요약
This lecture covered advanced quantum mechanics concepts focusing on electron behavior in periodic potentials and crystals.
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I'm only putting the...
이것은 싱글탐올로 플러스 아이베타입니다. 이 경우에는 플러스 아이베타 X 레이어를 볼 수 있습니다. 왼쪽으로 흔들리는 웨이브입니다. 마이네텍스에서 플러스X로 흔들리는 웨이브입니다. 저는 여기서 Iβ를 선택하고 있습니다. 혹시 기억하실지 모르지만, 전자공학과 전자공학에서 다른 공학적 개념인 것 같습니다. Iβ-Zr의 경우, 전자공학과 전자공학에서 다른 공학적 개념인 것 같습니다. Plus, uh, erasure. Is it?
In quantum mechanics, you have a t part, e to the minus i omega t part. This combination plus beta means you are actually going to the right. In section 3, I didn't put the parent going to the left. It is redundant if you do so. Later, once we see the solution, you will understand why I do this.
But now let's just start with this form. This beta and k from the Schrodinger equation, you can calculate what they are. Beta is defined as, as usual, 2mE-V0 over h-bar squared. We already have done this. This F and B means the forward and backward. It specifies the direction. Remember that a free electron has a direction depending on the sign.
The same procedure, we are going to apply boundary conditions.
So first of all, x equals minus a over 2. You probably remember, right? Psi continuous. Psi derivative continuous. You just do that.
So first, I'm just putting it into the sign expression itself.
And then, it's a derivative, right?
So nothing but simple mathematics. Let me just write down the whole equation here.
The stereo key continues, okay?
This procedure is almost the same as before for the bound state.
x plus a over 2 You probably know what's going on here. Just let me write down everything.
This equation, I won't go through all the details because we have done the same type of calculation previously.
I hope you have useful information here.
It has a function of a lot of parameters, which is proportional to AF. One second, let me write that down. E times I beta J.
It is just a number, a complex number. You know what that number is, right? In that expression, there is nothing you don't know. You know k, you know beta. You know A, so you know everything, what the complex number is, you can calculate. And then, what important information you get is AF and DX ratio. It's a fixed ratio.
So, what is this? This is a transmission coefficient. That complex expression is a transmission coefficient. Why? The AF is the incoming wave and DF is the transmitted wave if you do a physical interpretation. Let me draw the picture to show you more clearly.
Propagating this way, and then A backward is a left-hand side. And then, of course, you can see there is something going on. Within the well, two waves bouncing back and forth, but also you can just describe it as a standing waveform like sine and cosine.
Standing wave is a mixture of left travelling wave and right travelling wave. If you see sine and cosine, it is a form of a standing wave form, but you can equally express it as a wave form. You know how to express sine and exponential i theta form. You can go back and forth.
So that's what's going on in the middle. And then, now you can see why we only have this DF term. So this is a transmitted wave.
And then, this is an incoming wave.
The way we set up those 5 terms in the previous slide here, only these 5 things, right? It's exactly 1, 2, 3, 4, 5. Ok, it is 5.
So this is explaining what you have an incoming wave it hits the boundary and some of them is reflected back it's a b quotient that's what I present and then some energy leaks into the section 2 In between two boundaries, it has a reflection going back and forth. And then some of them will pass through and go to the right-hand side. But our process actually...
It doesn't have any more wall on the right-hand side, so there is no reflection from it. If you see the AF as an incoming wave, there is no left-traveling wave on the section 3. What happens if you put the arrow there? Nobody will stop you from putting some term there, right? But actually, as I mentioned earlier, you don't need to do it.
You are actually solving another independent problem then. If you draw an arrow, From the left-hand side, let's say you are drawing an error like something like this and then try to solve it and that's actually a stupid thing to do because you just make the problem more complicated.
But this problem, this problem, they are just independent, okay? Independent, and this is also symmetric, so you don't need to do the double amount of work.
So the general solution can be, if you include all the directions, it can be linear combination of this solution and this solution, and these two solutions, they actually can be handled separately. Since the Schrodinger equation is a linear system, Linear system is what? You have this input, you have an output, you have this input and output, and then what is that? They are nothing to do with each other, right?
The existence of the other input doesn't matter for this input and output, right? It doesn't change. That's the case. These two can be linearly superimposed. Actually, they are equivalent in this case. They are symmetric, so you don't need to solve it again.
That's why I only put 2, put DF.
In the middle, you have the wave bouncing back and forth. You will have a standing wave.
Transmission basically represents energy. How much energy is passed through if I send 100 watts and you receive 70 watts? That's a transmission, right? How much energy is transmitted?
How much fraction of energy...
Also, you can interpret it as a probability of an electron. If you throw an electron, the chance of getting it passed through the system, that's also transmission. You send 100 electrons and if the transmission is 0.7, you get 70 electrons.
The ratio of these two is read as absolute square. Remember, the way we interpret quantum mechanics was wave function absolute square is the chance of finding that particle. And then having that particle means an energy, right? The particle will carry an energy. So you have to take absolute square to account for the energy. This way, you always have a real number.
It is complicated. Let me just write down the full expression because it is analytically solvable.
It is an analytical expression and it is better to just see the shape.
