 Let's get the shoulder up here. So, yes, we have an exam coming up. And we will have a short review for about my office hours as well. So, again, the exam covers the materials through lecture notes number 7 that we completed yesterday or last lecture, sorry, Tuesday. And through the problems that start to have those solutions out, we're going to get this in as possible, all right? OK. So, after a month plus of establishing the foundations of quantum mechanics, what we're going to do now is try to put the mechanics back to quantum mechanics. We haven't really talked about mechanics, physics. We have this information theoretic foundation that we've established, but the most factor of the physical world, that's what we're going to do. That's what we're going to do for their main bear apart semester. So, I'm fond of this quote from Archon Paris, who's one of the great thinkers about the foundation of quantum mechanics. He was fond of saying, I think, roughly, quantum physics happens in the lab. It doesn't happen in the Hilbert space. So, we want to have a hook back to our space, our physical world. The Hilbert space is abstract, so there's a hook that connects these two together. And I think the best way to think about it is based on Noether's theorem. So, Emma Noether survived this theorem from a Bronte mechanics, but it follows, generally, all of physics. That if we want to think about space and time, we want to think about the symmetries of space and time. And associated to every one of those symmetries, the symmetries that are so-called continuous symmetries, that is to say, that you're described by the lead group, so we'll roughly explain what that means. Then, every one of those symmetries is associated with conserved physical quantities. We have conserved quantities in the world because there are symmetries. So, the reason that energy is conserved is to the degree to which the laws of physics are time-translationally inherent. So, time translation is associated with constant research of energy. And what Noether's theorem tells us is that those conserved quantities are the generators. So, that's what we want to define as well. Other symmetries are, we might have translational symmetry, stage translation, or the isotropy, the rotational symmetry of space that's associated with conservation of angular momentum. And that tells us that angular momentum is the generator of the group of variations. So, the hook between these space-time symmetries of the world and these physical quantities is through Wittler's theorem, which will derive next semester. Basically, what Wittler says is that all symmetries in quantum mechanics are unitary maps. So, there's our connection between the Hilbert space and the lab. It's through symmetries which have a space-time physical space interpretation and unitary maps which have a Hilbert space. There's our hook. And so, the mathematics, just the very basics of it, is that we're talking about symmetries. The set of symmetries form a group. And groups are sets that have a composition law. They have a composition law that's associative and it has identity elements and inverse elements. And we can represent the elements of the group as through unitary operators. So, for every element of the group, there's an associated unitary operator. And this is a representation of the group in that the composition law is just the matrix multiplication or the operator multiplication. Take two operators, multiply them together. That's the same thing as the operator that you would get if you did the group composition law. Yeah. It's an amazing, like, excuse me, question. But if you say you have time-translation symmetry, could that system not have space-translation symmetry? Yeah. I mean, we could have a circum-circum stance and, you know, we have walls here. And if we combine ourselves with these walls, this is not completely spatially symmetric. If I go over here, it's different than over here. Of course, what that means is the momentum is being taken up by the wall. So one symmetry can preclude any other symmetry? It cannot preclude, but they're independent symmetries. They don't have to have all of them. But the degree to which the overall universe is spatially the same as the laws of physics are the same in all parts of the universe. And there's no special direction of the universe, and the laws of physics are the same that all instant time is the degree to which these are concerned. OK. But as I was saying, those symmetries are the set of symmetries, the set of things that perform and say a time translation are a group. And we can, through Wittner's theorem, say that that group is represented by a set of unitary operators. And the fact that this is a representation says this. And that the identity element of the group is the identity operator. And the unitary associated with the inverse element of the group is the inverse of the operator. Which, of course, is unitary is the dagger. OK. And so what we're talking about now is time translation of that particular symmetry which is dynamics. OK. And so we have the group here, let's say, so I have time translation operation. And they are just a set of all times. OK. So this is every thing that translates by some amount of time is defined by the time by which I'm translating it. OK. So this set, of course, is a continuous set. It's the real line. This is what makes it a lead group. Is that the set of elements of the group forms a smooth surface. In this case, that surface is the one-dimensional one. OK. That's what makes it a lead group. It's a manifold, more precise. And the composition law here is just if I have two time translations, well, that's just I translate this addition to the composition law's addition. And the inverse, if I want to go backwards in time, I just have a minus sign. OK. And the identity element, 0, that's either. That means if I add no time to it, I haven't changed the translations at all. OK. So we're going to translate relative to some fixed time. So we're going to say if I want