1 00:00:00,540 --> 00:00:06,240 The quantum harmonic oscillator problem and its solution go far beyond the simple 2 00:00:06,240 --> 00:00:13,450 generalization of a particle and a spring problem in Classical Physics. 3 00:00:13,450 --> 00:00:18,700 And they appear in a variety of contexts in quantum physics, as already mentioned. 4 00:00:18,700 --> 00:00:23,584 For example electromagnetic radiation can be thought of as a collection of harmonic 5 00:00:23,584 --> 00:00:28,666 oscillators, the photons. And much more generally various complex 6 00:00:28,666 --> 00:00:33,503 structures in quantum mechanics. Oftentimes can be thought of as being 7 00:00:33,503 --> 00:00:39,690 built of the small building blocks being quantum harmonic oscillation. 8 00:00:39,690 --> 00:00:43,643 But in this particular case, the meaning of these individual oscillators is 9 00:00:43,643 --> 00:00:46,844 different. And today, I'm going to show you the 10 00:00:46,844 --> 00:00:50,402 appearance of these quantum harmonic oscillators. 11 00:00:50,402 --> 00:00:54,172 And the context of a very interesting problem that sort of underlies solid 12 00:00:54,172 --> 00:00:58,290 state physics or a condensed measure of physics. 13 00:00:58,290 --> 00:01:03,260 And this is a problem of excitations in crystals, which are collections of many, 14 00:01:03,260 --> 00:01:09,536 many ions that form a periodic structure. And these ions can oscillate, and these 15 00:01:09,536 --> 00:01:13,973 oscillations, quantum oscillations, are called phonons. 16 00:01:13,973 --> 00:01:18,194 And we will see how these phonons appear, and we will see also that they can be 17 00:01:18,194 --> 00:01:22,350 described. In terms of the creation-annihilation 18 00:01:22,350 --> 00:01:27,160 operators that we introduced in the first lecture this week. 19 00:01:27,160 --> 00:01:31,783 To understand the concept of a collective mode, such as a phonon for example, in a 20 00:01:31,783 --> 00:01:37,189 complex quantum system. We first can go back to classical physics 21 00:01:37,189 --> 00:01:41,864 and remind ourselves. The notion of a normal mode, which is 22 00:01:41,864 --> 00:01:45,302 equivalent to the quantum collective mode. 23 00:01:45,302 --> 00:01:49,270 And these normal modes appear in complex classical systems. 24 00:01:49,270 --> 00:01:54,390 So, here, I have a primitive classical system if you want this is a simpler 25 00:01:54,390 --> 00:01:59,332 harmonic oscillator. And in this case, as we know, the 26 00:01:59,332 --> 00:02:04,941 frequency of the oscillation is propor-, is equal to square root of the stiffness 27 00:02:04,941 --> 00:02:12,000 of this spring divided by the mass of this of this object. 28 00:02:12,000 --> 00:02:16,473 now if we make the system a little more complicated, and instead of just one 29 00:02:16,473 --> 00:02:20,868 oscillator. Consider let's say two masses attached to 30 00:02:20,868 --> 00:02:26,061 let's say three springs in this case. So, we have a system here with two 31 00:02:26,061 --> 00:02:31,451 degrees of freedom. And this system will have two different 32 00:02:31,451 --> 00:02:39,012 oscillation frequencies corresponding to different types of oscillatory motions. 33 00:02:39,012 --> 00:02:45,047 So, for instance this type of motion when the two masses sort of oscillate in in 34 00:02:45,047 --> 00:02:51,095 phase with one another. So, this let's see first type of motion 35 00:02:51,095 --> 00:02:56,590 is described by the frequency which is exactly equal to omega. 36 00:02:56,590 --> 00:03:01,078 But there is also a second type of motion that we can imagine this slightly more 37 00:03:01,078 --> 00:03:05,342 complicated system. In which case let's say these guys 38 00:03:05,342 --> 00:03:08,999 oscillate out of phase in this fashion, sort of. 39 00:03:08,999 --> 00:03:13,355 And this oscillation, this oscillation frequency let me call it two in this 40 00:03:13,355 --> 00:03:19,228 case, is equal to square root of times the basic frequency of these guys. 41 00:03:19,228 --> 00:03:22,552 So, let's assume they all have the same stiffness. 