 Okay, then today we're going to continue talking about parity, so let me remind you of the basic situation. We have discussed it with rotations as to say proper rotations. These are examples of space-time symmetries. In fact, proper rotations reflect the geometry of a three-dimensional space we live in. We have studied how these space-time symmetries are implemented on quantum mechanical systems by means of operators, in fact, military operators. The assumption of parity brings us into the area of improper rotations. In fact, every proper rotation gets mapped into an improper rotation of vice versa if you multiply by the spatial inversion operation. In fact, the parity operator, which acts on the quantum system, is the quantum analog of the spatial inversion operation, which takes place at the level of geometry and three-dimensional space. This is a basic overview of what we're doing. I might add that there's another space-time symmetry we'll consider probably later on this hour, which is a time reversal. Time reversal involves time, and all of these proper rotations, improper rotations, time reversal, and translations of it as well are examples of Lorenz transformations. In fact, the whole subject is united in a relativistic treatment, which we'll touch on at least to a certain extent next semester. The way that we are developing the parity operator and the quantum system is that we're first of all calling the operator pi, and we're placing a set of possibilities and requirements that it has to satisfy. First of all, that pi should be unitary in order that it should preserve probabilities. This is normally what you'd expect with symmetry operations. These properties, two and three, are reflections of the property of the spatial inversion operation at the level of classical geometry. The first one says that pi squared would be identity, and the third one says that pi commutes with all rotations, all proper rotations, that is. So, it bars the proper rotation by conjugating pi with such a rotation operator, you get pi, the parity operator back again. Now, to review briefly what we did last time, we're taking, well, to back up slightly. These are the requirements we impose on the parity operator, as we'll see the parity operator has different forms depending on the quantum mechanical system. So, we'll get different examples of the parity operator depending on the ket space. But these impose restrictions on what form the parity operator can take. As we'll see, they don't really disclassify the parity operator, but they narrow the possibilities rather much. So, there's not too much choice left when you're done. We started by taking the case of a particle moving in three-dimensional space, without spin, so the wave functions psi bar. We decided that the proper definition of parity should map this wave function psi of minus r. It's really a guess, but this is what we do. Or we can write this in ket language by saying that pi acts on the position I can get and maps them into another position I can get with the opposite position. And this we take as a definition of parity for a system with a spinless particle. Next, we took the case of the particles in which we had spin, but we ignored the spatial degrees of freedom. So, this is the spin part. In that case, the ket space is spanned by the basis cats. The Hilbert space is spanned by the basis cats as m, where s is the spin of the particle and m runs from minus s to plus s. And by considering, oh, I forgot to mention, property 3 here, which is the requirement that the parity operator should commute with rotations, really comes in two versions, because an operator commutes with rotations and call it a scalar operator. And, moreover, a scalar operator has to commute with all three components of angular momentum. So, this is conditions 3a and 3b, which are really equivalent to one another. So, if I require, I won't point back to the system with spin, but I require the parity commute with j sub z as one of these equations. What we found last time is that if you let parity act on one of these basis cats, you get something that has to be proportional to the same basis cat with a proportional up-downity factor of the right is e. This is saying that the basis states of a spin system are all, actually all eigenstates are parity and some eigenvalue e. Now, if you apply pi to both sides, or if you just use the fact that pi squared is equal to 1, you easily conclude that the square of this eigenvalue must be equal to 1, and therefore the eigenvalue itself must be plus or minus 1. If all you do is look at the jz equation, you wouldn't know whether or not this eigenvalue might depend on which of these factors you're using. That's to say, it might depend on the other. However, we'll recall that a scalar operator has eigenvalues that are independent of the magnetic quantum number. It's blessed normally that it's logically used in Hamiltonians, but it applies for parity as well. Or if you'd like to comply raising the lowering operators to both sides and show that the eigenvalue is actually independent of the magnetic quantum number. Thus, the eigenvalue, which is plus or minus 1 in this equation is eigenvalue is the same value that is to say either plus 1 or minus 1 for all of the states, all of the basis states. Thus, it's the same value in fact for the entire spin over its space. In fact, it's not really a characteristic of the state. It's a characteristic of the particle. That's