1 00:00:00,390 --> 00:00:02,380 Welcome back everyone. I'm Charles Clark. 2 00:00:02,380 --> 00:00:06,604 This is Exploring Quantum Physics, and we're going to begin a set of lectures 3 00:00:06,604 --> 00:00:11,290 today on the phenomenon of symmetry in quantum physics. 4 00:00:11,290 --> 00:00:14,767 It's a very important and pervasive concept and we'll try to give you a lot 5 00:00:14,767 --> 00:00:18,640 of interesting examples to think about. So, here we go. 6 00:00:21,040 --> 00:00:25,387 The forms of symmetry we'll be looking about fall into two classes a sa-, things 7 00:00:25,387 --> 00:00:28,915 that are associated with a single particle, or a, you know, a, an, an 8 00:00:28,915 --> 00:00:35,078 independent system those are like And then they also, you know, they have 9 00:00:35,078 --> 00:00:37,100 the implications for many particle systems as well. 10 00:00:37,100 --> 00:00:41,294 So parity, angular momentum. we'll return to the hydrogen atom and 11 00:00:41,294 --> 00:00:43,918 look at the four, the so called four dimensional symmetry that comes from the 12 00:00:43,918 --> 00:00:47,160 quantum mechanical treatment of the fun a lens factor. 13 00:00:48,610 --> 00:00:53,074 Then there'll be a break for about a week in which Ian Applebaum who's a professor 14 00:00:53,074 --> 00:00:57,104 here at the University of Maryland and an expert in spintronics, is going to be 15 00:00:57,104 --> 00:01:03,700 giving a detailed account of electron spin, which is a fabulous topic. 16 00:01:04,970 --> 00:01:09,301 we turn for brief discussion of things like isobaric spin, time reversal, and 17 00:01:09,301 --> 00:01:15,358 gauge invariance. And then, we'll conclude with discussions 18 00:01:15,358 --> 00:01:22,500 of some of the interesting effects of symmetry in many particle systems. 19 00:01:22,500 --> 00:01:26,854 It's a very important topic. So modern work on ultra-cold gasses has 20 00:01:26,854 --> 00:01:32,190 largely been made possible by the phenomenon of Bose-Eistein condensation. 21 00:01:32,190 --> 00:01:37,939 And the reason they're all around. Is because the the Fermi gas, Fermi-Dirac 22 00:01:37,939 --> 00:01:42,010 gas. the generic Fermi-Dirac gas has a very 23 00:01:42,010 --> 00:01:46,134 high pressure. Basically, electrons are prohibited from 24 00:01:46,134 --> 00:01:50,380 getting close to each other by the Pauli exclusion principle. 25 00:01:50,380 --> 00:01:52,763 So that, that has a, a major effect on our world. 26 00:01:52,763 --> 00:01:55,880 [INAUDIBLE] we'll just see and think about it. 27 00:01:55,880 --> 00:01:58,841 Now, the first thing on this list may seem a little bit strange to some of you, 28 00:01:58,841 --> 00:02:03,150 parity. It's a really important concept, but it's 29 00:02:03,150 --> 00:02:07,190 not really deserved. In fact, as I'll show you from some of 30 00:02:07,190 --> 00:02:11,478 the examples, parity is one of the most useful of the symmetries in in, well the 31 00:02:11,478 --> 00:02:15,446 study of ordinary matter from the standpoint of electronic structure and 32 00:02:15,446 --> 00:02:23,600 other things Because it's it, it works so well. 33 00:02:23,600 --> 00:02:29,051 It's easy to apply and it often produces a tremendous simplification in the 34 00:02:29,051 --> 00:02:35,085 difficulty of a problem. However it was, there was a theory 35 00:02:35,085 --> 00:02:42,400 developed by these two young men. Chen Ning Yang and Tsung Dao Lee. 36 00:02:42,400 --> 00:02:48,912 doctor Lee was only only thirty one years old when he won the Nobel prize for this 37 00:02:48,912 --> 00:02:53,040 work. They predicted. 38 00:02:53,040 --> 00:03:00,530 That, parity would be violated in some weak, decays due to weak interaction. 39 00:03:00,530 --> 00:03:03,200 I'll just say a little bit about that in the next slide. 