WEBVTT 00:00:00.000 --> 00:00:03.451 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:03.451 --> 00:00:15.923 align:middle line:90% 00:00:15.923 --> 00:00:17.090 align:middle line:90% DAVID KAISER: DA, thank you. 00:00:17.090 --> 00:00:17.990 align:middle line:90% It's an excellent point. 00:00:17.990 --> 00:00:18.990 align:middle line:90% You're absolutely right. 00:00:18.990 --> 00:00:22.340 align:middle line:84% So I think both statements stand. 00:00:22.340 --> 00:00:27.430 align:middle line:84% So Compton's experiments convinced 00:00:27.430 --> 00:00:29.530 align:middle line:90% even more of the community. 00:00:29.530 --> 00:00:31.330 align:middle line:84% To win the Nobel Prize then or now you 00:00:31.330 --> 00:00:36.040 align:middle line:84% have to convince three people in Stockholm, which is hard. 00:00:36.040 --> 00:00:39.760 align:middle line:84% To convince the larger international community, which 00:00:39.760 --> 00:00:44.290 align:middle line:84% even back then included several thousand people, 00:00:44.290 --> 00:00:45.820 align:middle line:84% that's a different-- that sometimes 00:00:45.820 --> 00:00:48.260 align:middle line:84% occurs on different timescales by different processes. 00:00:48.260 --> 00:00:50.920 align:middle line:84% So you're absolutely right, and I did not emphasize this. 00:00:50.920 --> 00:00:53.290 align:middle line:84% As many of you might know, and DA reminds us, 00:00:53.290 --> 00:00:55.450 align:middle line:84% Einstein did not win the Nobel Prize 00:00:55.450 --> 00:00:58.540 align:middle line:84% for any of the work on relativity 00:00:58.540 --> 00:01:02.020 align:middle line:84% even though he had achieved kind of renown, 00:01:02.020 --> 00:01:04.480 align:middle line:84% both within the profession and beyond, 00:01:04.480 --> 00:01:08.020 align:middle line:84% as we talked about briefly a few classes ago, 00:01:08.020 --> 00:01:10.000 align:middle line:84% because of his work on relativity, 00:01:10.000 --> 00:01:13.420 align:middle line:84% in particular, for the Eddington eclipse expedition 00:01:13.420 --> 00:01:17.560 align:middle line:84% stuff, which was released to the world in November of 1919. 00:01:17.560 --> 00:01:20.860 align:middle line:84% And yet, even after that, what he actually won the Prize for 00:01:20.860 --> 00:01:25.160 align:middle line:84% was indeed his heuristic suggestion about light quantum. 00:01:25.160 --> 00:01:27.830 align:middle line:84% And so I think the idea there, DA, to make sense of it-- 00:01:27.830 --> 00:01:28.250 align:middle line:90% and I agree with you. 00:01:28.250 --> 00:01:30.140 align:middle line:84% It is a little funny timing-- is that there 00:01:30.140 --> 00:01:35.930 align:middle line:84% was growing awareness his work was of value and of interest. 00:01:35.930 --> 00:01:38.840 align:middle line:84% And then Compton's work really convinced, so to speak, 00:01:38.840 --> 00:01:43.010 align:middle line:84% the remaining skeptics, of whom there were many. 00:01:43.010 --> 00:01:44.540 align:middle line:84% In fact, even Compton's work didn't 00:01:44.540 --> 00:01:46.220 align:middle line:84% convince every single remaining skeptic, 00:01:46.220 --> 00:01:48.560 align:middle line:84% but that did a lot of the work for the larger community 00:01:48.560 --> 00:01:49.220 align:middle line:90% beyond. 00:01:49.220 --> 00:01:50.762 align:middle line:84% And you can see that they were coming 00:01:50.762 --> 00:01:54.950 align:middle line:84% out kind of around the same time, but not literally one 00:01:54.950 --> 00:01:56.307 align:middle line:90% after the other. 00:01:56.307 --> 00:01:58.640 align:middle line:84% And I think a measure of that is how quickly Compton won 00:01:58.640 --> 00:02:00.500 align:middle line:90% his Nobel Prize for that work. 00:02:00.500 --> 00:02:05.660 align:middle line:84% So Einstein's prize was something like 16 years later, 00:02:05.660 --> 00:02:08.437 align:middle line:84% and Compton's was like 3 and 1/2 years later approximately, 00:02:08.437 --> 00:02:09.270 align:middle line:90% something like that. 00:02:09.270 --> 00:02:12.170 align:middle line:84% So in terms of the way that the community recognized 00:02:12.170 --> 00:02:13.880 align:middle line:84% the value of Compton's experiments, 00:02:13.880 --> 00:02:17.300 align:middle line:84% I think that does point to a broader kind of coalescing. 00:02:17.300 --> 00:02:19.430 align:middle line:84% But you're quite right that Einstein 00:02:19.430 --> 00:02:20.930 align:middle line:90% did receive the Nobel Prize. 00:02:20.930 --> 00:02:24.350 align:middle line:84% When he won it, he won it for the light quantum hypothesis. 00:02:24.350 --> 00:02:25.040 align:middle line:90% Excellent point. 00:02:25.040 --> 00:02:30.972 align:middle line:90% 00:02:30.972 --> 00:02:33.430 align:middle line:84% To be honest, I'd have to go back and check more carefully. 00:02:33.430 --> 00:02:38.560 align:middle line:84% My sense of that had always been Einstein's paper in 1905 that 00:02:38.560 --> 00:02:43.220 align:middle line:84% introduced the light quantum itself described three or four 00:02:43.220 --> 00:02:45.700 align:middle line:84% or five separate instances in which the light quantum 00:02:45.700 --> 00:02:47.110 align:middle line:90% effect-- 00:02:47.110 --> 00:02:49.610 align:middle line:84% light quantum hypothesis would help to make sense of things. 00:02:49.610 --> 00:02:52.390 align:middle line:84% So his paper on the light quantum 00:02:52.390 --> 00:02:54.340 align:middle line:84% covered several phenomena, including 00:02:54.340 --> 00:02:56.510 align:middle line:84% things like black-body radiation and other things 00:02:56.510 --> 00:02:59.800 align:middle line:84% and not only photoelectric phenomena 00:02:59.800 --> 00:03:00.730 align:middle line:90% like from [INAUDIBLE]. 00:03:00.730 --> 00:03:02.230 align:middle line:84% So I guess I had always thought they 00:03:02.230 --> 00:03:03.763 align:middle line:84% meant the light quantum which could 00:03:03.763 --> 00:03:05.680 align:middle line:84% be applied to things like photoelectric effect 00:03:05.680 --> 00:03:06.550 align:middle line:90% and other phenomena. 00:03:06.550 --> 00:03:07.870 align:middle line:84% But it's a great question you raise, Orson. 00:03:07.870 --> 00:03:09.160 align:middle line:90% To be honest, I'm not sure. 00:03:09.160 --> 00:03:09.790 align:middle line:90% OK. 00:03:09.790 --> 00:03:14.770 align:middle line:84% So today we're going to launch into-- we're 00:03:14.770 --> 00:03:19.570 align:middle line:84% going to move into what starts being known even in real time 00:03:19.570 --> 00:03:23.200 align:middle line:84% as quantum mechanics, not this or that portion of a kind 00:03:23.200 --> 00:03:24.398 align:middle line:90% of old quantum theory. 00:03:24.398 --> 00:03:25.690 align:middle line:90% And we're going to look today-- 00:03:25.690 --> 00:03:29.270 align:middle line:84% for kind of 2/3 of today's class, 00:03:29.270 --> 00:03:32.300 align:middle line:84% we'll be looking at work most closely associated 00:03:32.300 --> 00:03:33.470 align:middle line:90% with Werner Heisenberg. 00:03:33.470 --> 00:03:35.330 align:middle line:84% Again, a name you probably already knew. 00:03:35.330 --> 00:03:39.260 align:middle line:84% One of the readings for today was his very, very brief paper 00:03:39.260 --> 00:03:40.412 align:middle line:90% published in 1925. 00:03:40.412 --> 00:03:41.870 align:middle line:84% We're not going to start with that, 00:03:41.870 --> 00:03:43.860 align:middle line:90% we're going to warm up to it. 00:03:43.860 --> 00:03:47.360 align:middle line:84% So in fact, what we're going to do today is have one more look, 00:03:47.360 --> 00:03:49.220 align:middle line:84% a little more kind of detailed juicy 00:03:49.220 --> 00:03:53.270 align:middle line:84% look at the last gasp of what became 00:03:53.270 --> 00:03:55.130 align:middle line:90% known as old quantum theory. 00:03:55.130 --> 00:03:57.728 align:middle line:84% Not just because it's old, but I want 00:03:57.728 --> 00:03:59.270 align:middle line:84% to talk about it because it helps us, 00:03:59.270 --> 00:04:05.180 align:middle line:84% I think, understand really what was Heisenberg practicing. 00:04:05.180 --> 00:04:08.180 align:middle line:84% This is the work he himself was pursuing as a graduate student, 00:04:08.180 --> 00:04:11.390 align:middle line:84% working with some people we'll talk about for part one-- 00:04:11.390 --> 00:04:13.430 align:middle line:90% he and many of his peers. 00:04:13.430 --> 00:04:14.930 align:middle line:84% And also, in a sense, what finally 00:04:14.930 --> 00:04:17.670 align:middle line:84% made Heisenberg want to make a break with that. 00:04:17.670 --> 00:04:20.269 align:middle line:84% So I think it's worth sitting for this first part of today's 00:04:20.269 --> 00:04:24.470 align:middle line:84% class session on work that, again, was rapidly seen 00:04:24.470 --> 00:04:26.578 align:middle line:90% as part of the old not the new. 00:04:26.578 --> 00:04:29.120 align:middle line:84% Because I do think it helps us understand what was Heisenberg 00:04:29.120 --> 00:04:32.110 align:middle line:84% self-consciously trying to do when he did start 00:04:32.110 --> 00:04:35.830 align:middle line:84% composing this new work in the spring of 1925. 00:04:35.830 --> 00:04:36.970 align:middle line:90% So let's jump in with that. 00:04:36.970 --> 00:04:37.930 align:middle line:90% A quick reminder. 00:04:37.930 --> 00:04:40.570 align:middle line:84% I think this was likely familiar for many of you 00:04:40.570 --> 00:04:42.890 align:middle line:84% even before our previous class session. 00:04:42.890 --> 00:04:45.100 align:middle line:84% But nonetheless, we looked last time at things 00:04:45.100 --> 00:04:48.880 align:middle line:84% like the Bohr model of the atom first introduced in 1913. 00:04:48.880 --> 00:04:53.230 align:middle line:84% And Bohr had kind of finally in frustration-- 00:04:53.230 --> 00:04:57.040 align:middle line:84% he had first wanted to treat very complicated arbitrary 00:04:57.040 --> 00:04:58.750 align:middle line:90% electrons with many, many-- 00:04:58.750 --> 00:05:01.330 align:middle line:84% excuse me-- arbitrary atoms with many, many electrons. 00:05:01.330 --> 00:05:03.280 align:middle line:84% And in the end, finally settled to try 00:05:03.280 --> 00:05:05.590 align:middle line:84% to tackle just the simplest kind of atom-- 00:05:05.590 --> 00:05:07.730 align:middle line:84% hydrogen atoms with a single electron. 00:05:07.730 --> 00:05:11.170 align:middle line:84% And again, he had this kind of backwards/forwards, 00:05:11.170 --> 00:05:14.290 align:middle line:84% one step forward, one step back kind of approach 00:05:14.290 --> 00:05:17.500 align:middle line:84% to start with thoroughly familiar expressions 00:05:17.500 --> 00:05:22.060 align:middle line:84% from Newtonian's mechanics, from Coulomb's electrostatics-- 00:05:22.060 --> 00:05:24.100 align:middle line:84% really what we would now call classical physics. 00:05:24.100 --> 00:05:28.330 align:middle line:84% And then he kind of attached these new so-called quantum 00:05:28.330 --> 00:05:31.820 align:middle line:84% conditions and then try to see where that would lead him. 00:05:31.820 --> 00:05:35.240 align:middle line:84% And that led him, for example, to suggest that the motion 00:05:35.240 --> 00:05:36.860 align:middle line:84% of electrons in a hydrogen atom-- 00:05:36.860 --> 00:05:39.710 align:middle line:84% the motion of a single electron in a hydrogen atom-- 00:05:39.710 --> 00:05:43.590 align:middle line:84% might be restricted to very discrete states of motion. 00:05:43.590 --> 00:05:46.130 align:middle line:84% These radii or the average distances 00:05:46.130 --> 00:05:49.280 align:middle line:84% would not take any old value across a continuum, 00:05:49.280 --> 00:05:51.240 align:middle line:90% but would snap into place-- 00:05:51.240 --> 00:05:55.130 align:middle line:84% 1 squared, 2 squared, 3 squared and so on time some unit 00:05:55.130 --> 00:05:57.080 align:middle line:84% length we now call the Bohr radius. 00:05:57.080 --> 00:05:59.240 align:middle line:84% And when he did that, he was able then 00:05:59.240 --> 00:06:02.090 align:middle line:84% to have an explanation where really none had been 00:06:02.090 --> 00:06:04.850 align:middle line:84% in hand before for empirical phenomena 00:06:04.850 --> 00:06:07.860 align:middle line:84% like the so-called Balmer series, again, shown here. 00:06:07.860 --> 00:06:10.400 align:middle line:84% This is actual real data from a spectrograph-- not taken 00:06:10.400 --> 00:06:12.590 align:middle line:90% by me, taken from Google image. 00:06:12.590 --> 00:06:17.180 align:middle line:84% But someone performed an actual more modern investigation 00:06:17.180 --> 00:06:19.730 align:middle line:84% of the emission lines, the spectral lines that 00:06:19.730 --> 00:06:22.250 align:middle line:84% get emitted from a gas of hydrogen atoms 00:06:22.250 --> 00:06:23.990 align:middle line:90% when you add some extra energy. 00:06:23.990 --> 00:06:25.220 align:middle line:90% You excite the atoms. 00:06:25.220 --> 00:06:28.340 align:middle line:84% They relax by emitting not just light of any old color, 00:06:28.340 --> 00:06:31.047 align:middle line:84% but light of very particular colors, especially those 00:06:31.047 --> 00:06:33.380 align:middle line:84% to which our eyes are sensitive here in the visible part 00:06:33.380 --> 00:06:35.380 align:middle line:90% of the spectrum. 00:06:35.380 --> 00:06:37.870 align:middle line:84% Well, that was really very exciting 00:06:37.870 --> 00:06:39.370 align:middle line:84% and got a lot of people's attention. 00:06:39.370 --> 00:06:41.710 align:middle line:84% But, of course, the work wasn't over then. 00:06:41.710 --> 00:06:44.740 align:middle line:90% Even before Bohr's atom-- 00:06:44.740 --> 00:06:45.910 align:middle line:90% model of the atom-- 00:06:45.910 --> 00:06:48.880 align:middle line:84% it had been known by the same kinds of experts who were 00:06:48.880 --> 00:06:51.700 align:middle line:84% making very precise measurements of the exact color-- 00:06:51.700 --> 00:06:54.730 align:middle line:84% the real frequencies of this line and that line and that 00:06:54.730 --> 00:06:55.240 align:middle line:90% one-- 00:06:55.240 --> 00:06:58.180 align:middle line:84% that all kinds of things can happen to the pattern of light 00:06:58.180 --> 00:07:00.550 align:middle line:84% that comes out to the very specific values 00:07:00.550 --> 00:07:04.240 align:middle line:84% of the colors of light if the gas of atoms 00:07:04.240 --> 00:07:07.280 align:middle line:84% is placed in an external magnetic field. 00:07:07.280 --> 00:07:10.090 align:middle line:84% So what would appear as a single sharp line 00:07:10.090 --> 00:07:12.640 align:middle line:84% like this kind of bluish turquoise line, 00:07:12.640 --> 00:07:15.520 align:middle line:84% or this very sharp, single red line, 00:07:15.520 --> 00:07:17.650 align:middle line:84% when there's no external magnetic field, 00:07:17.650 --> 00:07:19.120 align:middle line:90% those would split. 00:07:19.120 --> 00:07:23.170 align:middle line:84% They would typically split into triplets of lines. 00:07:23.170 --> 00:07:26.890 align:middle line:84% So this is my simulated data using the Paint program 00:07:26.890 --> 00:07:27.760 align:middle line:90% in PowerPoint. 00:07:27.760 --> 00:07:30.580 align:middle line:84% Imagine we took this real measurement-- this line here. 00:07:30.580 --> 00:07:33.970 align:middle line:84% This is what it would look like in the absence 00:07:33.970 --> 00:07:35.840 align:middle line:90% of an external magnetic field. 00:07:35.840 --> 00:07:38.510 align:middle line:84% In the presence of an external magnetic field, 00:07:38.510 --> 00:07:41.930 align:middle line:84% that single line would split into three very slightly 00:07:41.930 --> 00:07:45.500 align:middle line:84% different colors, but very closely spaced. 00:07:45.500 --> 00:07:47.840 align:middle line:84% So again, if you look at a kind of spectrogram. 00:07:47.840 --> 00:07:49.350 align:middle line:84% They'd be very close to each other 00:07:49.350 --> 00:07:52.640 align:middle line:84% and just marginally different in the numerical value 00:07:52.640 --> 00:07:56.150 align:middle line:84% of their frequencies when the excited atoms were placed 00:07:56.150 --> 00:07:57.770 align:middle line:90% in an external magnetic field. 00:07:57.770 --> 00:07:59.480 align:middle line:84% Many researchers had noticed that. 00:07:59.480 --> 00:08:03.080 align:middle line:84% One of the people who was most careful and precise 00:08:03.080 --> 00:08:06.350 align:middle line:84% in quantifying this effect was a Dutch physicist 00:08:06.350 --> 00:08:09.590 align:middle line:84% named Pieter Zeeman, who became known as the Zeeman effect-- 00:08:09.590 --> 00:08:11.720 align:middle line:84% the splitting of single spectral lines 00:08:11.720 --> 00:08:14.000 align:middle line:84% into-- usually into triplets in the presence 00:08:14.000 --> 00:08:16.410 align:middle line:90% of an external magnetic field. 00:08:16.410 --> 00:08:20.570 align:middle line:84% So that set in motion some work by colleagues 00:08:20.570 --> 00:08:23.420 align:middle line:84% of people like Niels Bohr, who wanted 00:08:23.420 --> 00:08:26.570 align:middle line:84% to see could they make sense of this empirical information 00:08:26.570 --> 00:08:27.842 align:middle line:90% about spectral lines. 00:08:27.842 --> 00:08:29.300 align:middle line:84% One of the most significant in this 00:08:29.300 --> 00:08:32.840 align:middle line:84% was Arnold Sommerfeld, who was the senior professor, 00:08:32.840 --> 00:08:36.169 align:middle line:84% the single ordinarius professor of theoretical physics 00:08:36.169 --> 00:08:38.330 align:middle line:90% in Munich in Germany. 00:08:38.330 --> 00:08:40.730 align:middle line:84% And Sommerfeld is interesting for many, many reasons , 00:08:40.730 --> 00:08:45.290 align:middle line:84% among them he really took up the kind of mantle of Bohr's model, 00:08:45.290 --> 00:08:48.500 align:middle line:84% as we'll see for much of this first part of today's class, 00:08:48.500 --> 00:08:51.200 align:middle line:84% and he did some really, I think, very creative kind 00:08:51.200 --> 00:08:54.050 align:middle line:84% of conceptual moves we'll look at a little bit. 00:08:54.050 --> 00:08:57.200 align:middle line:84% He's also fascinating, I think, historically because he 00:08:57.200 --> 00:09:01.340 align:middle line:84% was a remarkably prolific and productive mentor 00:09:01.340 --> 00:09:02.930 align:middle line:90% of younger physicists. 00:09:02.930 --> 00:09:04.940 align:middle line:84% So coming out of the Sommerfeld school, 00:09:04.940 --> 00:09:08.060 align:middle line:84% his immediate advisees in Munich included 00:09:08.060 --> 00:09:10.550 align:middle line:84% many, many people who would go on to win the Nobel Prize, 00:09:10.550 --> 00:09:12.920 align:middle line:84% and otherwise, make a lasting impact 00:09:12.920 --> 00:09:14.930 align:middle line:90% on our understanding of matter-- 00:09:14.930 --> 00:09:18.050 align:middle line:84% people like Vera Heisenberg and Wolfgang Pauli and later 00:09:18.050 --> 00:09:19.230 align:middle line:90% Hans Bethe and others. 00:09:19.230 --> 00:09:22.820 align:middle line:84% So in Munich, Sommerfeld set up this really remarkable school 00:09:22.820 --> 00:09:25.940 align:middle line:84% to focus on what they began to call atomic structure-- 00:09:25.940 --> 00:09:28.260 align:middle line:90% atombau and spektralliniens. 00:09:28.260 --> 00:09:31.970 align:middle line:84% So what about these emissions of lines from excited atoms-- 00:09:31.970 --> 00:09:35.300 align:middle line:84% can they use that information in it's very fine-grained 00:09:35.300 --> 00:09:37.790 align:middle line:84% quantitative detail to try to learn more 00:09:37.790 --> 00:09:41.270 align:middle line:84% about the structure of, say, electrons motions in atoms? 00:09:41.270 --> 00:09:43.310 align:middle line:84% And they began, as we'll see, by trying 00:09:43.310 --> 00:09:46.670 align:middle line:84% to generalize Bohr's model to what they consider the more 00:09:46.670 --> 00:09:48.500 align:middle line:90% general state of motion-- 00:09:48.500 --> 00:09:49.767 align:middle line:90% elliptical orbits. 00:09:49.767 --> 00:09:51.350 align:middle line:84% And this was the first edition of what 00:09:51.350 --> 00:09:54.830 align:middle line:84% became a remarkably influential kind of research monograph-- 00:09:54.830 --> 00:09:58.010 align:middle line:84% not exactly a textbook-- but often used to train the younger 00:09:58.010 --> 00:09:59.510 align:middle line:90% generation. 00:09:59.510 --> 00:10:00.050 align:middle line:90% OK. 00:10:00.050 --> 00:10:03.200 align:middle line:84% So remember we saw last time, for simplicity 00:10:03.200 --> 00:10:07.310 align:middle line:84% when Bohr was putting together his first atomic model in 1913, 00:10:07.310 --> 00:10:10.700 align:middle line:84% he actually considered circular orbits for a single electron 00:10:10.700 --> 00:10:11.540 align:middle line:90% at a time. 00:10:11.540 --> 00:10:14.090 align:middle line:84% So he knew, as all of us would know, 00:10:14.090 --> 00:10:18.320 align:middle line:84% that in general, under a central force, whether we're 00:10:18.320 --> 00:10:20.060 align:middle line:84% talking about the motion of planets 00:10:20.060 --> 00:10:23.420 align:middle line:84% in the solar system or a lonesome electron 00:10:23.420 --> 00:10:26.630 align:middle line:84% around the positively charged nucleus of a hydrogen atom, 00:10:26.630 --> 00:10:29.960 align:middle line:84% the general state of motion will be ellipses. 00:10:29.960 --> 00:10:32.960 align:middle line:84% It will be not a circle necessarily. 00:10:32.960 --> 00:10:35.070 align:middle line:84% For simplicity, Bohr started with a circle. 00:10:35.070 --> 00:10:38.150 align:middle line:84% Well, Sommerfeld knew his celestial mechanics as well as 00:10:38.150 --> 00:10:38.720 align:middle line:90% anyone. 00:10:38.720 --> 00:10:41.240 align:middle line:84% He said the more general state of motion would be ellipses. 