to say, in the same way that the spin s is just a fixed number, which is characteristic of the particle. So this plus or minus 1 that comes out this way is called the intrinsic parity of the particle. There's a question now about whether or not this plus or minus 1 can be narrowed down further. You decide which of the two it is by using our postulates or using experimental data. Yes. Is it just by definition that parity is... No, it's not by definition. I went over this briefly last time. It comes from studying the commutation relation to pi with jz. The fact that pi can be used with jz means that if you apply jz to both sides, you find that pi acting on this is an eigenstate of jz with the same eigenvalue. But there is only one state that has that eigenvalue. That eigenvalue. That's as in itself. So they have to be proportionate. That's the logic. All right. So this e plus or minus 1 is a characteristic of the particle that's called intrinsic parity. Now, there's a question about whether a further postulate, these postulates are further postulates of the q-eco-arguments or experimental data would allow us to decide whether e is equal to plus 1 or minus 1. And the answer is that if you're dealing only with non-relativistic 1 mechanics, the answer is no. There's no way to decide whether it's plus 1 or minus 1. The reason is that in non-relativistic 1 mechanics, the number of particles is conserved. So in any transformation, the number of particles that are given tight is the same in the beginnings as it is at the ends, constant function of time. And so if you decided that some particles have a positive intrinsic parity, negative or if you changed your mind about that, it wouldn't in any way affect the balance of parity whether it's conserved or not. It only has two sides as a function of time on two sides of the reaction or something, or a scattering reaction or whatever. So it's a purely arbitrary choice as long as particles are either created or destroyed. However, as we'll see next semester, in all of this e quantum mechanics, it does make a difference because you can create particles. So whether you have three particles of odd parity makes a difference is different than having three particles of even parity. And if you can create one, it turns into four and that makes a difference too. So in fact, in non-relativistic quantum mechanics, you find that particles have this intrinsic parity and that's something that can be determined. Can is, in fact, determined experimentally. I lost him over some things and oversimplified things a little bit in regard to fermions. We'll worry about that next semester, but let me just leave it for now that a deeper understanding of parity requires a relativistic theory. But for now, as long as we're sticking with a non-relativistic theory, we might as well just take this e plus one here and that's, in fact, what we'll do for the rest of the semester. So we'll just say that parity acts on spin states and it just maps it in itself. It doesn't do anything to spin. Basic non-relativistic rule, parity doesn't do anything to spin. It does, however, affect the spatial part of the wave function as you see here. Now, by the way, this rule in non-relativistic quantum mechanics you don't let creative destroy particles is not quite always true. And the reason is that photons are particles that have zero mass. And so they can be created or destroyed in arbitrary numbers in arbitrarily low energy interactions. So ordinary matter, ordinary energies, you have a creation, the admission of absorption, the radiation going on all the time. And so when it comes to the question of diminishing and absorbing photons, parity does matter. And in fact, as we'll see, the intrinsic parity of photons minus one. And that means, for example, that when you do an atom, that's an electron, excuse me, when an atom is a photon in, let's say, an electric dipole transition, it means the parity of the atomic state changes. This is the way in which parity is conserved in the electric magnetic interactions. I'll elaborate on that a little more later. All right. Now, when we were doing proper rotations, we went through a discussion of how various operators transform under proper rotations. And we classified operators as scalar, spectres, and so on, depending on the transformation properties. Now we can do something similar in the case of parity. Let's go back to the case of a particle that has a spindle's particle moving in three dimensions as a simple example of the weight function of psi of r. If we consider, for example, taking the position operator and conjugating it by the parity operator, pi r, pi dagger, it would be a good guess that this should turn into a negative of the position operator because that's what the spatial reversion operation does to the position vector at the level of plasma geometry or the geometry of three-dimensional space. In fact, this is correct, and we can show that it's true. With this definition of parity, we can show that it's true rather easily. Note that pi dagger, pi here is unitary as part of our requirements, but since pi squared equals 1 is also a requirement, this implies that pi inverse is equal to pi, and since pi is unitary, it means it's also equal to pi dagger. So pi is actually not only unitary, it's also permission, as you see. So pi dagger here is actually the same thing as pi. I could drop the dagger if I wanted. Now let's take the weight function of psi of