Hi, everybody. Up to this point in the course Charles and Di have presented already a lot of material which included both material that can be found in traditional quantum mechanical textbooks. And also some nonstandard exposure to quantum physics. For example, super conductivity pet integrals. Some elements of quantum liquidation etc, etc. And I have to say that on my side the preference has been for the latter. I wanted to tell you more about things that you cannot find in traditional textbooks. But today is certainly going to be an exception because today I'm going to present a very traditional quantum mechanical problem that no reasonable quantum mechanical course can avoid. And this is the problem of an harmonic oscillator. So the reason I cannot avoid this problem is two-fold. So first of all quantum harmonic oscillator the problem of quantum harmonic oscillators relevant to a variety of various experimental systems ranging from atoms where atoms are trapped in a quadratic potential to let's say molecules who's low energy oscillations I described by [INAUDIBLE] oscillator to quantization of [INAUDIBLE] to quantization of oscillations in crystals. So for [INAUDIBLE] we are going to discuss them soon. And many, many other things. And second, so the problem on the theoretical side, the solution to the problem, a harmonic oscillator is very illuminating and useful in that it gives you very useful tools. So called creation and undulation operators that are useful in a variety of contexts across various fields of physics. And you're going to see these things very often in other courses if you pursue physics studies. But before solving the quantum harmonic oscillator problem, let me remind you the story behind the classical harmonic oscillator. And here I have an example of a classical system which exhibits these small oscillations in year and equilibrium position and these oscillations exactly are the harmonic oscillator motion. So, on the mathematical side what happens here, of course, is that the mass here experiences the so-called Hook's force which is proportional to the displacement, x, of the spring relative to its equilibrium position. So, let's say here, if this is my X-axis, and let's say we have an equilibrium position somewhere here, so this is my zero. So, this is X, and the force is proportional to this displacement, and in this case, at this moment in time, it's this Hook's force. In these directions, in the negative x direction. Now I can write this words of course as a minus gradient, in this case its just one dimensional gradient, or just the [INAUDIBLE] with respect to x of some potential. And to produce a linear force of course I need to differentiate the quadratic potential. So in this case it's kx squared over two and this is exactly the potential energy of a harmonic oscillator. That eventually is going to appear in our Schrodinger equation, the quantum version of this problem. Now, the classical side though, what I have to do is I have to solve the second, Newton law, represented here. And so the second Newton law, of course is just mass times acceleration. The second derivative of x is equal to all the forces. In this case, there is just one force, in this example, is if we ignore gravity. Which is not going to modify too much. But in this case it's going to be just one force which is the force due to the Hook's law. And if I write this equation, if I divide this equation by m and instead of writing the second derivative, I will write it in the compact form as with two dots on top. So I can write it as x to dust plus an ingenuous squared x is equal to 0 where Amanda is equal to the square root of key over m. And as solution to this differential equation is the sines or cosines to some arbitrary phases and it's up to us which solution to choose. So there will be a some amplitude here amplitude here so we can choose a sin or we can choose a cosine. And as long as we have two free parameters which will determine the initial condition, the initial position, the initial velocity these are all general solution of this classical problems. So let me choose for example the sin solution and present it here. So this is this solution to this harmonic oscillator. Now to move further, let me choose a particular initial condition which I can do which creates response to x of t equals 0 being 0, so let's say the initial moment of time are the position of our on harmonic oscillator was exactly the equilibrium point just for simplicity and in this case this is my solution. Now if I want to find the velocity of the oscillator as a function of time so all I have to do is to differentiate the position. And in this case it's going to be just x naught, some amplitude of the oscillations times omega times cosine of omega t. So what I can do now, I can write the energy of this system which is not a function of time as we shall see in accordance to the conservation law which would be a sum of the kinetic energy mv square over 2 and the potential energy v(x). Which is equal to one-half kx squared, k is the coefficient in the coefficient in the Hook's Law or expressing it through the mass and frequency. I can write it as so. One half, and my omega squared, x squared. And this is by the way, the form in which it typically appears in the literature. In particular, in quantum mechanic literature. So now indeed if we plug in this result into the