 We're looking at what happens when Hamiltonians can get into time reversal. We actually haven't defined yet what the time reversal operator is, but we've listed a series of properties that it's supposed to satisfy. These are conjugation relations involving position momentum and angular momentum. Also, it has to be an anti-unitary operator. And one of the things we found out was that if the Hamiltonian commutes with time reversal, this is the same thing as saying that if you conjugate with time reversal, it gives you the Hamiltonian back again. What we found was is that the time-reversed motion is the solution of the Schrodinger equation as long as the original notion, untimered risk, was a solution. Here, the time-reversed solution is defined psi r of t, the state vector is the function of time, is given by the time reversal operator acting on psi minus t. All right. So just to say closely, parallel is what we did in the classical case about time-reversed motion. Now, again, without saying exactly what the time reversal operator is, just using these properties, you can already say quite a few things about it. I'd like to start by giving us some examples of Hamiltonians that do commute with time reversal. And then we'll take it from there and look at cases where they develop. So allow me just to do this here. First of all, you'll recall that in the classical case, we found that particles moving in electrostatic field, their motion was time reversal and bearing. So let's start with that case. And then we've got particles moving in three dimensions. So that equals the potential Hamiltonian. In fact, let's make it 2 times phi. And phi is not the function of position for electrostatic field. I notice here that enough requirements be a central force motion. This is an arbitrary electric field here. But it's easy to see that this Hamiltonian is invariant in the time reversal, because if I conjugate it with theta, in the first place, the momentum, which appears to be kinetic energy, is odd under time reversal. But since it's squared here, it means the overall kinetic energy is even. And the position vector is also even, so it doesn't change. And the result is the whole Hamiltonian is invariant in time reversal. So this closely parallels what we saw in the classical case of electrostatic fields. I'll say again, this does not require this to be a central force Hamiltonian. Now, the easiest way to break time reversal invariance in a Hamiltonian like this is to put the system in an external magnetic field. Because if you do this, of course, the kinetic energy now becomes 1 over 2m. Then you have to move up from p minus q over c times the vector potential, which is the function of position in the whole thing squared. And if you now conjugate the time reversal, the momentum term will change sign as it's odd under time reversal, whereas the vector potential being a function of only a position is even. And so the Hamiltonian does not go into itself. Forget this, closely parallels what we saw in the classical case as well. Now, on the other hand, as I remarked earlier, if the magnetic field is internally generated, that's to say, if you include the charges and currents that produce the electric and magnetic fields as part of the definition of the quote unquote system, then time reversal invariance is restored. This is a reflection of the property of the fact that the electromagnetic interactions are invariant or time reversal. So to speak of an atomic problem, for example, let's talk about the spin arbitrary. The spin arbitrary has the form of the function of the radius times all dot s, as we've discussed previously, and there's a magnetic interaction that has to do with the magnetic energy of the electron as seen in its own, well, it's due to the magnetic field as seen in its own rest frame due to the motion of the nucleus around it as seen in the rest frame of the electron. Notice that F of r is certainly a parameter time reversal because the position vector is L and s are two angular momenta. The conjugation relations for j here actually applies to all types of angular momenta, orbital and spin. So both L and s are odd in the time reversal, but they're not product as even. And the result is that this Hamiltonian, even including the spin or return, is invariant in the time reversal. So this is an example of an incident that generated magnetic guilt. Similarly, by the way, if you have a hyperplot interactions, they are proportional to i dotted into s. The hyperplot interaction, for example, in the second hydrogen is the magnetic interaction of the electron spin with the magnetic field produced by the nucleus. And one of the terms that is proportional to i dotted s where i is the nuclear spin, then s is the electron spin. But you can see that this is also, even in the time reversal, the same thing is the same reason as L dot s. It involves not one of the two angular momenta. All right. Sorry, has its own magnetic term in it still violate its term? Excuse me, what's the question? L dot s, and it has the del h term? It had the a term? Del h term. Yeah. Well, the h term is coming from an external source. And yes, it violates time reversal. The a is coming from an internal source. And this is actually a function of the position, and it's really the velocity of the particle, you know. So then time reversal, in various cases, is an actual pulse for the whole system. You don't see this in atoms like hydrogen because the, well, let's see, what am I trying to say? The spin market interaction is, as I said, is the interaction of the, yeah. So the situation is that in an atom like hydrogen, you don't have any vector potential from the nucleus because it's stationary. So it doesn't make a magnetic field apart from that. It makes a magnetic dipole field because of the spin. But