40 00:03:03,200 --> 00:03:06,448 But this was sort of the, first of all, caused a, it was a downfall of a 41 00:03:06,448 --> 00:03:11,960 principle that had never been known to fail and was widely used and. 42 00:03:11,960 --> 00:03:15,288 You know, almost naturally expected in quantum mechanics. 43 00:03:15,288 --> 00:03:19,776 And it it, it led, it was the beginning of what's now called the electroweak 44 00:03:19,776 --> 00:03:23,934 theory, where we understand the electrici-, electricity and magnetism is 45 00:03:23,934 --> 00:03:29,438 coupled to weak interactions. Now the thing is, the parity 46 00:03:29,438 --> 00:03:35,343 transformation. Is the what you get when you invert all 47 00:03:35,343 --> 00:03:43,766 the coordinates in a system. So you might wonder, well, exactly how 48 00:03:43,766 --> 00:03:48,606 can you do that? and if the, the way it's done is a little 49 00:03:48,606 --> 00:03:52,326 bit indirect. But, I mean, I'll, I'll show that on the 50 00:03:52,326 --> 00:03:58,286 next slide. So once again, sometimes these various 51 00:03:58,286 --> 00:04:03,640 concepts, parity, angle, momentum, spin, are sort of linked together. 52 00:04:03,640 --> 00:04:06,768 And you have to understand several of them before the whole picture falls into 53 00:04:06,768 --> 00:04:09,614 place. But this, this is just a photograph of 54 00:04:09,614 --> 00:04:12,526 the experiment that was done. At the National Bureau of Standards, 55 00:04:12,526 --> 00:04:14,570 1956. And here is a, a schematic of what was 56 00:04:14,570 --> 00:04:21,069 observed. So, what was observed was the, the 57 00:04:21,069 --> 00:04:32,110 preferential decay of a radioactive nucleus by Beta emission. 58 00:04:32,110 --> 00:04:39,590 In one direction which was defined by the, the spin of the nucleus. 59 00:04:39,590 --> 00:04:44,075 And this is actually something that if parity is, is violated, if parity is 60 00:04:44,075 --> 00:04:50,234 conserved, can't possibly happen. There would have to be you know, on the 61 00:04:50,234 --> 00:04:55,742 average, an equal distribution of particles, of beta beta rays coming off 62 00:04:55,742 --> 00:05:01,940 on both sides of the nucleus. And that was, that was the asymmetry 63 00:05:01,940 --> 00:05:06,098 observed/g, and certainly a clear ,uh, sign that parity was not conserved in 64 00:05:06,098 --> 00:05:12,340 that interaction. here's a picture of one of the really 65 00:05:12,340 --> 00:05:14,940 great physicists of the twentieth century in my opinion. 66 00:05:16,560 --> 00:05:19,122 he won the Noble Prize in Physics a little bit later in life than some of 67 00:05:19,122 --> 00:05:24,540 these other fellows. But note the citation here makes 68 00:05:24,540 --> 00:05:32,306 particular, particular reference to his. His discovery and application of sym, 69 00:05:32,306 --> 00:05:37,610 symmetry principles. And he was, in large part, responsible 70 00:05:37,610 --> 00:05:44,330 for making symmetry sort of a, a theme of quantum physics. 71 00:05:44,330 --> 00:05:50,933 So Angular momentum in particular, is really a grand organizing principle of 72 00:05:50,933 --> 00:05:55,935 quantum physics. We'll see why that is in the next, in the 73 00:05:55,935 --> 00:06:00,970 next several lectures. But basically, it's it's extremely 74 00:06:00,970 --> 00:06:05,100 important and it, it reflects a rotational symmetry of a system, or the 75 00:06:05,100 --> 00:06:10,311 isotropy of space. And Wigner helped create a number of 76 00:06:10,311 --> 00:06:14,640 things that made the theory possible. Angular momentum algebra. 