1 00:00:00,25 --> 00:00:05,569 Superconductivity is an example of an intrinsically many bodied quantum method, 2 00:00:05,569 --> 00:00:10,960 which doesn't really have a single particle counterpart. 3 00:00:10,960 --> 00:00:13,235 So it cannot be really understood based on just a single particle Schrodinger 4 00:00:13,235 --> 00:00:16,868 equation. So in the context of superconductors, of 5 00:00:16,868 --> 00:00:19,712 course, these many particles are electrons. 6 00:00:19,712 --> 00:00:25,487 So that play together in some sense to create this amazing quantum coherent 7 00:00:25,487 --> 00:00:30,848 state. Now, to understand the very complicated 8 00:00:30,848 --> 00:00:38,2 array of many, Body quantum phenomena. The first step is to understand the 9 00:00:38,2 --> 00:00:42,496 concepts of quantum statistics. Which is an increasing quantum 10 00:00:42,496 --> 00:00:47,321 interaction between identical particles. It doesn't really have a classical 11 00:00:47,321 --> 00:00:50,530 counterpart. So unlike let's say the electrostatic 12 00:00:50,530 --> 00:00:55,340 let's say [UNKNOWN] which depends on distance between the two charges. 13 00:00:55,340 --> 00:00:59,100 So this statistical interaction between identical particles is quite different. 14 00:00:59,100 --> 00:01:03,585 It appears to never have identical particles and they feel each others 15 00:01:03,585 --> 00:01:07,665 presence. independently of the distance separating 16 00:01:07,665 --> 00:01:12,268 them, so this is very interesting. And so we're going to learn about it and 17 00:01:12,268 --> 00:01:16,492 also we are going to learn about the two fundamental classes of particles, with 18 00:01:16,492 --> 00:01:22,726 respect to their quantum statistics, that are called bosons and fermions. 19 00:01:23,840 --> 00:01:28,739 And those guys they lose energy space, [INAUDIBLE] was very much drastically 20 00:01:28,739 --> 00:01:32,530 different. And we will see that it's actually a 21 00:01:32,530 --> 00:01:37,915 transition in some sense from fermion like behavior, to boson like behavior. 22 00:01:37,915 --> 00:01:45,361 That is exactly what explains transition from a regular metal into a quantum 23 00:01:45,361 --> 00:01:51,86 superconductor. The first step in defining quantum 24 00:01:51,86 --> 00:01:54,920 statistics. Let us consider two identical quantum 25 00:01:54,920 --> 00:01:58,952 particles. This blue represented by these blue dots 26 00:01:58,952 --> 00:02:02,240 here. And let's assume that all intrinsic 27 00:02:02,240 --> 00:02:05,350 properties of these particles are exactly the same. 28 00:02:05,350 --> 00:02:09,610 And so the question I'm going to be asking is what happens if we braid two 29 00:02:09,610 --> 00:02:13,306 particles? So, in other words, if these two 30 00:02:13,306 --> 00:02:18,195 particles exchange places. So the first particle goes to the place 31 00:02:18,195 --> 00:02:24,766 of the second one and vice-versa. So what it means is that the wave 32 00:02:24,766 --> 00:02:28,580 functions. In this case, the wave function 33 00:02:28,580 --> 00:02:32,430 describing this two, this two particle system depends well, on the coordinates 34 00:02:32,430 --> 00:02:36,60 of, of these two particles, r1 and r2, in principle, some other quantum numbers 35 00:02:36,60 --> 00:02:40,872 that I'm not writing. And so the question we're asking is what 36 00:02:40,872 --> 00:02:45,860 happens with this wave function if we replace r1 with r2, and otherwise errors. 37 00:02:45,860 --> 00:02:50,892 So it turns out that upon this braiding the wave function requires quantum 38 00:02:50,892 --> 00:02:56,410 mechanical phase. So this is statistical phase phi. 39 00:02:56,410 --> 00:03:01,89 And well we can sort of get an idea of why it might happen. 