00:10:41.240 --> 00:10:43.100 align:middle line:84% So again, much like Bohr had done, 00:10:43.100 --> 00:10:45.140 align:middle line:84% Sommerfeld and his students go back 00:10:45.140 --> 00:10:48.650 align:middle line:84% to the perfectly familiar totally Newtonian 00:10:48.650 --> 00:10:51.260 align:middle line:84% kind of classical mechanical expression 00:10:51.260 --> 00:10:53.840 align:middle line:84% for the energy of a body of mass, m, 00:10:53.840 --> 00:10:57.590 align:middle line:84% in an elliptical orbit-- sorry-- an elliptical orbit, 00:10:57.590 --> 00:11:00.350 align:middle line:84% in this case, with a positively charged nucleus 00:11:00.350 --> 00:11:02.580 align:middle line:90% as one focus of the orbit. 00:11:02.580 --> 00:11:06.350 align:middle line:84% So whereas Bohr had neglected this middle term, 00:11:06.350 --> 00:11:07.850 align:middle line:84% we need to include that term when 00:11:07.850 --> 00:11:09.830 align:middle line:90% we describe elliptical motion. 00:11:09.830 --> 00:11:11.813 align:middle line:84% This is proportional to the angular momentum. 00:11:11.813 --> 00:11:14.480 align:middle line:84% I should say, by the way, one of my favorite books to make sense 00:11:14.480 --> 00:11:16.855 align:middle line:84% of this-- and I'm just going to give you a little taste-- 00:11:16.855 --> 00:11:18.497 align:middle line:84% is by a friend of mine, Suman Seth, 00:11:18.497 --> 00:11:20.330 align:middle line:84% a book came out a couple of years ago called 00:11:20.330 --> 00:11:23.810 align:middle line:84% Crafting the Quantum, and does a really deep dive 00:11:23.810 --> 00:11:25.352 align:middle line:84% into Sommerfeld and his whole school. 00:11:25.352 --> 00:11:26.685 align:middle line:90% I think it's really fascinating. 00:11:26.685 --> 00:11:28.830 align:middle line:84% Some of you might enjoy learning more about that. 00:11:28.830 --> 00:11:31.550 align:middle line:84% So I'm going to give you the short version of some 00:11:31.550 --> 00:11:34.660 align:middle line:84% of the things that we learned from Suman's work. 00:11:34.660 --> 00:11:35.160 align:middle line:90% OK. 00:11:35.160 --> 00:11:38.220 align:middle line:84% So whereas, a circular orbit has only one degree of freedom-- 00:11:38.220 --> 00:11:41.440 align:middle line:84% things can vary really only by changing the radius-- 00:11:41.440 --> 00:11:45.030 align:middle line:84% an elliptical orbit, in general, has two independent degrees 00:11:45.030 --> 00:11:45.558 align:middle line:90% of freedom. 00:11:45.558 --> 00:11:47.100 align:middle line:84% We could think of them as like radius 00:11:47.100 --> 00:11:48.420 align:middle line:90% and one of these angles-- 00:11:48.420 --> 00:11:50.220 align:middle line:90% a kind of azimuthal angle. 00:11:50.220 --> 00:11:54.150 align:middle line:84% So Sommerfeld proposed basically take Bohr's prescription 00:11:54.150 --> 00:11:57.960 align:middle line:84% and just do that for every independent degree of freedom. 00:11:57.960 --> 00:11:59.700 align:middle line:84% He's thinking very formally in terms 00:11:59.700 --> 00:12:01.410 align:middle line:90% of the classical mechanics. 00:12:01.410 --> 00:12:05.070 align:middle line:84% There are ways of characterizing all the ways that this body 00:12:05.070 --> 00:12:08.430 align:middle line:84% and motion can kind of wiggle-- all the independent states that 00:12:08.430 --> 00:12:12.240 align:middle line:84% would characterize its motion for each of those, subjected 00:12:12.240 --> 00:12:13.660 align:middle line:90% to this new quantum condition. 00:12:13.660 --> 00:12:16.060 align:middle line:84% So this is what had been Bohr's original one. 00:12:16.060 --> 00:12:19.500 align:middle line:84% This is where Bohr had written down mvr 00:12:19.500 --> 00:12:21.643 align:middle line:84% equals nh, which we saw last time. 00:12:21.643 --> 00:12:23.310 align:middle line:84% In Sommerfeld's hands, that just becomes 00:12:23.310 --> 00:12:26.760 align:middle line:84% one of potential whole series of quantum conditions. 00:12:26.760 --> 00:12:28.740 align:middle line:84% There is now an independent degree 00:12:28.740 --> 00:12:32.160 align:middle line:84% of freedom related to this angular motion, 00:12:32.160 --> 00:12:35.260 align:middle line:84% and it goes by the angular momentum capital 00:12:35.260 --> 00:12:38.020 align:middle line:90% L times this accumulated angle. 00:12:38.020 --> 00:12:40.420 align:middle line:84% What if that also could only take 00:12:40.420 --> 00:12:44.510 align:middle line:84% integer values set with a scale set by Planck's constant? 00:12:44.510 --> 00:12:46.900 align:middle line:84% So if you go through the same series of exercises 00:12:46.900 --> 00:12:50.530 align:middle line:84% as Bohr had done, but now for the more general expression, 00:12:50.530 --> 00:12:52.900 align:middle line:84% you arrive at a new expression for the energy, which 00:12:52.900 --> 00:12:55.300 align:middle line:84% depends on two of these integers, not only one. 00:12:55.300 --> 00:12:56.550 align:middle line:90% It's the same series of steps. 00:12:56.550 --> 00:12:59.050 align:middle line:84% It's just Sommerfeld's starting from the more general states 00:12:59.050 --> 00:12:59.840 align:middle line:90% of motion. 00:12:59.840 --> 00:13:02.620 align:middle line:84% And these integers have to obey certain kind of relationships 00:13:02.620 --> 00:13:04.190 align:middle line:90% to each other. 00:13:04.190 --> 00:13:06.440 align:middle line:84% Now, if one follows that through, 00:13:06.440 --> 00:13:08.660 align:middle line:84% Sommerfeld continues-- he's doing his work starting 00:13:08.660 --> 00:13:12.510 align:middle line:84% in 1915-16, very soon after Bohr's work, 00:13:12.510 --> 00:13:15.860 align:middle line:84% and finally brings it together in this monograph in 1919. 00:13:15.860 --> 00:13:20.780 align:middle line:84% If one introduces that second more kind of ellipse-specific 00:13:20.780 --> 00:13:24.830 align:middle line:84% quantity-- this is really telling us how much the orbital 00:13:24.830 --> 00:13:28.190 align:middle line:84% motion deviates from being merely a circle-- 00:13:28.190 --> 00:13:32.780 align:middle line:84% subject that to only take on integer units proportional 00:13:32.780 --> 00:13:36.410 align:middle line:84% to Planck's constant-- then the projection of that vector 00:13:36.410 --> 00:13:38.990 align:middle line:84% quantity along any other direction z 00:13:38.990 --> 00:13:40.460 align:middle line:90% will also be discrete. 00:13:40.460 --> 00:13:41.780 align:middle line:90% It will be quantized. 00:13:41.780 --> 00:13:45.290 align:middle line:84% So now you have another value that 00:13:45.290 --> 00:13:47.810 align:middle line:90% typically became labeled by M-- 00:13:47.810 --> 00:13:50.960 align:middle line:84% M sub l-- we still use that notation to this day-- 00:13:50.960 --> 00:13:55.250 align:middle line:84% that we have the original Bohr principal quantum number, 00:13:55.250 --> 00:14:00.170 align:middle line:84% the one associated with the momentum 00:14:00.170 --> 00:14:02.150 align:middle line:90% that's conjugate to the radius. 00:14:02.150 --> 00:14:04.260 align:middle line:84% We have now the second quantum number 00:14:04.260 --> 00:14:07.410 align:middle line:84% Sommerfeld introduces related to this kind of ellipse 00:14:07.410 --> 00:14:09.280 align:middle line:90% to the deviation from circles. 00:14:09.280 --> 00:14:12.120 align:middle line:84% And we have a third integer quantum number 00:14:12.120 --> 00:14:16.530 align:middle line:84% given by the projection of that kind of elliptical motion 00:14:16.530 --> 00:14:19.360 align:middle line:84% along any particular direction in space we might consider. 00:14:19.360 --> 00:14:21.280 align:middle line:90% We'll call that the z-axis. 00:14:21.280 --> 00:14:25.530 align:middle line:84% And so, for example, if you had one unit of deviation 00:14:25.530 --> 00:14:29.010 align:middle line:84% from circular motion, if this new quantum number associated 00:14:29.010 --> 00:14:32.070 align:middle line:84% with l Were equal to 1 instead of 0, 00:14:32.070 --> 00:14:35.070 align:middle line:84% then this projection could only take 00:14:35.070 --> 00:14:39.060 align:middle line:84% on one of three possible values-- either minus 1, 0, 00:14:39.060 --> 00:14:40.080 align:middle line:90% or plus 1. 00:14:40.080 --> 00:14:42.690 align:middle line:84% If you were kind of, so to speak, two units away 00:14:42.690 --> 00:14:45.000 align:middle line:84% from a pure circle, then this projection 00:14:45.000 --> 00:14:46.500 align:middle line:90% could take one of five values. 00:14:46.500 --> 00:14:49.590 align:middle line:84% The point was for any integer value n sub 00:14:49.590 --> 00:14:53.400 align:middle line:84% phi associated with that kind of ellipticity, 00:14:53.400 --> 00:14:56.495 align:middle line:84% there would be an odd number of these possible projections-- 00:14:56.495 --> 00:14:57.870 align:middle line:84% always an integer, and there will 00:14:57.870 --> 00:15:00.788 align:middle line:84% be an odd number of distinct values it could take. 00:15:00.788 --> 00:15:03.330 align:middle line:84% Some of this might be familiar to you from chemistry classes. 00:15:03.330 --> 00:15:06.210 align:middle line:84% We still use this in spectroscopy to this day. 00:15:06.210 --> 00:15:08.410 align:middle line:84% This is really where it starts to come from. 00:15:08.410 --> 00:15:10.290 align:middle line:84% So in place of only one quantum number, 00:15:10.290 --> 00:15:13.380 align:middle line:84% as in Bohr's very simple circular model, 00:15:13.380 --> 00:15:16.350 align:middle line:84% Sommerfeld and his students began considering three quantum 00:15:16.350 --> 00:15:16.920 align:middle line:90% numbers. 00:15:16.920 --> 00:15:19.110 align:middle line:84% The Bohr principle one, you might 00:15:19.110 --> 00:15:21.030 align:middle line:84% think about a kind of ellipticity one, 00:15:21.030 --> 00:15:23.220 align:middle line:84% and the projection, each of which 00:15:23.220 --> 00:15:24.760 align:middle line:84% could only take on integer values. 00:15:24.760 --> 00:15:27.900 align:middle line:84% So again, don't worry if that's brand new and seems confusing. 00:15:27.900 --> 00:15:29.910 align:middle line:84% I just want to focus briefly on what's 00:15:29.910 --> 00:15:32.200 align:middle line:90% the kind of form reasoning? 00:15:32.200 --> 00:15:35.400 align:middle line:84% What are the arguments that Sommerfeld's putting forward? 00:15:35.400 --> 00:15:37.050 align:middle line:84% Just like Bohr had been doing, he's 00:15:37.050 --> 00:15:39.690 align:middle line:84% going to start with these thoroughly familiar 00:15:39.690 --> 00:15:42.390 align:middle line:84% classical descriptions of the objects motions, 00:15:42.390 --> 00:15:44.550 align:middle line:84% familiar from things like celestial mechanics. 00:15:44.550 --> 00:15:46.110 align:middle line:84% He's not inventing that part new. 00:15:46.110 --> 00:15:47.580 align:middle line:90% He's taking it off the shelf. 00:15:47.580 --> 00:15:50.220 align:middle line:84% And then kind of midway through his calculation, 00:15:50.220 --> 00:15:54.090 align:middle line:84% he'll just snap on some new so-called quantum condition. 00:15:54.090 --> 00:15:55.430 align:middle line:90% He won't derive it. 00:15:55.430 --> 00:15:58.210 align:middle line:84% He won't really, so to speak, justify it or prove it. 00:15:58.210 --> 00:15:59.640 align:middle line:90% He's going to make-- 00:15:59.640 --> 00:16:01.470 align:middle line:84% he's going to do like Niels Bohr had done-- 00:16:01.470 --> 00:16:04.040 align:middle line:84% take classical descriptions of motion 00:16:04.040 --> 00:16:07.820 align:middle line:84% and then start adding on these kind of ad hoc constraints 00:16:07.820 --> 00:16:10.220 align:middle line:84% to force quantities that might have taken 00:16:10.220 --> 00:16:12.890 align:middle line:84% on any one of a continuum of values 00:16:12.890 --> 00:16:16.250 align:middle line:84% in the classical mechanics in the Newtonian case. 00:16:16.250 --> 00:16:18.890 align:middle line:84% Those will now snap into kind of integer values 00:16:18.890 --> 00:16:21.500 align:middle line:84% for a scale set by Planck's constant. 00:16:21.500 --> 00:16:23.060 align:middle line:90% Now, why would anyone do that? 00:16:23.060 --> 00:16:26.930 align:middle line:84% Whoever bought this crazy book, Atombau and Spektrallinien. 00:16:26.930 --> 00:16:29.210 align:middle line:84% Because Sommerfeld began giving-- 00:16:29.210 --> 00:16:31.130 align:middle line:84% showing some real payoffs for that. 00:16:31.130 --> 00:16:33.380 align:middle line:84% If one follows this seemingly, again, 00:16:33.380 --> 00:16:36.740 align:middle line:84% kind of complicated almost Baroque series of steps, 00:16:36.740 --> 00:16:39.320 align:middle line:84% then Sommerfeld could come back to the quantity 00:16:39.320 --> 00:16:42.410 align:middle line:84% he started with, the spectral lines, including things 00:16:42.410 --> 00:16:45.740 align:middle line:84% like the splitting of the Zeeman effect 00:16:45.740 --> 00:16:47.990 align:middle line:90% when these excited atoms-- 00:16:47.990 --> 00:16:51.720 align:middle line:84% gases were put into external magnetic fields. 00:16:51.720 --> 00:16:55.850 align:middle line:84% So again, he begins reasoning, begins with thoroughly familiar 00:16:55.850 --> 00:16:57.950 align:middle line:84% expressions from electromagnetism, 00:16:57.950 --> 00:17:01.160 align:middle line:84% even pre-Maxwell, going back to the days of Coulomb and Ampere 00:17:01.160 --> 00:17:01.790 align:middle line:90% and others. 00:17:01.790 --> 00:17:03.560 align:middle line:84% He goes to the early 19th century 00:17:03.560 --> 00:17:06.631 align:middle line:84% and then starts kind of bringing them into this new framework. 00:17:06.631 --> 00:17:09.089 align:middle line:84% So one of the first things that Sommerfeld and his students 00:17:09.089 --> 00:17:11.970 align:middle line:84% do with this new kind of generalized framework 00:17:11.970 --> 00:17:14.700 align:middle line:84% of the Bohr-Sommerfeld approach is 00:17:14.700 --> 00:17:17.490 align:middle line:84% to reason that any object with electric charge 00:17:17.490 --> 00:17:20.339 align:middle line:84% q-- any little body with some electric charge that's 00:17:20.339 --> 00:17:24.420 align:middle line:84% moving with some angular momentum in some orbital motion 00:17:24.420 --> 00:17:26.310 align:middle line:84% will have something called a magnetic moment. 00:17:26.310 --> 00:17:28.590 align:middle line:84% Again, that was something that the French scholars 00:17:28.590 --> 00:17:32.520 align:middle line:84% of electricity had worked out in the very early 1800s. 00:17:32.520 --> 00:17:35.730 align:middle line:84% This part was not new by the early 1900s. 00:17:35.730 --> 00:17:37.750 align:middle line:90% That's a classical effect. 00:17:37.750 --> 00:17:41.230 align:middle line:84% And likewise, had been worked out often 00:17:41.230 --> 00:17:44.380 align:middle line:84% in what we now call classical electrostatics is that 00:17:44.380 --> 00:17:49.120 align:middle line:84% in an external magnetic field, such a moving electric charge, 00:17:49.120 --> 00:17:51.670 align:middle line:84% the energy of that system will actually 00:17:51.670 --> 00:17:55.600 align:middle line:84% depend on the relative orientation between this vector 00:17:55.600 --> 00:17:57.370 align:middle line:84% quantity, the magnetic moment, which 00:17:57.370 --> 00:18:00.310 align:middle line:84% is proportional to this vector related to just angular 00:18:00.310 --> 00:18:02.890 align:middle line:84% momentum, the orientation between that vector 00:18:02.890 --> 00:18:04.870 align:middle line:90% and the vector-- 00:18:04.870 --> 00:18:07.720 align:middle line:84% excuse me-- the orientation of this vector field-- 00:18:07.720 --> 00:18:09.500 align:middle line:90% the external magnetic field. 00:18:09.500 --> 00:18:11.080 align:middle line:84% So in fact, the energy of a system 00:18:11.080 --> 00:18:13.810 align:middle line:84% will be lowered when the magnetic moment 00:18:13.810 --> 00:18:17.130 align:middle line:84% lines up perfectly in parallel with the external magnetic 00:18:17.130 --> 00:18:17.630 align:middle line:90% field. 00:18:17.630 --> 00:18:20.770 align:middle line:84% Then this quantity together would be maximally positive. 00:18:20.770 --> 00:18:22.510 align:middle line:84% There's an overall minus sign in front. 00:18:22.510 --> 00:18:24.700 align:middle line:84% You would reduce the energy of the system 00:18:24.700 --> 00:18:25.930 align:middle line:90% by the largest amount. 00:18:25.930 --> 00:18:30.040 align:middle line:84% The system will be most stable when the quantity called 00:18:30.040 --> 00:18:33.040 align:middle line:84% the magnetic moment lines up perfectly in line in parallel 00:18:33.040 --> 00:18:35.990 align:middle line:84% with the applied external magnetic field. 00:18:35.990 --> 00:18:38.150 align:middle line:84% Well, now Sommerfeld says, if that's 00:18:38.150 --> 00:18:40.730 align:middle line:84% true for any kind of objects even 00:18:40.730 --> 00:18:44.180 align:middle line:84% in the macroscopic or classical world, what if we 00:18:44.180 --> 00:18:48.890 align:middle line:84% bring those ideas now and mix them together 00:18:48.890 --> 00:18:53.330 align:middle line:84% with this Bohr-Sommerfeld notion of quantized states of motion 00:18:53.330 --> 00:18:55.950 align:middle line:84% for an electron in a hydrogen atom. 00:18:55.950 --> 00:19:00.300 align:middle line:84% So one now had quantized this state of motion, the angular 00:19:00.300 --> 00:19:03.080 align:middle line:84% momentum akin to the kind of ellipticity-- 00:19:03.080 --> 00:19:05.990 align:middle line:84% So if this thing can only take certain integer values 00:19:05.990 --> 00:19:07.800 align:middle line:84% proportional to Planck's constant, 00:19:07.800 --> 00:19:09.650 align:middle line:84% then this thing, the magnetic moment, 00:19:09.650 --> 00:19:11.900 align:middle line:84% will likewise now be quantized when 00:19:11.900 --> 00:19:14.760 align:middle line:84% we think about atomic systems or parts of atoms. 00:19:14.760 --> 00:19:16.400 align:middle line:90% So now, let's do the same trick. 00:19:16.400 --> 00:19:20.780 align:middle line:84% The energy levels of an electron as it whips around its nucleus, 00:19:20.780 --> 00:19:22.850 align:middle line:84% if that entire system is now placed 00:19:22.850 --> 00:19:26.240 align:middle line:84% in some external magnetic field, the energy associated 00:19:26.240 --> 00:19:29.240 align:middle line:84% with that electron's motion will now be split. 00:19:29.240 --> 00:19:32.570 align:middle line:84% In fact, it will be split into exactly M sub l 00:19:32.570 --> 00:19:33.980 align:middle line:90% distinct levels. 00:19:33.980 --> 00:19:36.920 align:middle line:84% Remember, the projection of this thing along some orientation 00:19:36.920 --> 00:19:39.950 align:middle line:84% space will also only take some integer values 00:19:39.950 --> 00:19:42.140 align:middle line:84% given by that third quantum number 00:19:42.140 --> 00:19:43.890 align:middle line:90% that Sommerfeld had introduced. 00:19:43.890 --> 00:19:46.880 align:middle line:84% So now, these Zeeman triplets, the splitting 00:19:46.880 --> 00:19:50.570 align:middle line:84% of what had otherwise been a single frequency of emission 00:19:50.570 --> 00:19:53.600 align:middle line:84% into three closely spaced lines, Sommerfeld 00:19:53.600 --> 00:19:57.120 align:middle line:84% and his student said, here's what must be happening. 00:19:57.120 --> 00:20:00.870 align:middle line:84% It must be that there was an electron in an elliptical state 00:20:00.870 --> 00:20:02.880 align:middle line:84% of motion-- we'll say one unit of this kind 00:20:02.880 --> 00:20:06.420 align:middle line:84% of elliptical motion, one quantum unit associated 00:20:06.420 --> 00:20:08.610 align:middle line:90% with the quantity l. 00:20:08.610 --> 00:20:11.460 align:middle line:84% Therefore, the projection along any direction in space 00:20:11.460 --> 00:20:16.110 align:middle line:84% could either be exactly anti-parallel 00:20:16.110 --> 00:20:20.273 align:middle line:84% to it, perpendicular, or exactly parallel with it. 00:20:20.273 --> 00:20:21.690 align:middle line:84% There's only three options when we 00:20:21.690 --> 00:20:24.420 align:middle line:84% project this discrete quantity onto a given 00:20:24.420 --> 00:20:25.813 align:middle line:90% direction in space. 00:20:25.813 --> 00:20:27.480 align:middle line:84% And so what must be happening is there's 00:20:27.480 --> 00:20:31.080 align:middle line:84% an electron with one unit, and therefore, three projections, 00:20:31.080 --> 00:20:34.020 align:middle line:84% that's dropping down to a state where 00:20:34.020 --> 00:20:36.360 align:middle line:84% this ellipticity vanishes and drops down 00:20:36.360 --> 00:20:38.880 align:middle line:90% to a circular orbit again. 00:20:38.880 --> 00:20:41.400 align:middle line:84% And so you had three different energies 00:20:41.400 --> 00:20:44.520 align:middle line:84% for that starting point before the electron jumped down 00:20:44.520 --> 00:20:45.870 align:middle line:90% to the lower energy state. 00:20:45.870 --> 00:20:48.750 align:middle line:84% That would correspond to three slightly different amounts 00:20:48.750 --> 00:20:51.900 align:middle line:84% of energy released in the form of the radiation. 00:20:51.900 --> 00:20:54.420 align:middle line:84% The spectral lines and therefore their colors, 00:20:54.420 --> 00:20:56.730 align:middle line:84% their frequencies, would be slightly 00:20:56.730 --> 00:20:58.980 align:middle line:84% different from each other because you 00:20:58.980 --> 00:21:03.150 align:middle line:84% had an electron in a more general state of motion 00:21:03.150 --> 00:21:04.980 align:middle line:90% than Bohr had considered. 00:21:04.980 --> 00:21:05.910 align:middle line:90% That's the idea. 00:21:05.910 --> 00:21:07.868 align:middle line:84% That's why people start paying attention to any 00:21:07.868 --> 00:21:10.290 align:middle line:90% of this kind of complexity. 