L4.4. Vectors and spinors
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[Slide 1] >> Welcome back to Unit 4 of our course on quantum transport. This is the fourth lecture. 

[Slide 2] Now in the last lecture we talked about how you could write down these, I guess, self-energy matrices, the sigma, the gamma, describing different types of magnetic contacts and in this context I said that, you know, something non-intuitive that's probably unfamiliar to most of you that you have to get used to is this concept of spinors as opposed to vectors, which you are all familiar with. And when you have this entity pointing along this direction n, theta, and phi, as a vector its component is described by these three real components. As a spinor you have these two complex components. This we are all familiar with, this is standard vector algebra. This is non-intuitive and for the moment let's assume this but what I'll try to do in this lecture is give you a little better feeling for where this came from. Now, for this purpose, it's, I guess, more convenient to think of a problem where the contacts are nothing, just both identical magnets and ordinarily you would think that this would inject a spin in this direction, which would then flow out freely into the other one and that's true if the H is non-magnetic, if the H has no spin-related properties. But, let us think of what we'll be discussing is H, in the channels described by H, which actually affect the spin so that even if the spin comes in pointing in this direction here by the time it leaves it has turned in some direction, okay? So what kind of property of H would make the spin do that as it travels through the channel? Now, usually the experiments we talk about are all steady state experiment. So you look at spatial variations but for this purpose it's more convenient to think of a short device, so the device is just one point but think of if you injected a spin here what would it do as a function of time, right? That's not the way the usual experiments are done, the ones we are talking about, but in terms of understanding the role of H that's more convenient. So let's think of just a, you know, a device with just two points so that the H is described by, like for a non-magnetic channel it would be epsilon zero, zero epsilon, epsilons on the diagonal and they're both equal. That's a non-magnetic channel. And then we'll talk about other possible types of H and ask the question, if you had a spin with components u and d at time equal to zero how would it evolve with time, okay? And, as I said, experimentally, that behavior with time in a long device would be manifested as an evolution in space because as the electron propagates it will turn and so on but this is the problem we'll be discussing in this lecture. So, let's first start with a trivial case, how does this affect the direction of spin? 

[Slide 3] Well, the basic, I took the i h-bar out from here and divided it over there so what you have is d/dt, up, down, epsilon over i h-bar, u, d. So this is basically two separate differential equations, d/dt  of the up component is epsilon over i h-bar times the up component, 1 times up plus zero times down and d/dt of the down component is the same thing times the down component. And this has a simple exponential solution so you could write the up component as a function of time is up at zero times this exponential of that quantity times t. Similarly, the down component is exponential of that quantity times t and these are identical. So what it means is that if you start out your spin looking like this as time goes on both up and down pick up the same phase factor. And one of the things I mentioned in the last lecture is that if you add any phase factor here that doesn't change the direction in which the spin points. It still points in the same direction. So the moral of this story is that if you had an ordinary Hamiltonian described by this epsilon times I, which is a non-magnetic channel, nothing much would happen with time. The spin, if it is pointing in some direction, will stay that way forever. 

[Slide 4] Now let us consider a non-trivial one where something will happen and this is where we say that the H has the form of plus 1 and minus 1 so for one of them it's positive epsilon z and the other case it's minus epsilon z. But here also there's no off-diagonal element so we could write it out as two separate differential equations. And if you solve that one you'll get up-- will evolve with time with that phase factor, the down will also evolve with time with the phase factor but if one is minus the other is plus. So now you see the up and down are picking up different phase factors as you go along in time and so the direction of spin will also change. How will it change? Well, you see, originally at t equals zero it was like this. Later times this picks up exponential minus i epsilon z and this picks up exponential plus i epsilon z so it is as if the theta stayed fixed and the phi changed with time. So phi at t over 2 is equal to phi at zero over 2 plus this epsilon zt over h-bar. You see, these two exponentials, when you multiply them the exponent adds. So the new phi over 2 is just the old phi over 2 times that factor. And, of course, it's the same thing appearing here. Here its minus, here its plus. So, the overall effect, then, of a Hamiltonian like this is to make the angle phi evolve with time, change linearly with time. So, physically what does that mean? So you have the spin pointing in some direction, theta with the z axis, with time the theta says the same but the phi keeps changing. So it means this thing is going around and around, around the z. So that's what you call precession. So if you had a Hamiltonian like this, one that gives you-- I mean, affects the up and down differently with this plus 1 and minus 1, then it will make the spin precess around the z direction and this is the kind of Hamiltonian you actually get when you put a magnetic field and that's something we'll talk about in the next lecture but-- and doesn't concern right now but I just mention it because you may have heard that in a magnetic field the spin wants to precess around the magnetic field. And what we kind of just showed is if you had a Hamiltonian like this the spin would, indeed, precess around the z direction. Now, and the rate at which it would precess, d phi dt, is this 2 epsilon z over h-bar. So this is what you could define as omega. Omega is the angular frequency of procession. 