77 00:06:14,640 --> 00:06:19,190 This Wigner-Eckert theorem is very important for understanding things like 78 00:06:19,190 --> 00:06:24,090 the whether particles can have permanent magnetic moments or electric dipolar 79 00:06:24,090 --> 00:06:29,563 moments of that sort. Isobaric spin which unified the 80 00:06:29,563 --> 00:06:34,151 description of the neutron and proton as two states of a common underlying 81 00:06:34,151 --> 00:06:39,553 particle turned out to be a terrific idea for rational understanding of the nuclear 82 00:06:39,553 --> 00:06:45,249 structure. some other things that have finally time 83 00:06:45,249 --> 00:06:49,360 reversal. So Wigner is the person who. 84 00:06:49,360 --> 00:06:53,080 Really first worked out the systematic theory of time reversal in quantum 85 00:06:53,080 --> 00:06:58,042 mechanics systems. Now, okay, parity seems, I mean, I hope 86 00:06:58,042 --> 00:07:03,080 it seems like somewhat exotic. Think how do you, how do you manage to 87 00:07:03,080 --> 00:07:06,310 reverse turn something completely inside out? 88 00:07:06,310 --> 00:07:10,020 but time reversal. how do we make time go backwards anyway? 89 00:07:10,020 --> 00:07:13,960 Well, there will be, we'll discuss that later. 90 00:07:13,960 --> 00:07:18,510 But first let's, let's just look at a, a simple example of time reversal, which 91 00:07:18,510 --> 00:07:24,150 has to do with motion of a charged particle in a magnetic field. 92 00:07:24,150 --> 00:07:27,932 So I hope that some of you, got to the point where you realized what this 93 00:07:27,932 --> 00:07:32,006 changing. Of the velocity does to a motion in a 94 00:07:32,006 --> 00:07:36,330 magnetic field. It causes a displacement of the orbit. 95 00:07:36,330 --> 00:07:40,614 So, we've been thinking about this in the context of a sort of an atomtronics 96 00:07:40,614 --> 00:07:47,530 application, and there's a wonderful device used in microwave engineering. 97 00:07:47,530 --> 00:07:51,885 When you want to let's say you have an input, an input channel and an output, 98 00:07:51,885 --> 00:07:56,307 and you're delivering high power loads, and you, you want, you want to avoid the 99 00:07:56,307 --> 00:08:00,729 problem of back reflection, of something coming from that, the channel you're 100 00:08:00,729 --> 00:08:08,530 pumping, coming back in to your input and then destroying the apparatus. 101 00:08:08,530 --> 00:08:13,000 It's a problem that often happens in power engineering. 102 00:08:13,000 --> 00:08:16,657 So, this the, this device which is known [INAUDIBLE] just called a circulator, and 103 00:08:16,657 --> 00:08:19,622 it's a time-reversal non-invariant device. 104 00:08:19,622 --> 00:08:23,652 And, wha, actually if you make up a system where you have a magnetic field in 105 00:08:23,652 --> 00:08:27,575 this region, and you send an electron beam. 106 00:08:27,575 --> 00:08:32,210 In this is what you get. So the, when the beam comes from this 107 00:08:32,210 --> 00:08:38,104 input port, it goes to the output port. Any part that's reflected, when it comes 108 00:08:38,104 --> 00:08:43,870 back, it has a time-reversed velocity that the input beam had here. 109 00:08:43,870 --> 00:08:46,164 But that means, it's reflected to the next port, where you could have a load 110 00:08:46,164 --> 00:08:48,690 dump or something like that. Like that. 111 00:08:48,690 --> 00:08:53,589 So this is, this is an example of the use of time-reversal noninvariance which 112 00:08:53,589 --> 00:08:59,980 requires the use of a magnetic medium, in the case of the micro-circulator. 113 00:08:59,980 --> 00:09:05,944 Wolfgang Pauli, another giant of 20th century physics his Nobel citation you 114 00:09:05,944 --> 00:09:11,742 know, consists in its entirety. For of, recognizing the discovery of the 115 00:09:11,742 --> 00:09:15,240 exclusion principle, called the Pauli Principle. 116 00:09:15,240 --> 00:09:20,734 This was a revolutionary concept because it was an intrinsically many particle 117 00:09:20,734 --> 00:09:28,497 phenomenon, and the indication of a simple symmetry rule that applies to all. 118 00:09:29,780 --> 00:09:33,928 well, as we now know, all particles with half integral spin, including the 119 00:09:33,928 --> 00:09:37,079 electron So, that, that those constitute sort of 120 00:09:37,079 --> 00:09:40,570 half the particles in the world. The others have integral spins. 