00:21:10.290 --> 00:21:13.980 align:middle line:84% Sommerfeld starts, again, scoring new wins 00:21:13.980 --> 00:21:17.010 align:middle line:84% empirically by trying to do the same kind of trick 00:21:17.010 --> 00:21:18.090 align:middle line:90% that Bohr had done-- 00:21:18.090 --> 00:21:19.950 align:middle line:84% thoroughly classical expressions, 00:21:19.950 --> 00:21:22.740 align:middle line:84% kind of snap on some new quantum conditions, 00:21:22.740 --> 00:21:25.410 align:middle line:84% and then reason about some empirical regularities. 00:21:25.410 --> 00:21:28.640 align:middle line:90% 00:21:28.640 --> 00:21:33.170 align:middle line:84% So now, this approach addressed what had already by this point 00:21:33.170 --> 00:21:35.270 align:middle line:84% become known as the ordinary Zeeman 00:21:35.270 --> 00:21:38.180 align:middle line:84% effect, the ordinary splitting, which was into triplets. 00:21:38.180 --> 00:21:43.010 align:middle line:84% There was also by this time, by the later 19-teens evidence 00:21:43.010 --> 00:21:46.050 align:middle line:84% of what became known as the anomalous Zeeman effect. 00:21:46.050 --> 00:21:48.230 align:middle line:84% So there was the ordinary and anomalous. 00:21:48.230 --> 00:21:51.110 align:middle line:84% For the anomalous Zeeman effect, it was not a splitting 00:21:51.110 --> 00:21:54.080 align:middle line:84% into triplets, it was a splitting into doublets. 00:21:54.080 --> 00:21:56.420 align:middle line:84% So again, with my simulated data, 00:21:56.420 --> 00:21:59.570 align:middle line:84% take that, say, that single purple line of the Balmer 00:21:59.570 --> 00:22:03.890 align:middle line:84% spectrum, in an external field sometimes these single lines 00:22:03.890 --> 00:22:06.890 align:middle line:84% will actually split not into three closely spaced lines, 00:22:06.890 --> 00:22:09.950 align:middle line:90% but is splitting into only two. 00:22:09.950 --> 00:22:12.140 align:middle line:84% And this now could not be explained 00:22:12.140 --> 00:22:15.170 align:middle line:84% by Sommerfeld's previous approach because remember, 00:22:15.170 --> 00:22:16.040 align:middle line:90% as you may-- 00:22:16.040 --> 00:22:16.940 align:middle line:90% I went by it quickly. 00:22:16.940 --> 00:22:18.320 align:middle line:90% You can check on the slides. 00:22:18.320 --> 00:22:20.420 align:middle line:84% Sommerfeld's early argument would only 00:22:20.420 --> 00:22:23.120 align:middle line:84% work for an odd number of splittings 00:22:23.120 --> 00:22:26.390 align:middle line:84% because the projections always went by 2n plus 1. 00:22:26.390 --> 00:22:29.400 align:middle line:84% There would always be an odd number of splittings. 00:22:29.400 --> 00:22:31.860 align:middle line:84% And therefore, an odd number of slightly different energy 00:22:31.860 --> 00:22:32.830 align:middle line:90% levels. 00:22:32.830 --> 00:22:34.890 align:middle line:84% So you could account for triplets or even 00:22:34.890 --> 00:22:36.810 align:middle line:90% five part splittings. 00:22:36.810 --> 00:22:39.060 align:middle line:84% You could never use that argument, 00:22:39.060 --> 00:22:41.910 align:middle line:84% as they recognized, to account for doublets. 00:22:41.910 --> 00:22:45.300 align:middle line:84% How do you get an even number of splittings-- split lines 00:22:45.300 --> 00:22:46.110 align:middle line:90% from that argument? 00:22:46.110 --> 00:22:47.220 align:middle line:90% You can't. 00:22:47.220 --> 00:22:50.490 align:middle line:84% So that became like the next big challenge for Sommerfeld 00:22:50.490 --> 00:22:53.520 align:middle line:84% and his most ambitious young students, one of whom 00:22:53.520 --> 00:22:56.760 align:middle line:84% was Wolfgang Pauli, a contemporary and a very close 00:22:56.760 --> 00:22:58.440 align:middle line:90% friend of Werner Heisenberg. 00:22:58.440 --> 00:23:01.800 align:middle line:84% They were students together in Sommerfeld's school in Munich. 00:23:01.800 --> 00:23:04.290 align:middle line:90% So Pauli finished his PhD. 00:23:04.290 --> 00:23:07.560 align:middle line:84% He took up a postdoc with Niels Bohr in Copenhagen. 00:23:07.560 --> 00:23:10.290 align:middle line:84% And this challenge was very much on his mind and many people's 00:23:10.290 --> 00:23:10.990 align:middle line:90% minds. 00:23:10.990 --> 00:23:14.340 align:middle line:84% And I love this story that Pauli recalled many years later. 00:23:14.340 --> 00:23:15.550 align:middle line:90% I'll just quote it for you. 00:23:15.550 --> 00:23:19.050 align:middle line:84% He says, a colleague who met me strolling rather aimlessly 00:23:19.050 --> 00:23:21.780 align:middle line:84% in the beautiful streets of Copenhagen, as shown here, 00:23:21.780 --> 00:23:24.750 align:middle line:84% said to me in a friendly manner, you look very unhappy. 00:23:24.750 --> 00:23:26.460 align:middle line:84% Whereupon I answered fiercely, how 00:23:26.460 --> 00:23:28.170 align:middle line:84% can one look happy when he's thinking 00:23:28.170 --> 00:23:29.940 align:middle line:84% about the anomalous Zeeman effect? 00:23:29.940 --> 00:23:31.830 align:middle line:84% With all the hard things on our minds, 00:23:31.830 --> 00:23:34.453 align:middle line:84% I'm sure we can recognize his dire frustration. 00:23:34.453 --> 00:23:36.120 align:middle line:84% Maybe we have worse things on our minds. 00:23:36.120 --> 00:23:38.980 align:middle line:84% The point is this really was just terribly frustrating, 00:23:38.980 --> 00:23:40.800 align:middle line:84% and many people were really concerned 00:23:40.800 --> 00:23:42.540 align:middle line:84% about the anomalous Zeeman effect. 00:23:42.540 --> 00:23:45.210 align:middle line:90% How could you explain doublets? 00:23:45.210 --> 00:23:47.430 align:middle line:84% As I mentioned, it was not only Sommerfeld 00:23:47.430 --> 00:23:49.830 align:middle line:84% and his very bright students like Pauli 00:23:49.830 --> 00:23:50.880 align:middle line:90% who worried about that. 00:23:50.880 --> 00:23:53.670 align:middle line:84% Many folks certainly across Central Europe 00:23:53.670 --> 00:23:55.150 align:middle line:84% who were in touch with each other, 00:23:55.150 --> 00:23:56.640 align:middle line:90% this was on many of their minds. 00:23:56.640 --> 00:23:58.650 align:middle line:84% So another set of very young students, 00:23:58.650 --> 00:24:01.050 align:middle line:84% still PhD students at the time, were also 00:24:01.050 --> 00:24:03.240 align:middle line:84% thinking hard about these anomalous effects, 00:24:03.240 --> 00:24:07.710 align:middle line:84% these doublet splittings for excited gases of atoms 00:24:07.710 --> 00:24:09.580 align:middle line:90% in an external magnetic field. 00:24:09.580 --> 00:24:12.330 align:middle line:84% So these now included students working very closely 00:24:12.330 --> 00:24:13.320 align:middle line:90% with Hendrik Lorentz. 00:24:13.320 --> 00:24:14.820 align:middle line:90% We know some of Lorentz's work. 00:24:14.820 --> 00:24:17.940 align:middle line:84% He was by now a very, very senior professor in Leiden 00:24:17.940 --> 00:24:21.000 align:middle line:84% in the Netherlands, and two young PhD students, George 00:24:21.000 --> 00:24:23.550 align:middle line:90% Uhlenbeck and Sam Goudsmit. 00:24:23.550 --> 00:24:25.650 align:middle line:84% They were working also to try to figure out 00:24:25.650 --> 00:24:27.560 align:middle line:90% this strange kind of anomaly. 00:24:27.560 --> 00:24:30.680 align:middle line:84% So they began reasoning kind of similarly 00:24:30.680 --> 00:24:33.338 align:middle line:84% to how Sommerfeld had been doing. 00:24:33.338 --> 00:24:34.130 align:middle line:90% They knew his book. 00:24:34.130 --> 00:24:35.963 align:middle line:84% His monograph had been out for years by now. 00:24:35.963 --> 00:24:38.240 align:middle line:84% This looked like the path forward. 00:24:38.240 --> 00:24:42.140 align:middle line:84% They realized that in analogy with the motion of planets 00:24:42.140 --> 00:24:46.760 align:middle line:84% around the sun, the Earth has two different kinds of angular 00:24:46.760 --> 00:24:47.450 align:middle line:90% momentum. 00:24:47.450 --> 00:24:50.660 align:middle line:84% It moves around the sun in its yearly orbit. 00:24:50.660 --> 00:24:53.810 align:middle line:84% That would be like the elliptical motion 00:24:53.810 --> 00:24:56.420 align:middle line:84% that Sommerfeld was building into his new kind 00:24:56.420 --> 00:24:57.830 align:middle line:90% of complicated models. 00:24:57.830 --> 00:25:00.260 align:middle line:84% But the Earth also spins on its own axis, 00:25:00.260 --> 00:25:01.490 align:middle line:90% as of course, we all know. 00:25:01.490 --> 00:25:03.230 align:middle line:84% That's what turns the day into night. 00:25:03.230 --> 00:25:05.120 align:middle line:84% So you have the orbital motion that 00:25:05.120 --> 00:25:06.620 align:middle line:84% accounts for the changing seasons 00:25:06.620 --> 00:25:09.650 align:middle line:84% over the course of the year in the analogy for the Earth. 00:25:09.650 --> 00:25:12.200 align:middle line:84% There's also the different spin, a different kind 00:25:12.200 --> 00:25:15.080 align:middle line:84% of angular momentum, of the Earth spinning on its own axis 00:25:15.080 --> 00:25:16.440 align:middle line:90% to turn day to night. 00:25:16.440 --> 00:25:18.470 align:middle line:84% So they began-- the younger students, 00:25:18.470 --> 00:25:21.080 align:middle line:84% Uhlenbeck and Goudsmit, realized-- 00:25:21.080 --> 00:25:23.930 align:middle line:84% reasoned that what if the electron also 00:25:23.930 --> 00:25:27.480 align:middle line:84% had some second kind of ingredient of angular momentum? 00:25:27.480 --> 00:25:31.650 align:middle line:84% What if it had some intrinsic spin along its own kind of axis 00:25:31.650 --> 00:25:35.280 align:middle line:84% separate from this kind of elliptical motion 00:25:35.280 --> 00:25:36.930 align:middle line:90% that Sommerfeld had focused on? 00:25:36.930 --> 00:25:39.390 align:middle line:84% And then they do the same trick that everyone was doing. 00:25:39.390 --> 00:25:44.910 align:middle line:84% If that second kind of intrinsic spin or angular momentum 00:25:44.910 --> 00:25:48.825 align:middle line:84% is associated with an electron, and if we snap the value 00:25:48.825 --> 00:25:51.900 align:middle line:84% that that spin could take into discrete values 00:25:51.900 --> 00:25:54.190 align:middle line:84% with a scale set by Planck's constant, 00:25:54.190 --> 00:25:56.970 align:middle line:84% then the electron would have an additional magnetic moment. 00:25:56.970 --> 00:25:58.710 align:middle line:84% In an external magnetic field, it 00:25:58.710 --> 00:26:01.410 align:middle line:84% would have a new vector quantity proportional 00:26:01.410 --> 00:26:04.830 align:middle line:84% to this new spin vector with a magnitude fixed 00:26:04.830 --> 00:26:06.580 align:middle line:90% by Planck's constant. 00:26:06.580 --> 00:26:09.930 align:middle line:84% So now they go back and, again, follow Sommerfeld's reasoning 00:26:09.930 --> 00:26:11.370 align:middle line:90% very, very closely. 00:26:11.370 --> 00:26:14.940 align:middle line:84% Place that entire assembly in an external magnetic field, 00:26:14.940 --> 00:26:18.930 align:middle line:84% there will again even just classically, one would expect, 00:26:18.930 --> 00:26:22.890 align:middle line:84% a shift to the energy of the entire system when 00:26:22.890 --> 00:26:27.510 align:middle line:84% the spinning charged object is placed in an external field. 00:26:27.510 --> 00:26:31.740 align:middle line:84% The energy associated with the magnetic field on the system 00:26:31.740 --> 00:26:35.490 align:middle line:84% will be minimized when the magnetic moment of the spin 00:26:35.490 --> 00:26:37.680 align:middle line:84% vector lines up perfectly in parallel 00:26:37.680 --> 00:26:39.160 align:middle line:90% with the external field. 00:26:39.160 --> 00:26:43.860 align:middle line:84% And now, Uhlenbeck and Goudsmit reasoned, that if this spin, 00:26:43.860 --> 00:26:47.490 align:middle line:84% if the intrinsic angular momentum could only ever line 00:26:47.490 --> 00:26:51.570 align:middle line:84% up exactly parallel or exactly anti-parallel, opposite 00:26:51.570 --> 00:26:54.270 align:middle line:84% direction, to that external field, 00:26:54.270 --> 00:26:55.950 align:middle line:90% then you would expect doublets. 00:26:55.950 --> 00:26:58.380 align:middle line:84% The same kind of argument that Sommerfeld and his students 00:26:58.380 --> 00:27:01.920 align:middle line:84% had made could now be applied to a two-valued offset 00:27:01.920 --> 00:27:05.490 align:middle line:84% in the energy instead of a three or an odd numbered-offset. 00:27:05.490 --> 00:27:09.030 align:middle line:84% And, in fact, they showed very compellingly 00:27:09.030 --> 00:27:13.170 align:middle line:84% that if you really do set the value of this magnitude 00:27:13.170 --> 00:27:18.360 align:middle line:84% of that intrinsic spin to be Planck's constant h divided 00:27:18.360 --> 00:27:22.515 align:middle line:84% by 4 pi, or as we saw last time, h bar, the common abbreviation, 00:27:22.515 --> 00:27:25.830 align:middle line:84% h divided by 2 pi itself divided by 2, 00:27:25.830 --> 00:27:28.420 align:middle line:84% then you would actually quantitatively match 00:27:28.420 --> 00:27:31.030 align:middle line:84% the actual values of the splittings 00:27:31.030 --> 00:27:32.550 align:middle line:90% of these frequencies. 00:27:32.550 --> 00:27:34.540 align:middle line:84% They didn't call this spin, interestingly. 00:27:34.540 --> 00:27:37.147 align:middle line:84% They call this space quantization, 00:27:37.147 --> 00:27:39.730 align:middle line:84% which tells you a bit of where their thinking was coming from. 00:27:39.730 --> 00:27:42.490 align:middle line:84% The idea was that this spin vector, this new vector 00:27:42.490 --> 00:27:46.180 align:middle line:84% S, this axis around which the electron was imagined 00:27:46.180 --> 00:27:50.260 align:middle line:84% to revolve, could only point in discrete directions 00:27:50.260 --> 00:27:52.960 align:middle line:84% in space, that it would have to snap into position 00:27:52.960 --> 00:27:56.560 align:middle line:84% either exactly parallel or exactly antiparallel 00:27:56.560 --> 00:27:57.950 align:middle line:90% to this external field. 00:27:57.950 --> 00:28:02.650 align:middle line:84% So the orientations in space along which that vector could 00:28:02.650 --> 00:28:04.090 align:middle line:90% point were quantized. 00:28:04.090 --> 00:28:06.610 align:middle line:84% They called the space quantization. 00:28:06.610 --> 00:28:09.550 align:middle line:84% And as I just mentioned, by fixing the magnitude they 00:28:09.550 --> 00:28:12.700 align:middle line:84% actually, again, could really get a remarkably close match 00:28:12.700 --> 00:28:16.330 align:middle line:84% to these very finely carefully measured optical results 00:28:16.330 --> 00:28:18.580 align:middle line:90% from the latest spectroscopy. 00:28:18.580 --> 00:28:20.930 align:middle line:90% But it wasn't so clear cut. 00:28:20.930 --> 00:28:23.650 align:middle line:84% And again, they recognized this themselves, to their credit. 00:28:23.650 --> 00:28:27.610 align:middle line:84% With that magnitude, by putting in this much, this amount 00:28:27.610 --> 00:28:30.940 align:middle line:84% of the kind of intrinsic spin, intrinsic angular momentum 00:28:30.940 --> 00:28:34.390 align:middle line:84% for the electron its own imagined little axis, 00:28:34.390 --> 00:28:38.470 align:middle line:84% then it started to sound pretty outlandish. 00:28:38.470 --> 00:28:40.390 align:middle line:84% A point on the electrons equator-- 00:28:40.390 --> 00:28:43.000 align:middle line:84% and they had ideas coming actually from Lorentz 00:28:43.000 --> 00:28:46.592 align:middle line:84% himself, their main advisor, about the kind typical size 00:28:46.592 --> 00:28:48.550 align:middle line:84% they would expect the electron to be-- they had 00:28:48.550 --> 00:28:51.130 align:middle line:84% a sense of a radius based on its charge distribution 00:28:51.130 --> 00:28:52.660 align:middle line:84% and all that, that if you actually-- 00:28:52.660 --> 00:28:55.030 align:middle line:84% if that little ball of a given size 00:28:55.030 --> 00:28:58.090 align:middle line:84% was spinning that quickly around its own axis, 00:28:58.090 --> 00:29:00.430 align:middle line:84% then a point on the equator would actually be spinning 00:29:00.430 --> 00:29:02.410 align:middle line:90% faster than the speed of light. 00:29:02.410 --> 00:29:06.100 align:middle line:84% By 1922 and '23 and '24, that seemed like a no-no. 00:29:06.100 --> 00:29:09.310 align:middle line:84% By that point, relativity was very widely accepted. 00:29:09.310 --> 00:29:11.560 align:middle line:84% Moreover, if you have something spinning that quickly, 00:29:11.560 --> 00:29:14.060 align:middle line:84% it looked like the mass of the whole electron would diverge. 00:29:14.060 --> 00:29:18.280 align:middle line:84% So they get some remarkable kind of clarity 00:29:18.280 --> 00:29:20.860 align:middle line:84% in terms of this very frustrating doublet 00:29:20.860 --> 00:29:22.840 align:middle line:90% of the anomalous Zeeman effect. 00:29:22.840 --> 00:29:25.480 align:middle line:84% But it starts to get harder and harder 00:29:25.480 --> 00:29:27.790 align:middle line:84% to make it all fit together in a single kind 00:29:27.790 --> 00:29:30.530 align:middle line:90% of visualizable model. 00:29:30.530 --> 00:29:34.450 align:middle line:84% And so by 1924 when these young students were pursuing that, 00:29:34.450 --> 00:29:38.380 align:middle line:84% it actually became kind of impossible at the same time 00:29:38.380 --> 00:29:40.900 align:middle line:84% to assume this is really a genuine angular 00:29:40.900 --> 00:29:46.000 align:middle line:84% momentum along some axis of rotation of a physically 00:29:46.000 --> 00:29:51.070 align:middle line:84% spinning tiny little object, and match both the empirical values 00:29:51.070 --> 00:29:53.020 align:middle line:84% they were aiming for, and be consistent 00:29:53.020 --> 00:29:56.770 align:middle line:84% with things like kinematics, like the motion 00:29:56.770 --> 00:29:58.300 align:middle line:90% of an object with mass. 00:29:58.300 --> 00:30:01.030 align:middle line:84% It became impossible to visualize. 00:30:01.030 --> 00:30:03.340 align:middle line:84% So they themselves worked that out. 00:30:03.340 --> 00:30:05.950 align:middle line:84% They talked about it with their main PhD advisor, 00:30:05.950 --> 00:30:07.517 align:middle line:90% a very senior Hendrik Lorentz. 00:30:07.517 --> 00:30:09.100 align:middle line:84% Lorentz said, actually, you know what? 00:30:09.100 --> 00:30:10.480 align:middle line:90% Let's not publish that yet. 00:30:10.480 --> 00:30:12.520 align:middle line:84% Don't publish that because it did 00:30:12.520 --> 00:30:14.410 align:middle line:84% seem to have such absurd conclusions 00:30:14.410 --> 00:30:17.780 align:middle line:84% or such unvisualizable conclusions. 00:30:17.780 --> 00:30:21.430 align:middle line:84% They had another advisor, much younger and much more brash, 00:30:21.430 --> 00:30:24.310 align:middle line:84% named Paul Ehrenfest also in Leiden. 00:30:24.310 --> 00:30:27.550 align:middle line:84% Behind their back, Ehrenfest actually submitted their paper 00:30:27.550 --> 00:30:30.160 align:middle line:84% to the journal without telling them, which is remarkable. 00:30:30.160 --> 00:30:32.022 align:middle line:84% And he supposedly told them, you're 00:30:32.022 --> 00:30:33.730 align:middle line:84% still young enough to afford a stupidity. 00:30:33.730 --> 00:30:36.040 align:middle line:84% And I have my little joke here that that's 00:30:36.040 --> 00:30:37.022 align:middle line:90% completely unethical. 00:30:37.022 --> 00:30:38.230 align:middle line:90% We would never do that today. 00:30:38.230 --> 00:30:40.760 align:middle line:84% Today, the advisor would steal their student's own work, 00:30:40.760 --> 00:30:42.760 align:middle line:84% put the advisor's name on it, and then submit it 00:30:42.760 --> 00:30:44.020 align:middle line:90% to a journal behind their back. 00:30:44.020 --> 00:30:44.560 align:middle line:90% That's a joke. 00:30:44.560 --> 00:30:45.393 align:middle line:90% We wouldn't do that. 00:30:45.393 --> 00:30:47.410 align:middle line:84% I just find this whole scenario quite amazing, 00:30:47.410 --> 00:30:48.640 align:middle line:84% that Ehrenfest was like, oh, yeah. 00:30:48.640 --> 00:30:49.598 align:middle line:90% I already sent that in. 00:30:49.598 --> 00:30:50.890 align:middle line:90% And they were like no, no. 00:30:50.890 --> 00:30:52.700 align:middle line:90% Lorentz told us not to. 00:30:52.700 --> 00:30:53.200 align:middle line:90% OK. 00:30:53.200 --> 00:30:55.870 align:middle line:84% Last little bit, and then we'll pause for some discussion 00:30:55.870 --> 00:30:56.720 align:middle line:90% for this first part. 00:30:56.720 --> 00:30:58.220 align:middle line:84% This is the longest part for today. 00:30:58.220 --> 00:31:01.180 align:middle line:84% So independent of these students in Leiden, 00:31:01.180 --> 00:31:03.070 align:middle line:84% independent of Goudsmit and Uhlenbeck, 00:31:03.070 --> 00:31:05.230 align:middle line:84% Wolfgang Pauli, that young graduate 00:31:05.230 --> 00:31:07.660 align:middle line:84% from Sommerfeld school, he was still 00:31:07.660 --> 00:31:10.990 align:middle line:84% wracking his brain against this anomalous Zeeman effect 00:31:10.990 --> 00:31:11.780 align:middle line:90% as well. 00:31:11.780 --> 00:31:14.740 align:middle line:84% He came up with what to him seemed like a really totally 00:31:14.740 --> 00:31:15.970 align:middle line:90% separate explanation. 00:31:15.970 --> 00:31:18.460 align:middle line:84% In time, people, including Pauli, 00:31:18.460 --> 00:31:21.070 align:middle line:84% began to put it together with this work which 00:31:21.070 --> 00:31:23.920 align:middle line:84% did see publication behind their backs 00:31:23.920 --> 00:31:25.660 align:middle line:90% by Goudsmit and Uhlenbeck. 