it's not a current in motion, not in a lab frame it isn't. And so there's no magnetic field in the lab frame due to this motion. However, you go to an atom like helium, and then both electrons are moving, and they produce magnetic fields. And then you get not only a spin-spin interaction, but a spin, it's called the spin-other orbit, spin-orbit, it's not the same orbit as a spin in one electron or the orbit of the other electron. You put all the pieces together, you'll find it's still invariant in the time reversal. All right, now. So now the question that arises is, are there any examples of Hamiltonians that involve electromagnetic fields which are purely internally generated, which do violate time-reversal invariance? Does that happen? And the answer is yes, I can show you an example of that. Let's take the case of a nucleus, you can just think about a nuclear spins. So the cut space for the nucleus, as we're used to, is the span of the spin-based states. That's him like this. And as we know, if you place this in the next term magnetic field, you get an interaction of minus v dot p. This is the energy of the y-hole in the next field. Nothing, I don't want to consider that because that's certainly not invariant in time reversal. Actually, I bet you part of it actually is invariant in time. Anyway, let me get to the point. The point is, is that let's consider the possibility of an electric dipole moment in the nucleus. The energy interaction is called a delta H for an electric dipole moment is minus d dot in the electric field, or d is the dipole moment in this case, in the nucleus. Now then, the electric field in question could be an external electric field for simplicity, but suppose it's that, replacing the nucleus in the next terminal electric field. Now, what about the dipole moment vector? As you are hopefully learning in this week's homework exercise, there's a lot in this Hilbert space here, this spin-Hilbert space, which is a single invisible subspace in the rotations. You find that all vector operators are proportionally governed. And in particular, for example, the magnetic moment operator has to be proportional to the spin. This is the usual thing we've seen earlier with g-factors. This is the explanation for it. It comes from the big record theory. Likewise, the dipole, electric dipole moment operator, if it exists, must also be proportional to the spin. Let's write it as k times the spin, okay, as some constant. It has to be so on this Hilbert space. And so this interaction becomes minus some constant times the spin dotted in the electric field. Now, if we conjugate this with time reversal, the electric field doesn't change because it's a function only in position. However, the spin does change and the result is this is an odd term. This is odd in a time reversal. Now, the result of this is that if time reversal is a good symmetry of the Hamiltonians, then the nucleus cannot have an electric dipole moment. And in fact, it's known that time reversal is a day to a very high degree of accuracy. In fact, as far as anybody knows, it's exact in the case of nuclei, as well as however, this also doesn't have to be a nucleus. It could be an elementary particle, such as the electron. And so there's a question also about the possibility of electric dipole moment for the electron. The argument is the same. If there is, the dipole moment operator must be proportional to the spin. And so this is the interaction in an external field and it's odd in a time reversal, not even. So time reversal invariance excludes the possibility of electron dipole moment, that there is a dipole moment of nuclei. This is quite a case of nuclei interacting in the next term of fields. We're interested primarily in the quadruple moment because that's the first non-vanishing term, assuming time reversal invariance. Now, of course, it's an experimental matter to detect whether or not the term like this exists. It can be said that first of all, such terms have never been discovered. Either a nuclear die or in so-called elementary particles, but it's not the lack of searching. In fact, there's been an extraneous effort to detect these, delicate experiments because they involve very high precision. Some of these have been carried out by Professor Cummins and also Professor Bunker in this department, especially Cummins, worked on this for a long time, establishing up the bounds on the electric dipole moment of the electron. If you saw us cloaking just a week ago, you heard something about this. In any case, the situation is that the standard model does not exclude such an electric dipole moment, but based on the numbers that are known from experimental data and the given standard model, if such an electric dipole moment exists, it's extremely small and way smaller than anything that's ever been measured up to this point. However, the standard model is certainly not the end of the story and there's a question about physics beyond the standard model and some of the hypotheses or guesses for such physics would be a rise to electric dipole moment so considered larger than what should be on the standard model. So these experiments are ways of probing physics beyond the standard model in which there's a great deal of interest these days. Let's just say that all that's been done from this point is established up the bounds. For the electron, I think the number is that the quadruple moment is, excuse me, the dipole moment is something like e times 10 to the right is 27 centimeters, sorry, let's put it this way. It's about 10 to the minus 27 e centimeters where e is the charge of the electron. My recollection is that's the order of magnitude of the number, it's something like that. So, all right, all right, so that's the likely quadruple moment. Now, as I said, normally go up to this