121 00:09:40,570 --> 00:09:45,045 Those, those are called bosons. But, this exclusion principle is really a 122 00:09:45,045 --> 00:09:48,262 symmetry principle because what it says is. 123 00:09:48,262 --> 00:09:55,282 That the wave function of two particles, including all variables, such as the 124 00:09:55,282 --> 00:10:02,518 space, the spin, or any sort of internal degree of freedom, changes sign when you 125 00:10:02,518 --> 00:10:09,981 exchange the particles. So this is really like a, a reflection 126 00:10:09,981 --> 00:10:13,920 that involves exchange of the position of two particles. 127 00:10:13,920 --> 00:10:19,140 And, a reflection in a, in a plane. So if we have a, we have a particle one 128 00:10:19,140 --> 00:10:23,204 here, and a particle two there. Then it's a reflection in the plane, that 129 00:10:23,204 --> 00:10:28,397 connects the, connects the two particles. And so that, that means for example, the 130 00:10:28,397 --> 00:10:32,020 wave function. Must vanish on the midpoint. 131 00:10:32,020 --> 00:10:37,052 And this is why the Pauli Exclusion Principle in, increases the energy of a 132 00:10:37,052 --> 00:10:41,714 system of particles compared to what it would be if the exclusion principle 133 00:10:41,714 --> 00:10:46,524 weren't valid, because it forces nodes in the wave function in between the 134 00:10:46,524 --> 00:10:51,600 particles. So that means they're, they're more 135 00:10:51,600 --> 00:10:55,010 localized and they had certain principles gives them higher energies. 136 00:10:55,010 --> 00:11:01,670 So indeed, this Pauli principle is, is often is the most important thing in the 137 00:11:01,670 --> 00:11:06,578 problem. And it can set the qualitative ordering 138 00:11:06,578 --> 00:11:10,934 of states. In a manner that's, that's not totally 139 00:11:10,934 --> 00:11:15,289 independent from the particular interactions that are at play but it sets 140 00:11:15,289 --> 00:11:19,443 the stage and they're things that really can't happen because of the Pauli 141 00:11:19,443 --> 00:11:25,090 Principle that, otherwise might be quite possible. 142 00:11:25,090 --> 00:11:29,914 So to conclude just mention a phenomena that's not going to be discussed at any, 143 00:11:29,914 --> 00:11:34,336 at any length in this course, but it's, you know, evidence for how symmetry sort 144 00:11:34,336 --> 00:11:41,428 of dominates thinking in physics. So here is the, the three laureates 145 00:11:41,428 --> 00:11:45,568 recognized for what is now known as The Standard Model, this fabulously 146 00:11:45,568 --> 00:11:50,150 successful. Modical of fundamental particles. 147 00:11:50,150 --> 00:11:52,685 Now it's pretty complicated, you know, they're fundamental, there's like, I 148 00:11:52,685 --> 00:11:54,805 don't know, there's 50 of these particles. 149 00:11:54,805 --> 00:12:06,950 so it isn't simple but it is based, it is based on a gauge symmetry. 150 00:12:06,950 --> 00:12:12,806 So it is very much a a theory whose structure is, is, you know, at least in 151 00:12:12,806 --> 00:12:18,278 form, and, and in many respects, dominated by a type of symmetry that 152 00:12:18,278 --> 00:12:24,685 exists. So you'll, you'll see more of this, I 153 00:12:24,685 --> 00:12:29,670 think, when Victor gives a lecture on topological insulators. 154 00:12:29,670 --> 00:12:33,288 You'll see that there are new developments in Gannett's matter, 155 00:12:33,288 --> 00:12:38,380 physics, that have, that have to do with gauge symmetries becoming available in 156 00:12:38,380 --> 00:12:44,260 those systems. It's also true for ultra cold atoms.