WEBVTT 00:00:00.240 --> 00:00:01.115 PROFESSOR: All right. 00:00:01.115 --> 00:00:05.460 We have this, so let's complete the Einstein discussion. 00:00:09.290 --> 00:00:13.640 And that's practical formulas that 00:00:13.640 --> 00:00:16.550 are relevant to some of the exercises 00:00:16.550 --> 00:00:17.980 you will find in homework. 00:00:21.470 --> 00:00:30.130 So when we set up the rates in Einstein's argument, 00:00:30.130 --> 00:00:34.000 we said that for spontaneous emission, 00:00:34.000 --> 00:00:44.530 Einstein's spontaneous emission, he put a rate that was-- 00:00:44.530 --> 00:00:47.560 he said the number of transitions 00:00:47.560 --> 00:00:50.980 that occur due to spontaneous emission 00:00:50.980 --> 00:00:53.350 is proportional to the number of atoms, 00:00:53.350 --> 00:00:58.630 Nb, the number of photons present, 00:00:58.630 --> 00:01:02.710 times some coefficient that I don't know, Bab. 00:01:02.710 --> 00:01:09.670 That's how the rate of atoms changing and going 00:01:09.670 --> 00:01:13.780 from the top level to the low level was. 00:01:13.780 --> 00:01:17.560 But this is the one per atom. 00:01:17.560 --> 00:01:21.070 This is the transition rate per atom, multiplied 00:01:21.070 --> 00:01:24.700 by the number of atoms present in the top level gave you 00:01:24.700 --> 00:01:28.030 the rate for spontaneous emission. 00:01:28.030 --> 00:01:34.780 This quantity is the one that is equal to that omega ab. 00:01:34.780 --> 00:01:43.180 So this is equal to omega ba, like this. 00:01:43.180 --> 00:01:47.710 Therefore, if you compare the Bab of Einstein, 00:01:47.710 --> 00:01:54.610 it's all except the U. So Bab is 4 pi 00:01:54.610 --> 00:02:02.785 squared over 3h squared dab vector squared. 00:02:07.210 --> 00:02:09.759 So Einstein could not do that because he 00:02:09.759 --> 00:02:11.620 didn't have quantum theory. 00:02:11.620 --> 00:02:18.070 But we do, and we now can calculate from first principles 00:02:18.070 --> 00:02:22.210 this quantity in an atomic transition. 00:02:22.210 --> 00:02:25.630 And the other good thing was that even though there 00:02:25.630 --> 00:02:39.050 was a rate for spontaneous emission, 00:02:39.050 --> 00:02:42.710 that rate was fixed by Einstein's argument 00:02:42.710 --> 00:02:49.340 to be h bar omega ba cubed over pi c cubed Bab. 00:02:54.780 --> 00:02:58.140 So if you knew the b coefficient, 00:02:58.140 --> 00:03:02.190 you knew the a coefficient for spontaneous emission-- many 00:03:02.190 --> 00:03:05.230 times were interested in spontaneous emission. 00:03:05.230 --> 00:03:12.750 So we calculate the stimulated emission coefficient, here, 00:03:12.750 --> 00:03:15.640 and that's given in terms of this matrix element. 00:03:15.640 --> 00:03:18.910 We might as well give the general formula. 00:03:18.910 --> 00:03:22.440 So you just substitute the Bab, and now we 00:03:22.440 --> 00:03:31.230 have that A is equal to 4/3 omega ba cubed over h bar 00:03:31.230 --> 00:03:47.630 c cubed Bab squared 4/3 omega ba cubed. 00:03:47.630 --> 00:03:49.980 There's one h bar in the bottom. 00:03:49.980 --> 00:03:50.880 There's the c cubed. 00:03:50.880 --> 00:03:53.360 Yep. 00:03:53.360 --> 00:03:57.770 So that completes that argument. 00:03:57.770 --> 00:04:04.370 We now have combined our development 00:04:04.370 --> 00:04:06.470 in time dependent perturbation theory 00:04:06.470 --> 00:04:08.405 with a statistical physics argument. 00:04:08.405 --> 00:04:11.810 [INAUDIBLE] to determine those rates. 00:04:14.410 --> 00:04:18.339 A little more is needed if you have 00:04:18.339 --> 00:04:22.390 to compute a practical rate, as you will 00:04:22.390 --> 00:04:24.820 have to do for the homework. 00:04:24.820 --> 00:04:29.200 It's just the concept of a lifetime. 00:04:29.200 --> 00:04:36.400 And a couple of selection rules. 00:04:36.400 --> 00:04:40.750 So let me make a few comments about it. 00:04:40.750 --> 00:04:43.180 We get, for example, a decay rate, 00:04:43.180 --> 00:04:48.540 A. It's a rate for spontaneous emission. 