The quantum harmonic oscillator problem and its solution go far beyond the simple generalization of a particle and a spring problem in Classical Physics. And they appear in a variety of contexts in quantum physics, as already mentioned. For example electromagnetic radiation can be thought of as a collection of harmonic oscillators, the photons. And much more generally various complex structures in quantum mechanics. Oftentimes can be thought of as being built of the small building blocks being quantum harmonic oscillation. But in this particular case, the meaning of these individual oscillators is different. And today, I'm going to show you the appearance of these quantum harmonic oscillators. And the context of a very interesting problem that sort of underlies solid state physics or a condensed measure of physics. And this is a problem of excitations in crystals, which are collections of many, many ions that form a periodic structure. And these ions can oscillate, and these oscillations, quantum oscillations, are called phonons. And we will see how these phonons appear, and we will see also that they can be described. In terms of the creation-annihilation operators that we introduced in the first lecture this week. To understand the concept of a collective mode, such as a phonon for example, in a complex quantum system. We first can go back to classical physics and remind ourselves. The notion of a normal mode, which is equivalent to the quantum collective mode. And these normal modes appear in complex classical systems. So, here, I have a primitive classical system if you want this is a simpler harmonic oscillator. And in this case, as we know, the frequency of the oscillation is propor-, is equal to square root of the stiffness of this spring divided by the mass of this of this object. now if we make the system a little more complicated, and instead of just one oscillator. Consider let's say two masses attached to let's say three springs in this case. So, we have a system here with two degrees of freedom. And this system will have two different oscillation frequencies corresponding to different types of oscillatory motions. So, for instance this type of motion when the two masses sort of oscillate in in phase with one another. So, this let's see first type of motion is described by the frequency which is exactly equal to omega. But there is also a second type of motion that we can imagine this slightly more complicated system. In which case let's say these guys oscillate out of phase in this fashion, sort of. And this oscillation, this oscillation frequency let me call it two in this case, is equal to square root of times the basic frequency of these guys. So, let's assume they all have the same stiffness. And we can make things more and more complicated, and let's say in this case have six six masses attached to springs. In which case we will have six different degrees of freedom. And six different oscilation frequencies. So, and this oscillation frequencies normal so called normal modes in classical physics. They have an analog in quantum physics. In particular I would say in this example, in this example of two harmonic oscillators. We just by knowing the results of the classical calculation. We can write down, as a matter of fact the spectrum of the quantum problem, or the corresponding quantum problem. So and the spectrum of the corresponding quantum problem will be. simply the energy is going to be equal to h omega 1 and 1 plus 1 half h plus h omega 2, and 2 plus 1 half. And its going to depends on two quantum numbers, n1 and n2 corresponding to two harmonic oscillators. But these are not going to be actually the harmonic oscillators built out of let's say the coordinates and momentum of the two the two particles. So these oscillators or the corresponding creation-annihilation operators are going to be a certain non-trivial linear combination of these x1's and p1's, x2 and p2. which sort of describe the generalization of these in-phase and out-of-phase motions to quantum physics. And in general, whenever we have a classical system with some normal modes. There is a corresponding quantum system with a spectrum associated with these normal modes. A very interesting example of a complex mini particle quantum system is a crystal. We know that crystalline structures of various types appear in a variety of solid state materials. And in this structure so what happens is that ions or atoms of which this solicited material sort of built. They arrange themselves into a unique order and pattern. And so here, for instance, I'm showing a fragment of a so called simple cubic lattice. And these blue balls in this picture, correspond to atoms sitting on their equilibrium position on their lattice size. But this picture is a bit deceptive, actually it is probably better to say incorrect in that the atoms here, in reality, are not static. So, they indeed have some preferential positions, and some equilibrium positions. But they move around all the time, so there, this position that they occupy, corresponds to a minimum in some effective potential they fill. But we can oscillate near the minimum of this potential and these oscillations are basically just grabbed locally as a essentially a harmonic oscillator motion. So, better illustration of what a crystal is that I put together well to simply sit here for a two dimensional case. So, it's presented here. So, instead of thinking about atoms which are sort of tied together in this crystalline structure by some rigid objects. We should think about atoms being tied together by some elastic springs with some stiffness. Which is fine at and so when we look at this system so my analogy to what I had discussed, what I had just discussed in the previous slide. We can think about this system as a many oscillator system with a moderate degrees of freedom. So, in reality in this material so they're not two, or three or six atoms, like in the previous slide, but they're min, billions and billions of atoms. And therefore, there are many frequencies, eigen-frequencies of normal modes that I classical normal modes. That appear and they correspond to various kinds of complicated oscillations in this system. And in the limit of an infinite number, basically a so called thermodynamic limit when we consider essentially an infinite crystal. So, these create normal modes, they merge into continuum curves which essential corresponded to elastic waves running through this system. And mathematically these waves are described by familiar expressions. So, if we think let's say about the displacement of an atom with regard to its equilibrium position, some delta r as a function of time, for a particular atom. Let's say nth atom, so it's proportional to the familiar exponential of key dot r, so where this atom is located, minus omega t. But this omega is itself a function of key. And so to find this so called dispersion, phonon dispersion is the main question that appears in the context of this problem. And this is what is going to be of interest to us. Of particular interest is the question of what are the low energy modes of this system. So, what it actually means is the following, suppose we have a completely static crystal. So, while as static is a guess, in quantum mechanics there is always zero point motion, but we can even consider classical system. Let's say we just built physically built this model by attaching a bunch of masses to the springs, and attach, and arranging them. In the, in equilibrium pattern and let's say we touch this system a little bit, we apply very weak perturbation. So the question I am asking about the law energy modes is. Have