40 00:03:01,89 --> 00:03:04,599 Because the wave function really is a tool for us to understand the observable 41 00:03:04,599 --> 00:03:10,84 phenomenon, most observable answer. Through the let's say the absolute value 42 00:03:10,84 --> 00:03:13,928 of the wave function or the various metric elements wave function phase is 43 00:03:13,928 --> 00:03:19,44 not something we worry about. And so we may imagine that even though 44 00:03:19,44 --> 00:03:22,908 from our perspective nothing has changed since two particles are identical which 45 00:03:22,908 --> 00:03:26,870 one is at which point doesn't really matter. 46 00:03:26,870 --> 00:03:31,424 So we still main principle get a quantum mechanical phase here, and indeed this 47 00:03:31,424 --> 00:03:36,81 happens, okay? So and so the question that we can ask, 48 00:03:36,81 --> 00:03:40,972 and that was asked a long time ago what are the possible values of this quantum 49 00:03:40,972 --> 00:03:47,680 statistical phase? And the answer was provided in this 50 00:03:47,680 --> 00:03:52,100 seminal paper entitled The Connection Between Spin and Statistics, by Wolfgang 51 00:03:52,100 --> 00:03:56,65 Pauly, who proved there exists a one-to-one correspondence between the 52 00:03:56,65 --> 00:04:00,550 internal angular momentum of a particle, so called Spin that you're going to learn 53 00:04:00,550 --> 00:04:05,360 about a little later in the course and it's quantum statistics so I want to give 54 00:04:05,360 --> 00:04:09,520 you a lecture on spin because there will be a separate one on that but that will 55 00:04:09,520 --> 00:04:13,485 just mention that their are just two types of particles allowed in nature 56 00:04:13,485 --> 00:04:23,790 which have either integer or half integer spin. 57 00:04:23,790 --> 00:04:31,290 And those differ by how their wave functions transform under rotation. 58 00:04:31,290 --> 00:04:35,580 So if we rotate our [INAUDIBLE] coordinate system by a certain angle of 59 00:04:35,580 --> 00:04:39,738 this rotation [INAUDIBLE] that we would encounter would look differently 60 00:04:39,738 --> 00:04:44,862 depending on what kind of spin of the particle has. 61 00:04:44,862 --> 00:04:49,418 And, interestingly enough let's say spin one half particles an electron is an 62 00:04:49,418 --> 00:04:54,356 example of such a particle. is going to be described by simple 63 00:04:54,356 --> 00:05:00,478 representation of this rotation. Which upon repeating it by 360 degrees, 64 00:05:00,478 --> 00:05:07,864 actually picks up a minus sign. As opposed to integer spin particles, or 65 00:05:07,864 --> 00:05:12,990 particles with no spins that don't really change. 66 00:05:12,990 --> 00:05:16,500 Whose wave function doesn't really change upon this rotation by 360 degrees. 67 00:05:16,500 --> 00:05:22,677 Now I'm just sort of giving you a flavor, but the outcome of this so-called spin 68 00:05:22,677 --> 00:05:29,600 statistics theory can be formulated in a very simple way. 69 00:05:29,600 --> 00:05:34,374 So if a particle with integer a internal angle and momentum including zero the 70 00:05:34,374 --> 00:05:40,140 phase here is really zero, so there is no change in the wave function. 71 00:05:40,140 --> 00:05:43,900 If we replace r one bl-, by r2, nothing changes. 72 00:05:43,900 --> 00:05:49,828 Well, for particles with a half integer spin this statistical [INAUDIBLE], phi is 73 00:05:49,828 --> 00:05:54,42 equal to pi. Or, in other words, this psi of r1 and r2 74 00:05:54,42 --> 00:05:58,940 is equal to minus of psi of r2 and r1. And these guys with half integer spins 75 00:05:58,940 --> 00:06:02,900 are half integer spin are called fermions. 76 00:06:02,900 --> 00:06:07,100 So, we'll just make one comment that it turns out that this classification is 77 00:06:07,100 --> 00:06:11,180 specific to a three dimensional relativistic quantum theory to the world 78 00:06:11,180 --> 00:06:15,260 that which we actually leave but if we consider two dimensional systems which 79 00:06:15,260 --> 00:06:22,568 appear as a matter of fact in, let's say in condensed measure context. 