r, and let's apply these three operations by the position operator in pi and sequence and see what we get. So the first pi turns this into psi of minus r. Then we multiply by the position operator. They both put a hat on it and indicate that it's the operator, not the c-numbers. That corresponds to just multiplying the weight function by the corresponding c-numbers, that's to say, without the hat. So psi of minus r goes into r times psi of minus r. And then we apply pi again, which changes the sign of all the r's. So this goes into minus r psi plus r now. And you can see the overall effect is to multiply the original weight function by minus r, which is what this says on this side. So this is a simple derivation of this conjugation relation for the position operator. Now, by a very similar analysis, you can show if you conjugate the momentum operator with parity, and this is, again, speaking of a spinless particle of three dimensions, that this goes over into minus the momentum. And so the position of momentum operators are, as you see, are odd with parity. Excuse me, I realized that it was something I meant to say a minute ago, but I didn't say. So let me put this question in the transformation of operators on a pole and go back to what I meant to say. We dealt with a case of a particle that had no spin, and then we dealt with a case of a particle where we didn't care about the spatial degrees of freedom. We were only looking at the spin. So if we combine those two together to deal with a particle where we care about both spatial degrees of freedom and spin, then how should we define parity? For the logical definition, you can take the basis states, which are rm like this, basis cats on the system. And since this is a product of a spatial trying to spin basis state, and we know what they do here, you see by these two rules, the logical definition is that this should be equal to the basis in position r, but the limit doesn't change. And if we do this, then the weight function, seen as a 2s plus 1 component spinner is going to get mapped into size of m minus r. The parity changes the spatial coordinates, but it doesn't do anything to spin. It's a simple rule. Now, let me go back to examples of how operators transform under parity. Again, dealing with the spinless particle and three dimensions. And we see that the both position momentum operators change sign of the parity. This means that if you take the orbital angular momentum operator and conjugate it by parity, since it's r cross p, you see the two signs cancel and you get plus l. So l goes into plus itself under parity. Actually, this had to be true because one of our requirements is that pie should be new because l is the angular momentum of the system of this kind. So it's really just a special case of 3a here, which you see explicitly when it works out. So some vectors you see change sign under parity and some don't. All of these things r, p, and l are vectors insofar. They transform as vectors insofar as proper rotations are concerned, but when you throw improper rotations, parity, the signs like this are called ordinary vectors or true vectors and those which transform with minus sign are called pseudo-vectors. And angular momentum is an outstanding example of a pseudo-vector. Other examples are magnetic fields. The electric fields are true vectors and the magnetic fields are pseudo-vectors. This, of course, generalizes to any angular vector of J, like this. Now, in addition to classify vectors, we can also classify scalars. Let's suppose K is a scalar. I'll remind you that our interest in scalar operators is that Hamiltonians for isolated systems are invariant under proper rotations because energy can't depend on orientation. But there's a question about what happens to such Hamiltonians under parity due to spatial inversion. Well, in any case, if we have a scalar K and we conjugate it with parity, and suppose it goes into plus K, and this is what we call a true scalar. On the other hand, if we conjugate it with what's called a S, another scalar, but with parity it goes into minus itself and it's called a pseudo-scalar. So the operators which are scalars in so far as proper rotations are concerned can be further classified as true or pseudo depending on how they behave in a parity. Notice the sign of a true scalar given a plus sign is the opposite of what we call a true vector. You get a minus sign for vector. The signs, as far as the true and pseudo get reversed when you go from vectors to scalars. Yes, it's sort of natural that a vector should change sign in an invariance what we call a true vector. All right. Now, so Hamiltonians, of course, are the most interesting operators. Let's ask how Hamiltonians transform in a parity. And in particular, are they invariant in a parity in the same circumstances as they are in proper rotations? Is being isolated enough to be invariant in a parity? That's a good question. Well, if we take a central force Hamiltonian for scarters as an example, of a 2M plus dr, it's easy to see that this does communicate with parity. And the reason this is that p squared is, of course, the dot product of the momentum vector with itself. Both of those are odd in parity as you see there. So you take the dot product that goes into itself with a plus sign. It makes it a true scalar in this. So, if you can catch us this later, I'm not going to have a very good physical understanding of what means a transverse similar