00:31:25.660 --> 00:31:27.250 align:middle line:84% So this came in really just weeks 00:31:27.250 --> 00:31:29.500 align:middle line:84% after Goudsmit and Uhlenbeck's own work 00:31:29.500 --> 00:31:32.020 align:middle line:84% had been submitted quite independently. 00:31:32.020 --> 00:31:36.880 align:middle line:84% So Pauli was well-trained in the Sommerfeld school. 00:31:36.880 --> 00:31:38.680 align:middle line:84% And by this point, Pauli was trying 00:31:38.680 --> 00:31:41.740 align:middle line:84% to abstract from these very mechanical kind 00:31:41.740 --> 00:31:44.290 align:middle line:84% of visualizable models of the solar system 00:31:44.290 --> 00:31:47.605 align:middle line:84% and was trying to think about a kind of algebraic approach. 00:31:47.605 --> 00:31:51.430 align:middle line:84% A kind of what are the numbers with which we can associate 00:31:51.430 --> 00:31:53.320 align:middle line:90% abstract states of motion? 00:31:53.320 --> 00:31:55.270 align:middle line:84% And one of the lessons that Pauli 00:31:55.270 --> 00:31:58.060 align:middle line:84% took from Sommerfeld's work was that they 00:31:58.060 --> 00:32:01.030 align:middle line:84% had to add more and more of these integer 00:32:01.030 --> 00:32:05.170 align:middle line:84% values, some new quantum numbers n, almost always 00:32:05.170 --> 00:32:08.050 align:middle line:84% proportional to Planck's constant with which 00:32:08.050 --> 00:32:12.280 align:middle line:84% to characterize states of motion of, say, an electron 00:32:12.280 --> 00:32:13.660 align:middle line:90% in an atom. 00:32:13.660 --> 00:32:16.120 align:middle line:84% Pauli, unlike his advisor Sommerfeld, 00:32:16.120 --> 00:32:18.610 align:middle line:84% was less and less concerned about coming up 00:32:18.610 --> 00:32:20.830 align:middle line:84% with an actual physical model of is it just 00:32:20.830 --> 00:32:22.450 align:middle line:90% like the Earth around the sun? 00:32:22.450 --> 00:32:25.720 align:middle line:84% He was just more like we have all these relationships 00:32:25.720 --> 00:32:29.350 align:middle line:84% involving quantum conditions, new integers, scale 00:32:29.350 --> 00:32:30.650 align:middle line:90% set by Planck's constant. 00:32:30.650 --> 00:32:33.080 align:middle line:84% And so he argued, very abstractly, 00:32:33.080 --> 00:32:35.650 align:middle line:84% not based on a kind of visual model, 00:32:35.650 --> 00:32:37.750 align:middle line:84% that if one included yet another quantum 00:32:37.750 --> 00:32:40.060 align:middle line:84% number, this fourth quantum number, that 00:32:40.060 --> 00:32:42.910 align:middle line:84% had what he called a classically indescribable 00:32:42.910 --> 00:32:45.610 align:middle line:84% or unvisualizable double valuedness-- 00:32:45.610 --> 00:32:48.550 align:middle line:90% it was an [GERMAN]. 00:32:48.550 --> 00:32:52.150 align:middle line:84% It was literally two values, like two-faced. 00:32:52.150 --> 00:32:55.550 align:middle line:84% If this new quantity could only take on one of two values, 00:32:55.550 --> 00:32:57.808 align:middle line:84% then one could account for the Zeeman effect 00:32:57.808 --> 00:32:59.350 align:middle line:84% for the rest of the reasons very much 00:32:59.350 --> 00:33:01.990 align:middle line:84% like what Goudsmit and Uhlenbeck had argued. 00:33:01.990 --> 00:33:05.177 align:middle line:84% Pauli was not talking-- thinking at first about a spin 00:33:05.177 --> 00:33:06.010 align:middle line:90% or angular momentum. 00:33:06.010 --> 00:33:09.010 align:middle line:84% He just said maybe there's another abstract property 00:33:09.010 --> 00:33:10.450 align:middle line:90% of this state of motion. 00:33:10.450 --> 00:33:11.680 align:middle line:90% We lock it in place. 00:33:11.680 --> 00:33:13.570 align:middle line:84% We quantize it with a new integer. 00:33:13.570 --> 00:33:15.100 align:middle line:84% This one can take one of two values. 00:33:15.100 --> 00:33:18.130 align:middle line:84% Whereas, the others could take different sets of values. 00:33:18.130 --> 00:33:21.010 align:middle line:84% This became known as the Pauli exclusion principle, which 00:33:21.010 --> 00:33:23.560 align:middle line:84% is a notion that, of course, we still take very seriously 00:33:23.560 --> 00:33:24.580 align:middle line:90% today. 00:33:24.580 --> 00:33:28.210 align:middle line:84% But what Pauli was arguing was that electrons in an atom 00:33:28.210 --> 00:33:31.690 align:middle line:84% should be described by a total of four distinct quantum 00:33:31.690 --> 00:33:33.670 align:middle line:84% numbers, as we still do to this day-- 00:33:33.670 --> 00:33:35.350 align:middle line:84% the original principal quantum number-- 00:33:35.350 --> 00:33:38.260 align:middle line:84% that was when Bohr introduced; the kind of elliptical part 00:33:38.260 --> 00:33:41.170 align:middle line:84% that Sommerfeld introduced; the projection part that Sommerfeld 00:33:41.170 --> 00:33:43.090 align:middle line:84% introduced of the orbital motions; 00:33:43.090 --> 00:33:46.960 align:middle line:84% and then this new one that we would now associate with spin. 00:33:46.960 --> 00:33:49.510 align:middle line:84% Pauli at first thought Uhlenbeck and Goudsmit's idea 00:33:49.510 --> 00:33:52.120 align:middle line:84% was ridiculous because of these visualizable things. 00:33:52.120 --> 00:33:54.700 align:middle line:84% He later realized, oh, maybe these really are relatable. 00:33:54.700 --> 00:33:57.450 align:middle line:84% And so the fourth quantum number becomes now associated 00:33:57.450 --> 00:33:59.230 align:middle line:90% with the spin. 00:33:59.230 --> 00:34:01.300 align:middle line:84% The next part that Pauli put forward 00:34:01.300 --> 00:34:02.890 align:middle line:84% was not just that we characterize 00:34:02.890 --> 00:34:04.990 align:middle line:84% an electron with these four separate integers, 00:34:04.990 --> 00:34:07.090 align:middle line:84% but that no two electrons can have 00:34:07.090 --> 00:34:09.760 align:middle line:84% the exact same set of those quantum numbers 00:34:09.760 --> 00:34:10.840 align:middle line:90% at the same time. 00:34:10.840 --> 00:34:12.730 align:middle line:90% That's the exclusion principle. 00:34:12.730 --> 00:34:16.750 align:middle line:84% He later recalled that he was inspired by this kind of no two 00:34:16.750 --> 00:34:20.960 align:middle line:84% can have the same set of values by watching can-can dancers, 00:34:20.960 --> 00:34:23.590 align:middle line:84% which was all the rage in the 1920s in many cities 00:34:23.590 --> 00:34:24.370 align:middle line:90% in Europe-- 00:34:24.370 --> 00:34:26.620 align:middle line:84% famously associated with, say, the Moulin Rouge 00:34:26.620 --> 00:34:28.690 align:middle line:84% in Paris, but in Copenhagen as well. 00:34:28.690 --> 00:34:31.000 align:middle line:84% And the idea was that these dancers were so skilled 00:34:31.000 --> 00:34:34.480 align:middle line:84% that one would have just left a little physical location 00:34:34.480 --> 00:34:36.429 align:middle line:84% on the dance floor immediately when 00:34:36.429 --> 00:34:39.100 align:middle line:84% the next-- when her partner had to step into that same spot. 00:34:39.100 --> 00:34:40.389 align:middle line:90% They never collided. 00:34:40.389 --> 00:34:41.800 align:middle line:90% They were close to each other. 00:34:41.800 --> 00:34:45.340 align:middle line:84% Their motions were similar, but no two of those dancers, 00:34:45.340 --> 00:34:49.719 align:middle line:84% much like no two electrons, occupied exactly the same state 00:34:49.719 --> 00:34:50.468 align:middle line:90% at the same time. 00:34:50.468 --> 00:34:52.510 align:middle line:84% So that becomes known as the exclusion principle. 00:34:52.510 --> 00:34:54.580 align:middle line:84% So let pause there, and we can take 00:34:54.580 --> 00:34:58.820 align:middle line:90% some questions and discussion. 00:34:58.820 --> 00:35:01.430 align:middle line:84% So DA asks, doesn't a circular orbit have angular momentum? 00:35:01.430 --> 00:35:02.850 align:middle line:90% It certainly does. 00:35:02.850 --> 00:35:06.590 align:middle line:84% And so the idea, DA, is that in the elliptical orbit 00:35:06.590 --> 00:35:09.602 align:middle line:84% there are two separate degrees of freedom, 00:35:09.602 --> 00:35:11.810 align:middle line:84% is the fancy way of saying it, the two different ways 00:35:11.810 --> 00:35:14.450 align:middle line:84% with which to characterize the orbital motion. 00:35:14.450 --> 00:35:17.600 align:middle line:84% Whereas, in a perfectly-- a uniform, circular orbit, 00:35:17.600 --> 00:35:20.720 align:middle line:84% there's only one quantity that is 00:35:20.720 --> 00:35:23.240 align:middle line:84% required to characterize-- to completely characterize 00:35:23.240 --> 00:35:24.230 align:middle line:90% the motion. 00:35:24.230 --> 00:35:26.600 align:middle line:84% And so that's why I think of this second one 00:35:26.600 --> 00:35:29.020 align:middle line:84% Sommerfeld introduces as more like the-- it's sort 00:35:29.020 --> 00:35:30.180 align:middle line:90% of like the ellipticity. 00:35:30.180 --> 00:35:32.180 align:middle line:84% It's not exactly that, but it's related to that. 00:35:32.180 --> 00:35:35.630 align:middle line:84% If the orbital motion is deformed away 00:35:35.630 --> 00:35:37.370 align:middle line:84% from a perfect circle, then you'll 00:35:37.370 --> 00:35:40.070 align:middle line:84% need at least two numbers to characterize that motion 00:35:40.070 --> 00:35:41.720 align:middle line:90% completely, not just one. 00:35:41.720 --> 00:35:45.140 align:middle line:84% So to DA's excellent point, the electron 00:35:45.140 --> 00:35:47.810 align:middle line:84% moving in a perfect circle certainly has angular momentum, 00:35:47.810 --> 00:35:49.640 align:middle line:84% but you can characterize that completely 00:35:49.640 --> 00:35:52.430 align:middle line:84% with only one number, one parameter. 00:35:52.430 --> 00:35:55.370 align:middle line:84% Once there is a generalization, you need at least two. 00:35:55.370 --> 00:35:58.440 align:middle line:84% So that's where Sommerfeld is going with that. 00:35:58.440 --> 00:35:59.780 align:middle line:90% And Jade also is correct. 00:35:59.780 --> 00:36:02.510 align:middle line:84% And in Jade's response, you put that whole assembly 00:36:02.510 --> 00:36:07.130 align:middle line:84% in an external magnetic field, and there would be no mechanism 00:36:07.130 --> 00:36:11.450 align:middle line:84% without accounting for these other states of motion 00:36:11.450 --> 00:36:15.820 align:middle line:84% to account for these what became known as fine structure 00:36:15.820 --> 00:36:16.320 align:middle line:90% splitting. 00:36:16.320 --> 00:36:17.300 align:middle line:90% That's right. 00:36:17.300 --> 00:36:20.300 align:middle line:84% Gary says, Heisenberg, de Broglie, and Einstein 00:36:20.300 --> 00:36:22.970 align:middle line:84% most noted work happened when still young scientists. 00:36:22.970 --> 00:36:23.720 align:middle line:90% That's right. 00:36:23.720 --> 00:36:25.705 align:middle line:84% So Gary, you'll be-- as an economist, 00:36:25.705 --> 00:36:27.260 align:middle line:84% as we've got to know-- economists 00:36:27.260 --> 00:36:30.590 align:middle line:84% have actually studied this quite a bit. 00:36:30.590 --> 00:36:34.260 align:middle line:84% And there are a number of studies-- 00:36:34.260 --> 00:36:36.890 align:middle line:84% I can send you some references afterwards-- suggesting-- 00:36:36.890 --> 00:36:39.380 align:middle line:84% although other counterstudies that muddy the waters. 00:36:39.380 --> 00:36:43.130 align:middle line:84% Some early studies suggest that in some fields of study 00:36:43.130 --> 00:36:45.980 align:middle line:84% there a kind of advantage to not knowing too much. 00:36:45.980 --> 00:36:48.260 align:middle line:84% That's how the economists had characterized it. 00:36:48.260 --> 00:36:51.410 align:middle line:84% So not having to feel like you have to answer every last thing 00:36:51.410 --> 00:36:53.870 align:middle line:84% you've tried to explain your whole career. 00:36:53.870 --> 00:36:56.750 align:middle line:84% Their explanation was a kind of conceptual freedom. 00:36:56.750 --> 00:36:58.370 align:middle line:84% That's how they tried to make sense 00:36:58.370 --> 00:37:02.183 align:middle line:84% of a modest statistical signal that at least in some fields, 00:37:02.183 --> 00:37:04.100 align:middle line:84% including quantum physics, is one of the areas 00:37:04.100 --> 00:37:05.840 align:middle line:84% that I know of from one of these studies, 00:37:05.840 --> 00:37:08.720 align:middle line:84% that there was a kind of early entrance kind of bonus, 00:37:08.720 --> 00:37:10.400 align:middle line:90% we might say-- early starts. 00:37:10.400 --> 00:37:13.880 align:middle line:84% Other data or studies have muddied 00:37:13.880 --> 00:37:15.380 align:middle line:90% that relationship a bit. 00:37:15.380 --> 00:37:18.110 align:middle line:84% But certainly, there are a bunch of economists in particular 00:37:18.110 --> 00:37:22.400 align:middle line:84% who are interested in life cycle and innovation is how 00:37:22.400 --> 00:37:23.730 align:middle line:90% they would describe it today. 00:37:23.730 --> 00:37:26.020 align:middle line:90% It's an excellent question. 00:37:26.020 --> 00:37:27.662 align:middle line:84% Lucas asked about space quantization-- 00:37:27.662 --> 00:37:29.620 align:middle line:84% very different from how we talk about spin now. 00:37:29.620 --> 00:37:30.880 align:middle line:90% It certainly is. 00:37:30.880 --> 00:37:34.090 align:middle line:84% We also have modern program of trying to quantize space time. 00:37:34.090 --> 00:37:34.960 align:middle line:90% Good. 00:37:34.960 --> 00:37:35.560 align:middle line:90% No. 00:37:35.560 --> 00:37:37.940 align:middle line:84% That's a-- so Lucas, it's a great question. 00:37:37.940 --> 00:37:40.600 align:middle line:84% And that's, again, we know too much, or at least 00:37:40.600 --> 00:37:43.120 align:middle line:84% we're exposed to too much, that was-- it is-- 00:37:43.120 --> 00:37:44.620 align:middle line:90% it's really funny vocabulary. 00:37:44.620 --> 00:37:49.790 align:middle line:84% I wanted to harp or emphasize Goudsmit and Uhlenbeck's 00:37:49.790 --> 00:37:50.660 align:middle line:90% first idea. 00:37:50.660 --> 00:37:53.300 align:middle line:84% Because they really agreed this was kind of nutty sounding. 00:37:53.300 --> 00:37:57.470 align:middle line:84% It was how could a three vector only point 00:37:57.470 --> 00:37:59.930 align:middle line:90% in one of two orientations. 00:37:59.930 --> 00:38:03.260 align:middle line:84% It was as if it was-- as if its orientations in space 00:38:03.260 --> 00:38:05.570 align:middle line:84% were quantized as opposed to space [INAUDIBLE].. 00:38:05.570 --> 00:38:08.570 align:middle line:84% They agreed that the x, y, and z in which we live 00:38:08.570 --> 00:38:10.880 align:middle line:84% would be continuous and have the kind 00:38:10.880 --> 00:38:13.132 align:middle line:84% of classical or relativistic behavior. 00:38:13.132 --> 00:38:14.840 align:middle line:84% So they weren't trying to challenge that. 00:38:14.840 --> 00:38:18.560 align:middle line:84% And the notion of trying to quantize space time, that 00:38:18.560 --> 00:38:22.430 align:middle line:84% begins to kick in by the late '20s, early '30s, 00:38:22.430 --> 00:38:24.770 align:middle line:84% but not quite-- that's not what these folks 00:38:24.770 --> 00:38:27.050 align:middle line:90% were thinking about just yet. 00:38:27.050 --> 00:38:29.090 align:middle line:84% Alex asks a question about conventions. 00:38:29.090 --> 00:38:30.757 align:middle line:84% Why is it m sub l and not n sub l? 00:38:30.757 --> 00:38:31.340 align:middle line:90% Ah, very good. 00:38:31.340 --> 00:38:33.150 align:middle line:90% I don't know. 00:38:33.150 --> 00:38:34.210 align:middle line:90% I have a suspicion. 00:38:34.210 --> 00:38:36.210 align:middle line:84% This third quantum number is, again, some of you 00:38:36.210 --> 00:38:38.040 align:middle line:84% might know from chemistry classes 00:38:38.040 --> 00:38:39.900 align:middle line:84% as well as from physics ones, it's 00:38:39.900 --> 00:38:42.300 align:middle line:84% nowadays often called a magnetic quantum number 00:38:42.300 --> 00:38:44.550 align:middle line:84% because it really is about the orientation 00:38:44.550 --> 00:38:47.820 align:middle line:84% of the orbital motion with respect 00:38:47.820 --> 00:38:49.900 align:middle line:90% to an external magnetic field. 00:38:49.900 --> 00:38:53.310 align:middle line:84% So it's now called the magnetic quantum number. 00:38:53.310 --> 00:38:55.470 align:middle line:84% And even in German, it would be [GERMAN].. 00:38:55.470 --> 00:38:58.650 align:middle line:84% So it wouldn't surprise me if it gets renamed as an m later. 00:38:58.650 --> 00:39:00.360 align:middle line:90% That's a good question. 00:39:00.360 --> 00:39:03.210 align:middle line:84% I don't actually own a copy of Sommerfeld's first edition. 00:39:03.210 --> 00:39:04.680 align:middle line:90% It's a very expensive book now. 00:39:04.680 --> 00:39:05.610 align:middle line:84% It would be interesting to go back. 00:39:05.610 --> 00:39:06.860 align:middle line:90% It's probably on Google Books. 00:39:06.860 --> 00:39:08.820 align:middle line:84% It's probably been scanned to see back 00:39:08.820 --> 00:39:13.410 align:middle line:84% in 1919 were people using N sub l or M sub l. 00:39:13.410 --> 00:39:14.680 align:middle line:90% I don't know. 00:39:14.680 --> 00:39:18.540 align:middle line:84% I suspect it gets-- if it wasn't put in that way from the start, 00:39:18.540 --> 00:39:21.320 align:middle line:84% I imagine why we use an M these days is 00:39:21.320 --> 00:39:23.820 align:middle line:84% because it is, as I say, called the magnetic quantum number. 00:39:23.820 --> 00:39:25.660 align:middle line:90% It's an interesting question. 00:39:25.660 --> 00:39:27.840 align:middle line:84% Other questions or thoughts about any of that stuff? 00:39:27.840 --> 00:39:33.960 align:middle line:90% 00:39:33.960 --> 00:39:34.460 align:middle line:90% OK. 00:39:34.460 --> 00:39:36.180 align:middle line:84% Keep the questions coming if you want. 00:39:36.180 --> 00:39:39.200 align:middle line:84% Let me press on now and see how are 00:39:39.200 --> 00:39:41.900 align:middle line:84% some very smart, young, and ambitious people reacting 00:39:41.900 --> 00:39:44.330 align:middle line:84% to that body of work, in particular, 00:39:44.330 --> 00:39:47.930 align:middle line:84% one of Pauli's closest friends, Werner Heisenberg. 00:39:47.930 --> 00:39:50.930 align:middle line:84% So Heisenberg and Pauli were almost exact contemporaries 00:39:50.930 --> 00:39:52.430 align:middle line:84% who are almost exactly the same age. 00:39:52.430 --> 00:39:56.332 align:middle line:84% They both finished their PhDs around the age of 22-- 00:39:56.332 --> 00:39:58.040 align:middle line:84% the PhD-- not their undergraduate degree. 00:39:58.040 --> 00:40:02.330 align:middle line:84% They're both very precocious young students of Sommerfeld. 00:40:02.330 --> 00:40:05.180 align:middle line:84% And by this time, Heisenberg, we know from his letters 00:40:05.180 --> 00:40:07.202 align:middle line:84% with people like Wolfgang Pauli-- many, 00:40:07.202 --> 00:40:08.660 align:middle line:84% many of the letters have survived-- 00:40:08.660 --> 00:40:11.360 align:middle line:84% he was also growing frustrated with what 00:40:11.360 --> 00:40:14.300 align:middle line:84% had become the pattern of this so-called old quantum theory, 00:40:14.300 --> 00:40:17.720 align:middle line:84% even from their own otherwise very beloved mentor, 00:40:17.720 --> 00:40:19.880 align:middle line:84% Sommerfeld-- this idea of starting 00:40:19.880 --> 00:40:22.520 align:middle line:84% with classical descriptions of motion 00:40:22.520 --> 00:40:26.990 align:middle line:84% and then kind of tacking on what seemed like ad hoc quantum 00:40:26.990 --> 00:40:27.900 align:middle line:90% conditions. 00:40:27.900 --> 00:40:33.780 align:middle line:84% So by 1924 at the tender age of 22 or 23, 00:40:33.780 --> 00:40:36.540 align:middle line:84% Heisenberg then began his first postdoc position. 00:40:36.540 --> 00:40:38.100 align:middle line:84% He began working even more closely 00:40:38.100 --> 00:40:41.220 align:middle line:84% with Niels Bohr in Copenhagen. And that's 00:40:41.220 --> 00:40:44.130 align:middle line:84% where he introduces this the paper 00:40:44.130 --> 00:40:45.968 align:middle line:84% that we had for part of our reading today. 00:40:45.968 --> 00:40:48.510 align:middle line:84% It was submitted-- a few months later, he moved to Gottingen. 00:40:48.510 --> 00:40:50.640 align:middle line:84% But he really did a lot of this work really fresh 00:40:50.640 --> 00:40:51.750 align:middle line:90% out of his PhD. 00:40:51.750 --> 00:40:54.930 align:middle line:84% So he now starts to set out a kind of program. 00:40:54.930 --> 00:40:58.710 align:middle line:84% The opening paragraph of this paper reads like a manifesto. 00:40:58.710 --> 00:41:00.840 align:middle line:84% We have to do something new, is how I read 00:41:00.840 --> 00:41:03.690 align:middle line:84% this very young author in that opening paragraph. 00:41:03.690 --> 00:41:06.360 align:middle line:84% What he's arguing for is a first principles 00:41:06.360 --> 00:41:10.110 align:middle line:84% treatment of the quantum realm, rather than what felt even then 00:41:10.110 --> 00:41:12.150 align:middle line:84% as a kind of kluge, or a grab bag, 00:41:12.150 --> 00:41:15.520 align:middle line:84% or a series of ad hoc rules of thumb. 00:41:15.520 --> 00:41:17.970 align:middle line:84% So he wanted to break this impasse of the kind 00:41:17.970 --> 00:41:19.620 align:middle line:90% of Bohr-Sommerfeld approach. 00:41:19.620 --> 00:41:23.100 align:middle line:84% And in fact, he drew very clear inspiration from Einstein 00:41:23.100 --> 00:41:24.380 align:middle line:90% in 1905. 