Welcome back everybody and this week we're going to go back to more traditional quantum mechanics, that is quantum mechanics described in terms of the Schrodinger equation. And today I'm going to solve the Schrodinger equation on a number of very important examples, where we'll show you how quantum potentials can capture quantum particles that result in so called bound states. So even though the solutions that we're going to see, the examples we're going to see are rather elementary from the quantum mechanical point of view. Well, otherwise, they will require some thinking. So these examples are going to give us insight into some rather complicated and fascinating problems such as for instance, theory of superconductivity. So in the end of the lecture today, we're going to discuss our solutions are relevant to the key phenomenon that is responsible for super conductivity that is so called Cooper pairs. But before we get to this point we need to go over the basics. And so let me start first with the problem of electron in a box. I will define it in a second. But even before that let me just talk about what kinds of problems we want to, we can't possibly study with the Schrodinger Equation. So here is a canonical Schrodinger equation. We have seen it already many times. It's a time dependent Schrodinger equation. As we will see later, today, so actually if the potential does not depend on time, there is really no need to start a time dependent Schrodinger equation. So as we'll see, it can be transformed into an eigenvalue problem for the [INAUDIBLE] that we will actually solve. And in any case, so this first term in the single particle mechanics anyways. So this first drum on this Hamiltonian is just the kinetic energy. There is essentially, it's a non-negotiable term. The second term is something which is sort of problem dependent which determines the potential in which our quantum particle moves. So amazingly the variety of all possible problems in single particle quantum mechanics are contained in this equation in a compact, and they differ simply by the choice of V of R. But this compactness of basically quantum mechanics is a bit deceptive because it can give you the impression that it can just solve all possible quantum mechanical problems in one go, just by solving this equation. Well, this equation is very rich equation and depending on the choice of V(r) of the potential at which the particle moves, we can get completely different physical situations. And understanding these physical situations would require, dramatically different mathematical approaches too. However, very roughly, what we can do, we can classify our quantum potential by putting it in one of two categories of either an attractive or a repulsive potential. Which, in the context of quantum physics, the former are called potential wells and the latter are called potential barriers. So here let me present, well sort of simple illustration of what a typical potential well looks like. So this is v of x. It's in 1D quantum mechanics and this is a 1D coordinate and potential barrier would like sort of opposite to it. It would look like a hill. And today, we're going to focus our attention on the physics of quantum potential wells. Basically to give you the result right from the onset, I think most of you probably already know this. In potential wells, we're going to see that the available energy levels for the particle to occupy With a negative energy. So they're going to be quantized, that is, the particle wouldn't be able to have any arbitrary energy just like in classical physics, where we could assign any energy. We could put, let's say, a ball at any level and let it oscillate between the turning points. In quantum mechanics, let's say if we got an electron in a potential well with this landscape, we cannot assign it an arbitrary energy. So the available energies are going to be quantized. And finding these quantized energy levels is one of the sort of canonical problems that we're going to solve. But this business about finding the quantized energy levels in an arbitrary quantum well is in principle rather complicated technically, and the complexity of this exercise really depends on the particular form of the potential we're studying. However, to see the appearance of these quantum levels, in general we can focus on the simplest examples of your facts and that's what we're going to do now. And perhaps the very simplest example Is a potential which has so called hard wall boundaries which implies the falling. So it basically means a potential were beyond certain points. Let's say x equals zero. And x equals l. So the value of the potential, the effects is equal to infinity. So there is no way the particle can move beyond this point. These are infinite walls. And in between these two points, the value of the potential is exactly equal to zero. So which means that the particle's sort of free to move between these two walls, but it cannot go outside. And this is what is known as the problem of an electron in the box. So as you probably can guess, the solution of this problem you're going to complete in the remainder of this segment is going to involve certain energy levels which are going to form a discrete series. So, it turns out that to understand the phenomenon of quantization, one does really need quantum mechanics at all. The quantization occur is in classical physics all over the place. And so here for instance, we have an example of a very familiar object, a guitar, which relies on quantization in some sense of the wave lengths and frequencies available in its oscillating strings. So here we have strings which are free to oscillate between two points where they're pinned down. And those two points in some sense correspond to the hard wall boundaries in the corresponding quantum mechanical problem that we're going to study a little later. The available wavelengths much depend on the distance between these points, these hard walls let's call it. And the frequencies are related to the