80 00:06:22,568 --> 00:06:27,160 So we have two dimensional metals and two dimensional super conductors. 81 00:06:27,160 --> 00:06:30,865 So there it turns out that the possibilities for phi are much more 82 00:06:30,865 --> 00:06:34,305 interesting. So apart from zero and pi we may have an 83 00:06:34,305 --> 00:06:38,100 extra arbitrary phase of pi, which are called in the particles like that, so the 84 00:06:38,100 --> 00:06:42,170 fictitious immersion particles like that are called anions this pi can be a matrix 85 00:06:42,170 --> 00:06:45,635 in which case they are called [UNKNOWN] in news but this is the only thing I'm 86 00:06:45,635 --> 00:06:52,170 going to mention about it. But it just, I want to want you to know 87 00:06:52,170 --> 00:06:57,290 that the parts of these well known possibilities for fundamental particles. 88 00:06:57,290 --> 00:07:02,610 So there are other situations which arise in a [UNKNOWN] manner of physics. 89 00:07:02,610 --> 00:07:11,423 But now let me focus on this case. So well, so we for irregular particles we 90 00:07:11,423 --> 00:07:15,942 have I, either a zero or pi. So let's assume we're dealing with 91 00:07:15,942 --> 00:07:18,290 fermions, let's say we're dealing with electrons. 92 00:07:18,290 --> 00:07:22,514 So what are the consequences of this minus sign that we pick up in front of 93 00:07:22,514 --> 00:07:26,116 the wave function. Does it really matter at all, so should 94 00:07:26,116 --> 00:07:29,524 we worry about it? So let us think about it. 95 00:07:29,524 --> 00:07:34,794 So what we're really saying again is that psi of r 1 and r 2 is equal to minus of 96 00:07:34,794 --> 00:07:42,424 psi of r 2 and r 1. And it immediately leads to a conclusion 97 00:07:42,424 --> 00:07:48,630 that for an important case r 1 equals r 2. 98 00:07:48,630 --> 00:07:52,710 So psi of r 1 and r 1 is equal to minus of psi of r 1 and r 1. 99 00:07:52,710 --> 00:07:57,630 Which means that the size is equal to 0. So what does it mean? 100 00:07:57,630 --> 00:08:01,985 It means that the probability of find two identical fermions in the same point is 101 00:08:01,985 --> 00:08:06,929 0, because the size is 0. So, well and It means that no two 102 00:08:06,929 --> 00:08:12,748 particles can be like no two fermions. Identical fermions cannot occupy the same 103 00:08:12,748 --> 00:08:15,121 point. And more generally we can actually 104 00:08:15,121 --> 00:08:19,510 generalize this for an arbitrary representation of the way function. 105 00:08:19,510 --> 00:08:23,101 We can prove and what this is about exclusion principle is that no two 106 00:08:23,101 --> 00:08:27,560 identical fermions cannot occupy the same quantum state. 107 00:08:27,560 --> 00:08:30,630 They cannot have exactly the same quantum numbers. 108 00:08:30,630 --> 00:08:35,12 But there is no such constraint for bosons okay. 109 00:08:35,12 --> 00:08:38,840 So, this has very important consequences so this statement again is called the 110 00:08:38,840 --> 00:08:42,994 Pauli exclusion principle and for bosons it doesn't apply. 111 00:08:42,994 --> 00:08:48,35 So, and that the consequences are that if we have certain quantum state, let's see 112 00:08:48,35 --> 00:08:52,863 an [UNKNOWN] quantum state i and we want to populate this state with fermions, 113 00:08:52,863 --> 00:09:00,460 there are only two possibilities for the occupation number of this state. 114 00:09:00,460 --> 00:09:03,550 So the occupation number can be either 0 or 1. 115 00:09:03,550 --> 00:09:07,562 Well for bosons, it can be anything, it can be any 0, 1, 2, can put any, any 116 00:09:07,562 --> 00:09:13,40 number of bosons we want in this quantum [INAUDIBLE] . 