1 00:00:01,960 --> 00:00:05,560 Hello everybody, and welcome back to Exploring Quantum Physics. 2 00:00:05,560 --> 00:00:08,464 I'm guest lecturer Ian Appelbaum, and I'll be telling you about Stern-Gerlach 3 00:00:08,464 --> 00:00:11,736 experiment. In the last lecture, we found that 4 00:00:11,736 --> 00:00:15,026 incorporating the correct relativistic and variance into the quantum mechanical 5 00:00:15,026 --> 00:00:18,183 wave equation. Constructing the Dirac equation, not the 6 00:00:18,183 --> 00:00:21,207 Schrodinger equation, left us with the requirement that the electron wave 7 00:00:21,207 --> 00:00:26,010 function has two components. Our hope is that understanding this 8 00:00:26,010 --> 00:00:28,960 degree of freedom will lead to an explanation of the anomalism effect, 9 00:00:28,960 --> 00:00:32,360 where gas discharge spectral lines are split into more than the three components 10 00:00:32,360 --> 00:00:37,012 of a Lorentz triplet. The Stern-Gerlach experiment, the subject 11 00:00:37,012 --> 00:00:40,738 of today's lecture, was instrumental in revealing the underlying physics of this 12 00:00:40,738 --> 00:00:44,883 so-called anomaly. The experimental geometry of the 13 00:00:44,883 --> 00:00:50,130 Stern-Gerlach Experiment was something like this figure. 14 00:00:50,130 --> 00:00:54,150 Where silver was heated in a vacuum so that individual atoms with thermal energy 15 00:00:54,150 --> 00:00:57,930 kBT of several thousand kelvin were emitted from a furnace, and collimated 16 00:00:57,930 --> 00:01:02,972 through two sequential slits. This beam then passed through a region of 17 00:01:02,972 --> 00:01:07,140 high magnetic field gradient perpendicular to its velocity. 18 00:01:07,140 --> 00:01:09,763 These were neutral atoms, so the magnetic field itself did not affect the 19 00:01:09,763 --> 00:01:13,511 trajectory. However, for atoms carrying a magnetic 20 00:01:13,511 --> 00:01:17,216 moment, the gradient of the magnetic field will impart a force and deflect the 21 00:01:17,216 --> 00:01:22,010 beam. We can predict the observed deflection by 22 00:01:22,010 --> 00:01:25,260 first calculating force. This is just the gradient of potential 23 00:01:25,260 --> 00:01:28,100 energy which we know from the dipole interaction energy. 24 00:01:29,230 --> 00:01:32,404 The displacement is given by the solution to the classical equation of motion for a 25 00:01:32,404 --> 00:01:36,147 constant force. Using the time of flight, length over 26 00:01:36,147 --> 00:01:39,202 thermal velocity, we then have a complete expression that we can use to obtain 27 00:01:39,202 --> 00:01:44,510 quantitative values of deflection. For a gradient of about ten Tesla per 28 00:01:44,510 --> 00:01:49,291 centimeter over a distance of three centimeters. 29 00:01:49,291 --> 00:01:53,439 We predicted deflection of 100 microns for a moment of one Bohr magnetron 30 00:01:53,439 --> 00:01:59,404 oriented parallel to the gradient. If the moment is oriented anti parallel, 31 00:01:59,404 --> 00:02:01,810 the deflection is in the opposite direction. 