energy. And so the energy is going to be equal to mv squared over two, so we're going to have m over 2, x naught squared, omega squared, plus cosine squared of omega t. From this velocity and potential energy plus m over 2, x squared, x naught squared, omega squared from here Sine squared of omega t. And then we can use the fact that cosine squared a, an arbitrary angle, plus sine squared of the same angle is equal to one. And so we see that indeed the energy is simply this. And the energy is conserved. Indeed, in accordance with the energy conservation law. But of course even though the energy E here is not a function of time. The two terms in this equation, the kinetic energy and the potential energy, are functions of time. So x is a function of time and the velocity is a function of time. But they work in unison with one another in such a way that this sum is a constant. And if we plot this, so let's say the x axis here is going to be m omega times the coordinate, and the y coordinate here is going to be the momentum. So we see that this trajectory, basically in the phase space, describes the motions of the harmonic oscillator. It's a motion on a circle. And the radius of the circle is determined by the energy of our harmonic oscillator that we, and this motion is a face portrait of this classical system. And by the way I will just mention in passing that there is a relation between the face portrait of classical system and the corresponding special quantum systems. But that's the only thing I'm going to say here in this regard. So now let us move closer to the main subject of today's lecture and make the oscillator quantum. The motivations to do so are many as I said, so for example we can think about instead of having a classical mass on a classical spring, we can talk about let's say two atom molecules and the oscillations of this molecule relative to the equilibrium position of the atoms. So at low energies these oscillations are going to be described by quantum harmonic oscillator. So of course, there are many more examples, as I mentioned, in the beginning of the lecture, let me just mention another one. So for instance, if we're talking about the behavior of a quantum particle in a rather arbitrary potential, v of x. Let's say something like this. So if we are interested in low energy behavior of the particles, and we're near the minimum of the potential, we can always expand the potential of the effects near the minimum. So it's going to be v of x not. So this point, plus v prime of x not, x minus x not. And this is zero, because this is the minimum plus the second area over 2x over minus x naught squared. And model of this overall constant which just shifts the minimum of energy and this shift of the equilibrium position and this shift the equilibrium position. So the remaining part of this potential is really, quadratic potential that we have in the from problem. So what I'm saying here is that you can always almost always approximate a reasonable potential well near the minimum of the well with the harmonic oscillator. You have a smooth function. Of course if you have a square quantum well we cannot do that but in most cases you have a smooth potential and in this case we can just replace it at low energies with this harmonic oscillator. Okay, so now let me erase this stuff, and move further. So how to formally go from the classical harmonic oscillator in this potential to quantum harmonic oscillator. So to do so, we have to pretty much do the following mathematical procedure. We can just put a hat here and that's it. So that's how we make the oscillator quantum. So, well seriously what do we have to do, we have to replace the energy of the oscillator, so the energy remember we derived in the previous slide for the classical harmonic oscillator. So now we replace the energy of the oscillator with the Hamiltonian which is an operator. The momentum which is just a variable in the classical description. Now we replace it with the operator that detects on the wave function and in principle we can put the hat also on top of the coordinate, although the action of the coordinate on a wave function in position space is pretty simple action. It's just a multiplication operator. But this pretty much what it means on the mathematical side to make the go from classical description with no operators involved to a quantum description with operators involved. So this is the. And the problem that we want to solve now is to find solutions to the eigenvalue problem. So, of course, it's a time independent problem, so there's no time involved. And what we want, we want to find a solution to this equation. So when solving this Shrodinger equation, and in particular when thinking about the allowed energy levels. So what we should keep in mind about this harmonic oscillator potential is one property that makes it very different from what we have seen before. Namely that this potential grows, the potential energy grows indefinitely to infinity as x becomes very large or very negative. So there is no room here for a plain wave-like solutions propagating to infinity. And the only thing we may have here are discrete states corresponding to particles localized in the vicinity of this x equals zero. And to determine these energy levels, this harmonic oscillator spectrum, is exactly the main problem we're going to be focusing on in the next couple of videos.