117 00:09:13,40 --> 00:09:17,596 Now what does it mean in the context of many body systems, when we have actually 118 00:09:17,596 --> 00:09:22,286 more than one and two particles, when we have billions and billions of identical 119 00:09:22,286 --> 00:09:26,909 particles as we usually do in quantum systems such as, let's say, metals and 120 00:09:26,909 --> 00:09:33,933 superconductors? So let us discuss this question first on 121 00:09:33,933 --> 00:09:38,855 for many bosons system. And what we're actually asking is what 122 00:09:38,855 --> 00:09:43,754 happens let's say if we have many energy levels that are initially empty and we 123 00:09:43,754 --> 00:09:47,756 want to know what happens if we sort of pour identical bosons into this 124 00:09:47,756 --> 00:09:52,517 Prescribe, landscape of quantum states, with some states I with some energy E, 125 00:09:52,517 --> 00:09:59,518 energy E sub I. And more specifically what we're asking 126 00:09:59,518 --> 00:10:04,230 is, what we're interested in is what is the lowest energy state, the so called 127 00:10:04,230 --> 00:10:10,570 ground state, that the system would want to, Form at low temperatures. 128 00:10:10,570 --> 00:10:14,724 So at low temperatures the excitations are going to sort of calm down and the 129 00:10:14,724 --> 00:10:18,900 system would want to form the lowest energy state. 130 00:10:18,900 --> 00:10:23,207 Now since any number of bosons can occupy any state, so clearly the lowest energy 131 00:10:23,207 --> 00:10:26,511 state would correspond to essentially a state in which all bosons, 132 00:10:26,511 --> 00:10:30,287 non-interacting bosons that is, are sitting just at the lowest possible 133 00:10:30,287 --> 00:10:36,366 level. Now for a free particle with dispersion, 134 00:10:36,366 --> 00:10:42,520 just having the kinetic energy dispersion p squared over 2 m. 135 00:10:42,520 --> 00:10:45,936 Some sort of parabolic dispersion here, I'm just plotting it in sort of an 136 00:10:45,936 --> 00:10:49,800 example of a two-dimensional system px, py and this is the energy basically all 137 00:10:49,800 --> 00:10:54,716 bosons would want to drop into the zero momentum state. 138 00:10:54,716 --> 00:11:01,266 Okay, and this kind of state is called Bose-Einstein condensate. 139 00:11:01,266 --> 00:11:05,74 The state in which a single quantum mechanical level in this case the state 140 00:11:05,74 --> 00:11:09,529 was zero momentum, zero velocity is microscopically occupied. 141 00:11:10,630 --> 00:11:16,0 So this kind of phenomenon has been known for a long time, since Einstein. 142 00:11:16,0 --> 00:11:21,544 But it was sort of, very observed in the clear way, only very recently in 1995 143 00:11:21,544 --> 00:11:25,154 first. Now it's observed routinely in various 144 00:11:25,154 --> 00:11:28,580 laboratories including our own joint quantum institute. 145 00:11:28,580 --> 00:11:32,864 But back in 1995, it was new. And, this discovery was actually awarded 146 00:11:32,864 --> 00:11:37,546 2001 Nobel Prize in physics. So here, I'm showing the [INAUDIBLE] . 147 00:11:37,546 --> 00:11:40,280 So, basically, what the experimentalists did. 148 00:11:40,280 --> 00:11:44,550 They trapped atoms, bosonic atoms in a confined geometry in this certain 149 00:11:44,550 --> 00:11:48,498 trapping potential. And cooled them down to ultra cool 150 00:11:48,498 --> 00:11:52,670 temperatures. And then let them remove the trap. 151 00:11:52,670 --> 00:11:56,719 And let them expand. And they just sort of took photographs of 152 00:11:56,719 --> 00:12:00,900 these atoms as time went by. And they saw this pic. 153 00:12:00,900 --> 00:12:03,330 So this is basically the so called time of flight extension. 154 00:12:03,330 --> 00:12:11,350 And this pic is shows the population. The density of particles in real space. 