00:41:24.380 --> 00:41:26.880 align:middle line:84% And again, we know that partly from his letters at the time, 00:41:26.880 --> 00:41:28.680 align:middle line:84% as well as his later recollections. 00:41:28.680 --> 00:41:31.620 align:middle line:84% He thought that much as Einstein had made a big deal, 00:41:31.620 --> 00:41:35.790 align:middle line:84% as we saw a few classes ago, about the mock-like 00:41:35.790 --> 00:41:36.990 align:middle line:90% positivism-- 00:41:36.990 --> 00:41:39.780 align:middle line:84% let's only focus on objects of positive experience, 00:41:39.780 --> 00:41:41.940 align:middle line:84% otherwise we'll get ourselves tied up in knots-- 00:41:41.940 --> 00:41:44.650 align:middle line:84% Heisenberg says we have to do the same thing all over again 00:41:44.650 --> 00:41:47.550 align:middle line:84% now for understanding the behavior of atoms 00:41:47.550 --> 00:41:48.585 align:middle line:90% and parts of atoms. 00:41:48.585 --> 00:41:50.460 align:middle line:84% And so in particular, Heisenberg says, again, 00:41:50.460 --> 00:41:53.010 align:middle line:84% in his opening paragraph, he says, 00:41:53.010 --> 00:41:56.800 align:middle line:84% we can never observe an electron in its orbit within an atom. 00:41:56.800 --> 00:41:59.040 align:middle line:84% So why are we trying to calculate properties 00:41:59.040 --> 00:42:01.200 align:middle line:84% of that motion that we could never 00:42:01.200 --> 00:42:03.010 align:middle line:90% observe in the first place? 00:42:03.010 --> 00:42:05.140 align:middle line:84% And here, again, as you've seen in the reader, 00:42:05.140 --> 00:42:09.090 align:middle line:84% he says it seems sensible to discard all hope of observing 00:42:09.090 --> 00:42:11.010 align:middle line:84% hitherto unobservable quantities, 00:42:11.010 --> 00:42:12.900 align:middle line:84% by which he means things like properties 00:42:12.900 --> 00:42:13.920 align:middle line:90% of an electrons orbit-- 00:42:13.920 --> 00:42:16.500 align:middle line:84% Keplerian or circular or otherwise. 00:42:16.500 --> 00:42:18.400 align:middle line:84% Instead, it seems more reasonable, 00:42:18.400 --> 00:42:21.660 align:middle line:84% he goes on, to try to establish a theoretical quantum 00:42:21.660 --> 00:42:25.260 align:middle line:84% mechanics-- his terms-- a theoretical quantum mechanics-- 00:42:25.260 --> 00:42:28.080 align:middle line:84% analogous to classical mechanics, but in which 00:42:28.080 --> 00:42:30.755 align:middle line:84% only relations between observable quantities appear. 00:42:30.755 --> 00:42:32.130 align:middle line:84% And that's really-- he has a kind 00:42:32.130 --> 00:42:36.120 align:middle line:84% of echo in his head of the once brash, young Albert 00:42:36.120 --> 00:42:38.580 align:middle line:90% Einstein and Mach. 00:42:38.580 --> 00:42:41.820 align:middle line:84% Previous approaches could be seriously criticized 00:42:41.820 --> 00:42:45.120 align:middle line:84% on the grounds that they contain as basic elements relationships 00:42:45.120 --> 00:42:47.730 align:middle line:84% between quantities that are apparently unobservable 00:42:47.730 --> 00:42:51.160 align:middle line:84% in principle-- it's just like textbook Ernest Mach here-- 00:42:51.160 --> 00:42:54.990 align:middle line:84% such as position and period of revolution of the electron. 00:42:54.990 --> 00:42:58.860 align:middle line:84% Supposedly, Heisenberg tells us in later recollections, 00:42:58.860 --> 00:43:01.170 align:middle line:84% he shared his excitement about this move 00:43:01.170 --> 00:43:06.840 align:middle line:84% with Einstein around this time, and around 1924 or '25. 00:43:06.840 --> 00:43:09.510 align:middle line:84% I was so inspired by your work, Einstein, that kind of thing. 00:43:09.510 --> 00:43:12.330 align:middle line:84% Einstein, by this point, was quite senior and very renowned. 00:43:12.330 --> 00:43:14.370 align:middle line:84% And supposedly Einstein's retort was 00:43:14.370 --> 00:43:16.800 align:middle line:84% a good joke shouldn't be repeated too often. 00:43:16.800 --> 00:43:19.320 align:middle line:84% As we'll see, Einstein was less convinced this is the way 00:43:19.320 --> 00:43:21.630 align:middle line:84% to go by the 1920s, even though it 00:43:21.630 --> 00:43:23.700 align:middle line:84% had been so critical to his own thinking 00:43:23.700 --> 00:43:27.630 align:middle line:84% in the early parts of around 1905. 00:43:27.630 --> 00:43:28.140 align:middle line:90% OK. 00:43:28.140 --> 00:43:29.910 align:middle line:84% So what does Heisenberg do with all this? 00:43:29.910 --> 00:43:32.370 align:middle line:84% It's one thing to have an opening paragraph about a kind 00:43:32.370 --> 00:43:33.780 align:middle line:90% of guiding philosophy. 00:43:33.780 --> 00:43:34.920 align:middle line:90% What does he do? 00:43:34.920 --> 00:43:36.870 align:middle line:84% So as we saw, he argues that physicists 00:43:36.870 --> 00:43:40.410 align:middle line:84% should focus on quantities that are in principle observable 00:43:40.410 --> 00:43:41.540 align:middle line:90% or empirical. 00:43:41.540 --> 00:43:43.290 align:middle line:84% And what he has in mind as he goes through 00:43:43.290 --> 00:43:45.450 align:middle line:84% in the rest of this brief paper are 00:43:45.450 --> 00:43:49.020 align:middle line:84% things like the frequencies, the numerical values of the colors, 00:43:49.020 --> 00:43:51.210 align:middle line:84% the frequencies of these spectral lines. 00:43:51.210 --> 00:43:53.220 align:middle line:84% How do we learn about the structure of atoms 00:43:53.220 --> 00:43:55.140 align:middle line:84% by measuring the stuff that comes out-- things 00:43:55.140 --> 00:43:57.900 align:middle line:84% that we can literally observe and very carefully measure, 00:43:57.900 --> 00:44:00.360 align:middle line:84% like the frequencies of spectral lines. 00:44:00.360 --> 00:44:02.640 align:middle line:84% And in particular, he goes back to say 00:44:02.640 --> 00:44:05.850 align:middle line:84% that these spectral lines obey a law of addition. 00:44:05.850 --> 00:44:09.720 align:middle line:84% Remember, we saw last time even in the 1880s, 00:44:09.720 --> 00:44:11.700 align:middle line:84% it had been clear that there were 00:44:11.700 --> 00:44:13.440 align:middle line:84% these kind of empirical relationships 00:44:13.440 --> 00:44:16.530 align:middle line:84% between the frequencies of the color lines-- really, 00:44:16.530 --> 00:44:19.020 align:middle line:84% like how many hertz, how many cycles per second, 00:44:19.020 --> 00:44:20.730 align:middle line:90% is this blue line? 00:44:20.730 --> 00:44:24.840 align:middle line:84% And then these combinations of inverse squares of integers. 00:44:24.840 --> 00:44:28.590 align:middle line:84% That was the Balmer relationship, 00:44:28.590 --> 00:44:30.150 align:middle line:84% and Bohr gave a kind of explanation 00:44:30.150 --> 00:44:32.440 align:middle line:90% for it with his atomic model. 00:44:32.440 --> 00:44:34.180 align:middle line:84% Well, if you start from that, you 00:44:34.180 --> 00:44:36.700 align:middle line:84% realize that the sum of the frequencies of two 00:44:36.700 --> 00:44:39.700 align:middle line:84% different spectral lines associated with transitions 00:44:39.700 --> 00:44:42.250 align:middle line:84% between different states of an electron inside, say, 00:44:42.250 --> 00:44:44.372 align:middle line:84% the hydrogen atom, they will sum. 00:44:44.372 --> 00:44:46.330 align:middle line:84% So the sum of any two of these frequencies that 00:44:46.330 --> 00:44:49.452 align:middle line:90% actually appear will give you-- 00:44:49.452 --> 00:44:51.160 align:middle line:84% add them together, you'll get a frequency 00:44:51.160 --> 00:44:55.100 align:middle line:84% that will also appear somewhere in the spectrum. 00:44:55.100 --> 00:44:57.190 align:middle line:90% And so that would be like-- 00:44:57.190 --> 00:44:58.940 align:middle line:84% and by the way, and you can keep going. 00:44:58.940 --> 00:45:00.982 align:middle line:84% This is not true only for adding any two of them. 00:45:00.982 --> 00:45:03.370 align:middle line:84% You could keep going for an arbitrary 00:45:03.370 --> 00:45:06.880 align:middle line:84% number of the actual numerical values of these frequencies 00:45:06.880 --> 00:45:08.270 align:middle line:90% of the spectral lines. 00:45:08.270 --> 00:45:10.750 align:middle line:84% And for all the colors that actually come out and can 00:45:10.750 --> 00:45:12.580 align:middle line:90% be measured, even beyond-- 00:45:12.580 --> 00:45:15.400 align:middle line:84% pardon me-- just the visible portion of the spectrum 00:45:15.400 --> 00:45:18.160 align:middle line:84% into the infrared, into the ultraviolet, 00:45:18.160 --> 00:45:20.530 align:middle line:90% they obey this law of addition. 00:45:20.530 --> 00:45:22.690 align:middle line:84% And so what that would be is like an electron could 00:45:22.690 --> 00:45:26.500 align:middle line:84% jump from say a principal quantum number of 6 to 1 00:45:26.500 --> 00:45:29.230 align:middle line:84% all at once, that will give a certain color 00:45:29.230 --> 00:45:30.880 align:middle line:90% of the corresponding emission. 00:45:30.880 --> 00:45:32.770 align:middle line:84% Or it could make a series of hops. 00:45:32.770 --> 00:45:34.760 align:middle line:84% It could jump from the principal quantum 00:45:34.760 --> 00:45:39.280 align:middle line:84% number 6 down to 3, 3 to 1, or 6 to 3, 3 to 2, 2 to 1, 00:45:39.280 --> 00:45:41.220 align:middle line:90% or 6 to 5 and 5, et cetera. 00:45:41.220 --> 00:45:43.940 align:middle line:84% Each of those jumps or transitions 00:45:43.940 --> 00:45:45.980 align:middle line:84% corresponds to an emission of light 00:45:45.980 --> 00:45:49.580 align:middle line:84% of a very specific frequency given by this relationship. 00:45:49.580 --> 00:45:51.580 align:middle line:84% And if you add up the frequencies of any 00:45:51.580 --> 00:45:53.330 align:middle line:84% of these jumps, you'd see they would all-- 00:45:53.330 --> 00:45:56.000 align:middle line:84% they could be arranged in this format. 00:45:56.000 --> 00:45:57.670 align:middle line:84% So Heisenberg began saying, we should 00:45:57.670 --> 00:45:59.600 align:middle line:90% be focusing on that stuff. 00:45:59.600 --> 00:46:04.660 align:middle line:84% These are our empirical objects of positive experience 00:46:04.660 --> 00:46:05.800 align:middle line:90% that we can see. 00:46:05.800 --> 00:46:07.030 align:middle line:90% We can measure. 00:46:07.030 --> 00:46:09.250 align:middle line:84% We can gain quantitative information 00:46:09.250 --> 00:46:12.080 align:middle line:84% about the frequencies of these spectral lines. 00:46:12.080 --> 00:46:14.830 align:middle line:84% So he begins trying to construct the kind of arrays 00:46:14.830 --> 00:46:16.960 align:middle line:84% of these lines that kind go together 00:46:16.960 --> 00:46:19.080 align:middle line:90% through this law of addition. 00:46:19.080 --> 00:46:20.910 align:middle line:84% In the midst of that work-- by now, 00:46:20.910 --> 00:46:23.190 align:middle line:84% this takes him from the fall into the spring of that 00:46:23.190 --> 00:46:26.010 align:middle line:90% academic year, spring of 1925-- 00:46:26.010 --> 00:46:30.990 align:middle line:84% Heisenberg actually has a very famously-- an unusually bad 00:46:30.990 --> 00:46:32.100 align:middle line:90% attack of hay fever. 00:46:32.100 --> 00:46:33.700 align:middle line:90% He had allergies all his life. 00:46:33.700 --> 00:46:36.540 align:middle line:84% He has a particularly bad attack of hay fever. 00:46:36.540 --> 00:46:38.370 align:middle line:84% So he actually had to leave Copenhagen. 00:46:38.370 --> 00:46:41.910 align:middle line:84% He travels to the nearby island of Heligoland, 00:46:41.910 --> 00:46:45.810 align:middle line:84% which is actually in the North Sea between Denmark 00:46:45.810 --> 00:46:47.160 align:middle line:90% and Germany, as I understand it. 00:46:47.160 --> 00:46:49.900 align:middle line:84% Over the years, both countries have claimed it. 00:46:49.900 --> 00:46:51.750 align:middle line:84% So the point is it's pretty close to where 00:46:51.750 --> 00:46:54.345 align:middle line:84% he was in Copenhagen, but gets out of town. 00:46:54.345 --> 00:46:56.160 align:middle line:84% And he's actually then basically on his own 00:46:56.160 --> 00:46:58.740 align:middle line:84% in this tiny little kind of touristy island 00:46:58.740 --> 00:47:02.350 align:middle line:84% for several weeks still trying to puzzle through this stuff. 00:47:02.350 --> 00:47:05.670 align:middle line:84% So he's even more isolated than he had been just as a postdoc. 00:47:05.670 --> 00:47:08.040 align:middle line:90% He literally leaves town. 00:47:08.040 --> 00:47:10.770 align:middle line:84% It's during that short stay that he continues 00:47:10.770 --> 00:47:12.900 align:middle line:84% trying to sit with this notion of the behavior 00:47:12.900 --> 00:47:16.400 align:middle line:84% of these frequencies of emitted spectral lines. 00:47:16.400 --> 00:47:20.250 align:middle line:84% Then he goes on to think now on his own these frequencies 00:47:20.250 --> 00:47:23.970 align:middle line:84% new refer to their characterizing 00:47:23.970 --> 00:47:25.170 align:middle line:90% the light that comes out. 00:47:25.170 --> 00:47:27.780 align:middle line:84% And he knows, again, even classically the way 00:47:27.780 --> 00:47:30.900 align:middle line:84% we would characterize electromagnetic waves, waves 00:47:30.900 --> 00:47:34.800 align:middle line:84% of light, would be some amplitude and some frequency, 00:47:34.800 --> 00:47:37.230 align:middle line:84% some phase, the phase being proportional 00:47:37.230 --> 00:47:38.700 align:middle line:90% to the color we see. 00:47:38.700 --> 00:47:40.920 align:middle line:84% The point is that the quantities he's now 00:47:40.920 --> 00:47:43.743 align:middle line:84% trying to build up in these arrays, these kind of new 1, 00:47:43.743 --> 00:47:48.000 align:middle line:84% 2, new 3, 4 and so on, they appear in the exponent 00:47:48.000 --> 00:47:50.880 align:middle line:84% when you actually try to mathematically describe 00:47:50.880 --> 00:47:53.100 align:middle line:90% the light that comes out. 00:47:53.100 --> 00:47:56.580 align:middle line:84% So then he goes on to just reason on the island. 00:47:56.580 --> 00:47:58.950 align:middle line:84% If these frequencies up here are adding, 00:47:58.950 --> 00:48:01.710 align:middle line:84% if the exponents are adding, that 00:48:01.710 --> 00:48:04.470 align:middle line:84% suggests that these amplitudes should multiply. 00:48:04.470 --> 00:48:07.008 align:middle line:84% Just think about even in elementary multiplication, 00:48:07.008 --> 00:48:08.550 align:middle line:84% if you have two numbers that we write 00:48:08.550 --> 00:48:10.500 align:middle line:84% in kind of scientific notation that 00:48:10.500 --> 00:48:12.610 align:middle line:84% have a coefficient and an exponent, 00:48:12.610 --> 00:48:16.330 align:middle line:84% if we multiply those two numbers together the exponents add. 00:48:16.330 --> 00:48:18.650 align:middle line:84% That's what he saw the frequencies doing. 00:48:18.650 --> 00:48:21.100 align:middle line:84% And that goes along with the coefficients 00:48:21.100 --> 00:48:23.200 align:middle line:90% of the amplitudes multiplied. 00:48:23.200 --> 00:48:25.060 align:middle line:84% So he says, well, maybe that's what goes on 00:48:25.060 --> 00:48:26.143 align:middle line:90% with these spectral lines. 00:48:26.143 --> 00:48:29.050 align:middle line:84% All the information we can glean about these spectral lines, 00:48:29.050 --> 00:48:31.420 align:middle line:84% the amplitude would be related to the intensity 00:48:31.420 --> 00:48:32.540 align:middle line:90% of the brightness. 00:48:32.540 --> 00:48:35.590 align:middle line:84% And so let's see if we can make similar arrays with more 00:48:35.590 --> 00:48:38.110 align:middle line:84% information we can glean of the structure of these atoms 00:48:38.110 --> 00:48:40.120 align:middle line:90% from the empirical information. 00:48:40.120 --> 00:48:43.390 align:middle line:84% So taking this analogy or this mathematical relationship, 00:48:43.390 --> 00:48:45.490 align:middle line:84% if these things add, these things should multiply. 00:48:45.490 --> 00:48:48.850 align:middle line:84% And that's what he finds all alone on Heligoland 00:48:48.850 --> 00:48:51.490 align:middle line:84% while suffering from hay fever, that somehow 00:48:51.490 --> 00:48:54.010 align:middle line:84% these arrays of the amplitudes start 00:48:54.010 --> 00:48:55.570 align:middle line:90% looking pretty funny to him. 00:48:55.570 --> 00:48:58.870 align:middle line:84% It turns out the order of multiplication changes 00:48:58.870 --> 00:49:03.340 align:middle line:84% the result, that if he just tries to multiply them, 00:49:03.340 --> 00:49:05.800 align:middle line:84% the order in which he multiplies them changes the answer. 00:49:05.800 --> 00:49:07.425 align:middle line:84% Whereas, of course, that doesn't happen 00:49:07.425 --> 00:49:08.590 align:middle line:90% if you add these things up. 00:49:08.590 --> 00:49:13.000 align:middle line:84% You can add them up, and the sum would not change. 00:49:13.000 --> 00:49:15.430 align:middle line:90% He writes up this brief paper. 00:49:15.430 --> 00:49:17.170 align:middle line:84% He submits it once he then actually moves 00:49:17.170 --> 00:49:20.050 align:middle line:84% to his new position in Gottingen back in Germany. 00:49:20.050 --> 00:49:21.850 align:middle line:84% And now, he's working not with Bohr, 00:49:21.850 --> 00:49:24.740 align:middle line:84% but most closely with the mathematical physicist, Max 00:49:24.740 --> 00:49:25.240 align:middle line:90% Born. 00:49:25.240 --> 00:49:26.440 align:middle line:90% So Born's more senior. 00:49:26.440 --> 00:49:28.360 align:middle line:84% He's more like Albert Einstein's age. 00:49:28.360 --> 00:49:31.450 align:middle line:84% Born, like Sommerfeld, was very, very well 00:49:31.450 --> 00:49:34.510 align:middle line:84% trained in classical mechanics, celestial mechanics, 00:49:34.510 --> 00:49:36.280 align:middle line:90% other mathematical techniques. 00:49:36.280 --> 00:49:39.010 align:middle line:84% And Born's reaction when this excited young Heisenberg 00:49:39.010 --> 00:49:41.710 align:middle line:84% arrives is basically saying, you're really an idiot. 00:49:41.710 --> 00:49:44.890 align:middle line:84% You're dummkopf, is what I imagine Born saying. 00:49:44.890 --> 00:49:46.750 align:middle line:90% You're studying matrices. 00:49:46.750 --> 00:49:49.840 align:middle line:90% The point is for the very-- 00:49:49.840 --> 00:49:52.240 align:middle line:84% even the very elite training of someone like Heisenberg 00:49:52.240 --> 00:49:55.090 align:middle line:84% and Pauli at this leading school of theoretical physics 00:49:55.090 --> 00:49:57.253 align:middle line:84% in Munich, throughout his entire training 00:49:57.253 --> 00:49:59.170 align:middle line:84% he'd never taken a course in what we would now 00:49:59.170 --> 00:50:01.930 align:middle line:84% call linear algebra, much like Einstein 00:50:01.930 --> 00:50:04.730 align:middle line:84% hadn't learned about non-Euclidean geometry, so on. 00:50:04.730 --> 00:50:09.490 align:middle line:84% So Born, now Max Born, had to tell and/or remind 00:50:09.490 --> 00:50:15.190 align:middle line:84% young Heisenberg that the notion that two arrays of numbers, 00:50:15.190 --> 00:50:16.900 align:middle line:84% when you multiply them, you get different 00:50:16.900 --> 00:50:18.370 align:middle line:90% answers depending on the order. 00:50:18.370 --> 00:50:20.560 align:middle line:84% That's what you should generically expect 00:50:20.560 --> 00:50:22.550 align:middle line:90% when you're handling matrices. 00:50:22.550 --> 00:50:25.330 align:middle line:84% So I've just taken two kind of random matrices 00:50:25.330 --> 00:50:28.420 align:middle line:84% just to remind ourselves 2 by 2 matrices, two 00:50:28.420 --> 00:50:30.910 align:middle line:84% arrays of numbers-- any random numbers would do, 00:50:30.910 --> 00:50:33.070 align:middle line:90% and any random components. 00:50:33.070 --> 00:50:36.910 align:middle line:84% Let's first multiply this one times that one in that order. 00:50:36.910 --> 00:50:39.490 align:middle line:84% And as I'm sure you all have learned how to do the matrix 00:50:39.490 --> 00:50:41.860 align:middle line:84% multiplication, this element here 00:50:41.860 --> 00:50:43.630 align:middle line:84% will result from the sum of this times 00:50:43.630 --> 00:50:45.048 align:middle line:90% this plus this times that. 00:50:45.048 --> 00:50:46.840 align:middle line:84% You know how to multiply matrices, I'm Sure 00:50:46.840 --> 00:50:48.440 align:middle line:84% So we can go through the exercise 00:50:48.440 --> 00:50:51.370 align:middle line:84% and multiply this matrix times that in that order. 00:50:51.370 --> 00:50:53.440 align:middle line:84% We'll get a new resultant matrix. 00:50:53.440 --> 00:50:57.160 align:middle line:84% It will also be a square matrix, in this case, 2 by 2. 00:50:57.160 --> 00:51:00.230 align:middle line:84% If we had multiplied these in the opposite order, 00:51:00.230 --> 00:51:03.430 align:middle line:84% if we multiplied B times A-- the exact same matrices multiply 00:51:03.430 --> 00:51:07.630 align:middle line:84% in the opposite order, again as all or most of you have likely 00:51:07.630 --> 00:51:09.940 align:middle line:84% seen by now, we get a different answer-- 00:51:09.940 --> 00:51:11.950 align:middle line:90% a perfectly legitimate answer. 00:51:11.950 --> 00:51:13.550 align:middle line:90% It's still a square matrix. 00:51:13.550 --> 00:51:15.800 align:middle line:84% But the actual matrix is quite different. 00:51:15.800 --> 00:51:17.690 align:middle line:90% The entries are quite different. 