1 00:00:04,600 --> 00:00:07,730 MARKUS KLUTE: Hello, and welcome back to 8.701. 2 00:00:07,730 --> 00:00:10,910 In this class, we'll talk about spin. 3 00:00:10,910 --> 00:00:14,270 If you remember the discussion on relativistic kinematics 4 00:00:14,270 --> 00:00:19,910 we had last week, you noticed that I 5 00:00:19,910 --> 00:00:26,090 discussed the decay of a pion, say a neutral pion, 6 00:00:26,090 --> 00:00:28,940 into an electron and-- 7 00:00:28,940 --> 00:00:32,659 a positron and an electron. 8 00:00:32,659 --> 00:00:35,270 And we were able to calculate the velocity of those two 9 00:00:35,270 --> 00:00:37,460 particles, the electron and the positron, 10 00:00:37,460 --> 00:00:40,790 quite easily by knowing the mass of the pion and the masses 11 00:00:40,790 --> 00:00:42,600 of the electron and the positron, 12 00:00:42,600 --> 00:00:45,040 which is served into the rest frame of the pion, 13 00:00:45,040 --> 00:00:47,780 and we can calculate our velocity. 14 00:00:47,780 --> 00:00:51,830 I also told you that this decay is highly suppressed 15 00:00:51,830 --> 00:00:54,210 because of the spins of the particles involved. 16 00:00:54,210 --> 00:00:58,280 The pion has spin zero, and the electron and the positron 17 00:00:58,280 --> 00:01:01,400 have spin 1/2, but it is not easily 18 00:01:01,400 --> 00:01:04,430 possible to align the electron and the positron 19 00:01:04,430 --> 00:01:08,720 such that the spins align to 0. 20 00:01:08,720 --> 00:01:12,246 So therefore, this decay is not usually possible. 21 00:01:12,246 --> 00:01:16,340 Now, let's dive a little bit into this. 22 00:01:16,340 --> 00:01:21,500 In quantum mechanics, the spin of a particle with a vector 23 00:01:21,500 --> 00:01:25,130 is quantized, and in terms of its length and its components. 24 00:01:25,130 --> 00:01:27,680 You calculate the length of the spin vector [? S. ?] 25 00:01:27,680 --> 00:01:30,350 You find that it's square root of f 26 00:01:30,350 --> 00:01:34,170 times s plus 1 in units of h-bar. 27 00:01:34,170 --> 00:01:37,110 The components, and along any axis, actually-- 28 00:01:37,110 --> 00:01:39,330 and in this case here, the d-axis-- 29 00:01:39,330 --> 00:01:44,530 have eigenvalues, and they are listed here. 30 00:01:44,530 --> 00:01:49,040 And we find that there is 2s plus 1 possible values. 31 00:01:49,040 --> 00:01:52,130 So I'll pick here, just arbitrarily, the z axis. 32 00:01:52,130 --> 00:01:55,550 But the question-- it's an obvious question-- 33 00:01:55,550 --> 00:01:58,885 which axis is a sensible choice for this problem? 34 00:01:58,885 --> 00:02:04,570 So I want you to actually stop here and think about this. 35 00:02:04,570 --> 00:02:06,250 What are sensible options? 36 00:02:09,039 --> 00:02:12,280 If you want to get an eigenvalue with the physical state 37 00:02:12,280 --> 00:02:16,480 of particles, which axis are the right ones to choose-- 38 00:02:16,480 --> 00:02:18,130 or, sensible? 39 00:02:18,130 --> 00:02:20,950 There's no right and wrong in this discussion. 40 00:02:20,950 --> 00:02:23,386 Let me motivate this. 41 00:02:23,386 --> 00:02:26,050 If you look at the orbital momentum of a particle, 42 00:02:26,050 --> 00:02:30,130 that's given by r cross p, where p is the momentum 43 00:02:30,130 --> 00:02:32,180 vector of the particle. 