42 00:03:22,552 --> 00:03:26,842 And we can make things more and more complicated, and let's say in this case 43 00:03:26,842 --> 00:03:34,011 have six six masses attached to springs. In which case we will have six different 44 00:03:34,011 --> 00:03:38,045 degrees of freedom. And six different oscilation frequencies. 45 00:03:38,045 --> 00:03:41,526 So, and this oscillation frequencies normal so called normal modes in 46 00:03:41,526 --> 00:03:46,800 classical physics. They have an analog in quantum physics. 47 00:03:46,800 --> 00:03:50,887 In particular I would say in this example, in this example of two harmonic 48 00:03:50,887 --> 00:03:54,476 oscillators. We just by knowing the results of the 49 00:03:54,476 --> 00:03:58,492 classical calculation. We can write down, as a matter of fact 50 00:03:58,492 --> 00:04:03,692 the spectrum of the quantum problem, or the corresponding quantum problem. 51 00:04:03,692 --> 00:04:09,959 So and the spectrum of the corresponding quantum problem will be. 52 00:04:09,959 --> 00:04:15,791 simply the energy is going to be equal to h omega 1 and 1 plus 1 half h plus h 53 00:04:15,791 --> 00:04:22,888 omega 2, and 2 plus 1 half. And its going to depends on two quantum 54 00:04:22,888 --> 00:04:28,512 numbers, n1 and n2 corresponding to two harmonic oscillators. 55 00:04:28,512 --> 00:04:32,292 But these are not going to be actually the harmonic oscillators built out of 56 00:04:32,292 --> 00:04:38,180 let's say the coordinates and momentum of the two the two particles. 57 00:04:38,180 --> 00:04:42,492 So these oscillators or the corresponding creation-annihilation operators are going 58 00:04:42,492 --> 00:04:46,076 to be a certain non-trivial linear combination of these x1's and p1's, x2 59 00:04:46,076 --> 00:04:50,730 and p2. which sort of describe the generalization 60 00:04:50,730 --> 00:04:55,316 of these in-phase and out-of-phase motions to quantum physics. 61 00:04:55,316 --> 00:05:00,215 And in general, whenever we have a classical system with some normal modes. 62 00:05:00,215 --> 00:05:04,505 There is a corresponding quantum system with a spectrum associated with these 63 00:05:04,505 --> 00:05:08,914 normal modes. A very interesting example of a complex 64 00:05:08,914 --> 00:05:12,490 mini particle quantum system is a crystal. 65 00:05:12,490 --> 00:05:17,110 We know that crystalline structures of various types appear in a variety of 66 00:05:17,110 --> 00:05:22,941 solid state materials. And in this structure so what happens is 67 00:05:22,941 --> 00:05:29,027 that ions or atoms of which this solicited material sort of built. 68 00:05:29,027 --> 00:05:33,188 They arrange themselves into a unique order and pattern. 69 00:05:33,188 --> 00:05:37,850 And so here, for instance, I'm showing a fragment of a so called simple cubic 70 00:05:37,850 --> 00:05:42,394 lattice. And these blue balls in this picture, 71 00:05:42,394 --> 00:05:47,364 correspond to atoms sitting on their equilibrium position on their lattice 72 00:05:47,364 --> 00:05:51,972 size. But this picture is a bit deceptive, 73 00:05:51,972 --> 00:05:55,428 actually it is probably better to say incorrect in that the atoms here, in 74 00:05:55,428 --> 00:06:00,319 reality, are not static. So, they indeed have some preferential 75 00:06:00,319 --> 00:06:03,530 positions, and some equilibrium positions. 76 00:06:03,530 --> 00:06:07,820 But they move around all the time, so there, this position that they occupy, 77 00:06:07,820 --> 00:06:13,226 corresponds to a minimum in some effective potential they fill. 78 00:06:13,226 --> 00:06:17,834 But we can oscillate near the minimum of this potential and these oscillations are 79 00:06:17,834 --> 00:06:23,840 basically just grabbed locally as a essentially a harmonic oscillator motion. 