point we haven't actually said what the time reversal operator actually is in any specific system, so let me get a turn to the simplest cases that explain what the black hole, excuse me, what the time reversal operator actually is. The simplest case to take is that of a spinless particle moving in three dimensions in which we just have a simple wave function psi of r like this. The time reversal operator, whatever it is, has to satisfy the given commutation equations. And as I discussed in the last lecture, we can factor this, what I call the LKD composition into the product of a linear operator times an anti-linear operator chosen for simplicity. And in fact, the anti-linear operators that were chosen in practice are usually these complex conjugation operators that apply in a particular representation. In the present case, the obvious representation uses position representations, but those are position eigencasts. So let's let K here, let's let K here in this context be the same thing as what I call K sub r, then satisfies the condition that maps the position eigencats into themselves like this. And I'll remind you that this doesn't mean that K is the identity, it's K is anti-linear. It does, however, mean that K is equal to K-bagger, is equal to K inverse. And so it's in fact an anti-unitry operator. Well, so let's begin the search for theta by just looking at the K part. But let's see what the K part does to the desired conjugation relations which are here, particularly the position of momentum. Position must be even momentum odd under times of times of momentum. So let's first of all look at this. Let's look at what the K rK-bagger is for r as the position operator. Actually, I want you to put a hat on this to indicate the operator here. The hat now needs operator and not unit vector. So we're interested in this product of operators. Well, one way to do this is to take a wave function of psi of r and just apply these operators in order of working from right to left. First of all, I K-bagger. K-bagger is the same thing as K because it's a complex conjugation operator. So this turns into wave function of psi of r, complex conjugated. Then we'll apply the position operator, r hat, that's equivalent to multiply by the position. So it just turns into r times psi of r, complex conjugated. And then we'll apply K again, which is just the complex conjugates that takes us in order of r times psi of r because the position vector is real, it doesn't change. And you see the effect of the product of these three operators is just the multiply the wave function by the position vector. That's the same thing as the effect of the unmodified position operator itself. And so you see the position vector actually is unchanged by the complex conjugation operation. As far as the momentum is concerned, this is of course the minus i h bar, d v r in the position representation. And since we've just decided that the position itself is invariant, all that I have to think about here is the i, since a time reversal is anti-unitary, this means the i changes sign. And the result is that you conjugate momentum in terms of the minus itself. And these are exactly the desired conjugation relations that we have for time reversal. By the way, to take a cross product of r times t, this gives you a meaning of k l k dagger for the orbital angle momentum of minus l. And we require that all types of angle momentum be odd over time reversal. Anyway, the net effectiveness is that the complex conjugation operation alone satisfies all the conjugation relations that we require in addition to its anti-unitary. So in this l k decomposition, we really don't need the value to set all this linear operator equal to 1 if you want. And we just end up with a very simple rule, which is that for such systems, spinless particles in three dimensions as the time reversal operator is just complex conjugation of the position of the representation. And it's actually the way functions is just given by a complex conjugation. The way function psi r goes over in this complex conjugate under time reversal. This is a very simple rule, which is easy to remember. And it applies to a lot of cases in practice. By the way, it's easy to generalize to multiple particle systems with no spin, but just complex conjugate the way function. That's all there is to it. To get the time reversal way function. All right. Now, I want to, in a moment, talk about time reversal and systems with spin. But before I do that, let me make some remarks about the time reversal of energy and eigenstates. Let's suppose we have a Hamiltonian which commutes with time reversal, or let's suppose it has an energy and eigenstates so that H psi equals V psi like this. And let's see what effect a time reversal, what time reversal can tell us, time reversal and variance of H can tell us about this. Let's take both sides of this equation, multiply through by the time reversal operator on the left. So on the left-hand side, I'll have theta times H. And then I'll insert theta, negative theta, which is equal to identity because theta has to be anti-unitary, acting on the side. So that's the left-hand side. But we're assuming that H is invariant of the time reversal. So this becomes H theta psi. And then if I'm theta to the right-hand side, I can pull the theta through the energy E because E is a real number, so it's complex, complex, equal to itself. And this becomes E times theta psi. And then putting in parentheses just to make the emphasis that I want, the result of this is that the time reverse state is also an energy eigenstate of the same energy. So this is the rule. It's easy to make easier to state the words, is to take an energy eigenstate of the time reversal of the invariant Hamiltonian, and you apply time reversal to it. You get another energy