to translate in time, bless you, some time t0 to time t. So really, I have to say what my initial time is. OK. So my element of the group is represented by a unitary operator, which takes me from t0 to t2. So this is the time translation operator that translates me from that time t0 to the time t. All right. So this is my element of the group. And we also have this composition law, which I can write in the following way. Suppose I go from t0, if I want to say here. So here's a composition law, which says I translate from t0 to t and then from t to t1. That's the same thing as translating from t0 if t1, if t is between t1. All right. So now what's special about lead groups is that they're continuous. And because they're continuous, there is a sense of differential calculus. That's important here. Which means we can talk about infinitesimal translations for infinitesimal elements of the group. That's to say there's a notion of a near identity element. So what I mean by that is the following. So the identity element was if I have this, this is the identity element. I don't do anything, right? Translate it like nothing. But if I go to t0 plus some dt, some very small differential. Well, this is something which is equal to some anti-unitary operator that's a function that's proportional to dt. Why does that have to be anti-unitary? Well, the composition law would work if that's a linear operator. So that's for sure. It's got to be linear or otherwise the composition law won't work. But it's got to be a unitary operator. So that terms cancel out? So it's got to be the case that you dagger u is the identity, right? And what is that? Well, you dagger, if that becomes a dagger, an a dagger is minus a. Because this is anti... Excuse me. Anti-permission. So if this is anti-permission, and I take the dagger of it, I get minus a, right? And this is then 1 plus dt, okay? And that's equal to a cross-terms dash. And then I get a dt squared. But order dt squared is zero if this is a differential. So in order for this to be a near-identity unitary operator, it's 1 plus something anti-unitary. And this thing is the generator of the group. Because it allows me to move away from doing nothing. It generates the transformation. Now, if it's anti-unitary, we can always write it as i or minus i under permission operator. And in this case, since this is time, it has units. That thing I'm going to call omega. So the anti-permission operator here is minus i times a permission operator. It has to be. And as we discussed, then, last time, notice theorem is the generator of time transformation. So that means that the generator must be proportional, that her mission generator must be proportional to energy. So in some proportionality, constant times the energy operator. In the energy operator, we always call the Hamiltonian. And we argued last time that this is some constant that has to have the units of energy 1 over the constant has to have the units of 1 over energy times time. Because you, for this differential map, this constant alpha has the unit of 1 over energy times time. Because this is dimensions. So what we have thus is that the generator of time transformation is the Hamiltonian and we want that to show a small distance from c0. That's the identity minus i times the Hamiltonian dg. So we want to solve for this time translation operator. Right now we have it as a differential. Let me write down a differential equation. So let's say I seek this map. The map that takes me from c0 to some finite time t. That's what I want. I have this. I can now have a differential equation for that. The way I do that is the the way in which the differential equation which says how this changes in time just from calculus is I think a small delta t. I translated by a small amount what it was before I would get that and divide by delta t. That's the definition of the derivative. What is this? Well, for so this time when I translated by a small amount delta t according to the composition law is first I go from c0 to t and then I step a little delta t. And this is a different, this is a near unitary map. I'm sorry. Near identity map. Very close to identity. Stepping by a little delta t. So this is one minus i over h bar that Hamiltonian delta t. So let's plug that in over here. I plug that in. This is during this out. I plug this in and I factored it out. I get back and this plug in. And when we plug that in and we put those together, what we get is this is equal to minus i over h bar Hamiltonian times u. And this is what we call the time one form of the time dependent equation. The first time we've mentioned the the Hamiltonian is the generator through this differential equation. Often this is written with a partial with respect to time. It doesn't really matter here because there's no other variable. But when we get to wave mechanics and we introduce position and as well as for momentum there's always time and it's often we have to emphasize that with other partitions. So I'll just write back that. So what is the solution to this equation? Whether or not it depends on time. So right now let's say this is just an operative that doesn't depend explicitly on time. I have to come back with that. That means anyway. Because this is the generator of time translation so why not we have h dependent time? So h given h assumed independent time. What is the solution to that differential equation? Exponential. So the solution u is the exponential of y. Yeah? But what about g? We got t0 here though. So we have to have it and we have a differential equation. We always have to think about the initial condition. It would be t minus t0. It would be because at time t equals t0 this has to be h. So that's the solution. How do we know that's a solution? Well we could check how can we check that. One way to check that is to think about this as a way we discuss. Hold on a second. The way we discuss in the homework where we looked at the solution of an operator, we can look at this as a power series. And then just take the