80 00:06:23,840 --> 00:06:28,600 So, better illustration of what a crystal is that I put together well to simply sit 81 00:06:28,600 --> 00:06:33,800 here for a two dimensional case. So, it's presented here. 82 00:06:33,800 --> 00:06:37,706 So, instead of thinking about atoms which are sort of tied together in this 83 00:06:37,706 --> 00:06:41,688 crystalline structure by some rigid objects. 84 00:06:41,688 --> 00:06:46,168 We should think about atoms being tied together by some elastic springs with 85 00:06:46,168 --> 00:06:50,740 some stiffness. Which is fine at and so when we look at 86 00:06:50,740 --> 00:06:54,452 this system so my analogy to what I had discussed, what I had just discussed in 87 00:06:54,452 --> 00:06:58,969 the previous slide. We can think about this system as a many 88 00:06:58,969 --> 00:07:02,600 oscillator system with a moderate degrees of freedom. 89 00:07:02,600 --> 00:07:06,692 So, in reality in this material so they're not two, or three or six atoms, 90 00:07:06,692 --> 00:07:12,970 like in the previous slide, but they're min, billions and billions of atoms. 91 00:07:12,970 --> 00:07:17,000 And therefore, there are many frequencies, eigen-frequencies of normal 92 00:07:17,000 --> 00:07:22,298 modes that I classical normal modes. That appear and they correspond to 93 00:07:22,298 --> 00:07:26,690 various kinds of complicated oscillations in this system. 94 00:07:26,690 --> 00:07:31,447 And in the limit of an infinite number, basically a so called thermodynamic limit 95 00:07:31,447 --> 00:07:36,090 when we consider essentially an infinite crystal. 96 00:07:36,090 --> 00:07:42,454 So, these create normal modes, they merge into continuum curves which essential 97 00:07:42,454 --> 00:07:49,127 corresponded to elastic waves running through this system. 98 00:07:49,127 --> 00:07:55,320 And mathematically these waves are described by familiar expressions. 99 00:07:55,320 --> 00:07:59,874 So, if we think let's say about the displacement of an atom with regard to 100 00:07:59,874 --> 00:08:04,566 its equilibrium position, some delta r as a function of time, for a particular 101 00:08:04,566 --> 00:08:09,650 atom. Let's say nth atom, so it's proportional 102 00:08:09,650 --> 00:08:14,130 to the familiar exponential of key dot r, so where this atom is located, minus 103 00:08:14,130 --> 00:08:22,014 omega t. But this omega is itself a function of 104 00:08:22,014 --> 00:08:25,162 key. And so to find this so called dispersion, 105 00:08:25,162 --> 00:08:29,130 phonon dispersion is the main question that appears in the context of this 106 00:08:29,130 --> 00:08:32,750 problem. And this is what is going to be of 107 00:08:32,750 --> 00:08:37,760 interest to us. Of particular interest is the question of 108 00:08:37,760 --> 00:08:42,600 what are the low energy modes of this system. 109 00:08:42,600 --> 00:08:46,824 So, what it actually means is the following, suppose we have a completely 110 00:08:46,824 --> 00:08:51,098 static crystal. So, while as static is a guess, in 111 00:08:51,098 --> 00:08:53,958 quantum mechanics there is always zero point motion, but we can even consider 112 00:08:53,958 --> 00:08:57,754 classical system. Let's say we just built physically built 113 00:08:57,754 --> 00:09:01,879 this model by attaching a bunch of masses to the springs, and attach, and arranging 114 00:09:01,879 --> 00:09:06,340 them. In the, in equilibrium pattern and let's 115 00:09:06,340 --> 00:09:11,872 say we touch this system a little bit, we apply very weak perturbation. 116 00:09:11,872 --> 00:09:15,969 So the question I am asking about the law energy modes is. 117 00:09:15,969 --> 00:09:19,387 Have we got inside any oscillation in this crystal ? 