00:04:48.540 --> 00:04:49.960 You have an atom. 00:04:49.960 --> 00:04:56.460 There's a decay rate, A. So if you have a population of atoms, 00:04:56.460 --> 00:05:09.298 N atoms, at t equals 0, and A being the decay rate per atom, 00:05:09.298 --> 00:05:15.700 it means that the number of states 00:05:15.700 --> 00:05:22.540 that are changing in a little time, dt, is minus A times dt-- 00:05:22.540 --> 00:05:26.200 the rate multiplied by the time per atom-- 00:05:26.200 --> 00:05:29.090 times the number of atoms. 00:05:29.090 --> 00:05:38.415 So dN dt is minus AN. 00:05:41.910 --> 00:05:49.470 And the number of atoms as a function of t go N at t 00:05:49.470 --> 00:05:55.200 equals 0 times e to the minus At. 00:05:55.200 --> 00:06:00.150 So since we always call processes 00:06:00.150 --> 00:06:07.290 that go e to the minus t over tau-- tau is the lifetime. 00:06:10.290 --> 00:06:17.040 That is nothing else than 1 over A. 00:06:17.040 --> 00:06:20.220 So you get the decay rate. 00:06:20.220 --> 00:06:23.070 It's synonymous to a lifetime. 00:06:23.070 --> 00:06:24.780 You just invert it. 00:06:24.780 --> 00:06:25.650 No big problem. 00:06:28.940 --> 00:06:31.110 Nevertheless, if you have an object, 00:06:31.110 --> 00:06:33.690 sometimes can decay in different ways. 00:06:33.690 --> 00:06:36.390 It can decay maybe to one type of state, 00:06:36.390 --> 00:06:39.430 another type of state, a third type of state. 00:06:39.430 --> 00:06:47.120 So the total A might be equal to A1 plus A2 00:06:47.120 --> 00:06:50.520 when there are various decay channels. 00:06:54.260 --> 00:06:57.800 You can decay-- a system can decay from an excited state. 00:06:57.800 --> 00:07:01.850 It can go to one state to another state to a third state. 00:07:01.850 --> 00:07:05.330 Each one contributes a total decay rate 00:07:05.330 --> 00:07:09.470 because a total disappearance of those atoms 00:07:09.470 --> 00:07:12.840 is because they either go into one or the other or the third. 00:07:12.840 --> 00:07:17.440 So this is true that the total decay rate is that. 00:07:17.440 --> 00:07:20.180 So this gives you a relation. 00:07:20.180 --> 00:07:24.320 The total lifetime, which is 1 over this, 00:07:24.320 --> 00:07:29.140 is the sum of the inverses of the partial lifetimes. 00:07:33.930 --> 00:07:37.500 So if you have a process that can go in various ways, 00:07:37.500 --> 00:07:40.950 you may want to combine them to get a total lifetime. 00:07:45.540 --> 00:07:51.530 So the last thing we should say about this-- 00:07:51.530 --> 00:07:54.290 the last thing we're going to do is 00:07:54.290 --> 00:07:56.870 mention what happens when you really 00:07:56.870 --> 00:08:01.490 try to calculate these things for hydrogen levels. 00:08:01.490 --> 00:08:03.800 You're going to have the ability to calculate 00:08:03.800 --> 00:08:09.200 the lifetime of an excited state of hydrogen now. 00:08:09.200 --> 00:08:11.730 That's quite the thing, to be able to get 00:08:11.730 --> 00:08:17.880 a number that is correct up to factors of 2 and is measurable. 00:08:17.880 --> 00:08:23.510 So what do you have to consider? 00:08:23.510 --> 00:08:28.970 Well, the only thing we haven't quite done 00:08:28.970 --> 00:08:35.630 is computed this matrix element of the dipole operator. 00:08:35.630 --> 00:08:39.320 You see, everything else that could be done was done. 00:08:39.320 --> 00:08:44.660 We integrated over the incoherent radiation, 00:08:44.660 --> 00:08:49.130 we found the Fermi golden rule, we have everything simplified. 00:08:49.130 --> 00:08:52.730 But still, the matrix element is something 00:08:52.730 --> 00:08:55.450 that you have to calculate. 00:08:55.450 --> 00:08:57.595 So there's some selection rules. 00:09:02.050 --> 00:09:03.010 Rules. 00:09:03.010 --> 00:09:07.750 And this will be discussed in recitation. 00:09:07.750 --> 00:09:11.880 It's pretty important stuff. 00:09:11.880 --> 00:09:13.700 I'll just mention the results. 00:09:13.700 --> 00:09:17.860 It's very readable in Griffiths. 