we got inside any oscillation in this crystal ? Or this weak perturbation is not going to have any effect? Or, in other words, do we have to overcome any threshold, or gap as condensed measure people call it. In order to excite the minimal perturbation in the crystal. And as we will see in a later, in, in today's lecture, so the answer will be that, it doesn't matter how weak the perturbation is. There is always a very low energy mode which corresponds to long wave length. Or small key wave running through the samples, through this crystal. and this type of an excitation is called an acoustic mode or it corresponds to sound. Sound modes running through through this crystal. And it turns out that this appearance of this acoustic sound modes. Has a very profound explanation which goes as far as relativistic quantum physics actually where. There is a theorem called Goldstone theorem which guarantees the appearance of certain low energy modes. And even though I'm talking about classical system, let me nevertheless try to connect it to this theorem. But I will present sort of a poor man's version of this Goldstone theorem which is relevant to our problem. And then hopefully, we will be able to understand how they are connected. So this going to be a little bit of detail work. So, the poor man's version of the Goldstone theorem that I'm presenting is related to the notion of symmetry in quantum physics that we already discussed. So, suppose we have a quantum system which is invariant or symmetric with respect to certain class of transformations. What it means is that there exists transformations, let me call them u. And this operator commutes with the Hamiltonian which describes this system. So, this is what it means or in other words U, H, U minus 1 is equal to H itself. And let me consider this situation, when there is not just one or two symmetry operations that exist. But as a matter a fact we have a certain, this whole family of such operations let's see labeled by some parameter. Let me call it alpha. So, an example of such a symmetry for instance, could be symmetry with respect to translation. So, let's say if we, if our Hamiltonian is symmetric with respect with translating everything, every single particle in space, let's say by certain distance. So, it could be a distance of 1 meter or 1 centimeter, 1 kilometer. So, there is a whole continuum of distances and directions we can translate our system. In which case is alpha, it's actually a three dimensional vector of translations. And while another example could be symmetry with respect to rotations. In which case alpha could represent a, an angle that we can choose, by which we can rotate our system. But in principle, can be anything, so we just basically postulate that the existence of some symmetry. What is called symmetry group but I'm not going to go into this details. with respect to H the Hamiltonian of the system is invariant. But a very interesting fact is that, even though the model itself can be invariant under a certain symmetry operations. The quantum states, the for instance lowest energy state of the ground state, does not necessarily have to be invariant. under the same symmetry U. So, what it means is that the system will find itself in a state, let's say, let me call it psi naught. Such that well, U acting on this psi naught is not equal to psi naught. So, by translating let's say our system in space, we do not necessarily represent the same state. And in this case we call, we say that the symmetry is broken, spontaneously broken. But the key statement, which is also the statement of this, of this Goldstone theorem in this context, is the following. If once again we have a situation where the Hamiltonian is a variant under a certain continuum of symmetries. But some or all of these symmetries are broken in this state in which the system finds itself. In this state let's say psi naught, then, to each broken continuum symmetry. There corresponds low energy mode, a sound-like mode if you want, that we just discussed. and well, this statement, this correspondence, is again the statement of this Goldstone theorem. So, I realize that there has been, you know, a lot of words that I've said. But perhaps the best way to understand the actual meaning of these words would be to provide an example. Which also will explain why this discussion is relevant to what we talked about before in the previous slides. About hormones, crystals and all these things. Now, let us look at a crystal structure once again. So we have here a periodic arrangement of atoms. But nobody puts this crystal structure by hand, so if we have a piece of some solid, solid material, let's say a piece copper. So, nobody builds it by hand like atom by atom. We don't we don't have to do it so that nature does it itself. So, this structure, they form spontaneous and there is some wave function if you want psi naught. We describe the, describe, well very complicated wave functions which describes these atom. So, how do these atom's form in such a pattern? So, are they you know, this, psi naught, is it a solution to a certain Hamilitonian that describes those atoms. And these Hamilitonian , it has a well kinetic energy of all the particles involved in the problem, all these atoms. plus an energy describing its interactions. And very importantly here, the interact, well, it depends only on the distance between the atoms, so here n and m, they labeled various atoms. Now, if you look at this Hamiltonian, so you, you see that, if we translate the whole thing. If we, let's say change r to r plus some constant distance in, let's go, let me go d, then the Hamiltonian's not going to change. Since we have here only the distance between particles, which is the position of fth particle minus the position of mth particle. If we translate both of them by the same distance, the distance between them is not going to change, right? So, so therefore, in other words what I'm saying here is that here the Hamiltonian isn't variant in translations in real space. By an arbitrary it's very important, by an arbitrary distance. So, but if you look at the crystalline arrangement here, so, it is no longer invariant under an arbitrary translation. So, if I remove my particles let's say, by a fraction of the slightest constant. I'm not going to reproduce the crystals. So, the crystal is still symmetric, but it's it has a symmetry which is sort of lower. It will, it has only discreet symmetry, while the original symmetry of the problem was a continuum symmetry. And there's a crucial difference and therefore here we have a broken, spontaneously broken. If you want three continuum symmetries in the x, y, z directions. And therefore there must exist three modes or three or threefold degenerate mode or at least three low energy modes. In this state and these modes are exactly the formats that we're going to find. Of course in our solution and we'll, we'll see them automatically and they're not going to be related to spontaneaous symmetry breaking itself. But it's very interesting this connection is very interesting because you can sort of predict the existance of such low energy modes. Just by the fact that certain, certain types of structures form out of symmetric systems. So, in the rest of the lectures, today I'm not going to be alluding to these Goldstone theories. So, this was just remark for those of you who are interested in advanced understanding or advanced subjects. But I just wanted to make this remark because I find it quite illuminating. Because you can see the appearance of this is very powerful. The application of this is very powerful theorem in a very simple example.