eigenstate of the same energy. All right? Now, it doesn't change the energy in other words. Now, it's logical that it should do that. Now, let's make this a more special case. Let's suppose that this is a non-degenerate state, an energy eigenstate. Then in that case, since theta psi has the same, it's an eigenstate of the same energy, it must be proportional to the original state, some proportionality factor on call C, because it's only one initial eigenspace. Now, what can we say about this constant C? We can determine it, we can get a condition. It turns out it's a phase factor. We can show this by squaring both sides of this equation. This will be a little exercise in using anti-linear operators. The squaring the left-hand side of the equation, I first convert this cat here into a bra. To do that, I run it backwards like this and dagger everything. And I also need to keep track of the parentheses, because now, if theta is acting to the left, theta-dagger is acting to the left. As far as the original cat is concerned, this is theta psi like this, and I'll put parentheses here to indicate that theta acts to the right. Squaring the right-hand side, we just get the absolute value of C squared, assuming that psi is normalized, which I will assume that's both psi-sides equal to 1. Then we just know mod C squared. Now, on the left-hand side, this theta-dagger acts to the left. If I reverse the direction at which this acts, which is what I'd like to do is to make it move over this way, then I need a complex conjugate to make itself work. That's one of the rules of anti-linear operators. So the left-hand side becomes psi like this, and then theta-dagger theta applied to psi, with no parentheses indicating that everything works that way. But then I have the complex conjugated, and that's the other way to write the left-hand side. However, theta is just the identity, so this just turns into the square of psi, which is just 1. So the complex conjugate just turns it into 1. So the whole left-hand side is just equal to 1. And the result is, as I said, this is the C becomes the phase factor. So for a non-degenerative energy eigenstate, a time reversal of gradient Hamiltonian, we say that theta after the eigenstate gives us a phase factor, which will not write as e to the i alpha, instead of C, multiply psi. Now, allow me to take this equation, multiply both sides from the left, like e to the minus alpha over 2. When I take the e to the minus alpha over 2 and communicate it past the theta, it becomes complex conjugated, and it becomes e to the plus i alpha over 2. So the left-hand side now becomes theta times e to the plus i alpha over 2 times psi. And then the right-hand side, e to the minus alpha over 2, combines e to the plus i alpha, and gets us e to the minus i alpha over 2 times psi. Now what you see in this is that this original eigenstate multiplied by e to the i alpha over 2 is what you might call an eigenstate of a time reversal. It's mapped into itself by time reversal. You mean plus e to the minus alpha over 2? It should be s, thank you, should be plus s, thank you. So if I call this, I just call this state phi like this, then what we have is theta acting on phi, and so the statement here is that if you have a non-regenerate energy eigenstate of a time reversal of a varying system, then by multiplying by an overall phase, you can make the eigenstate itself a very different time reversal. In a particular case of one of our spinless particles moving in any, actually it doesn't have to decrease the any number of dimensions, this would say, this result would be in the next case to call the time reversal operation is just complex conjugation. This result would say you should have only a non-regenerate energy eigenstate real by multiplying by an overall phase factor. This is a fairly trivial conclusion because in those cases, the Schrodinger equation, the time, the eigenvalue problem, the time independent Schrodinger equation, that is a purely real equation so naturally you can find real eigenfunctions. This puts this in a more general context for any system that's time reversal and varying. All right, so that's actually kind of a small result. Now, let me turn now to the question of time reversal in the context of spin. I'm going to move on the board for this. Simplistically, we'll start with the case in which we'll ignore the spatial degrees of freedom, let only the spin degrees of freedom, the spatial degrees of freedom are not important. So this means, again, that we've got a whole different space which is the span of states. That's an unusual spin state like this. All right, now, here there's a spin operator. The spin operator is the angular momentum in the system. And like all angular momentum operators, this is required to be odd in the time reversal so if you're conjugated, it goes in the minus itself. So this is a basic relation, this basic conjugation or conjugation relation that we have. It's also of interest to conjugate the raising and lowering operators with a time reversal. If I take the raising operator, for example, as xs plus isy, and if I conjugate this with time reversal, the spin components sx and sy will change sign. But the i also changes sign because of complex conjugation. And the result is this turns into minus sx plus isy, which is the same thing as minus s minus. So to combine this together, we can say that minus s minus is equal to theta s plus theta dagger. We only box that part of it because that's the part of long years. All right, now, let's take these basis states as m and explore their properties under time reversal. In particular, let's apply the time reversal operator to the basis state, sm. We don't actually have to have a definition of time reversal yet. We're going to work on that, but we're starting with the required