parity, Well, there's really two faces of it. One is in the level of classical geometry where you invert a vector through the origin. So you want to see what's happening on the other side. The invert vector through your heart is the center to use the origin. It swaps right and left. It's not an operation you can do on a rigid body, but not tearing all the atoms apart and moving them around. It's not like a proper rotation where you can only turn a book. You can't parity. You can't spatially invert a book without making a new book. It's written the other way around. But anyway, that's what it is at the level of spatial geometry. I think your question was can I clarify what this means in a quantum system? It might help in the case of the wave function here. Just to see what this says in the wave function. It just flips the position vector. So if I have a wave function like this, x, y, and z, let's say that it's concentrated here in a blob with a plus y axis. This is the state psi. In the state parity active of the psi is concentrated in a blob with a minus y axis. It's floating through your heart. Does that help? Yes, a mirror reflection is not the same as parity, but it's very closely related. A mirror reflection is also an improper rotation. The reason the mirror reflection is not the same as the parity is because it just reflects one plane through like this, where the parity reflects to the origin. However, if you have a plane like this, then the mirror reflection of the plane is the same thing as a total spatial inversion followed by an ordinary rotation by an angle of pi in the plane. So in other words, I can write this if I call, let's see, in the mirror operation. This is equal to parity times the rotation perpendicular to the plane by an angle of pi. This is a proper rotation. So the product of the two is a new proper rotation. The mirror inversion is a special example of proper rotation. You can see this because first you invert through the origin, that's what pi does and maybe I should call this p, or they indicate that this is at the level of the class or three-dimensional geometry. The p inverts all vectors through the origins of flips x, y, and z. Whereas if I then followed by a rotation of the plane at this sort of z-axis, what that would do is change the signs of x and y back again so they become pluses, but in these days of arguments, okay? So it's very closely related to parity, mirror reflections. All right. Thanks for these questions. I really know what I think. Okay, so yeah, so I was getting to the question of how do real Hamiltonians transform under parity, and in particular, is it true that isolated systems are invariant under parity? Well, if we have a central force Hamiltonian, it's clear that it isn't invariant under parity because the kinetic energy is p dot p. These are two true vectors. They change sign under parity, but when you take the dot product, the signs are through scalar, as you see. And similarly, the radius r is the square root of r squared, and r squared is the dot product of r dotted with r, both of which are odd under parity so that, again, the square is even. And so this entire Hamiltonian is even under parity. This means that all central force Hamiltonians commute under parity. This is just for a single particle interacting by means of scalar potential. If you generalize this with multi-particle systems that are variable, actually, as we know, this Hamiltonian that I just wrote down here is, let's say, a single particle system. As we know, they typically arise with two body interactions where we go to the center of mass frame and separate that out. And the requirement is that the potential of the function only at the distance between the particles. Then you get this with the center of mass or the relative motion. Now, if you go to a multi-particle system, you will see how long the potential is a function only at the distance between the particles. This is what you have if you have only electrostatic interactions between particles. Charged particles don't just interact by electrostatic interactions, but it's a lot of times a good approximation. Then it's used all the time as a model of molecular, condensed matter physics, for example. So all such Hamiltonians commute to parity. Now, what about if we include spin and null-fibistic effects? One of these we've seen already is where we've got some function of r and we're going to spin an orbit term, L, Y, S. What does that do to parity? Well, Bell and S are pseudo-vectors. They are even in parity. And so when we take a dot product, we get a scalar, which is also even in parity, and that's why it's a true scalar. And so the L, Y, S terms are spin orbit terms. And if they occur, in atomic physics, still can serve parity. What happens if we add more electromagnetic effects, such as null-fibistic effects, retardation, emission and absorption of radiation, and so on, does it still conserve parity? The answer is yes, except that when you're talking about emission and absorption of radiation, you have to take into account the parity of the totons as well. You can include that as part of the system. And if you do that, then you find that parity is conserved. This is part of the statement that the electromagnetic forces conserve parity. Now, what about the strong interactions which are responsible for holding the nuclei together? The interactions and strong forces between particles, like the proton, the neutron, the thymus, and