44 00:02:32,180 --> 00:02:35,230 So now, if you're looking at the total momentum, 45 00:02:35,230 --> 00:02:38,170 we have to look add the angular momentum 46 00:02:38,170 --> 00:02:40,790 and the spin of the particle together. 47 00:02:40,790 --> 00:02:44,430 So as shown in this picture here, 48 00:02:44,430 --> 00:02:47,880 you see that the parallel component-- 49 00:02:47,880 --> 00:02:51,150 the component in parallel to the slide direction-- 50 00:02:51,150 --> 00:02:55,050 is 0 by definition, because the cross-product is defined-- 51 00:02:57,407 --> 00:02:59,490 the angular momentum is defined as a cross-product 52 00:02:59,490 --> 00:03:01,260 with the momentum. 53 00:03:01,260 --> 00:03:06,810 So this is a nice choice of coordinate system or of axis-- 54 00:03:06,810 --> 00:03:10,600 namely, that is the choice of the momentum of the particle. 55 00:03:10,600 --> 00:03:13,710 So you find that the total momentum perpendicular 56 00:03:13,710 --> 00:03:16,920 is the spin of the particle perpendicular, 57 00:03:16,920 --> 00:03:21,688 and the transverse component is its angular momentum, 58 00:03:21,688 --> 00:03:24,105 the orbital angular momentum, and the spin of the particle 59 00:03:24,105 --> 00:03:27,280 in the transverse direction. 60 00:03:27,280 --> 00:03:30,340 This, then, immediately gets us to a new definition, 61 00:03:30,340 --> 00:03:31,940 then of helicity. 62 00:03:31,940 --> 00:03:34,570 You can define the helicity of a particle 63 00:03:34,570 --> 00:03:37,750 as the spin of the particle dotted 64 00:03:37,750 --> 00:03:39,410 with the momentum of the particle, 65 00:03:39,410 --> 00:03:41,980 and then normalized by the momentum. 66 00:03:41,980 --> 00:03:46,810 So basically, for a fermion, which has a spin 1/2, 67 00:03:46,810 --> 00:03:52,120 you get plus 1/2 if the spin points in the momentum 68 00:03:52,120 --> 00:03:54,850 direction and minus 1/2 if it points 69 00:03:54,850 --> 00:03:57,020 in the opposite direction. 70 00:03:57,020 --> 00:04:01,630 So now, if you go back to our particle here, off by here, 71 00:04:01,630 --> 00:04:11,050 decaying into an electron and a positron, spin is at 1/2, 72 00:04:11,050 --> 00:04:15,140 but you find that an electron is a left-handed particle, 73 00:04:15,140 --> 00:04:18,730 so its helicity will point in this direction. 74 00:04:18,730 --> 00:04:23,070 And for the positron-- sorry-- for the positron, 75 00:04:23,070 --> 00:04:25,120 it points in the same connection. 76 00:04:25,120 --> 00:04:31,000 So the pion here is spin 0, and if you 77 00:04:31,000 --> 00:04:33,280 discuss this in the rest frame of the pion, 78 00:04:33,280 --> 00:04:35,830 the electron and the positron fly off 79 00:04:35,830 --> 00:04:41,490 in opposite directions, which means that the spin doesn't 80 00:04:41,490 --> 00:04:43,600 in align to 0. 81 00:04:43,600 --> 00:04:47,110 So that's why this is highly suppressed. 82 00:04:47,110 --> 00:04:49,627 It's not 0, because you can find-- 83 00:04:49,627 --> 00:04:52,210 you can put [INAUDIBLE] into the rest frame, where you're just 84 00:04:52,210 --> 00:04:54,250 basically looking at both particles 85 00:04:54,250 --> 00:04:57,340 from one side that was coming to you, and in that case, 86 00:04:57,340 --> 00:04:58,000 it's allowed. 87 00:04:58,000 --> 00:05:02,850 But the spintronic configuration is highly [INAUDIBLE]..