conjugation relations to see what we supply. So if they back on sm, what can we say about that? Well, one thing we do is allow jz back on that state and see what we get. If we do, we can pull the jz through the theta, but if you cut, it's not jz, it's sc, excuse me. Sc, we can pull the sc through the theta, but if we do it, it changes sign. So this turns into theta times the minus theta times s of z acting on sm. And now the s of z brings out m, so we get minus m times theta acting sm, sm like this. And so what you see is that theta acting on sm is an eigenvalue of s of z with an eigenvalue of minus m. And the result is that the time reversal operation changes the sign of the magnetic quantum number. Logical, because it changes the sign of the operator sc. And so theta acting on sm, I mean in this overt space, there is only one vector with an eigenvalue of minus m. So this has to be proportional to that, but some proportionality factor, all called c, that's kind of minus m. It must have a relation like this. And in fact, basically what I said so far, there's no reason why the c couldn't depend on the particular magnetic quantum number. So I'll put it in a sub-script on that. And in fact, we'll find out in a minute if it really does. By the way, it's easy to show by squaring this both sides of this equation. The cm is a phase factor, so it's absolute value is equal to 1. I won't go through that again, but we will use the phase factors. Now we can get further information about these c's if we use raising and lowering operators. A lot of me to take is finally this last equation here. Let's apply s plus, the raising operator to both sides. So on the left-hand side, we get s plus times theta acting on sm. And I'll work on the left-hand side for a while. If we pull the s plus through the beta, it becomes minus s minus. So this becomes minus s minus theta times s minus acting on sm. That becomes minus theta times square root, which is s plus m times s minus m plus 1, acting on the state s comma m minus 1, in function of water type 1. And now we can bring the theta through the square root, which is real, without changing the square root. We're going to do the theta is adjacent to s comma m minus 1. But from the equation above, that's going to bring out a constant c, cc of m minus 1. And it will also change the magnetic boundary. So this becomes equal to an overall minus sign in the same square root. And then we get c sub m minus 1 times the state s comma minus m plus 1. Having time reverse the state. Now that's working out the consequences of applying s plus to the left-hand side. If I apply s plus to the right-hand side, c m is constants. I'll bring it through. And so this is equal to cm times s plus applied to s comma minus m. And if you look that up as cm, then you get another square root here. And the square root is the square root of s plus m times s minus m plus 1. And then we have to raise the magnetic problem number. So it becomes s comma minus m plus 1. Now, if we compare this result here, which I'll circle, excuse me, this version came out of the left-hand side. With this version, it came out of the right-hand side. And those who are equal, you see the square roots are the same. So I can cancel them. You can see the kets are the same, the same state. That's comma minus m plus 1. So I'll cancel them. What's left over is a relation of the magnetic coefficient. c m is equal to c m minus 1 to the right-hand side. I'll put it up here, write it like this. c sub m minus 1 is equal to minus c sub m. And so as I said, these phase factors, which occur here, apply controversial these states do depend on them. And whatever they are, they change sign as n goes up and down. So by recursion, every time you're moving them up and down, you get a factor of minus 1 for every step. That means in particular, if it's starting from the lower stretch state where n is equal to minus s, we can say that c sub m is equal to c sub minus s times the factor which is minus 1 to the s plus m power. You can see this is the right solution because if I set n equals minus s to the minus 1 term, it cancels and I get c minus s to the c minus s. And you can see otherwise, it goes to minus 1 and they have to change sign for every end of the way. Allow me to write this as c sub minus s times i square root of minus 1 to the power of 2s plus 2m. I'll do this for a reason I'll explain in a minute. But if I do then, this becomes c minus s times i to the 2s times i to the 2m, where I factor it into the part that depends on s and the part that depends on m. I want to write it as i like this because the magnetic one number can be half an inch or, and it's convenient to have only one inch or powers because there's a half power and you worry about the sign that's square root. So this keeps track of the problem. There's no ambiguity about what this means. No square root ambiguity. In any case, the first factor here depends only on spin of the particle and the second factor depends on the magnetic one number. Allow me to write this as part just to call it eta just to give it a name like this. And if we now put these pieces together, here's what we get for time reversal acting on these basis states of s n. It's a phase factor eta times i to the 2m times s times minus s n. And this is the basic conclusion we get by using nothing but the required conjugation relations of time reversal with the spin operator. And this gives us information about the phase factor that's introduced when the magnetic one number is split by the application of theta. This factor eta is, I just indicated, depends only on s. In a sense, it's a characteristic of a particle. But it's actually not even as important as that, because it turns out that it's completely non-physical. The first thing to say is that it doesn't affect any of