[Slide 5] So, in terms of this omega then we could write the equation we had as d/dt of up down is omega over 2 i times this matrix, which we defined earlier as the sigma z. That's the z component of the poly spin matrix, u d. And, what this does is it makes the individual component change with time according to these two-phase factors, which amounts to I guess the phi continually changing with time. Now, supposing we now want to describe in terms of vectors, so what I mean is this equation tells you how the up and down component of the spinner changes with time but suppose we want to write an equation telling us how this vector itself changes with time. Well, you could do it like this. You know the x component of n is given by sin theta cosine phi. So how does it change with time? Well, if I take the derivative I'll get minus sin theta sin phi and then d phi dt. Similarly, if I want to know how the y component changes with time I'll get sin theta cosine phi, I'm taking the derivative of that, d/dt of phi. So, now we'll notice that sin theta sin phi is, of course, ny. So you can write minus ny omega because d/dt of phi is the omega. Similarly here, this is plus nx omega. So, and of course, if you look at the d/dt of nz there isn't any because there's no phi here, so it's just theta. So d/dt of nz is zero. So you could collect all of this and write it in the form of a matrix. d/dt of nx is equal to minus ny times omega, right? So zero times nx minus 1 times ny, 0 times nz, that's minus ny omega. Similarly, d/dt of ny is this nx times omega, that's exactly what you have here and d/dt of nz is zero. So, what we just showed was that if we were working in terms of the components of the vector, the ones we learn in vector algebra, then you would have written an equation like this, you see? And this matrix we could call the rotation matrix Rz. What it does is it takes that vector and rotates it around z. On the other hand, what we have is the spinor with two complex components and this matrix has this same effect on it in terms of making it rotate. And if there's omega there we have this omega over 2i here, but this has the same effect, it basically describes this rotation. Now these two matrices, as you notice, will look very different. Now, next question you could ask is what matrices would actually make it rotate around the x axis, let's say? 

[Slide 6] Now, in terms of vectors that's easy to write because from here I could easily say well, if you really want to rotate around the x axis what we should do is kind of move this thing down so that the x kind of becomes like the z with all zeroes here and this minus 1 just slides down here, the 1 slides down there and you have the new matrix. So basically you're just changing the x, y, z directions. So knowing this, knowing the matrix that rotates things around the z axis, you can easily write down a matrix that rotates things around the x axis but here it's not at all obvious what to do, okay? And the correct answer is that the matrix that rotates around the x axis is actually the sigma x, which is this other poly spin matrix with 0 1 1 0. And there's no simple logical argument that will tell you how to look at this and write down this. So what is it then about these matrices that, you know, make this behave kind of like this and this behave like that? And although mathematically it looks very different and this is what you'd learn, I guess, if you read description of this in a standard quantum mechanics text. The point you'd make here is that see these rotation matrices basically obey-- they have this very important property that Rx Ry minus Ry Rx, namely these matrices, when you multiply two of them,  xy minus yx you get Rz. And so, this is-- reflects the basic algebra of rotations. That is, when you take an object and rotate it around the x axis a little bit and around the z axis a little bit that is different from if you rotated it in the reverse order and the difference is exactly like rotating around the Z axis so Ry followed by Rx is different from Rx followed by Ry and the difference is Rz and this is a very important property of this general algebra of rotations and so if you want to have a consistent theory of these rotatable objects then it is important that these sigmas, when you're trying to work with two complex components, also have the same property namely just as Rx Ry minus Ry Rx was Rz you need to have sigma x over 2i, sigma y over 2i and the 2i is because of this difference of this omega over 2i here minus sigma y over 2i sigma x over 2i should be sigma z over 2i, which amounts to this. So usually, as I said, if you read a quantum mechanics text that's how the argument would go. It would start from this group theory of rotations and it would say that well, this is the basic algebra of rotations so if you want to represent a rotatable object with two complex components, first thing you need to do is find a set of 2 by 2 complex matrices which obey this property and those are the properties that are obeyed by the poly spin matrices. 

[Slide 7] So these poly spin matrices that I had written down earlier, they have this basic property, sigma x sigma y minus sigma y sigma x is 2i sigma z. Similarly, sigma y sigma z minus sigma z sigma x-- sigma y, is 2i sigma x and so on. So, cyclically, they have that property and so, and once you have these matrices then the way you interpret them is this matrix rotates around the z axis, this matrix rotates around the x axis, this matrix rotates around the y axis. And how do I get these components that I had mentioned earlier? You know, I said that something pointing in some direction has certain components? Well, the idea is that if you want a spinor that points along the z axis that spinor will be a eigenvector of sigma z. Why? Because if something points along the z axis then when you rotate it around the z axis nothing happens to it, right? And that is what an eigenvector is like, so, it's like this operating on a spinor that's pointing in the z direction has no effect on it, so it doesn't, and so spinor pointing along z is an eigenvector of sigma z and, of course, this is a 2 by 2 matrix so there are 2 eigenvectors, one is the plus Z and the other is the minus z. Similarly, sigma x, if you want the spinor pointing along the X direction, well it's an eigenvector of sigma x and the two eigenvalues again, plus 1 and minus 1, the plus 1 gives you the plus x, minus 1 gives you the minus x and the same here. So, in general then if you want to write a spinor pointing along plus n or minus n, what you should do is construct this matrix sigma dot n so if n is z you get sigma z, if it's x you get sigma x. If it is in some arbitrary direction it's some combination of these sigmas and you take that matrix, find its eigenvectors and that is how you normally deduce these results that I stated earlier. So that's usually how it is developed in a standard quantum mechanics text. That first point is that if-- in order to have a consistent theory of this rotatable object you need to discover matrices which obey this commutation relation. These sigmas have that property and then how do you find the components? Well, you look at the eigenvectors of these sigmas and the plus 1 gives you a spin pointing in that direction, minus 1 gives you the spin pointing in the other direction. 

[Slide 8] So, that's a quick introduction then to this notion of vectors and spinors and the differences in the relationship and what we'll now do is, in the next lecture, talk about spin-orbit coupling, this very important effect in channels, which causes spins to recess inside a channel and which can be used to control the flow of spins in spintronic devices.