Those rates have the funny situation 00:01:34.050 --> 00:01:37.380 that the calculation can be somewhat 00:01:37.380 --> 00:01:39.930 involved and interesting. 00:01:39.930 --> 00:01:43.230 And the answers, generally, by the time you simplify them, 00:01:43.230 --> 00:01:45.650 are pretty simple and pretty nice. 00:01:45.650 --> 00:01:47.730 So it's a good idea. 00:01:47.730 --> 00:01:50.550 You have to have patience with those calculations 00:01:50.550 --> 00:01:55.150 to simplify it till the end, and that's pretty instructive. 00:01:55.150 --> 00:02:00.270 So we begin with harmonic perturbations. 00:02:00.270 --> 00:02:05.160 So we did constant perturbations already. 00:02:05.160 --> 00:02:09.389 So now harmonic perturbations. 00:02:13.660 --> 00:02:18.160 So our situation is that in which age H of t 00:02:18.160 --> 00:02:23.230 is equal to a known Hamiltonian plus delta H of t. 00:02:23.230 --> 00:02:29.620 And this time, delta H of t is conventionally 00:02:29.620 --> 00:02:34.900 written as 2 H prime cosine omega t 00:02:34.900 --> 00:02:40.930 for some t between t0 and 0, and 0 otherwise. 00:02:49.230 --> 00:02:53.220 All of us wonder why the 2 here. 00:02:53.220 --> 00:02:56.940 One reason for it-- it's all convention, of course. 00:02:56.940 --> 00:03:01.410 You have your perturbation, and what you call E H prime 00:03:01.410 --> 00:03:05.010 or what you call 2 H prime is your choice. 00:03:05.010 --> 00:03:09.120 But this 2 has the advantage that when 00:03:09.120 --> 00:03:13.560 you describe the cosine in terms of exponentials-- 00:03:13.560 --> 00:03:17.480 e to the i omega t plus e to the minus i omega t-- the over 2, 00:03:17.480 --> 00:03:19.620 it cancels this one. 00:03:19.620 --> 00:03:26.100 And that makes Fermi's golden rule, that will follow also 00:03:26.100 --> 00:03:29.820 and will be valid for these perturbations, 00:03:29.820 --> 00:03:33.840 take exactly the same form as it did 00:03:33.840 --> 00:03:36.030 for the case of constant perturbation. 00:03:36.030 --> 00:03:42.450 So it's fairly convenient to put that 2, and we'll put it in. 00:03:42.450 --> 00:03:48.830 Some books don't, and then they have different looking formulas 00:03:48.830 --> 00:03:52.190 for Fermi golden rule depending to which case you're 00:03:52.190 --> 00:03:53.300 talking about. 00:03:53.300 --> 00:03:58.050 Of course, when we mean that this is the time dependence, 00:03:58.050 --> 00:04:02.490 we are implying that H prime is time independent. 00:04:05.160 --> 00:04:07.190 Because the time dependence is this one. 00:04:07.190 --> 00:04:11.000 That's what we're interested in considering. 00:04:11.000 --> 00:04:14.720 Of course-- this has been asked sometimes-- 00:04:14.720 --> 00:04:18.500 H prime can depend on all kinds of other thing-- position 00:04:18.500 --> 00:04:21.170 coordinates, some other quantities. 00:04:21.170 --> 00:04:25.960 But we're focusing on time here, so we'll leave it there. 00:04:25.960 --> 00:04:29.030 Moreover, for reasons of convention, 00:04:29.030 --> 00:04:32.480 just let's always thing of omega as positive. 00:04:32.480 --> 00:04:35.690 It wouldn't make a difference if it would be negative here 00:04:35.690 --> 00:04:41.210 with the cosine function, but let's just set by convention 00:04:41.210 --> 00:04:43.590 that omega is positive. 00:04:43.590 --> 00:04:46.340 Finally, we're going to do transitions again 00:04:46.340 --> 00:04:49.220 from an initial to a final state. 00:04:49.220 --> 00:04:51.080 So we will consider the case when 00:04:51.080 --> 00:04:58.440 we go from an initial state to a final state. 00:04:58.440 --> 00:05:03.650 And therefore, we will work in this language 00:05:03.650 --> 00:05:07.610 with this constant coefficients Cn's, these 00:05:07.610 --> 00:05:12.190 coefficients that multiply the states in psi tilde. 00:05:12.190 --> 00:05:22.180 Psi tilde is equal to Cn n of t, sum over n. 00:05:22.180 --> 00:05:29.930 And these Cn's, at time equals 0, 00:05:29.930 --> 00:05:38.090 will be equal to delta ni, which means that they are all 0 00:05:38.090 --> 00:05:42.500 except when we're talking about Ci at 0 is equal to 1 00:05:42.500 --> 00:05:47.040 because we start with an initial state. 00:05:47.040 --> 00:05:52.770 We had a general formula for the transition coefficient. 