others? Is there a question here? Hi, again, do you think the phrase conserve parity? Is that completely equivalent to having a true scalar Hamiltonian? Yes, it means that the community of parity. You know, what you call a conserved quantity, what you call a classical mechanics, would be one whose time is under the Hamiltonian. And in classical mechanics, that's equivalent to saying that it's plus on record that the Hamiltonian is zero. There's an analogous state, but in quantum mechanics, if you look, for example, the Heisenberg equation of motion, the time evolution is zero if it commutes from the Hamiltonian. Or, another point of view is expectation values, if you expect arbitrary states are constant at time if the operator commutes from the Hamiltonian. So normally you talk about a constant in motion, but some of the commutes from the Hamiltonian. This would be the case of parity and these examples that I'm talking about. All right. So in particular, the parity of an initial state would be the same as the parity of the final state. As it turns out, in strong interactions, parity is conserved there as well. This is a fact that was originally established on the basis of experimental data coming out in the 1930s and 1940s. And that it all observed strong interactions of parity in the initial state was equal to that of the final state. In order to make this work, you have to include the intrinsic parities and particles that are involved. It became evident that the pi meson has a negative parity. But what you do that, the rules work out. And so the result was is that for a long time, people thought that parity was a fundamental symmetry of nature and that all isolated Hamiltonians would conserve parity. Well then, in the 1950s, it was suggested by Lee and Yang that in the case of the weak interactions, that parity might be violated. And in fact, they proposed an experiment that involves spin polarization of the decay, of the electron decay and of the invader decay, electrons invader decay. And this experiment was quickly carried out and it turned out that indeed parity is violated in the weak interactions. In fact, it's massively violated. It has the worst possible parity violation that you could get given the matrix element you work with. So this is a notable case in which parity is violated and I say that the system is in the case of the weak interactions. It's just a property of the weak interactions. Now, under normal circumstances, the weak interactions are extremely small. I mean, they're incredibly amazingly small in ordinary low energy, you know, electron-multiple scale interactions that you find in atoms and molecules in that matter of physics. And so the result is that in all such systems that arise in a practical matter, in low energy, non-multiple systems of this form, you can assume from an extremely high degree of accuracy that parity is conserved. This is what one would normally do. In fact, parity violations are so small that they're extremely hard to see even if you're looking at them, which quite a few people in this department or years or so have been doing. Gene Cummins reported that he's tired of being the booker of both an adult and experiments of all the parity violations in atomic physics. In any case, these are really hard experiments. So anyway, that's part of the story of parity violations of the weak interactions. The weak interactions become relatively more important as the energy increases and so the higher energies the weak interactions play a role. All right. So that's some comments about weak interactions and places in which parity is conserved where it isn't. But even if we're not going into the weak interactions it's certainly easier to write down Hamiltonians if they don't conserve parity and the easiest way of doing it is if you don't make a Hamiltonian isolated you'll put it in an external field. Suppose, for example, I go back to my kinetic plus potential Hamiltonian except I'll make it a I'll throw out the scalar term because I don't care about that but let's make the potential of Q5 like this but let's say the potential is not rotationally invariant. There's a uniform electric field in the C direction that's how I'd e to e0z that so that phi the scalar potential is minus e0 times c and so this becomes Q5 times e0 times z the coordinate of the z direction and now you see if you conjugate this Hamiltonian with parity, the z-chain goes from plus c to minus c so the external electric field is broken of the parity conservation. Actually, you'll find that external magnetic fields don't break parity conservation but external electric fields do. This means that to detect parity violation, you have to be particularly aware of stray electric fields if you mess up the experiment. The definition of by the way, whether a system concerns parity or not depends on the definition of quote unquote the system if we include in the system here we're just thinking the external electric field is given but if we include in the quote unquote system the charges which are producing the field maybe there's plus and minus charges on a capacitor plate making a uniform electric field if we include those charges in the system then when we apply parity the two plates reverse and the plus and minus signs change so the electric field changes and that compensates for this and restores parity conservation again so you get parity violation if you don't include the external fields as part of the