00:09:17.860 --> 00:09:22.130 And it has to do with the matrix elements. 00:09:22.130 --> 00:09:26.980 So d is the dipole operator, so it boils down to r. 00:09:26.980 --> 00:09:36.760 So if you really knew n l prime m prime, the matrix 00:09:36.760 --> 00:09:45.460 elements of nlm, if you knew all of this, you would be done. 00:09:45.460 --> 00:09:48.020 You could calculate anything. 00:09:50.650 --> 00:09:54.580 So the question is, when do these elements exist, 00:09:54.580 --> 00:09:56.560 and when are they nonzero? 00:09:56.560 --> 00:10:01.670 You need them to be nonzero to get a nonzero transition rate. 00:10:01.670 --> 00:10:09.400 So when do they not vanish? 00:10:09.400 --> 00:10:14.380 So the only cases these-- 00:10:14.380 --> 00:10:16.330 I'll summarize the answer-- 00:10:16.330 --> 00:10:24.810 are nonzero, these can be nonzero-- 00:10:27.630 --> 00:10:31.080 there can still be accidents and they could cancel-- 00:10:31.080 --> 00:10:39.530 only if delta m, the change in m from one side to another, 00:10:39.530 --> 00:10:47.310 which is m prime minus m, is either equal to 0 or plus 00:10:47.310 --> 00:10:48.940 or minus 1. 00:10:48.940 --> 00:10:54.300 You cannot change that quantum state of a system with a dipole 00:10:54.300 --> 00:11:01.590 interaction unless m changes at most by 1. 00:11:01.590 --> 00:11:04.910 And moreover, the one that is quite interesting 00:11:04.910 --> 00:11:11.940 as well is that delta l, which is l prime minus l, 00:11:11.940 --> 00:11:14.870 must only be plus or minus 1. 00:11:19.370 --> 00:11:22.630 So those changes have to happen. 00:11:22.630 --> 00:11:27.150 So now you can at least get a qualitative understanding 00:11:27.150 --> 00:11:31.530 of the hydrogen atom stability. 00:11:31.530 --> 00:11:35.550 You have the ground state here, 1S. 00:11:35.550 --> 00:11:47.530 Here is 2S, 2P, 3S, 3P, and 3D. 00:11:47.530 --> 00:11:50.940 Let's see who can go to who. 00:11:50.940 --> 00:11:53.760 Well, the ground state cannot go anywhere. 00:11:53.760 --> 00:11:56.700 That's at the bottom. 00:11:56.700 --> 00:11:58.750 So OK. 00:11:58.750 --> 00:12:01.500 2S going to 1S. 00:12:01.500 --> 00:12:02.570 Is it possible? 00:12:02.570 --> 00:12:03.507 AUDIENCE: No. 00:12:03.507 --> 00:12:04.090 PROFESSOR: No. 00:12:04.090 --> 00:12:05.350 Very good. 00:12:05.350 --> 00:12:09.160 It violates delta l. 00:12:09.160 --> 00:12:10.570 So this doesn't happen. 00:12:13.130 --> 00:12:17.210 So if this doesn't happen, this 2S 00:12:17.210 --> 00:12:22.130 can almost go nowhere because it doesn't have-- 00:12:22.130 --> 00:12:24.510 it has to go by spontaneous emission. 00:12:24.510 --> 00:12:26.790 Something has to go to something lower, 00:12:26.790 --> 00:12:28.930 so it almost doesn't go anywhere. 00:12:28.930 --> 00:12:32.280 So actually, it has a long lifetime, 00:12:32.280 --> 00:12:38.790 and it can decay via two photons or more complicated decays. 00:12:38.790 --> 00:12:40.110 And we will not consider it. 00:12:40.110 --> 00:12:42.120 But this is fairly stable. 00:12:42.120 --> 00:12:48.970 Now, this one, 2P, can go here because it 00:12:48.970 --> 00:12:52.210 can go from l equal 1 to l equal 0, 00:12:52.210 --> 00:12:54.715 and it can change m by some value. 00:12:57.530 --> 00:13:02.030 The 3S you would say, oh, it cannot go here, cannot go here, 00:13:02.030 --> 00:13:11.060 but it can go laterally in that direction because it can go 00:13:11.060 --> 00:13:14.030 change l from 1 to 0. 00:13:14.030 --> 00:13:19.450 So it's a change of l, and that is possible. 00:13:19.450 --> 00:13:23.060 The 3P can go here. 00:13:23.060 --> 00:13:26.990 And my diagram starting to get cluttered. 00:13:26.990 --> 00:13:29.915 And it could go here as well, all the way down. 00:13:35.720 --> 00:13:40.330 And the 3D can only go here. 00:13:40.330 --> 00:13:43.460 Cannot go up to here because that would be l equal 2. 00:13:43.460 --> 00:13:51.520 So that is your diagram of possible decays of hydrogen. 00:13:51.520 --> 00:13:55.110 And you're going to be computing a couple of those.