00:05:52.770 --> 00:06:00.702 And Cm of 1 at time equal t-- 00:06:00.702 --> 00:06:03.130 or I'll put t0-- 00:06:03.130 --> 00:06:09.270 is equal to sum over n, integral from 0 to t0 e 00:06:09.270 --> 00:06:22.550 to the i omega mn t prime, delta H mn of t prime over i h bar, 00:06:22.550 --> 00:06:28.110 Cn at time equals 0, dt prime. 00:06:30.980 --> 00:06:35.880 This was our general formula for transition coefficients. 00:06:35.880 --> 00:06:42.590 The Cm's is the coefficient or the amplitude for the state 00:06:42.590 --> 00:06:47.190 to be found in the m eigenstate at time t0 00:06:47.190 --> 00:06:50.230 to first order in perturbation theory. 00:06:50.230 --> 00:06:53.090 And it depends on where you started on. 00:06:53.090 --> 00:06:57.830 That's why the sum over n here with initial states. 00:06:57.830 --> 00:07:01.010 But this sum is going to collapse because we 00:07:01.010 --> 00:07:03.950 know we start with the state i. 00:07:03.950 --> 00:07:08.430 So when we substitute Cn equal to this, 00:07:08.430 --> 00:07:12.410 the sum only works when n is equal to i. 00:07:12.410 --> 00:07:15.740 So we'll put for ni's. 00:07:15.740 --> 00:07:19.640 And, of course, we're going to also take for the final state 00:07:19.640 --> 00:07:21.140 to be f. 00:07:21.140 --> 00:07:29.830 So the formula now reads Cf 1 at t0 00:07:29.830 --> 00:07:37.220 is equal to integral from 0 to t0 e to the I omega f-- 00:07:37.220 --> 00:07:43.730 m was f-- i t prime. 00:07:43.730 --> 00:07:49.250 And now the delta H. The delta H is this whole quantity, 00:07:49.250 --> 00:07:51.610 so we have to substitute it. 00:07:51.610 --> 00:07:58.830 So 2 H prime is the only part that has matrix elements. 00:07:58.830 --> 00:08:01.060 The cosine omega t is just a function. 00:08:01.060 --> 00:08:12.825 So it's H prime fi cosine omega t prime, and dt prime. 00:08:20.060 --> 00:08:21.455 There's the i h bar. 00:08:24.600 --> 00:08:27.150 So I think I got everything right. 00:08:27.150 --> 00:08:30.110 The sum collapsed. 00:08:30.110 --> 00:08:35.240 mn is being replaced by the right labels. 00:08:35.240 --> 00:08:40.580 mn here, this is the expectation value between m and n. 00:08:40.580 --> 00:08:43.130 And that becomes between f and i. 00:08:43.130 --> 00:08:45.620 And it affects this whole thing, but it just 00:08:45.620 --> 00:08:50.570 ends up affecting the Hamiltonian H prime here. 00:08:50.570 --> 00:08:52.240 So I think we're OK. 00:08:52.240 --> 00:08:54.530 We have everything there. 00:08:54.530 --> 00:09:01.560 And Hfi, of course, doesn't have time dependence. 00:09:01.560 --> 00:09:06.900 So we said H prime has no time dependence. 00:09:06.900 --> 00:09:10.440 So that thing can go out of the integral. 00:09:10.440 --> 00:09:14.410 So this will go out. 00:09:14.410 --> 00:09:17.890 And the integral is simple because you have now 00:09:17.890 --> 00:09:24.200 Hfi prime over i h bar. 00:09:24.200 --> 00:09:26.470 And the 2, we leave it for the cosine. 00:09:26.470 --> 00:09:28.370 So we get two integrals. 00:09:28.370 --> 00:09:42.100 t0 e to the i omega fi plus omega t prime-- 00:09:42.100 --> 00:09:45.520 from the first exponential in the cosine-- 00:09:45.520 --> 00:09:52.930 plus an e to the i omega fi minus omega t prime-- 00:09:52.930 --> 00:09:55.570 from the second exponential in cosine-- 00:09:55.570 --> 00:09:58.400 dt prime. 00:09:58.400 --> 00:09:59.830 Well, that's very nice. 00:09:59.830 --> 00:10:01.630 This is all doable. 00:10:01.630 --> 00:10:04.480 The Hfi doesn't give us any trouble. 00:10:04.480 --> 00:10:05.500 It's a constant. 00:10:05.500 --> 00:10:07.660 It's out of the integral. 00:10:07.660 --> 00:10:10.300 It's all pretty nice and simple. 00:10:10.300 --> 00:10:14.840 So we can do these two integrals. 00:10:14.840 --> 00:10:16.790 They're integrals of exponentials, 00:10:16.790 --> 00:10:20.900 so it's just an exponential divided by those coefficients. 00:10:20.900 --> 00:10:27.860 So I'll just do it and evaluated it between t0 and 0. 00:10:27.860 --> 00:10:29.120 So what do we get? 00:10:29.120 --> 00:10:37.230 Minus i Hfi prime over h bar, e to the i omega 00:10:37.230 --> 00:10:48.530 fi plus omega t0, minus 1, over omega fi plus omega. 00:10:48.530 --> 00:10:51.440 You can imagine that e to the i omega 00:10:51.440 --> 00:10:55.710 t integrates to e to the i omega t over omega. 