system, external electric fields. To go back to the system let me go back to the central force Hamiltonian because it's common in practice and it's an important case that we're dealing with a lot let's just go back to this it's under force Hamiltonian we know that there's a complete set of commuting observables here which is the Hamiltonian L squared and Lz and this is reflected in the energy guidance space usually by this NLM like this but now what we see is that there's another operator we can throw in the mix here which is parity and as you see it also commutes with L squared and Lz because parity commutes with egg and momentum all three components and so many function of it and thus it is possible to organize the energy eigenstates of any central force Hamiltonian so that they're also eigenstates of parity so the question is what's involved with that the easy way to answer that is just to look at what P pi does to one of these standard energy eigenstates in the central force Hamiltonian which is this too but this will be easier to analyze if you look at the wave function the wave function of R and L of R times the YLM with K and 5 like this there is a property of the YLMs that I didn't mention before but it is this is that if you take the YLM and listen to the development of the theory of YLMs that we developed several lectures ago which is given in the notes it's implicit in that development that if you take the YLM and multiply it by the radius of the health power which you get as a homogeneous polynomial of the coordinates X, Y and Z for example with L is 3 so you're dealing with cubic polynomials you'll find that this is some linear combinations for the K cell equals 3 it was some linear combination of polynomials that are like X cubed, X squared, Y X, Y, Z, etc all cubic polynomials in the sense when we conjugate the position vector under parity they change the sign it means that X, Y and Z all change sign and so such a polynomial is odd under parity likewise if L is even even polynomial so the result is goes into minus 1 to the L times the cell on the other hand the radial wave function depends on the radius which is a very linear parity and so the effect of the tunnel wave function is simply to bring out a factor of minus 1 to the L parity, the eigenstates of H also in LC that we normally use as it turns out are automatically also eigenstates of parity and the eigenvalue is minus 1 to the L this is a useful rule it gets applied quite often in simple force problems notice that the eigenvalue does not depend on the magnetic quantum number this is because the parity applies as a scalar operator that the eigenvalues can't depend on the L they do depend on the L it's minus 1 to the L so that's an important rule here's a few more considerations about parity frequently in practice if you do any research in analogue physics and chemistry or anything you end up doing a lot of numerical work it's all state physics well a lot of numerical work is diagonalizing Hamiltonians because there's no theory for doing it analytically so let me say make some remarks about what's involved in diagonalizing the Hamiltonian perhaps numerically if the Hamiltonian can use with parity it equally does parity has two eigenvalues the eigenvalues of parity which I'm calling E area plus and minus 1 parity is a remission operator and so this means that if E is the Hilbert space it means that the Hilbert space can be decomposed into a direct sum of two Hilbert spaces let me call them the odd and the even Hilbert spaces and the subspaces which are orthogonal to one another in which parity is minus 1 in the odd space and plus 1 in the even space these are the eigenspaces of parity now if we want to diagonalize the Hamiltonian the idea is you take the Hamiltonian you choose some convenient basis let's call the basis vectors N and M this is going to be a matrix you call H and M and then you just diagonalize this on a computer however the Hamiltonian can use with parity it's going to be wise not to choose just any old basis which is also an eigenbasis of parity the idea is this if we take the even and odd subspaces we diagonalize parity first and then we take the even and odd subspaces and we choose an orthonormal basis in E odd and another orthonormal basis in E even let's call these bases E prime again here where E is equal to plus or minus 1 tells you which of these two spaces you're in and then N just labels the basis vectors N and then you begin the subspace like this and so now what you want to do is a set of matrix that looks like this E in on one side H in the middle and then it's what's called E prime on the left side and then E in on the right side this is a matrix element in the Hamiltonian in this basis this is by the way called a symmetry adapted basis because it's a basis that reflects the symmetry of the Hamiltonian but you can easily show in fact that you're parity this is diagonal in the D prime times something else that depends on on everything else what this means is that we set up the matrix for the Hamiltonian and we go out like this and we break it up to the even block and the odd block or a big odd first odd and even like this odd and even what you find is is that because of this product or delta zero and you only get matrix elements in the Hamiltonian amongst itself connecting odd states or connecting even states the language what we say is often times say is that Hamiltonian does not connect states of opposite