the conjugation relations because those all have all theta theta dagger. And so the eta is going to cancel out. And there's actually no physical consequences to this eta. You can set it even to anything you want, any phase factor you want. In fact, there are different conventions in the literature. I don't care which point you want to use for any of your applications. You can set it even to one if you want to. But in any case, this is the action of time reversal with spin states. I'd like to approach this by the same problem of time reversal with spin states from another point of view. It's useful to have both of them. This is the first point of view here, using conjugation relations, connotation relations. Here's a second point of view. Again, it's the same Hilbert space but it's just a spin space. It's not an abortion problem. Let's consider an LK decomposition of the time reversal operator as we get earlier with the case of the final spin. The K is the complex conjugation operation on the wave function in some representation. And we need to say what that is. The logical one here is the S and Z representation because that's just the basis states we're using up there. So let's do this. Let's take the complex conjugation in the S and Z basis and let's see how far we get if we apply it to the, where there are the desired connotation relations. Well, for a purely spin system, there is no positional momentum operator but there is an angular momentum. So this is the main one we concentrate on. And of course, it's supposed to be hand unitary. So let's just work on the K itself in complex conjugation operation and see what we get when we conjugate J or we identify the spin. So that's the strategy. So let's do this. Just for gravity's sake, let's call this K, well, I'll do it this way. Let's take this case of S and Z and let's take the operator's S, Y, and S, C, and the range of the column like this. And in case of S and Z, that's the other case. All of these complex conjugation optimizations are all of them are anti-unitary. All anti-unitary and effective squares equal to 1. Clearly, complex conjugation twice you get identity. Now, what happens when we conjugate the components of spin with this complex conjugation operation? The answer is easy to see if you think about the matrix representations of these operators in the standard angular momentum basis. The matrices for S, X, and S, Z are purely real, whereas those for S, Y are purely imaginary. This is a consequence of the phase conventions that we established in the standard angular momentum basis. And the result of this is another of this conjugation, this complex conjugation. S, X, and S, Z go into themselves, but S, Y changes sign, so you've got a minus sign with S, Y. Well, this doesn't give us what we want. This is what we want is that theta acting on S, multiply by theta dagger should take you to minus S. In fact, all three components should change their sign. And so in the case of spin systems, the time reversal operator cannot be simply complex conjugation. It's got to be more to it. We need a linear operator that's going to make things come out right. Such a linear operator is not hard to find. If we draw spin space like this, S, X, S, Y, and S, Z, the effect of the K, as you see here, was to change the sign to S of Y so it flips over like this, the S, Y, change sign. But S, X, and S, Z did not change. However, if one lay the S, Y alone, not change that result, which is still good, yes. I'm sorry, I have an idea. Because the matrix for S, Y is purely imaginary. Oh, okay. In the standard phase conventions, yeah. Whereas those for S, X, and S, Z are purely real. You can see that with the poly matrices, but it's also true for any value of spin, not only one half, but all the others as well. So we flip the sign of S, Y, but now we need to flip the sign of S, X, and S, Z, as well. And one way to do that is to perform a notation about the Y axis by an angle of pi, which I think you can see we'll flip those as well. So what this gives us in is we'll identify the linear operator L here with rotational U of Y hat comma pi, which is equal to V to the minus i pi Y over the spin divided by H of Y, divided out explicitly. And so we'll do this with spin systems. We'll say let's take this system, purely spin systems. We'll say theta is equal to V to the minus i pi over H of R times the complex conjugation in the case of Z basis. And this is actually a satisfactory definition of time reversal operator for a purely spin system. Now on the basis of what was done here on the upper board, this time reversal operator here, the theta, must have this form up here for some choice of the phase factor theta. In a little extra work, you can actually do that and find out what the eta is. I won't go through it because it's really not important. The point is that the two approaches we actually give you is to give you a compatible result. All right. Now, having obtained this for a pure spin system, it's now easy to guess at least what the time reversal operator should be for a system where you have both spatial and spin degrees of freedom. Now on the basis of that, the usual ones would usually use would be R and M like this, which is the tension product of the position eigencat times the sm eigencats for the spin. And in the case of the pure position, in the case in which the position operator along this complete set, we just use complex conjugation and the position representation. So it's logical that for a system like this, we should take this time with this theta here and just multiply by complex conjugation and the position representation. Another thing we're going to do is complex conjugate the entire wave function, both position and s of c. So this would mean that theta here for a system like this would be equal to