32 00:02:01,810 --> 00:02:05,158 Now if the magnetic moments of the neutral atoms are distributed randomly 33 00:02:05,158 --> 00:02:08,344 upon emission from the furnace, as expected from a classical degree of 34 00:02:08,344 --> 00:02:11,977 freedom. This deflection will vary continuously 35 00:02:11,977 --> 00:02:15,260 and merely result in a broadened beam of width times D. 36 00:02:15,260 --> 00:02:22,150 So what happened? What did Stern and Gerlach actually see? 37 00:02:22,150 --> 00:02:25,165 Here's what Stern and Gerlach saw on the glass slide removed from the apparatus, 38 00:02:25,165 --> 00:02:28,090 after converting the nearly transparent deposited silver into black silver 39 00:02:28,090 --> 00:02:31,066 sulfide. On the left is the deposited pattern from 40 00:02:31,066 --> 00:02:34,126 the silver beam without the perpendicular field gradient, and on the right is the 41 00:02:34,126 --> 00:02:38,197 pattern with the field gradient. Overlaying the pole geometry used to 42 00:02:38,197 --> 00:02:41,577 create the gradient shows that that horizontal deflection is greatest in the 43 00:02:41,577 --> 00:02:45,540 middle, simply because the field gradient is higher there. 44 00:02:46,810 --> 00:02:48,754 The length scale shows that this deflection is in the order of our 45 00:02:48,754 --> 00:02:52,594 calculated 100 microns. But the most remarkable thing about the 46 00:02:52,594 --> 00:02:55,762 pattern with the magnetic gradient is that it is not simply smeared as expected 47 00:02:55,762 --> 00:02:58,930 from randomly oriented magnetic moments, but that they're split into two well 48 00:02:58,930 --> 00:03:02,758 resolved beams. This postcard was sent to Bohr and it 49 00:03:02,758 --> 00:03:05,600 says below something like, congratulations on confirmation of your 50 00:03:05,600 --> 00:03:09,452 atomic theory. Stern and Gerlach thought that they were 51 00:03:09,452 --> 00:03:12,990 measuring the orbital magnetic moment predicted by Bohr's model. 52 00:03:12,990 --> 00:03:16,730 But this was 1922 before the Schrodinger equation, and the realization that the L 53 00:03:16,730 --> 00:03:20,820 equals zero S dates have zero angular momentum. 54 00:03:20,820 --> 00:03:24,670 It doesn't measure the orbital component, they were measuring the magnetic moment 55 00:03:24,670 --> 00:03:28,870 of the intrinsic angular momentum of the unpaired 5S electron. 56 00:03:28,870 --> 00:03:32,235 This is rather unfortunately called spin for historical reasons. 57 00:03:32,235 --> 00:03:35,367 And the twofold splitting of the atomic beam is due to the two possible 58 00:03:35,367 --> 00:03:39,490 Eigenvalues of the Z component, plus or minus h bar over two. 59 00:03:40,620 --> 00:03:43,462 It's then natural to identify an additional quantum number, the spin 60 00:03:43,462 --> 00:03:46,402 magnetic quantum number M sub S, that takes the value of plus or minus one 61 00:03:46,402 --> 00:03:50,564 half. A so-called g factor, nominally equal to 62 00:03:50,564 --> 00:03:53,981 two, due to a relativistic effect called Thomas procession from Lorentz boost, 63 00:03:53,981 --> 00:03:59,730 must be included in the conversion from angular momentum to magnetic moment. 64 00:03:59,730 --> 00:04:03,492 We therefore identify the two components of the electron part of the Dirac wave 65 00:04:03,492 --> 00:04:07,280 function as the amplitudes of spin up and spin down. 66 00:04:08,690 --> 00:04:11,679 For those interested in the serendipity behind this experiment, and the 67 00:04:11,679 --> 00:04:15,710 historical development of the correct interpretation. 