155 00:12:11,350 --> 00:12:16,700 Large number of particles sort of don't, move too much, as time goes by. 156 00:12:16,700 --> 00:12:21,460 And this speaks sort of [COUGH] in real space is [UNKNOWN] of condensation and 157 00:12:21,460 --> 00:12:25,586 [UNKNOWN] space. So this corresponds to particles with 158 00:12:25,586 --> 00:12:28,610 zero velocity so the just sort of stay still. 159 00:12:28,610 --> 00:12:33,670 So now I would like to mention a very important circumstance namely that. 160 00:12:33,670 --> 00:12:38,914 This Bose-Einstein condensates may host a very interesting phenomenon of 161 00:12:38,914 --> 00:12:43,930 superfluidity where a condensate of interacting bosons form a superfluid 162 00:12:43,930 --> 00:12:51,582 which has exact 0 viscocity and which can flow therefore without resistance. 163 00:12:51,582 --> 00:12:55,870 So, here for example I'm showing a photo of superfluid. 164 00:12:55,870 --> 00:13:00,217 Helium in a cup, the helium which can actually escape the cup climbing up the 165 00:13:00,217 --> 00:13:03,916 walls. So let me mention that superfluidity, the 166 00:13:03,916 --> 00:13:09,426 phenomenon of superfluidity has actually been known for a very very long time. 167 00:13:09,426 --> 00:13:15,46 Since 1937 it was discovered then in Helium by Pyotr Kapitsa. 168 00:13:15,46 --> 00:13:19,768 And, and Kapitsa got his Nobel prize for this discovery of 1978. 169 00:13:19,768 --> 00:13:24,920 Well, it would be too late, 40 years after the initial discovery. 170 00:13:24,920 --> 00:13:28,896 So the mathematical theory of superfluidty was put together by Lev 171 00:13:28,896 --> 00:13:35,840 Landau a famous Russian theorist who got his Nobel prize for this theory in 1962. 172 00:13:35,840 --> 00:13:39,458 So actually before Kapitsa and hey, let me just mention that the You know this 173 00:13:39,458 --> 00:13:44,810 noble prize was awarded to him after Lev Landau got into a very serious accident. 174 00:13:44,810 --> 00:13:48,93 Everybody thought he wouldn't survive but by some miracle he lived for a few more 175 00:13:48,93 --> 00:13:51,21 years. And this allowed the Nobel community to 176 00:13:51,21 --> 00:13:55,426 award him long overdue Nobel prize. So, in any case, I'm not going to 177 00:13:55,426 --> 00:13:57,918 mention. I'm not going to talk too much about the 178 00:13:57,918 --> 00:14:02,48 theory of superfluidity now because it actually requires understanding of the 179 00:14:02,48 --> 00:14:05,588 interactions in Bose-Einstein condensates, and this is a non-trivial 180 00:14:05,588 --> 00:14:11,788 business interacting many-body states. But I will just that, superfluidity sort 181 00:14:11,788 --> 00:14:16,530 of rings a bell in the context of superconductivity because. 182 00:14:16,530 --> 00:14:21,3 If we want to explain the main property of a superconductor, namely the zero 183 00:14:21,3 --> 00:14:25,973 resistance state of the superconductor, we would want essentially a superfluid of 184 00:14:25,973 --> 00:14:33,70 electrons, which would move with zero viscosity and without resistance. 185 00:14:33,70 --> 00:14:37,38 This is exactly what the doctor had as prescribed, but there is one little 186 00:14:37,38 --> 00:14:41,998 problem with this explanation of super, superconductivity, namely well electrons 187 00:14:41,998 --> 00:14:46,28 in the metal are not [INAUDIBLE], they have half of half of the [INAUDIBLE], so 188 00:14:46,28 --> 00:14:49,314 they spin one half, so they're [INAUDIBLE], so they do not form a 189 00:14:49,314 --> 00:14:56,78 superfluid. So, what do they do? 190 00:14:56,78 --> 00:15:00,294 And, to answer, to answer this question, I need to consider the ground state of 191 00:15:00,294 --> 00:15:04,910 many-fermion system now. We talked about the ground state of 192 00:15:04,910 --> 00:15:08,300 bosons. Now let me talk about what happens with 193 00:15:08,300 --> 00:15:13,270 fermions if I pour them into a given energy landscape. 