Transmission looks like this. It is a function of an electron's energy. In this case, it goes up to one quickly and then oscillates. And then there are specific points. Transmission is one. It is a function of energy. Electron's energy, in this case, can be continuous, just like a free particle.
It is not bound. In general, the unbounded electron can have a continuous spectrum of energy. As a function of energy, you have different transmission. If you are familiar with optics or electromagnetics or RF circuit, it is somewhat similar to Fabry-Perot resonator or Atala.
You have two boundaries and then you send the light. And then for particular wavelengths, it actually has 100% transmission. It is somewhat similar to that. You can see that it is natural because if you study that problem, you will realize that the equation looks exactly the same.
Plane comes in, reflect it, in the middle there is another standing wave, and then go, pass through. The Fabric Pearl Epsilon, you solve the problem exactly the same way as this. The basis needs to be somewhat complicated expression that you have both bound states and then...
Arbitrary superposition you actually need to take into account the both discrete energies and continuous energies.
So these are for bond states.
It is above the barrier. And this unbound state... This is something we want to learn for the crystal. That's why I introduced this problem. For the well, you have an energy above the well. Then the solution, you can think of it as some sort of a plane wave going left and right. It's all about that.
In case of 1D problem, plane wave going left and right. The electron, you should think of electron this way, in crystal. Since it is a wave, now you realize that it is actually possible that they are moving around. Electrons are waves, so waves can go this way and that way. Bound states are tied up to your atom.
If you have learned other semiconductor physics, you always learn that there are these electrons that can freely move around their boundaries. There are bound electrons in the inner shell. They don't contribute to the conductivity. But there are electrons that flow around. It doesn't belong to a specific atom.
Let's solve more specifically those problems. This installs Electron State as a plot theorem.
Let me introduce him with a picture. He is a great physicist. American physicists passed away a long time ago.
Please name it after him. He is a Nobel laureate.
It should not block the NMR, Nuclear Magnetic Resonance.
Which is the base field for building MRI machine. You can find MRI machine in any hospital. This is more relevant to R-class and also quantum.
Crystals.
So, NMR itself is quantum mechanics. NMR is what you guys have learned. It's a precession. You apply magnetic field, you have this precession, and we learned that the precession is a function of B-field. In MRI machine, when you do imaging, if you have a gradient of B-field, B-field is different as a function of space. And then, even for the same spin, as a function of space, you have a different precession frequency.
If you have a frequency-resolving spectroscopic measurement, you can actually see a specific spot. That's MRI. So it's quantum, NMR, MRI, or quantum, so he knows other quantum as well. So he contributed a bloodstain. So let's see what bloodstain is.
Let me bring this picture again, this picture you have seen previously. Somewhat more realistic picture.
Let's say the periodicity is a and then I'm assuming that there's no such thing as this is an infinite Periodic Potential. Infinitely Periodic Potential. infinitely long periodic potential for electron. What is the solution? That's what we want to study.
With infinite 1D potential.
And then he claims that Blackwood, the solution of this problem, needs to satisfy this boundary condition. It is the same as the translated wave function, except for some phase, which we can write as this.
So the magnitude is the same. It has to be the same. The phase can be different. It actually makes sense, right? The magnitude cannot be different, actually, if you think about it. It is infinitely periodic. If you are standing here, standing at zero, standing at A, you shouldn't be able to distinguish the physical environment nearby, so the chance of finding the electron cannot be different.
So that magnitude has to be the same when you express it this way. But phase doesn't matter because when you calculate the probability, you always have absolute square. So that e to the i k a is the magnitude is 1. It's not going to influence your probability.
So you are allowed to have that face. And then we can express it as e to the i k a. For any face, you are allowed to write it that way. And then actually you can interpret that as a plane wave. So, there is a proof. Let me introduce the proof of this.
Translation operator In the parity, we had an exchange operator. It is another operator. It does something to the function as a result. There is an operator that makes this translation.
It is possible to prove that they are commuting with each other. And that Hamiltonian is of course giving you what? Energy eigenstate will give you this relationship. It will extract energy from the energy eigenstate. We are assuming psi is the energy eigenstate.
So let's do the calculation together.
We apply two operators in a row to an energy eigenstate, and then of course, You can write it this way.
And then E is just a number, you can take it out, and you have this.
You can write it as this. Now I'm converting E to H.
We prove that they are commuting with each other. We have learned this theorem that you can find the simultaneous eigenstates of both operators. because they are commuting, and then it's infinitely periodic.
Which is the chance of finding that electron at that point, right? It's going to be for whatever end. If it's an integer multiple of n, it's going to be the same as that. Physically, you cannot distinguish from each other at that exact location. With periodicity of A, it's all the same. Wherever you go, it's the same. There's nothing different physically in your environment.
It's a business called discrete translational symmetry. Every A, you are repeating the exact same thing. And then if we apply our eigenvalue problem form there, then this is the same as you apply this n times to this.
Figure it out. What is the eigenvalue for the translational operator, lambda? Lambda is just nothing but what? Magnitude 1 of any phase. Theta can be any number. The theta, I can choose to write it as this by expressing theta as kA. And then this theta, kA, they are all real numbers.
Graphically, I can draw this.
Incomplete function analysis.
Theta will be some point here, basically. This can be lambda e to the i theta.