even the minus i pi S y over h bar, the one with the usual rotation about the y axis times the complex conjugation and the position and s of c representation. In fact, this is the satisfactory, this is a satisfactory definition of time reversal for a spinning particle. Notice that this is only a spin rotation. It's not a total rotation of the system. We don't rotate the orbital degrees of freedom only the spin. You know, there's a question here. Time reversal itself doesn't have anything to do with any particular orbit system of directions in space. You might wonder why in the world we have a rotation about the y direction and what's special about that. So the answer is there's nothing special about the y direction except for the fact that it was picked out by our phase conventions that we use in setting up these standard angular momentum bases. There's eigenbasis of s and z but there's also phase conventions involved. Remember these complex conjugation operations depend on the phase conventions from the basis states. In any case, the result is that although the linear operator and the anti-linear operator in which we pass through theta do involve the directions in this case y and z the product of them does not as actually in a sense rotational or it is rotational. It's independent of any orientation in space. All right. Now, this result of a single spinning particle of spin is as easy to generalize even further to include multi-particle systems with spin. The particles don't have to be identical. They can have different particles that have different spins. Some can be fermions, some can be bosons. Think of the helium amount of two protons and two neutrons in the nucleus and two electrons. Well, that's six fermions if you want. All right. The other thing, the alpha particles are single particles. One boson and two fermions and two electrons. All right. In any case, this is easy to generalize. All you do is you just apply a spin rotation on all the spins. They just use the same formula here except that sy is now interpreted as the sum of all the particles of the syi. It's a total y component of the spin. And the k here is interpreted as a complex conjugation of the wave function in a combined position in s and z for all the particles just by reinterpreting this equation. Okay. This now leads to an interesting conclusion which I have a space to do with here. You'll recall that a moment ago I explained that if you have a time reversal in variant Hamiltonian and an energy ion state, let's call it psi, so that H psi equals H psi is that we found that if you apply time reversal to that state you get another you get another you get another you get another ion state in Hamiltonian with the same energy. All right. So in particular this is equal to c. So let's see. So if you have the same energy in particular this is non-degenerate. If this is non-degenerate then theta acting on psi must be some constant time psi itself. Now you showed that it's a phase factor. It's like what we call it a phase factor right e to the i alpha 4. It's a repetition of this a few moments ago. Now let's take this the second equation and let's apply time reversal the second time to it is multiplied through by time reversal. So on the left-hand side we get theta squared acting on psi. So to recapitulate a little bit this applies to any non-degenerate ion state of a Hamiltonian which is time reversal in variant. Now let's apply theta again so on the left-hand side we get theta squared. On the right-hand side this turns into theta times e to the i alpha times psi. We can bring the theta through the e to the i alpha but if we do we have to complex conjugate it so it goes to e to the minus i alpha times theta acting on psi. Theta acting on psi gives me e to the i alpha times psi so the result is this just turns into psi itself. And so what we see is in the system theta squared acting on psi and brings back psi itself psi as a variant of theta squared. Alright. For any non-degenerate energy ion state of the time reversal in variant system. Now on the other hand we have this expression for the time reversal operator in general for now spending any number of spending part of this. Let's take the square of this let's take the theta squared let's write this as it becomes e to the minus i i s y over h bar times this k I'll just write k for it times e to the minus i pi s y over h bar times k is part of the product of these these operatives. The k operator actually commutes with this linear unitary rotation. The reason is that the matrix for s y is purely imaginary but up in the exponent it gets multiplied by i and so this is an example of the y rotation this is one of the lower case d matrices which are purely real so the k will commute through that and if we do a k squared the k squared is an identity because your complex congregates y squared back to the identity. So all this left is the product of these two rotations and we can end up with e to the minus 2 pi i s y over h bar this is a spin rotation by an angle of 2 pi about the y axis for all the particles. Well the spin of quantum number is an integer that a rotation by 2 pi brings you back to where you started from it's just 1. If the spin is a half integer then a rotation by 2 pi gives you a minus 1 this is this minus 1 in rotating electrons by an angle of 2 pi and so this operator is equal to minus 1 to the certain power which I'll call, let's call it a nu here where nu is equal to the number of fermion arms in the system that has an even number of fermion arms the square of the time reversal operator is plus 1 otherwise it's minus 1 but over here we said that if we had a non-pagenetic energy eigenstate about a time reversal in a very Hamiltonian then theta squared applied to that would give us psi back again. You see that's inconsistent