available wavelengths by the speed of sound. Let's now think about, what is the longest possible wavelength? That we can induce in a finite string. So these are basically our hard walls again. So this is our 0 and this is our L and the hard walls in this context essentially mean that the strings cannot oscillate here. So the displacement, let's call it u, from this horizontal axis is exactly zero. X equals zero and X equals L. In other words, so, we must have notes at these end points. And the longest possible wave length that achieves that is lambda equals 2L. And this is sort of the fundamental oscillation, the longest possible wavelength that we can induce on this string. Because if we try to make this wavelength even longer, it would imply essential that we would have either no note in this point or no note in this point. And this will violate our boundary. But there are of course many more wave lengths that are possible that would satisfy the proper boundary conditions and we can achieve we can sort of find these additional wave lengths or higher harmonics as they're called by putting additional nodes in between these end points. And so for instance this example gives us a wavelength which is exactly equal to the L to the distance between the n points. And we can continue this procedure and generate many more wavelengths. And these wavelengths are going to follow this quantization rule if you want with the corresponding solution, while if you look at the snapshot of an oscillating string at a certain time. So the solution is simply going to be given by the sin, well, some amplitude which is not really important. Sin of 2 pi x over lambda sub n, and of course, if x is equal to zero, we're going to have the boundary, we're going to satisfy the boundary condition, x equals zero. And if x is going to be equal to L, this quantization rule is going to enforce the other boundary conditions, namely that the displacement vanishes at sequence L, where we have a node here. Now if we look at the corresponding quantum mechanical problem. So it essentially is very similar to the problem of this oscillating string. With the only difference being that instead of putting an elastic string between the two points, we put an electron wave in between these two points. But the wavelengths that are available for the electrons, so the electron cannot really move beyond this hard wall rules right where the probability of finding the electron there is equal to zero. Therefore, we must demand that, well the probability of psi-squared of x=0 and L is identically equal to zero, Okay? So, or in other words psi itself is equal 0 and this is much similar to having a node at the endpoints of this string. So if we solve this problem for the electron, well, which will involve in this case actually solving the Schroedinger equation, we're going to find exactly the same wavelength. And the wave function is actually going to look exactly the same as the displacement of the string. So, there's absolutely no difference. I just copy and pasted it exactly the same equations, so only replacing u sub n which is a displacement by psi of n which is the wave function of the electron. So this shows you that actually there is a lot of analogy in quantization of electron wave lengths and quantization of wave length of classic objects, elastic objects. So the only difference, and an important difference here, would be in how the frequency, or the energy in the case of electron, scale with the wavelength. So here we discuss that the frequency of the oscillation, which by the way determines the sound that we actually hear, scales linearly with the wavelength. On the other hand we know that the energy of the quantum particle of just well kinetic energy is essentially a free particle moving in between these two points is going to be p squared over 2n and if we recall the de Broile relation between the momentum and the wave length which is 2 pi h bar over lambda is equal to p. So then we can regularly combine these two results of the quantization of the wave length and the scaling of the energy to get the quantization of energy. So if you put everything together it is quantization rule and the de Broglie relation we're going to get the quantized energy levels, e sub n where n is a positive integer equal to pi squared, h squared, n squared divided by the l squared, the distance here that corresponds to the momentum squared times 1 over twice the mass. So, in August, that interestingly we solved the Schrodinger equation without writing it down. The only thing we used, is basically was the boundary conditions, and same analogy with classical physics, along with the de Brogile relation. But of course, one can solve it formally and we'll show later how it works. So, sadly one can not always solve the string equation in a simplistic way without writing it down. So to understand why it happens we can again use this guitar sort of string analogy. So if we push down on the string, so we effectively shorten the distance between this ends points and the effective end points here and by doing so we of course change the fundamental wavelength and the quantization of the wavelength and the frequencies. This sort of results in different sounds that we, produce this way. But if we push not very hard, but if we just touch the string here, so this creates in sort of an infinite wall, impenetrable wall for the string to propagate. Beyond this point, it creates a finite barrier. And to solve the wave equation, for this string, the classical wave equation in the presence of such a small perturbation is actually more complicated then solving this equation for two hard walls. And likewise, we're going to see actually that the problem of a quantum mechanical particle, when instead of the hard walls we have let's say finite barrier here is more complicated and it requires actually a serious mathematical calculations in solving the actual Schroedinger equation which is a differential equation. We're going to discuss in the following segment.