68 00:04:15,710 --> 00:04:19,155 I rem, recommend this popular account by Friedrich and Herschbach in Physics 69 00:04:19,155 --> 00:04:22,905 Today, very highly. Incidentally, long after this extremely 70 00:04:22,905 --> 00:04:27,530 important experiment, Stern won the Nobel Prize during World War II. 71 00:04:27,530 --> 00:04:30,698 But Gerlach was snubbed, in part because he stayed in Germany to work on wartime 72 00:04:30,698 --> 00:04:34,990 weapons development for the Nazis. I said it was unfortunate that the word 73 00:04:34,990 --> 00:04:38,878 we use for electronic intrinsic magnetic moment is spin. 74 00:04:38,878 --> 00:04:42,358 And this is because, although it's natural to try to understand how an 75 00:04:42,358 --> 00:04:46,498 individual particle can carry intrinsic angular momentum, there's no classical 76 00:04:46,498 --> 00:04:50,354 analog to it. Still, one can find figures like this 77 00:04:50,354 --> 00:04:55,720 propagating a false notion in text books. In fact, the idea of spin was at first 78 00:04:55,720 --> 00:04:59,048 ridiculed for the very reason that the geometrical interpretation leads to 79 00:04:59,048 --> 00:05:03,210 nonsense such as this. If an electron is a spinning charged 80 00:05:03,210 --> 00:05:07,680 sphere, then what's the velocity at the electron surface? 81 00:05:07,680 --> 00:05:11,298 We can calculate the classical electron radius, by equating the electrostatic 82 00:05:11,298 --> 00:05:14,430 potential energy needed to assemble charge E into the sphere with the 83 00:05:14,430 --> 00:05:19,598 electron rest mass. And then use it to determine the velocity 84 00:05:19,598 --> 00:05:24,246 necessary to generate one Bohr magneton. The result is the speed of light divided 85 00:05:24,246 --> 00:05:28,415 by the fine structure constant. A dimensionalist quantity equal to about 86 00:05:28,415 --> 00:05:33,126 one over 137, far less than unity. The velocity then would have to be over 87 00:05:33,126 --> 00:05:36,740 two orders of magnitude higher than the speed of light. 88 00:05:36,740 --> 00:05:40,532 Clearly nonsense. This issue played an important role in 89 00:05:40,532 --> 00:05:43,488 history. Goudsmit and Uhlenbeck are credited with 90 00:05:43,488 --> 00:05:47,392 the correct interpretation of the Stern-Gerlach experiment in 1925, as 91 00:05:47,392 --> 00:05:51,960 being due to the intrinsic electron angular momentum. 92 00:05:51,960 --> 00:05:55,846 But perhaps only because they were lucky enough to have an open-minded advisor, 93 00:05:55,846 --> 00:06:00,730 Ehrenfest, who said, well, that's a nice idea, though it may be wrong. 94 00:06:00,730 --> 00:06:04,480 But you don't yet have a reputation, so you have nothing to lose. 95 00:06:04,480 --> 00:06:07,817 Whereas the same idea had apparently been suggested several years before by Kronig, 96 00:06:07,817 --> 00:06:10,919 who was thoroughly discouraged by his mentor Pauli, who said, it's indeed very 97 00:06:10,919 --> 00:06:14,785 clever, but of course has nothing to do with reality. 98 00:06:14,785 --> 00:06:19,075 Llewellyn Thomas, who introduced the Thomas G factor, wrote a humorous letter 99 00:06:19,075 --> 00:06:24,080 to Goudsmith after his paper's publication saying the following. 100 00:06:24,080 --> 00:06:27,230 I think you and Uhlenbeck have been very lucky to get your spinning electron 101 00:06:27,230 --> 00:06:31,270 published and talked about before Pauli heard of it. 102 00:06:31,270 --> 00:06:34,260 It appears that more than a year ago Kronig, believed in the spinning electron 103 00:06:34,260 --> 00:06:37,866 and worked out something. The first person he showed it to was 104 00:06:37,866 --> 00:06:41,023 Pauli. Pauli ridiculed the whole thing so much 105 00:06:41,023 --> 00:06:44,677 that the first person became also the last, and no one else heard anything of 106 00:06:44,677 --> 00:06:47,636 it. Which all goes to show that the 107 00:06:47,636 --> 00:06:52,614 infallibility of the Deity does not extend to his self-styled vicar on Earth.