with this result if you have an odd number of fermions and so the assumption that we had a non-pagenetic energy eigenstate must be false and so the conclusion is that in any time reversal in your system with an odd number of fermions is that the energy eigenstates are necessarily degenerate. In fact with a little more effort one can show that degeneracy is always an even number this is called promise degeneracy and it's a rather I mean time reversal is of all the symmetries time reversal is the odd ball because it's any unitary so there's nothing like time to have promise degeneracy for the other symmetries but this is one one of the main results you get from time reversal. If you want an example of promise degeneracy it occurs in this week's homework soccer I asked you about this problem you have a spin 3 halves particle notice that it's an odd let it say half integer spin a spin 3 halves a nucleus in the presence of an external electric field and the problem is to work out the energy levels of that and do the quadruple interaction and then you ask is there a degeneracy well it's a 4 by 4 matrix so it's 4 eigenvalues which you'll find they come in envelopes there's 2 pairs and this is a nice example it's a single fermion and it's also time reversal and varying because it's a quadruple interaction in an external electric field so it's an example of promise degeneracy giving another example if we took I'll give you another example of promise degeneracy if we took let's say a ordinary hydrogen atom 7 Hamiltonian is p squared of 2m and then a minus z squared over r like this and then a hydrogen atom now this is in an electrostatic model for hydrogen so we're ignoring the spin but the spin is really there the electron is a spin 1 half particle so promise degeneracy would say all of the energy levels should come in doublets this is true if you count the spin it's kind of obvious because the Hamiltonian doesn't depend on the spin so spin up and spin down has to have the same energy that's why it's always a doublet to make this a little less trivial we can add the spin order counter which really does bring in the spin and in that case spin up and spin down down to 2 components of spin are coupled however it's still true that the energy levels come in double let's say time reversal in variance we can fix this we can make this even more elaborate let's put in an external electric field put in a minus e times a an external potential which can be an arbitrary external potential function of position doesn't have to be a uniform field the energy levels will still be in doublets because of the promise degeneracy so these are examples of this I have one final remark about time reversal in that this is the question of CPT and CP violation CPT and CPNT are three symmetries the parity of T is time reversal that's what stands for CPT and C stands for charge conjugation we talked about time reversal and parity but not charge conjugation the reason is that you need relativistic quantum mechanics to properly understand charge conjugation nevertheless the basic idea is is that that's an operation that adapts particles into their antiparticles and therefore changes the charge of the electrons of the positrons for example it is believed in quantum field theory that all quantum fields should be relativistically covariant of the Lorenz transformations and also that they should only involve local interactions and on the basis of these assumptions it was proven by Paulian in 1940 so it's called the CPT theorem that says product charge conjugation parity and time reversal all together is an exact symmetry of any such quantum field and so CPT invariance is widely believed to be an absolute absolute balance symmetry of nature on the other hand as I explained in a previous lecture it's known that parity is highlighted in the weak interactions which was shown in 1954 so what about the other symmetries such as C and T particularly what about time reversal well ten years later in 1964 it was shown experimentally that there's a violation of the product of CP the CP product in the k mesons the neutral k mesons there's a whole story about that there was an interesting piece of physics in other words CP violation was discovered well if CPT really is a true symmetry and you've got CP violation as it's been proven experimentally then it follows it must be also time reversal violation recently there's been some more experimental work on CP violation in the beam mesons that's to be so-called the bar of de-factory that Stanford has been a good deal greater insight obtained into CP violation in the recent years so it's indirect evidence through one's belief in CPT there's also a time reversal violation of time reversal now I just mentioned one more thing before we go which is that there's a current speculation that it was violation of time reversal in variance in the early universe that gives rise to the imbalance between matter and any matter the universe that we live in of course we know based on anything anybody knows there are no any matter galaxies out there there's some questions about that but you certainly can't have a mixture of the two because they wouldn't violate in any case so the current belief is that the matter of the universe was created in the very early moments of the Big Bang basically it came out of gravitational energy and the matter and any matter were created in almost equal amounts with a small a small difference between them due to T violation and then as the universe cooled off matter in any matter annihilated one another leaving behind a small residue of matter which is everything that we have around us including ourselves so this is the current speculation on the state of the early universe and time reversal in variance plays a role in that okay that's all for today