00:51:17.690 --> 00:51:19.420 align:middle line:90% There's no 6 over here. 00:51:19.420 --> 00:51:21.370 align:middle line:84% There's no 1 over here and so on. 00:51:21.370 --> 00:51:24.315 align:middle line:84% A times B does not equal B times A in general 00:51:24.315 --> 00:51:25.690 align:middle line:84% when you're multiplying matrices. 00:51:25.690 --> 00:51:29.170 align:middle line:84% Heisenberg didn't know that in 1925. 00:51:29.170 --> 00:51:33.340 align:middle line:84% And so Max Born, his newest kind of mentor and colleague 00:51:33.340 --> 00:51:36.220 align:middle line:84% in Gottingen had to explain to him what most of us 00:51:36.220 --> 00:51:38.390 align:middle line:84% now learn very early in our training. 00:51:38.390 --> 00:51:40.820 align:middle line:84% Most of us learn that because of Heisenberg's work. 00:51:40.820 --> 00:51:43.720 align:middle line:84% We now have many reasons to want to be 00:51:43.720 --> 00:51:47.380 align:middle line:84% able to manipulate matrices in all kinds of investigations. 00:51:47.380 --> 00:51:49.510 align:middle line:84% So matrices-- the fancy way to say this 00:51:49.510 --> 00:51:51.430 align:middle line:90% is that matrices do not commute. 00:51:51.430 --> 00:51:54.820 align:middle line:84% That just means that the outcome of an operation 00:51:54.820 --> 00:51:58.490 align:middle line:84% like multiplication depends on the ordering. 00:51:58.490 --> 00:52:01.160 align:middle line:84% So to get-- that's, again, a bit abstract. 00:52:01.160 --> 00:52:03.350 align:middle line:84% And if you have not spent a lot of time with that, 00:52:03.350 --> 00:52:05.810 align:middle line:90% that's OK, of course. 00:52:05.810 --> 00:52:08.160 align:middle line:84% I find the following example pretty helpful. 00:52:08.160 --> 00:52:10.410 align:middle line:84% I always mess this up when I try to do this in person. 00:52:10.410 --> 00:52:12.650 align:middle line:84% So this is the one benefit of doing this over Zoom 00:52:12.650 --> 00:52:14.750 align:middle line:84% is I have this picture to guide me. 00:52:14.750 --> 00:52:16.760 align:middle line:84% One kind of matrix we might consider 00:52:16.760 --> 00:52:19.250 align:middle line:84% would be the rotation matrix in three dimensions. 00:52:19.250 --> 00:52:23.060 align:middle line:84% We saw rotation matrices when we talked about Minkowski 00:52:23.060 --> 00:52:24.930 align:middle line:90% and special relativity. 00:52:24.930 --> 00:52:26.550 align:middle line:90% Now, imagine we want to have-- 00:52:26.550 --> 00:52:29.270 align:middle line:84% describe rotations in three dimensions of space-- 00:52:29.270 --> 00:52:30.350 align:middle line:90% x, y, and z. 00:52:30.350 --> 00:52:32.120 align:middle line:90% So imagine I have a book-- 00:52:32.120 --> 00:52:33.320 align:middle line:90% this little rectangle here. 00:52:33.320 --> 00:52:35.810 align:middle line:84% In class, I would do holding a book. 00:52:35.810 --> 00:52:37.610 align:middle line:84% And let's say I want to first rotate 00:52:37.610 --> 00:52:39.740 align:middle line:90% this book in the yz plane. 00:52:39.740 --> 00:52:42.170 align:middle line:84% I'm going to rotate it down first. 00:52:42.170 --> 00:52:45.230 align:middle line:84% Then separately, my next operation 00:52:45.230 --> 00:52:47.640 align:middle line:84% is to rotate that book in the xy plane. 00:52:47.640 --> 00:52:50.330 align:middle line:84% So I'm going to rotate it down by 90 degrees 00:52:50.330 --> 00:52:54.950 align:middle line:84% and then over by 90 degrees, and my result of applying those two 00:52:54.950 --> 00:52:56.180 align:middle line:90% matrices-- 00:52:56.180 --> 00:52:59.610 align:middle line:84% the matrix corresponding to rotation only in the yz plane 00:52:59.610 --> 00:53:03.330 align:middle line:84% by 90 degrees, then applying a separate matrix only rotation 00:53:03.330 --> 00:53:05.080 align:middle line:90% in the xy plane by 90 degrees. 00:53:05.080 --> 00:53:06.630 align:middle line:84% Here's the result. So now, the book 00:53:06.630 --> 00:53:08.280 align:middle line:84% is pointing in a certain orientation 00:53:08.280 --> 00:53:10.660 align:middle line:90% different from where it started. 00:53:10.660 --> 00:53:13.920 align:middle line:84% Let me apply the exact same two rotation matrices 00:53:13.920 --> 00:53:15.610 align:middle line:90% but in the opposite order. 00:53:15.610 --> 00:53:18.160 align:middle line:84% So let me start with the book in the same starting position. 00:53:18.160 --> 00:53:22.180 align:middle line:84% Now, I'll rotate in the xy plane first by 90 degrees 00:53:22.180 --> 00:53:25.290 align:middle line:84% and then in the yz plane by 90 degrees. 00:53:25.290 --> 00:53:27.400 align:middle line:84% You see the orientation is different. 00:53:27.400 --> 00:53:29.740 align:middle line:84% This is, for me at least, I find that helpful, 00:53:29.740 --> 00:53:32.700 align:middle line:84% especially when I don't mess it up in person, as a reminder 00:53:32.700 --> 00:53:36.090 align:middle line:84% that the outcome or the product of the multiplication 00:53:36.090 --> 00:53:38.040 align:middle line:84% of matrices depends on their order. 00:53:38.040 --> 00:53:40.770 align:middle line:84% It really can be as simple as that thinking about moving 00:53:40.770 --> 00:53:44.040 align:middle line:84% things around in an ordinary three-dimensional space, let 00:53:44.040 --> 00:53:47.380 align:middle line:84% alone any abstract matrices we might be manipulating. 00:53:47.380 --> 00:53:50.670 align:middle line:84% So this becomes known, therefore, as matrix mechanics. 00:53:50.670 --> 00:53:54.030 align:middle line:84% The work that Heisenberg introduces in 1925, 00:53:54.030 --> 00:53:55.740 align:middle line:84% and soon he works very closely with Max 00:53:55.740 --> 00:53:58.620 align:middle line:84% Born and another colleague in Gottingen, Pascual Jordan, 00:53:58.620 --> 00:54:01.380 align:middle line:84% they develop an entire first principles 00:54:01.380 --> 00:54:04.830 align:middle line:84% mechanics of the quantum realm-- a quantum mechanics 00:54:04.830 --> 00:54:06.760 align:middle line:90% built around these matrices. 00:54:06.760 --> 00:54:09.030 align:middle line:84% So this first version is called matrix mechanics 00:54:09.030 --> 00:54:11.780 align:middle line:90% dating from 1925 originally. 00:54:11.780 --> 00:54:12.710 align:middle line:90% Any questions on that? 00:54:12.710 --> 00:54:18.020 align:middle line:90% 00:54:18.020 --> 00:54:20.510 align:middle line:84% I still find it just stunning that 00:54:20.510 --> 00:54:24.140 align:middle line:84% this super hotshot young kid, Werner Heisenberg, 00:54:24.140 --> 00:54:27.260 align:middle line:84% didn't know what matrices were and hadn't realized that you 00:54:27.260 --> 00:54:29.240 align:middle line:84% could actually multiply arrays of numbers 00:54:29.240 --> 00:54:31.657 align:middle line:84% and worry about the order-- have to worry about the order. 00:54:31.657 --> 00:54:37.643 align:middle line:90% 00:54:37.643 --> 00:54:39.810 align:middle line:84% Any questions about the framework or the motivation? 00:54:39.810 --> 00:54:41.670 align:middle line:84% We can see, again, he's clearly trying 00:54:41.670 --> 00:54:43.830 align:middle line:84% to do the kind of young Einstein, 00:54:43.830 --> 00:54:45.510 align:middle line:90% Ernst Mach kind of thing. 00:54:45.510 --> 00:54:47.200 align:middle line:84% That's what sets him down that path. 00:54:47.200 --> 00:54:49.200 align:middle line:84% And he's literally removed from the conventions. 00:54:49.200 --> 00:54:52.980 align:middle line:84% He has to move to the island of Heligoland for a short break. 00:54:52.980 --> 00:54:54.426 align:middle line:90% I find that pretty interesting. 00:54:54.426 --> 00:55:00.260 align:middle line:90% 00:55:00.260 --> 00:55:00.920 align:middle line:90% No questions? 00:55:00.920 --> 00:55:02.120 align:middle line:90% It's all perfectly obvious. 00:55:02.120 --> 00:55:04.040 align:middle line:90% All of you would have done that? 00:55:04.040 --> 00:55:05.543 align:middle line:90% I wouldn't have done that. 00:55:05.543 --> 00:55:07.710 align:middle line:84% Jesus asks, did Einstein have any particular reasons 00:55:07.710 --> 00:55:09.150 align:middle line:84% for thinking Heisenberg should be careful of positive-- 00:55:09.150 --> 00:55:10.200 align:middle line:90% oh, thank you, Jesus. 00:55:10.200 --> 00:55:11.430 align:middle line:90% Great question. 00:55:11.430 --> 00:55:12.270 align:middle line:90% Really good. 00:55:12.270 --> 00:55:14.460 align:middle line:84% And I think the answer is-- well, the answer is yes. 00:55:14.460 --> 00:55:16.320 align:middle line:84% I think Einstein did have reasons. 00:55:16.320 --> 00:55:18.960 align:middle line:84% And it really tells us, so to speak, more about Einstein 00:55:18.960 --> 00:55:21.690 align:middle line:84% than about the value of that kind of argument. 00:55:21.690 --> 00:55:26.130 align:middle line:84% By 1925, Einstein's thinking was really quite different 00:55:26.130 --> 00:55:28.230 align:middle line:84% than it had been in 1905, and a lot of it 00:55:28.230 --> 00:55:31.440 align:middle line:84% had to do with his experience in between with what became 00:55:31.440 --> 00:55:33.580 align:middle line:90% known as general relativity. 00:55:33.580 --> 00:55:36.240 align:middle line:84% So we saw that in 1905, Einstein was 00:55:36.240 --> 00:55:39.420 align:middle line:84% very concerned with this kind of Machian positivism 00:55:39.420 --> 00:55:41.280 align:middle line:84% with what we might call operationalism. 00:55:41.280 --> 00:55:44.940 align:middle line:84% Tell me exactly how I measure the time-- 00:55:44.940 --> 00:55:47.040 align:middle line:84% I have my little hand on my watch-- 00:55:47.040 --> 00:55:50.220 align:middle line:84% all that kind of spelling things out empirically and kind 00:55:50.220 --> 00:55:51.660 align:middle line:90% of operationally. 00:55:51.660 --> 00:55:54.600 align:middle line:84% And he gets more and more kind of separated from that, 00:55:54.600 --> 00:55:57.180 align:middle line:84% or drifts away from that, let's say, 00:55:57.180 --> 00:56:00.330 align:middle line:84% as he begins working more and more on general relativity, 00:56:00.330 --> 00:56:02.100 align:middle line:84% where he gets more and more involved 00:56:02.100 --> 00:56:05.950 align:middle line:84% with pretty fancy, but also pretty abstract mathematics. 00:56:05.950 --> 00:56:07.570 align:middle line:84% He's thinking eventually not only 00:56:07.570 --> 00:56:10.750 align:middle line:84% about what would an observer see sealed up 00:56:10.750 --> 00:56:13.030 align:middle line:84% in an elevator or a spaceship, but actually 00:56:13.030 --> 00:56:15.940 align:middle line:84% about global properties of an entire universe that 00:56:15.940 --> 00:56:17.990 align:middle line:90% might be warping and so on. 00:56:17.990 --> 00:56:22.310 align:middle line:84% So by 1915, especially by 1917 and later, 00:56:22.310 --> 00:56:23.980 align:middle line:84% meaning before Heisenberg's work, 00:56:23.980 --> 00:56:27.280 align:middle line:84% Einstein has moved to thinking about things like cosmology, 00:56:27.280 --> 00:56:30.220 align:middle line:84% entire universes, which certainly not all of which 00:56:30.220 --> 00:56:33.790 align:middle line:84% could be subject to the kind of Mach-like empiricism 00:56:33.790 --> 00:56:35.770 align:middle line:90% that had driven his 1905 work. 00:56:35.770 --> 00:56:40.720 align:middle line:84% So he becomes more driven by both a kind of beauty and power 00:56:40.720 --> 00:56:43.960 align:middle line:84% of mathematical reasoning, which had really not been clearly 00:56:43.960 --> 00:56:47.110 align:middle line:84% a driver for him in 1905, and also, just 00:56:47.110 --> 00:56:50.422 align:middle line:84% moving away from the kind of litmus test of, 00:56:50.422 --> 00:56:51.130 align:middle line:90% is it measurable? 00:56:51.130 --> 00:56:52.760 align:middle line:90% Can I smell it or taste it? 00:56:52.760 --> 00:56:55.660 align:middle line:84% So I think Einstein's comment-- if it ever happened. 00:56:55.660 --> 00:56:57.850 align:middle line:84% We learned that only through Heisenberg's much later 00:56:57.850 --> 00:56:58.790 align:middle line:90% recollections. 00:56:58.790 --> 00:57:00.850 align:middle line:90% It might be an apocryphal story. 00:57:00.850 --> 00:57:02.068 align:middle line:90% But it sort of adds up. 00:57:02.068 --> 00:57:03.610 align:middle line:84% Even if it didn't really happen, it's 00:57:03.610 --> 00:57:07.870 align:middle line:84% plausible because of Einstein's own kind of shift 00:57:07.870 --> 00:57:11.840 align:middle line:90% in his intellectual approaches. 00:57:11.840 --> 00:57:14.110 align:middle line:84% So I take that to be more a comment about Einstein 00:57:14.110 --> 00:57:17.895 align:middle line:84% than about is Machian positivism good or bad for science? 00:57:17.895 --> 00:57:20.020 align:middle line:84% I think we can still debate that on separate terms. 00:57:20.020 --> 00:57:24.350 align:middle line:84% I think it tells us about Einstein's thinking. 00:57:24.350 --> 00:57:25.190 align:middle line:90% I agree with Alex. 00:57:25.190 --> 00:57:28.780 align:middle line:84% It's hard to believe a time when leading scholars, let 00:57:28.780 --> 00:57:32.640 align:middle line:84% alone anyone at all, would never have heard about matrices. 00:57:32.640 --> 00:57:33.140 align:middle line:90% I agree. 00:57:33.140 --> 00:57:36.460 align:middle line:84% I think that's part of what's so kind of-- 00:57:36.460 --> 00:57:38.560 align:middle line:84% I don't know-- kind of touching about this story. 00:57:38.560 --> 00:57:40.300 align:middle line:84% Heisenberg was both like this brash kid, 00:57:40.300 --> 00:57:42.280 align:middle line:84% saying everyone-- all my teachers are wrong. 00:57:42.280 --> 00:57:44.410 align:middle line:84% That's not unique in the history of science. 00:57:44.410 --> 00:57:46.060 align:middle line:90% Young people often say that. 00:57:46.060 --> 00:57:49.240 align:middle line:84% But also, in some sense, kind of just so naive. 00:57:49.240 --> 00:57:53.650 align:middle line:84% So like, I'm going to set off on my own, literally leave town. 00:57:53.650 --> 00:57:56.730 align:middle line:84% Everything that's been done [INAUDIBLE] for these 22 years 00:57:56.730 --> 00:57:57.980 align:middle line:90% that I've been on the planet-- 00:57:57.980 --> 00:58:01.180 align:middle line:84% he's really young-- that's all just a dead end. 00:58:01.180 --> 00:58:04.212 align:middle line:84% And yet, also just be like, how are you 00:58:04.212 --> 00:58:05.920 align:middle line:84% going to do your next steps there, buddy? 00:58:05.920 --> 00:58:07.240 align:middle line:90% Have you heard about matrices? 00:58:07.240 --> 00:58:10.350 align:middle line:84% It's a remarkable kind of moment to me. 00:58:10.350 --> 00:58:12.935 align:middle line:84% And so I just find that story really compelling. 00:58:12.935 --> 00:58:14.310 align:middle line:84% Fisher asks, is there is anything 00:58:14.310 --> 00:58:16.810 align:middle line:84% in this class has taught me that you can never [INAUDIBLE].. 00:58:16.810 --> 00:58:17.340 align:middle line:90% Fair enough. 00:58:17.340 --> 00:58:21.960 align:middle line:84% So the course 18 rolls will now swell from this class. 00:58:21.960 --> 00:58:23.820 align:middle line:84% Taking more and more math is probably 00:58:23.820 --> 00:58:25.740 align:middle line:84% good for one's soul anyway, and certainly can 00:58:25.740 --> 00:58:27.553 align:middle line:90% be helpful in unexpected ways. 00:58:27.553 --> 00:58:29.220 align:middle line:84% By the way, we'll see this kind of thing 00:58:29.220 --> 00:58:32.303 align:middle line:84% happen up happen even later this term. 00:58:32.303 --> 00:58:33.720 align:middle line:84% Well after the middle of the term, 00:58:33.720 --> 00:58:37.105 align:middle line:84% we'll see examples from later in the 20th century, where 00:58:37.105 --> 00:58:38.730 align:middle line:84% I think the same kind of thing happens, 00:58:38.730 --> 00:58:42.450 align:middle line:84% where some physicists kind of accidentally 00:58:42.450 --> 00:58:43.980 align:middle line:84% or for other reasons find that they 00:58:43.980 --> 00:58:47.310 align:middle line:84% get really interested in ways of classifying 00:58:47.310 --> 00:58:48.900 align:middle line:90% large groups of objects-- 00:58:48.900 --> 00:58:50.250 align:middle line:90% what's like with like-- 00:58:50.250 --> 00:58:52.530 align:middle line:84% without having studied things like group theory, which 00:58:52.530 --> 00:58:55.110 align:middle line:84% was by then a very, very well-established branch 00:58:55.110 --> 00:58:57.340 align:middle line:84% of pure mathematics, lead groups and all the rest. 00:58:57.340 --> 00:59:00.750 align:middle line:84% So there's a not uncommon kind of pattern 00:59:00.750 --> 00:59:03.240 align:middle line:84% that's not only limited to Einstein or Heisenberg 00:59:03.240 --> 00:59:05.820 align:middle line:90% or these other folks. 00:59:05.820 --> 00:59:07.990 align:middle line:84% It feels like the flip scenario of the TRIPOS, where 00:59:07.990 --> 00:59:08.980 align:middle line:84% those students could do so much math, 00:59:08.980 --> 00:59:09.805 align:middle line:90% but didn't typically learn. 00:59:09.805 --> 00:59:10.060 align:middle line:90% OK. 00:59:10.060 --> 00:59:11.268 align:middle line:90% Yeah, the tables [INAUDIBLE]. 00:59:11.268 --> 00:59:11.950 align:middle line:90% That's right. 00:59:11.950 --> 00:59:13.010 align:middle line:90% That's a good example. 00:59:13.010 --> 00:59:15.580 align:middle line:84% So Heisenberg clearly was not a Cambridge student. 00:59:15.580 --> 00:59:18.800 align:middle line:84% I think it's an excellent thing to keep in mind. 00:59:18.800 --> 00:59:20.750 align:middle line:84% And that's not to make fun of Heisenberg. 00:59:20.750 --> 00:59:21.940 align:middle line:90% He was very well-trained. 00:59:21.940 --> 00:59:25.390 align:middle line:84% He was trained at one of Central Europe's finest academies 00:59:25.390 --> 00:59:27.700 align:middle line:90% for training young scientists. 00:59:27.700 --> 00:59:30.670 align:middle line:84% He and so many of his cohort would went on 00:59:30.670 --> 00:59:33.340 align:middle line:84% to-- would go on to earn extraordinary recognition 00:59:33.340 --> 00:59:34.090 align:middle line:90% in the field. 00:59:34.090 --> 00:59:36.880 align:middle line:84% It was just a different kind of training-- different choices 00:59:36.880 --> 00:59:39.490 align:middle line:84% about what to prioritize even for very young students. 00:59:39.490 --> 00:59:43.450 align:middle line:84% I find that fascinating, the kind of different options 00:59:43.450 --> 00:59:45.440 align:middle line:84% that are being prioritized at different times. 00:59:45.440 --> 00:59:47.065 align:middle line:84% I think that's one lesson we can learn. 00:59:47.065 --> 00:59:49.030 align:middle line:84% We can step back and take in a kind 00:59:49.030 --> 00:59:55.070 align:middle line:84% broader view, a sample across a few different institutions. 00:59:55.070 --> 00:59:57.648 align:middle line:84% So it seems like physicists are getting younger too. 00:59:57.648 --> 00:59:59.440 align:middle line:84% He was in the Swiss Polytech as a teenager, 00:59:59.440 --> 01:00:01.107 align:middle line:84% where the Wranglers were into their 20s. 01:00:01.107 --> 01:00:01.820 align:middle line:90% Ah. 01:00:01.820 --> 01:00:04.248 align:middle line:84% So I'm not sure that's quite right, Fisher. 01:00:04.248 --> 01:00:06.290 align:middle line:84% So I think the idea that there is a lot of people 01:00:06.290 --> 01:00:08.250 align:middle line:90% will just never leave Cambridge. 01:00:08.250 --> 01:00:09.380 align:middle line:90% It's still true, actually. 01:00:09.380 --> 01:00:12.200 align:middle line:84% So the actual people competing to be Wranglers 01:00:12.200 --> 01:00:14.030 align:middle line:84% were basically undergraduates and usually 01:00:14.030 --> 01:00:16.800 align:middle line:84% on what was a kind of canonical age for that. 01:00:16.800 --> 01:00:19.850 align:middle line:84% But then a bunch of them would stay on in town. 01:00:19.850 --> 01:00:22.940 align:middle line:84% The Wranglers would stay, sometimes get fellowships 01:00:22.940 --> 01:00:25.538 align:middle line:84% to then do almost like a research fellowship, 01:00:25.538 --> 01:00:27.830 align:middle line:84% which sometimes would be like grad school and sometimes 01:00:27.830 --> 01:00:28.610 align:middle line:90% wouldn't. 01:00:28.610 --> 01:00:30.530 align:middle line:84% There's a whole tradition of never bothering 01:00:30.530 --> 01:00:32.170 align:middle line:90% to get a PhD from Cambridge. 01:00:32.170 --> 01:00:34.700 align:middle line:84% They would be called Mr. They were at this point 01:00:34.700 --> 01:00:36.380 align:middle line:84% exclusively-- almost exclusively-- men. 01:00:36.380 --> 01:00:39.020 align:middle line:84% They'd be called Mr. so-and-so, even 01:00:39.020 --> 01:00:41.540 align:middle line:84% though they were on they were paid to do research full time. 01:00:41.540 --> 01:00:44.480 align:middle line:84% Or they would kind of tutor TRIPOS students 01:00:44.480 --> 01:00:46.830 align:middle line:84% while pursuing their own later degrees. 01:00:46.830 --> 01:00:49.340 align:middle line:84% So the Wranglers-- many of them would hang around longer. 01:00:49.340 --> 01:00:51.470 align:middle line:84% And they'd have the Wrangler title for life. 01:00:51.470 --> 01:00:54.530 align:middle line:84% But the actual-- they had to sit for their exam in the Senate 01:00:54.530 --> 01:00:57.650 align:middle line:84% house when they were approximately 21 or 22, 01:00:57.650 --> 01:00:58.940 align:middle line:90% typically. 01:00:58.940 --> 01:01:02.510 align:middle line:84% So Heisenberg had skipped grades, in that sense. 01:01:02.510 --> 01:01:08.180 align:middle line:84% He was finishing his brief PhD around the age of 22. 01:01:08.180 --> 01:01:09.380 align:middle line:90% Yeah, anyway. 