derivative of this. And when you take that derivative it works. Yeah, Stephen, you had a question? You answered my question. Yeah, okay. All right. I want to make one point here that every unitary operator is exponential of some unitary operator. I can write it equivalently as minus i times some k where this is Are you trying to give spoilers to your next lecture? What's that? You keep saying anti-unitary. It's because symmetries can be either unitary or anti-unitary. And that's just stuck in my head. We can show it one way. If I have an operator of this form and I take its adjoint what do I get? Either plus i k because k dagger is k. And that is v dagger. So these are my unitary operators and the things in the exponent are the generators. All right. So now it was alluded to the question what if h itself is time dependent? What would that mean anyway? Yeah, so let's first of all let's, I want to just say something about the physics part. Suppose the Hamiltonian is an explicit function of time. A little bit of a weird thing anyway, what does that mean? How could that possibly be? Yeah? Maybe the magnetic field changes in time? Right. It's a good example. When we have a Hamiltonian as a function of time, what that means say for example as he suggested the Hamiltonian is a functional of something like for example a magnetic field and that magnetic field itself is a function of time. Now this is a little bit weird and I want to explain why it says this is weird because what this is saying is that the Hamiltonian is a function of some variables which we're taking as classical variables. We're taking this as a classical field. We are not quantizing the magnetic field here. We're treating the magnetic field as a classical parameter which is explicitly a function of time. Now when you can do that and when that's fair is a very subtle and deep question. We're not going to get into it for a little while maybe not ever this year but we'll try. But for the moment we have when we control quantum systems we shine laser beams on them we shine microwaves we put beads on semiconductors and put currents and we treat those variables as classical fields and that's how we drive the system so in some sense it's an open quantum system but it's not open in the sense that we talked about last time where we treated the environment as quantum degrees of freedom. This is classical degrees of freedom so this evolution is still unitary especially if this is deterministic I mean if the magnetic feels a little bit noisy well then we lose some information but to the degree to which we know what this field is and we control it very well this is still a unitary evolution even though we are interacting with it from the outside there's a way in which we can very subtle points but we know about dates and time alright so given that with that said we just have to be some mathematics now we say my differential equation is that u dt is equal to minus i h bar the Hamiltonian which is now itself some time dependent operator the operator itself times u so now how are you going to solve this one and we can do that same power series again and try to put it in but I can't just make that concept let's do a little something suppose I have a classical function which is what's the solution to that differential equation can you solve it yeah exactly so I can I can say this is the f dt over f of t times g of t dt right I then integrate this from say t0 to t make this a t prime because I it's a dummy variable now and then this the left hand side here is the log of this integral and so then if I just then exponentiate that I get that f of t is f of t0 times the exponential so I could just say well let's just try that as the solution to this in this case that would say u at t and t0 is u with well that at time t0 that's the identity and then I have e to the minus i over h bar t0 to t dt prime the integral of the Hamiltonian that would seem to be the generalization of what we wrote over here as the simple case where h were time independent so let's check that in the same way let's write it in terms of the power series okay so as a power series this is equal to the sum over all the powers that's just expanding that in a power series but there is a real problem here because when I write let's look at for example let's write out the first few terms so this is equal to 1 minus i over h bar and then I have plus a half over h bar squared integral v0 dt okay so far but if I try to take this derivative I'm not just going to get the same thing one power down because there's a question of what order is this in the question is does this operator commute with this operator if they commute I can take the derivative it'll be perfectly fine but if they don't commute check for yourself I won't get back this I only get this if they commute with one another so this is only the solution the Hamiltonian commutes with itself at different times and this is why actually solving for the time evolution of quantum systems is hard and why we need Feynman diagrams to solve for you know functional theory whatever this is a very hard problem so we'll come back to that next semester when we talk about time-dependent perturbation theory this is only the solution if they commute with each other very good so that's the general solution but often instead of just solving for the whole time evolution operator at how the state itself evolves as a function of time Hamiltonian of the state how are we going to do that? well the state let's just consider a pure state or almost always for the rest of the semester talk about just pure states we have some state at some time t we can think about that as the propagation from time t0 to t that's what the time evolution operator does it takes you from time t0 to time t so now I want to look at how does this thing evolve as a function of time so to do that I'm going to solve for differential equation and that is equal to the time derivative that's the only thing that depends on t and this we think that the state is generated