194 00:15:13,270 --> 00:15:16,556 Let's say I have, Has some available levels and they have a many fermion 195 00:15:16,556 --> 00:15:20,54 system that would want to occupy those levels and the question is, How will I 196 00:15:20,54 --> 00:15:24,302 minimize energy? So due to the power of exclusion 197 00:15:24,302 --> 00:15:27,956 principle I cannot allow my fermions to be sitting on the same lowest energy 198 00:15:27,956 --> 00:15:31,781 state. So I can only at minimized energy, can 199 00:15:31,781 --> 00:15:37,35 only sort of stack them up one by one to the lowest energy states as so until they 200 00:15:37,35 --> 00:15:42,502 reach the the certain level called the fermion level or fermion energy up here 201 00:15:42,502 --> 00:15:47,685 and basically this is the picture of fermion ground state, which is quite 202 00:15:47,685 --> 00:15:57,304 different from bosonic ground state. Now in the context of a metal and three 203 00:15:57,304 --> 00:16:02,122 dimensional, three dimensional metal, so what we have Is well our energy landscape 204 00:16:02,122 --> 00:16:07,180 is given by the usual disbursement relation. 205 00:16:07,180 --> 00:16:11,570 The energy versus momentum is equal to p squared over 2m. 206 00:16:11,570 --> 00:16:17,264 So what I need to do in order to sort of satisfy the same picture with this energy 207 00:16:17,264 --> 00:16:24,418 is that I will occupy all momentum stays. So here I have k x, k y,k z ,or I could 208 00:16:24,418 --> 00:16:28,800 have written p x, p y, and p z. So below certain momentum is called 209 00:16:28,800 --> 00:16:32,628 fermion momentum I will have all states occupied just like I have here all these 210 00:16:32,628 --> 00:16:36,960 levels occupied and while these levels are all empty. 211 00:16:36,960 --> 00:16:41,160 So and the threshold momentum it is just ignore this red and white dots for the 212 00:16:41,160 --> 00:16:45,140 moment. So below this threshold momentum. 213 00:16:45,140 --> 00:16:49,329 I have all states occupied above this threshold momentum I have all states 214 00:16:49,329 --> 00:16:55,160 empty and this is exactly the picture which describes electrons in, in metals. 215 00:16:55,160 --> 00:16:59,220 Actually it works amazingly well for metals. 216 00:16:59,220 --> 00:17:03,55 And it's also known that the typical Fermi temperature in metal, the Fermi 217 00:17:03,55 --> 00:17:07,67 temperature being the threshold momentum here The p, pF square divided by 2m this 218 00:17:07,67 --> 00:17:11,20 is the Fermi energy divided by the Bols, Bolsman constants or constant which 219 00:17:11,20 --> 00:17:17,794 converts energy into the temperature. So if we do this calculation the typical 220 00:17:17,794 --> 00:17:21,738 Fermi temperature in the metal would be around 10000k, which is huge so even if 221 00:17:21,738 --> 00:17:25,914 the room temperature left alone at 4 kelvin so this energy scale is normal as 222 00:17:25,914 --> 00:17:32,416 compared to all other energy. [UNKNOWN] So and what the temperature, 223 00:17:32,416 --> 00:17:37,108 well Fermi temperature usually does is it excited electrons from below the thermo 224 00:17:37,108 --> 00:17:40,238 surface. For instance I can not excite electron 225 00:17:40,238 --> 00:17:43,130 from here to here. I can not move, so let me just draw this 226 00:17:43,130 --> 00:17:46,323 paralysis and then I will say it's not allowed. 227 00:17:47,420 --> 00:17:50,287 So have, let's say, an electron here, I can move it to this point, it's not a 228 00:17:50,287 --> 00:17:53,870 lot. Because both the stays are occupied. 