118 00:09:19,387 --> 00:09:23,301 Or this weak perturbation is not going to have any effect? 119 00:09:23,301 --> 00:09:27,135 Or, in other words, do we have to overcome any threshold, or gap as 120 00:09:27,135 --> 00:09:32,819 condensed measure people call it. In order to excite the minimal 121 00:09:32,819 --> 00:09:37,626 perturbation in the crystal. And as we will see in a later, in, in 122 00:09:37,626 --> 00:09:42,111 today's lecture, so the answer will be that, it doesn't matter how weak the 123 00:09:42,111 --> 00:09:47,242 perturbation is. There is always a very low energy mode 124 00:09:47,242 --> 00:09:53,190 which corresponds to long wave length. Or small key wave running through the 125 00:09:53,190 --> 00:09:59,434 samples, through this crystal. and this type of an excitation is called 126 00:09:59,434 --> 00:10:04,520 an acoustic mode or it corresponds to sound. 127 00:10:04,520 --> 00:10:07,770 Sound modes running through through this crystal. 128 00:10:08,960 --> 00:10:13,655 And it turns out that this appearance of this acoustic sound modes. 129 00:10:13,655 --> 00:10:17,933 Has a very profound explanation which goes as far as relativistic quantum 130 00:10:17,933 --> 00:10:22,332 physics actually where. There is a theorem called Goldstone 131 00:10:22,332 --> 00:10:27,550 theorem which guarantees the appearance of certain low energy modes. 132 00:10:27,550 --> 00:10:31,369 And even though I'm talking about classical system, let me nevertheless try 133 00:10:31,369 --> 00:10:36,147 to connect it to this theorem. But I will present sort of a poor man's 134 00:10:36,147 --> 00:10:40,654 version of this Goldstone theorem which is relevant to our problem. 135 00:10:40,654 --> 00:10:44,902 And then hopefully, we will be able to understand how they are connected. 136 00:10:44,902 --> 00:10:48,400 So this going to be a little bit of detail work. 137 00:10:48,400 --> 00:10:52,690 So, the poor man's version of the Goldstone theorem that I'm presenting is 138 00:10:52,690 --> 00:10:56,650 related to the notion of symmetry in quantum physics that we already 139 00:10:56,650 --> 00:11:01,876 discussed. So, suppose we have a quantum system 140 00:11:01,876 --> 00:11:05,410 which is invariant or symmetric with respect to certain class of 141 00:11:05,410 --> 00:11:09,819 transformations. What it means is that there exists 142 00:11:09,819 --> 00:11:15,123 transformations, let me call them u. And this operator commutes with the 143 00:11:15,123 --> 00:11:21,796 Hamiltonian which describes this system. So, this is what it means or in other 144 00:11:21,796 --> 00:11:26,730 words U, H, U minus 1 is equal to H itself. 145 00:11:27,920 --> 00:11:31,632 And let me consider this situation, when there is not just one or two symmetry 146 00:11:31,632 --> 00:11:35,721 operations that exist. But as a matter a fact we have a certain, 147 00:11:35,721 --> 00:11:40,820 this whole family of such operations let's see labeled by some parameter. 148 00:11:40,820 --> 00:11:43,788 Let me call it alpha. So, an example of such a symmetry for 149 00:11:43,788 --> 00:11:48,760 instance, could be symmetry with respect to translation. 150 00:11:48,760 --> 00:11:52,298 So, let's say if we, if our Hamiltonian is symmetric with respect with 151 00:11:52,298 --> 00:11:56,507 translating everything, every single particle in space, let's say by certain 152 00:11:56,507 --> 00:12:01,171 distance. So, it could be a distance of 1 meter or 153 00:12:01,171 --> 00:12:05,790 1 centimeter, 1 kilometer. So, there is a whole continuum of 154 00:12:05,790 --> 00:12:08,815 distances and directions we can translate our system. 155 00:12:08,815 --> 00:12:12,888 In which case is alpha, it's actually a three dimensional vector of translations. 156 00:12:12,888 --> 00:12:16,701 And while another example could be symmetry with respect to rotations. 