01:01:09.380 --> 01:01:10.730 align:middle line:90% So some of you might know-- 01:01:10.730 --> 01:01:13.173 align:middle line:84% we'll come soon actually in coming-- well, not so soon. 01:01:13.173 --> 01:01:14.840 align:middle line:84% A couple of weeks we'll come to the work 01:01:14.840 --> 01:01:17.780 align:middle line:84% of the preeminent physicist Freeman Dyson, 01:01:17.780 --> 01:01:20.990 align:middle line:84% who actually just passed away this past spring in his 90s. 01:01:20.990 --> 01:01:24.860 align:middle line:84% Dyson was one of these British guys who just never got a PhD. 01:01:24.860 --> 01:01:27.230 align:middle line:84% He was hired as a full professor at Cornell 01:01:27.230 --> 01:01:30.380 align:middle line:84% at like the age of 26, and he was Mr. Dyson. 01:01:30.380 --> 01:01:33.688 align:middle line:84% So I mean, that was a very kind typical British-- like, 01:01:33.688 --> 01:01:34.730 align:middle line:90% almost like in your face. 01:01:34.730 --> 01:01:37.280 align:middle line:84% Like, I'm so good I don't even need that stupid grad school 01:01:37.280 --> 01:01:38.000 align:middle line:90% thing. 01:01:38.000 --> 01:01:39.500 align:middle line:84% That's a separate kind of tradition 01:01:39.500 --> 01:01:41.417 align:middle line:84% we can talk about when we come back to Freeman 01:01:41.417 --> 01:01:42.330 align:middle line:90% Dyson in a few weeks. 01:01:42.330 --> 01:01:42.830 align:middle line:90% OK. 01:01:42.830 --> 01:01:43.130 align:middle line:90% Let me go. 01:01:43.130 --> 01:01:45.005 align:middle line:84% And for the last part for today, unless there 01:01:45.005 --> 01:01:47.720 align:middle line:84% are any more questions on Heisenberg and matrix 01:01:47.720 --> 01:01:48.920 align:middle line:90% mechanics. 01:01:48.920 --> 01:01:51.830 align:middle line:84% And again, it probably is already obvious to you, 01:01:51.830 --> 01:01:55.880 align:middle line:84% but just to say if parts of the kind of maneuvers 01:01:55.880 --> 01:01:57.620 align:middle line:84% that Heisenberg does in that paper that 01:01:57.620 --> 01:02:00.988 align:middle line:84% is on the reading for today, if it's hard to follow all the ins 01:02:00.988 --> 01:02:03.530 align:middle line:84% and outs of that paper, first of all, you're in good company. 01:02:03.530 --> 01:02:05.810 align:middle line:84% It's hard even for today's physicists 01:02:05.810 --> 01:02:07.940 align:middle line:84% because it's just not how we do things today. 01:02:07.940 --> 01:02:12.290 align:middle line:84% So it looks weird even to the pros, so that's OK. 01:02:12.290 --> 01:02:14.480 align:middle line:84% And for our purposes, I think what's 01:02:14.480 --> 01:02:17.690 align:middle line:84% most fascinating is really just the first opening 01:02:17.690 --> 01:02:20.030 align:middle line:84% steps, this kind of manifesto, the kind 01:02:20.030 --> 01:02:22.580 align:middle line:90% of Mach-like manifesto. 01:02:22.580 --> 01:02:26.270 align:middle line:84% And then just this concentration on frequencies-- 01:02:26.270 --> 01:02:28.490 align:middle line:84% obey this empirical relation, frequencies 01:02:28.490 --> 01:02:29.690 align:middle line:90% appear in an exponent. 01:02:29.690 --> 01:02:31.160 align:middle line:90% Let me follow that down. 01:02:31.160 --> 01:02:33.080 align:middle line:84% If that much is clear, that's what 01:02:33.080 --> 01:02:34.455 align:middle line:84% I hope we'll get out of the paper 01:02:34.455 --> 01:02:36.497 align:middle line:84% even though he goes into a little bit more detail 01:02:36.497 --> 01:02:37.270 align:middle line:90% and gets pretty-- 01:02:37.270 --> 01:02:40.550 align:middle line:84% a little more complicated after page two. 01:02:40.550 --> 01:02:43.360 align:middle line:84% Don't worry about that if it's hard to follow. 01:02:43.360 --> 01:02:46.663 align:middle line:84% Jesus says, it's cool how in the late 19th century in Britain, 01:02:46.663 --> 01:02:48.080 align:middle line:84% a lot of leading physics was based 01:02:48.080 --> 01:02:49.850 align:middle line:84% on complicated mathematical models. 01:02:49.850 --> 01:02:51.718 align:middle line:84% Whereas, Einstein and Heisenberg, 01:02:51.718 --> 01:02:54.260 align:middle line:84% who seemed to have cared less about math than the Wranglers-- 01:02:54.260 --> 01:02:55.880 align:middle line:84% certainly at that stage-- focus on 01:02:55.880 --> 01:02:57.400 align:middle line:84% unintuitive non-mathematical ideas 01:02:57.400 --> 01:02:58.080 align:middle line:90% as the basis for their work. 01:02:58.080 --> 01:02:58.913 align:middle line:90% I think that's true. 01:02:58.913 --> 01:03:01.310 align:middle line:84% Or at least there's a kind of spectrum. 01:03:01.310 --> 01:03:03.710 align:middle line:84% I think then, as now, there's a range of ways 01:03:03.710 --> 01:03:06.440 align:middle line:84% that have proven to be productive in trying 01:03:06.440 --> 01:03:07.700 align:middle line:90% to learn about nature. 01:03:07.700 --> 01:03:08.867 align:middle line:90% And isn't that a good thing? 01:03:08.867 --> 01:03:11.325 align:middle line:84% And now, I think we hopefully have learned that lesson even 01:03:11.325 --> 01:03:13.490 align:middle line:84% more so, if I may get on a soapbox for a second. 01:03:13.490 --> 01:03:15.140 align:middle line:84% Shouldn't we be working even harder 01:03:15.140 --> 01:03:18.320 align:middle line:84% to get even more perspectives and life experiences 01:03:18.320 --> 01:03:20.690 align:middle line:90% and people in the mix? 01:03:20.690 --> 01:03:22.565 align:middle line:84% Because there's not one way to do this right. 01:03:22.565 --> 01:03:25.065 align:middle line:84% And so shouldn't we work even harder than frankly we've been 01:03:25.065 --> 01:03:26.900 align:middle line:84% able to today-- than we have to date-- 01:03:26.900 --> 01:03:29.510 align:middle line:84% to increase the type-- the numbers of people 01:03:29.510 --> 01:03:31.010 align:middle line:84% and the types of people from whence 01:03:31.010 --> 01:03:33.680 align:middle line:84% they come with the whole grab bag of experience. 01:03:33.680 --> 01:03:35.120 align:middle line:84% Just for this lesson, even we only 01:03:35.120 --> 01:03:38.480 align:middle line:84% look at very narrowly demographically a range of ways 01:03:38.480 --> 01:03:40.560 align:middle line:84% to succeed, doesn't that lesson suggest 01:03:40.560 --> 01:03:42.045 align:middle line:90% even broader [INAUDIBLE]? 01:03:42.045 --> 01:03:44.420 align:middle line:84% Hopefully, we'll keep that in mind not just for this term 01:03:44.420 --> 01:03:45.020 align:middle line:90% but beyond. 01:03:45.020 --> 01:03:46.380 align:middle line:90% But it's a great point. 01:03:46.380 --> 01:03:48.590 align:middle line:84% I agree, that there was more than one way 01:03:48.590 --> 01:03:53.830 align:middle line:84% to do pretty interesting and original work circa 1910, 1920. 01:03:53.830 --> 01:03:55.580 align:middle line:84% And that's a lesson, I think, again, we'll 01:03:55.580 --> 01:03:57.120 align:middle line:90% see throughout this term. 01:03:57.120 --> 01:03:57.620 align:middle line:90% OK. 01:03:57.620 --> 01:03:58.120 align:middle line:90% Good. 01:03:58.120 --> 01:04:01.192 align:middle line:84% Let me go through this last part because now we want to see 01:04:01.192 --> 01:04:03.650 align:middle line:84% what does Heisenberg do once he knows that there are things 01:04:03.650 --> 01:04:05.073 align:middle line:90% called matrices in the world? 01:04:05.073 --> 01:04:06.740 align:middle line:84% He keeps thinking about this, of course. 01:04:06.740 --> 01:04:08.360 align:middle line:90% He's not done. 01:04:08.360 --> 01:04:11.530 align:middle line:84% And so with Max Born's help, he's 01:04:11.530 --> 01:04:13.930 align:middle line:84% clarified some of the basic mathematics, what 01:04:13.930 --> 01:04:17.020 align:middle line:84% he had kind of stumbled into thinking 01:04:17.020 --> 01:04:20.560 align:middle line:84% about were these arrays of numbers, like matrices. 01:04:20.560 --> 01:04:23.650 align:middle line:84% He then had learned kind of bumblingly on his own 01:04:23.650 --> 01:04:28.000 align:middle line:84% and with more formal help from Born in Gottingen 01:04:28.000 --> 01:04:30.130 align:middle line:84% that the outcome of transformations 01:04:30.130 --> 01:04:32.180 align:middle line:84% depends on the order of operations. 01:04:32.180 --> 01:04:33.970 align:middle line:90% These matrices don't commute. 01:04:33.970 --> 01:04:35.830 align:middle line:84% Now, Heisenberg, again, is still trying 01:04:35.830 --> 01:04:38.620 align:middle line:84% to make sense of that kind of physically or conceptually. 01:04:38.620 --> 01:04:42.080 align:middle line:84% It still doesn't kind of sit with him all at once. 01:04:42.080 --> 01:04:46.390 align:middle line:84% So a few years later now in the spring of 1927, just two years 01:04:46.390 --> 01:04:50.788 align:middle line:84% after his trip to Heligoland, he returns to Born's 01:04:50.788 --> 01:04:52.580 align:middle line:84% Institute in Copenhagen for another visit-- 01:04:52.580 --> 01:04:55.360 align:middle line:84% a basically, kind of longer-- another postdoctoral visit. 01:04:55.360 --> 01:04:57.010 align:middle line:84% And now he's trying-- he's still kind 01:04:57.010 --> 01:05:01.210 align:middle line:84% of trying to make sense of what this noncommuting matrix 01:05:01.210 --> 01:05:03.970 align:middle line:84% stuff might mean physically or conceptually. 01:05:03.970 --> 01:05:08.320 align:middle line:84% And he returns to, again, a kind of simple thought experiment, 01:05:08.320 --> 01:05:11.170 align:middle line:84% one that he can kind of picture or visualize in his mind. 01:05:11.170 --> 01:05:13.868 align:middle line:84% It comes to be known as the gamma ray microscope. 01:05:13.868 --> 01:05:15.660 align:middle line:84% And he thinks about it really in these kind 01:05:15.660 --> 01:05:16.860 align:middle line:90% of cartoon-like forms. 01:05:16.860 --> 01:05:19.878 align:middle line:84% So my cartoon here is, I think, at the level 01:05:19.878 --> 01:05:21.420 align:middle line:84% that I think Heisenberg was beginning 01:05:21.420 --> 01:05:23.100 align:middle line:84% to grapple with is when he comes back 01:05:23.100 --> 01:05:26.770 align:middle line:84% to Copenhagen spring, summer 1927. 01:05:26.770 --> 01:05:29.850 align:middle line:84% So let's imagine we're trying to identify or measure 01:05:29.850 --> 01:05:31.890 align:middle line:90% the location of an electron. 01:05:31.890 --> 01:05:33.450 align:middle line:90% Obviously, electrons are small. 01:05:33.450 --> 01:05:35.340 align:middle line:90% We can't just look at them. 01:05:35.340 --> 01:05:37.320 align:middle line:84% We have to use something like a microscope. 01:05:37.320 --> 01:05:40.170 align:middle line:84% How do we see any small things or characterize 01:05:40.170 --> 01:05:40.930 align:middle line:90% their positions? 01:05:40.930 --> 01:05:42.060 align:middle line:90% We use a microscope. 01:05:42.060 --> 01:05:44.230 align:middle line:90% So electrons are really small. 01:05:44.230 --> 01:05:47.980 align:middle line:84% And so we have to use light of a particularly small wavelength, 01:05:47.980 --> 01:05:50.100 align:middle line:84% which is to say, a very high frequency. 01:05:50.100 --> 01:05:52.650 align:middle line:84% And so we'd have to use something like gamma rays-- 01:05:52.650 --> 01:05:54.690 align:middle line:84% rays of light, little beams of light, 01:05:54.690 --> 01:05:57.330 align:middle line:84% that have particularly small wavelengths 01:05:57.330 --> 01:05:59.400 align:middle line:84% on the-- comparable to the size of the thing 01:05:59.400 --> 01:06:01.360 align:middle line:84% we're actually trying to measure. 01:06:01.360 --> 01:06:03.670 align:middle line:84% So how does a microscope work in general? 01:06:03.670 --> 01:06:05.930 align:middle line:84% We have some object we're trying to resolve-- 01:06:05.930 --> 01:06:09.610 align:middle line:84% we're trying to make an image of to measure its location. 01:06:09.610 --> 01:06:12.160 align:middle line:84% We shine light on it and collect the scattered light 01:06:12.160 --> 01:06:13.060 align:middle line:90% in our microscope. 01:06:13.060 --> 01:06:15.748 align:middle line:84% That's what we do, sort of generally. 01:06:15.748 --> 01:06:17.290 align:middle line:84% So if we want to measure the position 01:06:17.290 --> 01:06:20.290 align:middle line:84% of this really tiny thing like an electron, we'll do that. 01:06:20.290 --> 01:06:21.550 align:middle line:90% We'll bounce light off of it. 01:06:21.550 --> 01:06:24.190 align:middle line:84% We'll collect some of that scattered light in our device. 01:06:24.190 --> 01:06:25.550 align:middle line:90% The electron's very small. 01:06:25.550 --> 01:06:28.210 align:middle line:84% So we have to be a little sensible about the kind 01:06:28.210 --> 01:06:29.260 align:middle line:90% of light we'll use. 01:06:29.260 --> 01:06:31.780 align:middle line:84% We'll have to use light of a wavelength that is at least as 01:06:31.780 --> 01:06:34.370 align:middle line:84% small as the object we're trying to measure 01:06:34.370 --> 01:06:37.370 align:middle line:84% and as we have to have reasonable resolution. 01:06:37.370 --> 01:06:40.270 align:middle line:84% We also have to keep in mind there's something called 01:06:40.270 --> 01:06:43.660 align:middle line:84% the resolving power of any actual optical instrument. 01:06:43.660 --> 01:06:45.610 align:middle line:84% It's related to diffraction, or sometimes it's 01:06:45.610 --> 01:06:47.170 align:middle line:90% called the diffraction limit. 01:06:47.170 --> 01:06:48.837 align:middle line:84% Some of you might have learned this even 01:06:48.837 --> 01:06:51.880 align:middle line:84% in kind of classical E&M or optics coursework. 01:06:51.880 --> 01:06:54.910 align:middle line:84% So any of the light that is scattered into our collecting 01:06:54.910 --> 01:06:57.370 align:middle line:84% device, into our microscope, has to fall 01:06:57.370 --> 01:07:00.430 align:middle line:84% within some kind of angular cone or aperture 01:07:00.430 --> 01:07:02.680 align:middle line:90% with some angle we call theta. 01:07:02.680 --> 01:07:05.890 align:middle line:84% And it turns out if we try to squeeze down this aperture 01:07:05.890 --> 01:07:08.740 align:middle line:84% to be smaller and smaller, if we squeeze theta to be a smaller 01:07:08.740 --> 01:07:12.520 align:middle line:84% and smaller angle, we actually-- we lose the ability to make 01:07:12.520 --> 01:07:14.140 align:middle line:90% a sharp picture. 01:07:14.140 --> 01:07:17.050 align:middle line:84% You have to get the balance right between the wavelength 01:07:17.050 --> 01:07:18.250 align:middle line:90% of light you use-- 01:07:18.250 --> 01:07:19.370 align:middle line:90% you want that to be small. 01:07:19.370 --> 01:07:21.100 align:middle line:84% So in principle, it could differentiate 01:07:21.100 --> 01:07:23.110 align:middle line:90% small features of your target. 01:07:23.110 --> 01:07:25.450 align:middle line:84% But you can't make your aperture too small 01:07:25.450 --> 01:07:29.270 align:middle line:84% or you'll get a fuzzed-out, diffraction-limited image. 01:07:29.270 --> 01:07:31.120 align:middle line:84% There's a trade-off, an inverse trade-off 01:07:31.120 --> 01:07:34.110 align:middle line:84% that was known, again, even kind of classically in optics. 01:07:34.110 --> 01:07:35.860 align:middle line:84% If that's not familiar to you, by the way, 01:07:35.860 --> 01:07:37.990 align:middle line:90% you're in good company. 01:07:37.990 --> 01:07:41.123 align:middle line:84% Heisenberg almost failed his PhD qualifying exam 01:07:41.123 --> 01:07:43.540 align:middle line:84% because he couldn't remember the diffraction limit for how 01:07:43.540 --> 01:07:45.320 align:middle line:90% actual microscopes work. 01:07:45.320 --> 01:07:48.700 align:middle line:84% So he had forgotten this quite familiar classical result. 01:07:48.700 --> 01:07:51.820 align:middle line:84% It stuck with him after that, and so he was back on his mind 01:07:51.820 --> 01:07:53.320 align:middle line:90% in 1927. 01:07:53.320 --> 01:07:56.290 align:middle line:84% There's this trade-off in what we call the resolving power 01:07:56.290 --> 01:07:58.125 align:middle line:84% proportional to the wavelength, inverse 01:07:58.125 --> 01:08:00.520 align:middle line:84% or proportional with a measure of the kind of aperture 01:08:00.520 --> 01:08:02.620 align:middle line:90% or angular size. 01:08:02.620 --> 01:08:04.630 align:middle line:84% So the aperture will collect light, 01:08:04.630 --> 01:08:08.960 align:middle line:84% the scattered light, again, with a range of scattered momentum. 01:08:08.960 --> 01:08:13.120 align:middle line:84% So not any old scattered light beam will fall into our device. 01:08:13.120 --> 01:08:16.240 align:middle line:84% There'll be momenta, whose vector quantities fall 01:08:16.240 --> 01:08:18.080 align:middle line:84% within this kind of angular cone-- 01:08:18.080 --> 01:08:20.180 align:middle line:90% a cone of angular size theta. 01:08:20.180 --> 01:08:21.939 align:middle line:84% So that means these scattered photons 01:08:21.939 --> 01:08:25.270 align:middle line:84% in order to fall into our device could have any component 01:08:25.270 --> 01:08:26.200 align:middle line:90% in this direction. 01:08:26.200 --> 01:08:29.090 align:middle line:84% I'll call this vertical direction now the x direction. 01:08:29.090 --> 01:08:31.510 align:middle line:84% We're trying to measure the position of the electron 01:08:31.510 --> 01:08:33.160 align:middle line:90% in this direction. 01:08:33.160 --> 01:08:37.750 align:middle line:84% What's the component of the scattered photon's momentum 01:08:37.750 --> 01:08:39.790 align:middle line:84% that would enable that photon to be 01:08:39.790 --> 01:08:42.399 align:middle line:90% captured by us in our device? 01:08:42.399 --> 01:08:48.670 align:middle line:84% Any component delta px that falls within the aperture, 01:08:48.670 --> 01:08:52.270 align:middle line:84% and that's just the sine theta component of our hypotenuse 01:08:52.270 --> 01:08:53.500 align:middle line:90% here. 01:08:53.500 --> 01:08:55.210 align:middle line:90% Well, now we've just-- 01:08:55.210 --> 01:08:57.729 align:middle line:84% as Compton had already learned, we've now just 01:08:57.729 --> 01:09:01.899 align:middle line:84% basically produced a collision, a kind of two-body collision, 01:09:01.899 --> 01:09:05.020 align:middle line:84% between an electron and a very high-energy photon. 01:09:05.020 --> 01:09:08.250 align:middle line:84% So there'll be some recoil of the electron. 01:09:08.250 --> 01:09:10.080 align:middle line:84% There'll be some scatter of the electron 01:09:10.080 --> 01:09:12.870 align:middle line:84% because it's just been smacked by some very 01:09:12.870 --> 01:09:14.160 align:middle line:90% high-energy photon. 01:09:14.160 --> 01:09:17.340 align:middle line:84% So whatever the uncertainty in the scattered momentum 01:09:17.340 --> 01:09:21.210 align:middle line:84% of the photon is, we'll have a comparable uncertainty 01:09:21.210 --> 01:09:24.160 align:middle line:84% in the resulting momentum of the electron. 01:09:24.160 --> 01:09:26.760 align:middle line:84% It will recoil just like in Compton's scattering. 01:09:26.760 --> 01:09:28.779 align:middle line:90% What's its recoil momentum? 01:09:28.779 --> 01:09:31.649 align:middle line:84% Well, we can't identify that with perfect accuracy 01:09:31.649 --> 01:09:34.149 align:middle line:84% because we've clumped together, so to speak, 01:09:34.149 --> 01:09:37.050 align:middle line:84% we've lost the ability to distinguish or differentiate 01:09:37.050 --> 01:09:38.640 align:middle line:90% among scattered momenta-- 01:09:38.640 --> 01:09:40.170 align:middle line:84% components of the scattered momenta 01:09:40.170 --> 01:09:44.319 align:middle line:84% within some cone of angular size or with theta. 01:09:44.319 --> 01:09:48.540 align:middle line:84% So we have some uncertainty in the post-scattering recoil 01:09:48.540 --> 01:09:50.970 align:middle line:84% momentum of the electron from a kind of Compton 01:09:50.970 --> 01:09:52.090 align:middle line:90% scattering argument. 01:09:52.090 --> 01:09:54.420 align:middle line:84% But now we know what that component is. 01:09:54.420 --> 01:09:55.890 align:middle line:90% We can actually fill this in. 01:09:55.890 --> 01:09:58.530 align:middle line:84% Because using Einstein's arguments from photoelectric 01:09:58.530 --> 01:10:00.780 align:middle line:84% effect and so on-- again, we saw this with the Compton 01:10:00.780 --> 01:10:02.760 align:middle line:84% scattering in the previous class, 01:10:02.760 --> 01:10:05.160 align:middle line:90% or maybe two classes ago-- 01:10:05.160 --> 01:10:07.830 align:middle line:84% the actual momentum of that photon 01:10:07.830 --> 01:10:11.670 align:middle line:84% is given by Planck's constant divided by the wavelength 01:10:11.670 --> 01:10:12.990 align:middle line:90% of the associated wave. 01:10:12.990 --> 01:10:15.960 align:middle line:84% That's how Einstein began reasoning, 01:10:15.960 --> 01:10:19.140 align:middle line:84% as we saw, combining the kind of poynting 01:10:19.140 --> 01:10:21.540 align:middle line:84% vector classical electromagnetism 01:10:21.540 --> 01:10:24.630 align:middle line:84% with his notion that photons are discrete bundles of energy 01:10:24.630 --> 01:10:26.010 align:middle line:90% with [INAUDIBLE]. 01:10:26.010 --> 01:10:29.220 align:middle line:84% So now, Heisenberg says let's combine those things. 01:10:29.220 --> 01:10:31.420 align:middle line:84% We can't make the resolving power, 01:10:31.420 --> 01:10:36.300 align:middle line:84% which is say how fine a picture or how fine a determination 01:10:36.300 --> 01:10:39.390 align:middle line:84% of the location in space that electron is-- we 01:10:39.390 --> 01:10:42.660 align:middle line:84% can't make that arbitrarily small while also making 01:10:42.660 --> 01:10:46.110 align:middle line:84% the uncertainty in the scattered momentum arbitrarily small. 01:10:46.110 --> 01:10:47.970 align:middle line:84% In fact, if we multiply them together, 01:10:47.970 --> 01:10:51.780 align:middle line:84% their simultaneous values cannot be made zero. 01:10:51.780 --> 01:10:54.240 align:middle line:84% In fact, the product is going to be roughly 01:10:54.240 --> 01:10:56.640 align:middle line:90% of order h, Planck's constant. 