parity it's a way of saying that there's zeroes here let me show you how they impact what this has on numerics there are many algorithms for diagonalizing matrices but a common one which is called a household algorithm is in Q where n is the size of the matrix that's the amount of labor computer time involved but if you can block diagonalizing Hamiltonian by using a symmetry adaptive basis now you have two matrices there's twice as many matrices as before but they're only half as big so the labor now is n over 2 Q times 2 which is in Q over 4 and you see using the symmetry adaptive basis cuts the work done by a factor of 4 and so this is a simple example of the symmetry adaptive basis but it shows you some of the advantages of using it by the way if you're talking about ordinary rotations and Hamiltonian unions with rotations there's a symmetry adaptive basis for that too and that's what we've been calling the standard random momentum basis the idea here is you diagonalize j spring and j z first and then you worry about finishing the diagonalization and that's once again you get block diagonal structure with the matrices for the Hamiltonian and although there's now more matrices they're also much smaller and the total work is much reduced by using symmetry adaptive basis yes Q is not a little bit faster than you think because the E is even related to the minus 1 of the L that we were talking about it would depend on the Hamiltonian actually this wouldn't have to be a standard Hamiltonian you could have a 1D Hamiltonian for example but it may be out of potential and it's satisfied to be a minus s and then the the rule here is that if you're going to use a basis because you can't otherwise diagonalize a Hamiltonian use a basis in which the basis vectors and our eigenstakes of parity are even non ones I think we care for you automatically because the even ends are even and the odd ends are odd so if you had an even potential of a finite class of their basis it might be a good one you'd get this block diagonal structure if you did so when we break up the a s maybe how did no one break up that how do you know if you can break up that diagonal space which diagonal space how do I know I can break up the whole bit space like this because pi is a permission operator and it has two eigenvalues which are plus or minus one and any time you have a permission operator the eigen spaces are orthogonal spaces the entire Hilbert space breaks up into a orthogonal subspace in this case it was only two and if the Hilbert space is infinite dimensional which it usually is the eon and the even are both infinite dimensional and it's still in the true bit now the last thing I want to say about parity of all selections to give you an example of this the most important example let's talk about selection rules in electric dipole transitions in a hydrogen like atom as I've explained previously the relevant matrix elements then this is going to spin the particles so we just use a clearly modeled for the atom and we put the position back to the level we talked about this matrix element earlier and we're going to talk about rotations this is an example in which the bigger aircraft there can be applied the operator in the middle is k equals one you reduce the intensity of the operator and so the selection rules come from the bigger aircraft there and say that this matrix element is equal to zero unless the null prime the angular minimum on the left-hand side is an angular minimum that's reachable by combining null with one the one here represents the spin of the photon and providing L with one you can of course get L minus one or L or L plus one there's three choices like this so as far as the bigger aircraft there is concerned these are the restrictions on the angular momentum that allows the matrix element to be non-zero there's also one there's also a selection rule of the magnetic one which I'm not writing down but I'm going to focus on L now however at parity that we mixed if the system also commutes with parity which it will because it's a second force simultaneous then there's additional selection rules that are not obtained by the bigger aircraft but come from parity that works like this let's take this to use the fact that the position operator is odd under conjugation of my parity so n prime null prime m prime first pi dagger r pi nlm given that this operator in the middle is minus r this has to be the same thing as minus nlm prime and the left position vector r and nlm on the right so this just comes with the transformation properties of the position operator however we can level up the pi over here acting right with the pi dagger acting left so what we have here is minus 1 to the L plus L prime times the original matrix element so minus 1 to the L plus L prime equals minus 1 this has not vanished so this is by by parity by parity we see that this is equal to 0 or less basically the rule is L plus L prime minus 1 minus 1 is even that's the rule and this is actually the reason it's stated by saying which is equal to L prime minus L this even is another way to say it I can turn this plus sign into a minus sign because it's the same thing adding 2 plus L and that's an even number so the result here is just that to go back to the original selection rules to make it so that it is equal to 0 plus L is even excuse me I'm saying this wrong L plus L prime minus 1 is even so L has to be odd because of the war and so the matrix element is 