229 00:17:53,870 --> 00:17:57,376 So this stay is already occupied. So power exclusion wouldn't allow me to 230 00:17:57,376 --> 00:18:00,173 do this thing. So, but what I can do if I have 231 00:18:00,173 --> 00:18:04,394 temperature, I can, I have some energy to spend, so I can excite an electron let's 232 00:18:04,394 --> 00:18:10,255 say from here to here. And so the y point here is an empty space 233 00:18:10,255 --> 00:18:15,550 that the electron leaves behind. It's called a hole and the red one is the 234 00:18:15,550 --> 00:18:19,495 actual electronic excitation. And this is exactly the picture of a 235 00:18:19,495 --> 00:18:23,960 metal and so this guys, basically this red guys and this white guys. 236 00:18:23,960 --> 00:18:29,412 these guys form basically they are the linked leaders in a typical metal. 237 00:18:29,412 --> 00:18:33,434 But they do not condense, they don't form a super fluid. 238 00:18:33,434 --> 00:18:37,460 They just move around and experience scattering you know from one point to a 239 00:18:37,460 --> 00:18:41,630 different point and this result in a finite resist. 240 00:18:41,630 --> 00:18:46,229 So how do you reconcile this picture of a metal with the presence of super 241 00:18:46,229 --> 00:18:51,90 conductor is that is observed at low temperatures. 242 00:18:51,90 --> 00:18:54,190 So this was again a major mystery for many many years, was not at all obvious 243 00:18:54,190 --> 00:18:58,0 how to do this. But of course there is a hint about it 244 00:18:58,0 --> 00:19:02,752 due to the presence of superfood so sort of in retrospect it's pretty natural to 245 00:19:02,752 --> 00:19:07,144 ask whether or not it's possible to somehow convert to electronic liquid 246 00:19:07,144 --> 00:19:12,940 [UNKNOWN] liquid Into a bosonic superfluid. 247 00:19:12,940 --> 00:19:14,920 So this guys are fermions, so they're not bosons. 248 00:19:14,920 --> 00:19:20,274 So can we make a bosonic superfluid out of a fermionic gas such as here? 249 00:19:20,274 --> 00:19:26,309 Well, it's difficult unless, so unless I allow my electrons to form composite 250 00:19:26,309 --> 00:19:31,567 objects. Of two electrons states whose total spin 251 00:19:31,567 --> 00:19:35,975 is either one or a zero. Remember the speed of a single electron 252 00:19:35,975 --> 00:19:39,165 is one half but if I combine the two of them together, so it's going to be an 253 00:19:39,165 --> 00:19:42,320 integer. So they are going to be boson. 254 00:19:42,320 --> 00:19:47,94 And so if I somehow find a way to say force objects well I could have actually 255 00:19:47,94 --> 00:19:52,638 drawn it like this but well knowing the results, I'm drawing these objects this 256 00:19:52,638 --> 00:19:57,720 week. But this station doesn't matter. 257 00:19:57,720 --> 00:20:02,200 So if I allow the formation of to electron states, this, these 2 electrons 258 00:20:02,200 --> 00:20:06,939 states are going to be bosons. And these bosons would be able to 259 00:20:06,939 --> 00:20:09,650 condense. And these guys would be able to form a 260 00:20:09,650 --> 00:20:15,583 super-fluid now, a charge super-fluid. And this is great because this state will 261 00:20:15,583 --> 00:20:19,370 superconduct. And this is exactly what happens and this 262 00:20:19,370 --> 00:20:23,390 is bound states of two electrons called Cooper burst. 263 00:20:23,390 --> 00:20:29,620 And from this perspective superconductor is a Bose-Einstein condensate of cooper 264 00:20:29,620 --> 00:20:32,860 burst. But the main question remains is why 265 00:20:32,860 --> 00:20:36,836 would this Cooper burst would form? Why would such states appear? 266 00:20:36,836 --> 00:20:40,866 And this is the question that we're actually going to answer in the remaining 267 00:20:40,866 --> 00:20:44,570 two segments. And we're going to solve the cooper 268 00:20:44,570 --> 00:20:48,30 pairing problem. And this was also established connection 269 00:20:48,30 --> 00:20:50,870 between this lecture and the previous lecture. 270 00:20:50,870 --> 00:20:54,563 When we discussed, [INAUDIBLE] states and weak potentials.