01:10:56.640 --> 01:10:59.200 align:middle line:84% If we make the aperture too small, 01:10:59.200 --> 01:11:02.140 align:middle line:84% then we fuzz out our location information. 01:11:02.140 --> 01:11:05.290 align:middle line:84% However, by making the aperture not small enough, 01:11:05.290 --> 01:11:10.060 align:middle line:84% we allow in scattered momenta of a sizable range of components. 01:11:10.060 --> 01:11:12.820 align:middle line:84% So we have this kind of seesaw trade-off. 01:11:12.820 --> 01:11:15.640 align:middle line:84% The wavelength appears opposite to each other 01:11:15.640 --> 01:11:19.390 align:middle line:84% between the resolving power and the scattered momentum. 01:11:19.390 --> 01:11:22.540 align:middle line:84% The effect of the aperture width is one is downstairs. 01:11:22.540 --> 01:11:23.380 align:middle line:90% One is upstairs. 01:11:23.380 --> 01:11:28.540 align:middle line:84% So we can't simultaneously make both delta x and delta p 01:11:28.540 --> 01:11:31.098 align:middle line:84% arbitrarily small at the same time. 01:11:31.098 --> 01:11:32.890 align:middle line:84% And it gets there from thinking about where 01:11:32.890 --> 01:11:34.690 align:middle line:84% operating on these things, he thinks 01:11:34.690 --> 01:11:37.080 align:middle line:84% about this inspired by the fact that his matrices don't 01:11:37.080 --> 01:11:37.580 align:middle line:90% commute. 01:11:37.580 --> 01:11:39.490 align:middle line:84% So the order in which we do things should matter. 01:11:39.490 --> 01:11:41.490 align:middle line:84% What does it mean to be acting on this electron? 01:11:41.490 --> 01:11:44.030 align:middle line:84% And he goes to this physical kind of cartoon model. 01:11:44.030 --> 01:11:47.560 align:middle line:84% So at first, Heisenberg says this is really us being clumsy. 01:11:47.560 --> 01:11:50.380 align:middle line:84% Heisenberg's own interpretation of his own uncertainty 01:11:50.380 --> 01:11:53.140 align:middle line:84% principle, which he publishes in the spring of 1927 01:11:53.140 --> 01:11:55.880 align:middle line:84% while visiting in Copenhagen, is basically 01:11:55.880 --> 01:11:59.020 align:middle line:84% that we humans are big, clumsy animals. 01:11:59.020 --> 01:12:02.170 align:middle line:84% We can't help but disturb tiny little things like electrons. 01:12:02.170 --> 01:12:04.210 align:middle line:84% After all, we're the ones who released 01:12:04.210 --> 01:12:07.210 align:middle line:84% this kind of enormously energetic light beam 01:12:07.210 --> 01:12:09.728 align:middle line:84% on our poor little electron because we 01:12:09.728 --> 01:12:11.020 align:middle line:90% wanted to make a sharp picture. 01:12:11.020 --> 01:12:13.630 align:middle line:84% But the energy scale of the light 01:12:13.630 --> 01:12:15.850 align:middle line:84% was comparable to the kind of relevant energies 01:12:15.850 --> 01:12:17.420 align:middle line:90% or momentum of the electron. 01:12:17.420 --> 01:12:18.370 align:middle line:90% So we've intervened. 01:12:18.370 --> 01:12:20.150 align:middle line:90% We've disturbed this object. 01:12:20.150 --> 01:12:22.870 align:middle line:84% So of course, we can't make a clear picture. 01:12:22.870 --> 01:12:25.810 align:middle line:84% He's talking about this in Copenhagen with Bohr. 01:12:25.810 --> 01:12:28.090 align:middle line:84% And Bohr strongly, strongly disagrees. 01:12:28.090 --> 01:12:30.310 align:middle line:84% And here again, I want to think back 01:12:30.310 --> 01:12:34.630 align:middle line:84% to that really quite, I think, very compelling piece by Megan 01:12:34.630 --> 01:12:36.553 align:middle line:84% Shields Fermato that I put on the reader, 01:12:36.553 --> 01:12:38.095 align:middle line:84% I think, just for the previous class. 01:12:38.095 --> 01:12:41.350 align:middle line:84% I think this actually makes even more sense here. 01:12:41.350 --> 01:12:42.910 align:middle line:84% Again, this tells us about how Bohr 01:12:42.910 --> 01:12:47.530 align:middle line:84% was operating all the time-- his entire career, often 01:12:47.530 --> 01:12:51.670 align:middle line:84% with these young, sometimes very brash kind of alpha male style 01:12:51.670 --> 01:12:52.540 align:middle line:90% young assistants. 01:12:52.540 --> 01:12:56.680 align:middle line:84% But also, even more often with his very patient wife 01:12:56.680 --> 01:12:57.490 align:middle line:90% Margrethe. 01:12:57.490 --> 01:12:59.410 align:middle line:90% He's always working in dialogue. 01:12:59.410 --> 01:13:02.440 align:middle line:84% He's always talking these things through orally 01:13:02.440 --> 01:13:04.360 align:middle line:84% and sharpening his arguments like-- 01:13:04.360 --> 01:13:06.220 align:middle line:84% it's like a championship debater. 01:13:06.220 --> 01:13:09.190 align:middle line:84% And Margrethe was in the mix from the start. 01:13:09.190 --> 01:13:11.920 align:middle line:84% Bohr's also always doing that, especially with his younger 01:13:11.920 --> 01:13:13.720 align:middle line:84% assistants, people like Heisenberg. 01:13:13.720 --> 01:13:17.350 align:middle line:84% That's what they do throughout the spring and summer of 1927. 01:13:17.350 --> 01:13:20.560 align:middle line:84% Heisenberg comes excitedly to his advisor and mentor 01:13:20.560 --> 01:13:22.810 align:middle line:84% Bohr, almost a kind of father figure for Heisenberg 01:13:22.810 --> 01:13:25.480 align:middle line:90% by this point, Niels Bohr. 01:13:25.480 --> 01:13:26.920 align:middle line:84% Look, I've learned this new thing. 01:13:26.920 --> 01:13:29.080 align:middle line:84% There's this trade-off because we humans 01:13:29.080 --> 01:13:31.060 align:middle line:84% can't treat the atomic realm with kind 01:13:31.060 --> 01:13:36.130 align:middle line:84% of enough kind of daintiness or with enough sufficient care. 01:13:36.130 --> 01:13:38.710 align:middle line:84% And Bohr says no, you've misunderstood your own work. 01:13:38.710 --> 01:13:40.060 align:middle line:90% There are shouting matches. 01:13:40.060 --> 01:13:42.670 align:middle line:84% There are crying fits, mostly Heisenberg, not Bohr. 01:13:42.670 --> 01:13:43.870 align:middle line:90% There's name-calling. 01:13:43.870 --> 01:13:47.530 align:middle line:84% This becomes intensely emotional as they continue 01:13:47.530 --> 01:13:51.550 align:middle line:84% this kind of behind the scenes discussion in Copenhagen 01:13:51.550 --> 01:13:54.670 align:middle line:84% really for weeks and weeks, not just for a day or two. 01:13:54.670 --> 01:13:56.860 align:middle line:84% It continues throughout the spring and summer. 01:13:56.860 --> 01:13:58.885 align:middle line:84% So Heisenberg and Bohr tussle over 01:13:58.885 --> 01:14:00.760 align:middle line:84% how to make sense of the exact same equation. 01:14:00.760 --> 01:14:02.552 align:middle line:84% Again, we've seen this over and over again. 01:14:02.552 --> 01:14:04.647 align:middle line:84% Bohr is not saying your math is wrong. 01:14:04.647 --> 01:14:06.730 align:middle line:84% Bohr is not saying I don't believe your expression 01:14:06.730 --> 01:14:09.010 align:middle line:84% or you're off by a factor of 2 pi. 01:14:09.010 --> 01:14:12.460 align:middle line:84% He says, you're not interpreting your own mathematics correctly. 01:14:12.460 --> 01:14:14.410 align:middle line:84% I make meaning of your expression 01:14:14.410 --> 01:14:16.450 align:middle line:90% differently than you do. 01:14:16.450 --> 01:14:20.230 align:middle line:84% And Heisenberg doesn't-- it is not a comfortable discussion. 01:14:20.230 --> 01:14:25.510 align:middle line:84% What Bohr argues instead is that this expression, what we now 01:14:25.510 --> 01:14:27.460 align:middle line:84% call the uncertainty principle, is not 01:14:27.460 --> 01:14:31.180 align:middle line:84% a result of our interventions, of our clumsy interventions 01:14:31.180 --> 01:14:34.060 align:middle line:84% with small little-- a fragile, quantum world. 01:14:34.060 --> 01:14:36.400 align:middle line:84% But Bohr says it's actually about the quantum objects 01:14:36.400 --> 01:14:37.990 align:middle line:90% independent of ourselves. 01:14:37.990 --> 01:14:41.770 align:middle line:84% It's something deep about how nature works on the scale 01:14:41.770 --> 01:14:43.900 align:middle line:84% of atoms, he's now convinced himself, 01:14:43.900 --> 01:14:48.040 align:middle line:84% that they simply do not and cannot have simultaneously 01:14:48.040 --> 01:14:50.920 align:middle line:84% sharp values for certain pairs of properties, 01:14:50.920 --> 01:14:53.690 align:middle line:84% for complementary pairs of properties. 01:14:53.690 --> 01:14:57.950 align:middle line:84% One set of those pairs is position and momentum. 01:14:57.950 --> 01:15:00.890 align:middle line:84% That's the component of momentum in that same direction-- 01:15:00.890 --> 01:15:03.940 align:middle line:90% so delta p sub x here. 01:15:03.940 --> 01:15:05.740 align:middle line:84% That that's one of these examples 01:15:05.740 --> 01:15:08.740 align:middle line:84% of a complementary pair of properties-- delta 01:15:08.740 --> 01:15:10.240 align:middle line:90% x delta p sub x. 01:15:10.240 --> 01:15:13.000 align:middle line:84% They go on together to work out-- 01:15:13.000 --> 01:15:15.640 align:middle line:84% a similar one would be the energy involved 01:15:15.640 --> 01:15:18.580 align:middle line:84% in a certain interaction, delta e, and the time 01:15:18.580 --> 01:15:20.350 align:middle line:84% over which that interaction takes place. 01:15:20.350 --> 01:15:24.070 align:middle line:84% That becomes another pair of complementary properties 01:15:24.070 --> 01:15:28.810 align:middle line:84% or quantities that can't be simultaneously identified 01:15:28.810 --> 01:15:30.167 align:middle line:90% with arbitrary precision. 01:15:30.167 --> 01:15:32.500 align:middle line:84% So there's another version of the uncertainty principle. 01:15:32.500 --> 01:15:37.240 align:middle line:84% Delta e delta t is of order or not less than roughly h. 01:15:37.240 --> 01:15:38.740 align:middle line:84% So to Bohr, he says that's something 01:15:38.740 --> 01:15:40.660 align:middle line:84% we have to learn about nature not 01:15:40.660 --> 01:15:42.730 align:middle line:84% about our own kind of clumsiness, 01:15:42.730 --> 01:15:44.650 align:middle line:84% that the electron even before we smacked it 01:15:44.650 --> 01:15:47.500 align:middle line:84% with this high-energy photon simply did not 01:15:47.500 --> 01:15:51.580 align:middle line:84% have on its own simultaneously sharp values of position 01:15:51.580 --> 01:15:53.410 align:middle line:84% and momentum along that direction 01:15:53.410 --> 01:15:56.930 align:middle line:84% even before it was smacked by the photon. 01:15:56.930 --> 01:15:59.600 align:middle line:84% So now, that begins to lead to even broader 01:15:59.600 --> 01:16:02.030 align:middle line:84% kind of sprawling discussions, much more 01:16:02.030 --> 01:16:04.280 align:middle line:84% kind of philosophically freighted discussions which 01:16:04.280 --> 01:16:07.940 align:middle line:84% they're still working out often during strolls in the gardens 01:16:07.940 --> 01:16:11.420 align:middle line:84% right near Bohr's Institute in Copenhagen. Really, again, 01:16:11.420 --> 01:16:15.810 align:middle line:84% this occupies them throughout the spring and summer of 1927. 01:16:15.810 --> 01:16:18.620 align:middle line:84% So if the uncertainty principle holds for quantum objects 01:16:18.620 --> 01:16:21.650 align:middle line:84% even on their own, not just when we interact with them, 01:16:21.650 --> 01:16:23.300 align:middle line:84% then they begin to puzzle through 01:16:23.300 --> 01:16:27.590 align:middle line:84% together there could be no such thing as a sharp trajectory-- 01:16:27.590 --> 01:16:31.370 align:middle line:84% a specific path through space and time for quantum objects. 01:16:31.370 --> 01:16:33.560 align:middle line:90% After all, what is a trajectory? 01:16:33.560 --> 01:16:35.420 align:middle line:84% It's a collection at every moment 01:16:35.420 --> 01:16:37.700 align:middle line:84% in time of exactly where an object is 01:16:37.700 --> 01:16:38.990 align:middle line:90% and exactly where it's going. 01:16:38.990 --> 01:16:41.870 align:middle line:84% And that's precisely what Heisenberg's uncertainty 01:16:41.870 --> 01:16:44.630 align:middle line:84% principle seems not to allow us to attribute 01:16:44.630 --> 01:16:45.950 align:middle line:90% to things like an electron. 01:16:45.950 --> 01:16:48.170 align:middle line:84% We make a trajectory by saying at this moment, 01:16:48.170 --> 01:16:50.210 align:middle line:84% the electron is right here and it's 01:16:50.210 --> 01:16:52.310 align:middle line:84% moving in this specific direction at this rate 01:16:52.310 --> 01:16:52.910 align:middle line:90% and so on. 01:16:52.910 --> 01:16:54.890 align:middle line:90% That's what a trajectory is. 01:16:54.890 --> 01:16:56.790 align:middle line:84% And the ingredients for a trajectory 01:16:56.790 --> 01:17:01.230 align:middle line:84% are what we're now told by this unfolding work is simply not 01:17:01.230 --> 01:17:01.920 align:middle line:90% available to us. 01:17:01.920 --> 01:17:04.380 align:middle line:84% Because in nature, at least according to Bohr, 01:17:04.380 --> 01:17:07.660 align:middle line:84% these things were never sharp enough at any given moment. 01:17:07.660 --> 01:17:10.955 align:middle line:84% So to Bohr at least, and eventually to Heisenberg 01:17:10.955 --> 01:17:12.330 align:middle line:84% through this kind of-- to call it 01:17:12.330 --> 01:17:15.780 align:middle line:84% a dialogue is to paper over the emotion involved 01:17:15.780 --> 01:17:17.160 align:middle line:90% and the real struggle. 01:17:17.160 --> 01:17:19.140 align:middle line:84% But eventually, Heisenberg is kind of, 01:17:19.140 --> 01:17:22.020 align:middle line:84% so to speak, converted as well to this Bohr line. 01:17:22.020 --> 01:17:24.810 align:middle line:84% It becomes known as the kind of Copenhagen interpretation 01:17:24.810 --> 01:17:27.700 align:middle line:84% because so much of it is being worked out in Bohr's Institute 01:17:27.700 --> 01:17:29.460 align:middle line:84% through these long walks in the gardens, 01:17:29.460 --> 01:17:33.180 align:middle line:84% often with Margrethe and with Heisenberg and Pauli 01:17:33.180 --> 01:17:36.400 align:middle line:84% and the younger postdocs basically cycled 01:17:36.400 --> 01:17:37.680 align:middle line:90% through Bohr's Institute. 01:17:37.680 --> 01:17:41.760 align:middle line:84% So the real lesson of Heisenberg's own equation 01:17:41.760 --> 01:17:44.940 align:middle line:84% is that given some position of some object 01:17:44.940 --> 01:17:47.610 align:middle line:84% at some initial time, say, x at T0, 01:17:47.610 --> 01:17:49.860 align:middle line:84% like we would do even with Newtonian mechanics, 01:17:49.860 --> 01:17:53.970 align:middle line:84% we can't know its position at some later time with certainty. 01:17:53.970 --> 01:17:55.590 align:middle line:84% Because to do that, we'd have to know 01:17:55.590 --> 01:17:58.560 align:middle line:84% both its position and its simultaneous momentum 01:17:58.560 --> 01:18:00.490 align:middle line:90% to arbitrary accuracy. 01:18:00.490 --> 01:18:02.940 align:middle line:84% So that's not just interesting to Bohr. 01:18:02.940 --> 01:18:05.940 align:middle line:84% This heralds an amazing conceptual revolution. 01:18:05.940 --> 01:18:07.650 align:middle line:84% Bohr is always talking in this period 01:18:07.650 --> 01:18:10.260 align:middle line:84% in these kind of grand philosophical kind 01:18:10.260 --> 01:18:11.490 align:middle line:90% of sweeping statements. 01:18:11.490 --> 01:18:13.800 align:middle line:84% And eventually, people like Heisenberg and Pauli 01:18:13.800 --> 01:18:15.970 align:middle line:90% come along to that as well. 01:18:15.970 --> 01:18:18.450 align:middle line:84% It suggests the fall of determinism-- 01:18:18.450 --> 01:18:22.860 align:middle line:84% this grand plan that stretches back to the 18th century, that 01:18:22.860 --> 01:18:26.490 align:middle line:84% given the present state of a system and knowledge of all 01:18:26.490 --> 01:18:29.530 align:middle line:84% the things that are impacted that used to seem, 01:18:29.530 --> 01:18:32.130 align:middle line:84% at least in principle, to be sufficient to protect 01:18:32.130 --> 01:18:34.230 align:middle line:84% with certainty-- to predict with certainty what 01:18:34.230 --> 01:18:35.880 align:middle line:90% would happen in the future. 01:18:35.880 --> 01:18:37.590 align:middle line:84% And instead, what Bohr eventually 01:18:37.590 --> 01:18:40.110 align:middle line:84% convinces Heisenberg of and many others, 01:18:40.110 --> 01:18:43.590 align:middle line:84% is that that's simply no longer possible, that the whole notion 01:18:43.590 --> 01:18:46.980 align:middle line:84% of causality, or of A causing B, or of determinism-- 01:18:46.980 --> 01:18:49.060 align:middle line:84% I know this now, therefore, I can with certainty 01:18:49.060 --> 01:18:50.310 align:middle line:90% what's going to happen later-- 01:18:50.310 --> 01:18:54.900 align:middle line:84% that starts to fall into an enormous amount of challenge 01:18:54.900 --> 01:18:57.720 align:middle line:84% in the quantum realm given these arguments about things 01:18:57.720 --> 01:18:59.205 align:middle line:90% like the uncertainty principle. 01:18:59.205 --> 01:19:00.080 align:middle line:90% So let me stop there. 01:19:00.080 --> 01:19:02.038 align:middle line:84% We have a moment or two at least for questions. 01:19:02.038 --> 01:19:07.460 align:middle line:84% Sorry I went on a little bit long So let's see. 01:19:07.460 --> 01:19:09.530 align:middle line:84% DA says, isn 't this work really around the time 01:19:09.530 --> 01:19:11.120 align:middle line:90% of de Broglie's work? 01:19:11.120 --> 01:19:11.940 align:middle line:90% Oh, very good. 01:19:11.940 --> 01:19:13.500 align:middle line:90% So very good. 01:19:13.500 --> 01:19:17.150 align:middle line:84% So Heisenberg knew a little bit about de Broglie's work 01:19:17.150 --> 01:19:19.275 align:middle line:84% but was not terribly influenced by it. 01:19:19.275 --> 01:19:20.900 align:middle line:84% They were actually-- the first inklings 01:19:20.900 --> 01:19:23.720 align:middle line:84% were coming out independently and in parallel. 01:19:23.720 --> 01:19:25.880 align:middle line:84% So when Heisenberg was working on what 01:19:25.880 --> 01:19:27.590 align:middle line:84% becomes known as matrix mechanics, 01:19:27.590 --> 01:19:30.128 align:middle line:84% de Broglie's thesis had been submitted to the Sorbonne 01:19:30.128 --> 01:19:32.420 align:middle line:84% or whatever the Parisian University was but was not yet 01:19:32.420 --> 01:19:33.620 align:middle line:90% widely known. 01:19:33.620 --> 01:19:35.480 align:middle line:84% And in the coming months, they get in touch 01:19:35.480 --> 01:19:37.040 align:middle line:90% and they begin comparing notes. 01:19:37.040 --> 01:19:39.020 align:middle line:84% And we'll actually talk more about the impact 01:19:39.020 --> 01:19:41.510 align:middle line:84% of de Broglie's work in our next class. 01:19:41.510 --> 01:19:43.422 align:middle line:84% And meanwhile, DA, you say, Bohr's argument 01:19:43.422 --> 01:19:44.630 align:middle line:90% contrary to his earlier work? 01:19:44.630 --> 01:19:46.020 align:middle line:90% Yes, it certainly was. 01:19:46.020 --> 01:19:49.340 align:middle line:84% So Bohr was not stuck at one moment in time either. 01:19:49.340 --> 01:19:52.550 align:middle line:84% Bohr's work in 1913 depended on having 01:19:52.550 --> 01:19:54.380 align:middle line:84% very, very specific trajectories. 01:19:54.380 --> 01:19:57.930 align:middle line:84% In fact, very simple ones-- an exact uniform circular motion. 01:19:57.930 --> 01:20:02.690 align:middle line:84% He was convinced by Heisenberg a decade later that these things 01:20:02.690 --> 01:20:04.760 align:middle line:90% are not subject to observation. 01:20:04.760 --> 01:20:07.270 align:middle line:84% That's kind of a leap that gets us into a model. 01:20:07.270 --> 01:20:10.770 align:middle line:84% So Bohr's own thinking was not kind of fixed only in 1913 01:20:10.770 --> 01:20:12.930 align:middle line:84% either maybe, you might say, to his credit. 01:20:12.930 --> 01:20:18.810 align:middle line:84% And then he dug in even more over the course of that more 01:20:18.810 --> 01:20:20.760 align:middle line:84% recent work with people like Heisenberg 01:20:20.760 --> 01:20:23.940 align:middle line:84% to think that there was a bigger conceptual lesson that they all 01:20:23.940 --> 01:20:25.320 align:middle line:90% had missed, including himself. 01:20:25.320 --> 01:20:27.840 align:middle line:84% So he's making his own break from his own work from 1913 01:20:27.840 --> 01:20:28.900 align:middle line:90% as well. 01:20:28.900 --> 01:20:30.900 align:middle line:84% Jesus asks, what's the competing interpretations 01:20:30.900 --> 01:20:32.550 align:middle line:84% of quantum mechanics other than Copenhagen? 01:20:32.550 --> 01:20:32.850 align:middle line:90% Yeah. 01:20:32.850 --> 01:20:33.390 align:middle line:90% Very good. 01:20:33.390 --> 01:20:33.890 align:middle line:90% OK. 01:20:33.890 --> 01:20:36.708 align:middle line:84% So let me pause on that both because there's no time. 01:20:36.708 --> 01:20:39.000 align:middle line:84% That's-- and also because we will talk a bit about some 01:20:39.000 --> 01:20:40.390 align:middle line:90% of that in the coming classes. 01:20:40.390 --> 01:20:42.670 align:middle line:84% It's a huge question, Jesus, a great question, 01:20:42.670 --> 01:20:44.190 align:middle line:90% one that's very dear to me. 01:20:44.190 --> 01:20:47.130 align:middle line:84% And we'll get a little sampling at least in a few class 01:20:47.130 --> 01:20:49.310 align:middle line:90% sessions-- a great question. 01:20:49.310 --> 01:20:59.000 align:middle line:90%