157 00:12:16,701 --> 00:12:20,201 In which case alpha could represent a, an angle that we can choose, by which we can 158 00:12:20,201 --> 00:12:24,214 rotate our system. But in principle, can be anything, so we 159 00:12:24,214 --> 00:12:27,812 just basically postulate that the existence of some symmetry. 160 00:12:27,812 --> 00:12:31,400 What is called symmetry group but I'm not going to go into this details. 161 00:12:31,400 --> 00:12:35,768 with respect to H the Hamiltonian of the system is invariant. 162 00:12:35,768 --> 00:12:41,928 But a very interesting fact is that, even though the model itself can be invariant 163 00:12:41,928 --> 00:12:48,828 under a certain symmetry operations. The quantum states, the for instance 164 00:12:48,828 --> 00:12:53,448 lowest energy state of the ground state, does not necessarily have to be 165 00:12:53,448 --> 00:12:59,450 invariant. under the same symmetry U. 166 00:12:59,450 --> 00:13:04,142 So, what it means is that the system will find itself in a state, let's say, let me 167 00:13:04,142 --> 00:13:09,371 call it psi naught. Such that well, U acting on this psi 168 00:13:09,371 --> 00:13:15,402 naught is not equal to psi naught. So, by translating let's say our system 169 00:13:15,402 --> 00:13:19,777 in space, we do not necessarily represent the same state. 170 00:13:19,777 --> 00:13:26,010 And in this case we call, we say that the symmetry is broken, spontaneously broken. 171 00:13:26,010 --> 00:13:30,426 But the key statement, which is also the statement of this, of this Goldstone 172 00:13:30,426 --> 00:13:34,403 theorem in this context, is the following. 173 00:13:34,403 --> 00:13:38,681 If once again we have a situation where the Hamiltonian is a variant under a 174 00:13:38,681 --> 00:13:44,164 certain continuum of symmetries. But some or all of these symmetries are 175 00:13:44,164 --> 00:13:48,530 broken in this state in which the system finds itself. 176 00:13:48,530 --> 00:13:52,768 In this state let's say psi naught, then, to each broken continuum symmetry. 177 00:13:52,768 --> 00:13:56,576 There corresponds low energy mode, a sound-like mode if you want, that we just 178 00:13:56,576 --> 00:13:59,814 discussed. and well, this statement, this 179 00:13:59,814 --> 00:14:04,608 correspondence, is again the statement of this Goldstone theorem. 180 00:14:04,608 --> 00:14:09,605 So, I realize that there has been, you know, a lot of words that I've said. 181 00:14:09,605 --> 00:14:14,093 But perhaps the best way to understand the actual meaning of these words would 182 00:14:14,093 --> 00:14:19,537 be to provide an example. Which also will explain why this 183 00:14:19,537 --> 00:14:25,460 discussion is relevant to what we talked about before in the previous slides. 184 00:14:25,460 --> 00:14:28,100 About hormones, crystals and all these things. 185 00:14:28,100 --> 00:14:32,039 Now, let us look at a crystal structure once again. 186 00:14:32,039 --> 00:14:35,770 So we have here a periodic arrangement of atoms. 187 00:14:35,770 --> 00:14:39,475 But nobody puts this crystal structure by hand, so if we have a piece of some 188 00:14:39,475 --> 00:14:43,233 solid, solid material, let's say a piece copper. 189 00:14:43,233 --> 00:14:45,952 So, nobody builds it by hand like atom by atom. 190 00:14:45,952 --> 00:14:49,770 We don't we don't have to do it so that nature does it itself. 191 00:14:49,770 --> 00:14:53,321 So, this structure, they form spontaneous and there is some wave function if you 192 00:14:53,321 --> 00:14:56,580 want psi naught. We describe the, describe, well very 193 00:14:56,580 --> 00:14:59,830 complicated wave functions which describes these atom. 194 00:14:59,830 --> 00:15:04,100 So, how do these atom's form in such a pattern? 