0 unless L is odd and this comes from parity and as a result of this the minimal possibility where L equals L prime which would be allowed by proper rotations totally is actually excluded by parity so the net matrix element electric cycle transitions on the L prime member on the delta L plus people plus or minus 1 so for example in the case of hydrogen where we have the 1S ground state 2S first excited state the 2B 3S 3B like this the transitions that are allowed 3B to 2S as this is delta L it is minus 1 if we go 3S to 2P that's delta S equals plus 1 2B to 1S those are all allowed if this one 2S to 1S is not allowed by parity this is an interesting fact because it means that if the hydrogen finds itself in the 2S state it cannot decay at the ground state by the emission of the dipole electric dipole photon there are two mechanisms for that decay that they have much smaller matrix solvents so the transition is much smaller so even though the 2S and the 2P levels are both in N equals 2 and even though they both have the same energy their transition rate at the ground state is radically different it's a difference of about 10 to 10 to the 8th in decay rate so the 2B state in the 1S state very fast in 20 seconds 2S to 1S takes about 10 to the 2nd 100 million times longer alright this rule by the way that delta L is equal to plus or minus 1 is sometimes called the Quartz rule it was it was discovered experimentally in the early days of spectroscopy before the theoretical explanation became available that explanation was provided by Wigner and it's really just what I just showed you using conjugation by parity here it's a simple argument Wigner later on said that this was of all of his results this was the one he was most proud of explaining the Quartz rule on the basis of symmetry alright that's the end of time reversal excuse me of parity that's all I wanted to say about it and now I'd like to make at least to begin with time reversal I said it wrong excuse me I do want to make it beginning with time reversal I do want to make it beginning with time reversal maybe it was making a problem maybe not that how do you get it yeah okay alright I'm sparing you for so many months that's a little bit bad okay alright so time reversal so that takes care of parity and now I'm going to turn to another discrete symmetry which is important which is time reversal let me begin by saying time reversal is harder than parity it involves more formalism let me say something about time reversal time reversal in classical mechanics it's supposed to be a particle moving in three dimensional space under some forces and there's a trajectory that's given by the position vector as a function of time so it's moving through space like this as a function of time so you can think of this as a movie or you turn it on to watch the flow of it now if we run the movie backwards we get something that follows that goes to the same region of space that goes in the opposite direction that's r minus d we'll call this the time-reversed motion let's suppose that the initial motion is the solutions of Newton's laws and some force of electric field perhaps out here the question is is the time-reversed motion also a solution of Newton's laws in other words is it physically allowable there's a math thing here that takes us from the original motion of the time-reversed motion and the question is does it convert a physically allowable motion into another physically allowable motion the answer to that question depends on what the dynamics is it depends on the nature of the forces for example the purely electric forces so then the mass times the acceleration suppose it's a charged particle moving an electric field is given by the charge times the electric field the value of the position of the particle like this well if we change the sine of sine on time because the acceleration most second grid is on time the left-hand side goes into itself but the right-hand side and the right-hand side is also and so the result is is that in electric field the time-reversed motion actually is a physically allowed motion it can take place on the other hand it's a magnetic field then we begin it up the force equals mass times acceleration so the mass times acceleration is the force but that's the same thing now as q over c times the velocity v which all by the v-r-e-t cross into the magnetic field now if we now if we so let's say r of t satisfy this equation does r minus t satisfy the same equation the answer is no the left-hand side is 2 the second time the derivative so it goes into itself where t goes to the minus t but the velocity of it changes sine because it's the first derivative and so the left-hand side is invariant the right-hand side is not and the answer is not the difference you know if you just think of a charged particle in a uniform magnetic field the direction of the particle it goes in a circle the charged particle in a uniform magnetic field the direction of the motion is counter-clockwise if the particle is negative then it's clockwise if the particle is positive so the direction of the motion depends on the sine of the charge and not on the initial conditions so if you change the initial conditions of the velocity it doesn't change that right into the left in any case, the motion of the magnetic forces is not time-reversed for the classical effect I hear the thing with the bell so I guess I'll stop with that then it will I'll be on with time reversal