00:10:55.710 --> 00:10:58.430 So that's why that works. 00:10:58.430 --> 00:11:03.620 And the two limits are t0 and 0. 00:11:03.620 --> 00:11:13.540 Plus e to the i omega fi minus omega t0, minus 1 00:11:13.540 --> 00:11:20.080 again, over omega fi minus omega. 00:11:20.080 --> 00:11:22.170 Great. 00:11:22.170 --> 00:11:23.470 Our integral is there. 00:11:23.470 --> 00:11:24.070 It's done. 00:11:24.070 --> 00:11:27.730 And now it's time to appreciate what it tells us, 00:11:27.730 --> 00:11:30.745 because it tells us something very important, this formula. 00:11:41.180 --> 00:11:44.270 So you look at this and you say, well, OK. 00:11:44.270 --> 00:11:50.290 This is the transition amplitude to state omega-- 00:11:50.290 --> 00:11:53.810 I'm sorry-- state f, final state. 00:11:53.810 --> 00:11:57.670 And it depends on omega fi. 00:11:57.670 --> 00:12:09.150 And omega fi just Ef minus Ei over h bar. 00:12:09.150 --> 00:12:12.830 So if I know my final discrete state Ef, 00:12:12.830 --> 00:12:18.440 I can figure out what is the transition probability. 00:12:18.440 --> 00:12:23.750 Now, these denominators are intriguing because maybe 00:12:23.750 --> 00:12:26.580 if you adjust the frequency omega-- 00:12:26.580 --> 00:12:30.770 suppose you have an initial state and a final state here. 00:12:30.770 --> 00:12:35.660 You may adjust the frequency omega to match them, 00:12:35.660 --> 00:12:43.650 and in that case maybe make the denominators equal to 0. 00:12:43.650 --> 00:12:49.960 And that's exactly what kind of happens here. 00:12:49.960 --> 00:12:53.110 So, first of all, if you look at this expression, 00:12:53.110 --> 00:12:59.850 it's a sum of two terms that are added together and multiplied 00:12:59.850 --> 00:13:01.560 by a constant. 00:13:01.560 --> 00:13:05.850 As t0 really goes to 0, completely 00:13:05.850 --> 00:13:12.100 goes to 0, t0, each factor, actually, 00:13:12.100 --> 00:13:16.090 if you see the Taylor expansion of the exponential, 00:13:16.090 --> 00:13:19.990 you cancel the 1 and you then cancel the linear term 00:13:19.990 --> 00:13:21.410 with the denominator. 00:13:21.410 --> 00:13:24.420 This is just i t0. 00:13:24.420 --> 00:13:26.440 And this is also i t0. 00:13:26.440 --> 00:13:32.980 So they're comparable as time is really going to 0. 00:13:32.980 --> 00:13:36.670 But time going to 0 is never of interest for us. 00:13:36.670 --> 00:13:40.120 For us, we need to be the time a little big 00:13:40.120 --> 00:13:43.180 already so that our calculations, 00:13:43.180 --> 00:13:50.160 as we did in the constant transitions that had lobes that 00:13:50.160 --> 00:13:54.730 decreases constants over t0-- 00:13:54.730 --> 00:13:57.640 we needed the time to be sufficiently large 00:13:57.640 --> 00:14:00.400 so that the lobes are narrow. 00:14:00.400 --> 00:14:01.870 And that, we could guarantee. 00:14:01.870 --> 00:14:05.180 So t0 going to 0 is not very interesting. 00:14:05.180 --> 00:14:08.200 We need t0 a little bigger. 00:14:08.200 --> 00:14:12.550 Not too big that the rate of a process 00:14:12.550 --> 00:14:15.640 overwhelms the probability. 00:14:15.640 --> 00:14:17.800 But we need a little big. 00:14:17.800 --> 00:14:20.610 So, in that case, the numerators are 00:14:20.610 --> 00:14:22.450 going to be bounded numbers. 00:14:22.450 --> 00:14:25.580 You see, you have an exponential minus 1. 00:14:25.580 --> 00:14:28.630 So that varies from-- 00:14:28.630 --> 00:14:34.570 the magnitude of this thing varies from 2 to 0, basically. 00:14:34.570 --> 00:14:36.400 In fact, in these numerators, you 00:14:36.400 --> 00:14:42.680 can see, if the phase is 0 for some particular value-- 00:14:42.680 --> 00:14:47.020 if the exponential has a phase that's proportional to 2 pi, 00:14:47.020 --> 00:14:48.090 this is 0. 00:14:48.090 --> 00:14:50.500 And then sometimes this exponential is minus 1, 00:14:50.500 --> 00:14:52.820 so it gets to minus 2. 00:14:52.820 --> 00:14:54.610 So it's finite. 00:14:54.610 --> 00:14:56.890 And the same is here. 00:14:56.890 --> 00:14:59.710 So these are bounded numerators. 00:14:59.710 --> 00:15:03.040 On the other hand, you may have the possibility 00:15:03.040 --> 00:15:06.400 that these things become 0. 00:15:06.400 --> 00:15:09.460 And those are the cases that are of interest to us, 00:15:09.460 --> 00:15:13.500 the cases when those terms are going to be 0.