195 00:15:04,100 --> 00:15:06,932 So, are they you know, this, psi naught, is it a solution to a certain 196 00:15:06,932 --> 00:15:13,088 Hamilitonian that describes those atoms. And these Hamilitonian , it has a well 197 00:15:13,088 --> 00:15:19,506 kinetic energy of all the particles involved in the problem, all these atoms. 198 00:15:19,506 --> 00:15:24,720 plus an energy describing its interactions. 199 00:15:24,720 --> 00:15:29,343 And very importantly here, the interact, well, it depends only on the distance 200 00:15:29,343 --> 00:15:35,650 between the atoms, so here n and m, they labeled various atoms. 201 00:15:35,650 --> 00:15:39,810 Now, if you look at this Hamiltonian, so you, you see that, if we translate the 202 00:15:39,810 --> 00:15:44,286 whole thing. If we, let's say change r to r plus some 203 00:15:44,286 --> 00:15:48,446 constant distance in, let's go, let me go d, then the Hamiltonian's not going to 204 00:15:48,446 --> 00:15:52,030 change. Since we have here only the distance 205 00:15:52,030 --> 00:15:55,255 between particles, which is the position of fth particle minus the position of mth 206 00:15:55,255 --> 00:15:58,898 particle. If we translate both of them by the same 207 00:15:58,898 --> 00:16:04,180 distance, the distance between them is not going to change, right? 208 00:16:04,180 --> 00:16:08,724 So, so therefore, in other words what I'm saying here is that here the Hamiltonian 209 00:16:08,724 --> 00:16:13,680 isn't variant in translations in real space. 210 00:16:13,680 --> 00:16:17,790 By an arbitrary it's very important, by an arbitrary distance. 211 00:16:17,790 --> 00:16:21,040 So, but if you look at the crystalline arrangement here, so, it is no longer 212 00:16:21,040 --> 00:16:25,915 invariant under an arbitrary translation. So, if I remove my particles let's say, 213 00:16:25,915 --> 00:16:30,850 by a fraction of the slightest constant. I'm not going to reproduce the crystals. 214 00:16:30,850 --> 00:16:33,865 So, the crystal is still symmetric, but it's it has a symmetry which is sort of 215 00:16:33,865 --> 00:16:36,164 lower. It will, it has only discreet symmetry, 216 00:16:36,164 --> 00:16:39,870 while the original symmetry of the problem was a continuum symmetry. 217 00:16:39,870 --> 00:16:43,814 And there's a crucial difference and therefore here we have a broken, 218 00:16:43,814 --> 00:16:48,932 spontaneously broken. If you want three continuum symmetries in 219 00:16:48,932 --> 00:16:53,183 the x, y, z directions. And therefore there must exist three 220 00:16:53,183 --> 00:16:59,300 modes or three or threefold degenerate mode or at least three low energy modes. 221 00:16:59,300 --> 00:17:03,360 In this state and these modes are exactly the formats that we're going to find. 222 00:17:03,360 --> 00:17:08,188 Of course in our solution and we'll, we'll see them automatically and they're 223 00:17:08,188 --> 00:17:13,900 not going to be related to spontaneaous symmetry breaking itself. 224 00:17:13,900 --> 00:17:17,915 But it's very interesting this connection is very interesting because you can sort 225 00:17:17,915 --> 00:17:22,180 of predict the existance of such low energy modes. 226 00:17:22,180 --> 00:17:27,576 Just by the fact that certain, certain types of structures form out of symmetric 227 00:17:27,576 --> 00:17:31,680 systems. So, in the rest of the lectures, today 228 00:17:31,680 --> 00:17:35,720 I'm not going to be alluding to these Goldstone theories. 229 00:17:35,720 --> 00:17:39,006 So, this was just remark for those of you who are interested in advanced 230 00:17:39,006 --> 00:17:43,840 understanding or advanced subjects. But I just wanted to make this remark 231 00:17:43,840 --> 00:17:47,908 because I find it quite illuminating. Because you can see the appearance of 232 00:17:47,908 --> 00:17:51,854 this is very powerful. The application of this is very powerful 233 00:17:51,854 --> 00:17:54,348 theorem in a very simple example.