but I don't want to for the moment let us again just take the we'll consider time independent Hamiltonians pretty much for now so this is equal to over here minus i over h r the Hamiltonian the time evolution operator is plugging in what we have and this is so we have another form of the time independent equation it says that the way in which the state is changing as a function of time is given by the Hamiltonian not the time independent if we wanted to write the time evolution for an arbitrary state we'd be mixed according to Hamiltonian dynamics to do that there's lots of ways to do it anybody have any suggestions yep can you maybe try to take the adjoint of this equation? sure we could do that so let's just say for example this guy is a statistical mixture of states this is one way to do it for sure and there's the adjoint let's take the partial derivative of this so that's the partial derivative of that times that plus this times the partial derivative of that using the product and this is that and this is the adjoint and the adjoint is plus i now in terms of the termination operators it is equal to so this is equal to I can factor out the Hamiltonian this is the Hamiltonian i over h bar with a minus sign the Hamiltonian acting on the density of the derivative that's this first term just to take it out and then I can take the Hamiltonian out the other way and so what we have is that the partial derivative with respect to time of rho is equal to minus i over h bar the commutator of the Hamiltonian this is another form of the time dependent this is Hamiltonian evolution of the state but it could be initially mixed this preserves the purity preserves the adjoint right no this is not the Heisenberg equation of motion we are going to maybe you have seen that before in the Heisenberg equation we are going to talk about that next lecture if you have seen it this is the state the state evolves so now of course once we are wanting to solve the sugar equation very special states that play a very special role and those are the eigenstates of the Hamiltonian they we care about them because they are the thing that tell us all about dynamics so in dynamics special role of the eigenstates or eigenvectors so the Hamiltonian the Hermitian operator Hermitian operators have energy values or energy Hermitian operators have eigenvectors let's call them U this is the energy eigenvalue you know they are real numbers because this is a Hermitian operator and the eigenvalues of any Hermitian operator are real we don't know whether they are positive or negative this is not actually a positive operator and the set of possible eigenvalues can be anything of any rule number at this point unless we know something more about the future of the Hamiltonian and these states or this equation is known as the time independent now for many physicists and quantum chemists this is all about quantum mechanics the rest of it that we've been doing I never think about it is it going to solve the Schrodinger equation for some big molecule or some solid state material of course this is a piece of the story in quantum mechanics but it's not the whole story but it's an important part because once we have the energy eigenstates then in some sense we can derive everything about dynamics first of all let's say one thing about what's one thing that's special about the energy eigenstates suppose at time t equals 0 the state of the system is one of the energy eigenstates that's suppose that's true what is the state of the system at a later time well that's given by the time of new from now on I'm just going to take let me take t0 so here t0 is time t0 so according to the rule we translate the state in time according to our unitary translation and this is equal to e to the minus i over h bar and have a tonian times time so what is that the Hamiltonian these states are defined as eigenvectors of the Hamiltonian so when I act this on this what do I get h acting on u is the energy you have a power series how does it get initial to the energy you're exactly right this is the power series so this is equal to e to the minus i over h bar so here's a little aside over here if I have a function of an operator acting on that that is equal to that right? so you just have to replace the operator by a tiny value that's a rule keep in mind okay we said that the overall phase of the state is irrelevant right? because probability depends on it so this is irrelevant the overall phase the overall phase doesn't change any probability for any measurement outcome which means that the probability of any measurement we do is independent of time that's why these things are called stationary states you've heard that term before in other words if you are an energy eigenstake you are a stationary state nothing would ever change you would measure the same thing every time no matter what time I call it okay these were the Bohr Organs now suppose on the other hand I am in a superposition of energy eigenstates so suppose I had somehow managed to prepare that state at some time then I'm going to call t equals 0 then what is the state I'm related to time so what is that like this on this guy and what do I get and I get the phase right minus I in 1 t plus c2 is in line with I h2t over h bar is this a stationary state no if you are not degenerate if this energy is different from that energy then there is a relative phase I can factor out if I like this one phase I can do that if I choose to what we see here is that there is a relative phase and that can make all the difference in the world you can turn this state into a state that's at a later time but they are falling to this state if this became a minus sign okay so what this says is that dynamics depends the overall energy of the system doesn't it can be set wherever I like what the ground state energy actually is doesn't affect the dynamics what affects the dynamics is how different energies are relative to some initial fixed energy one other thing once I have the energy I get states there is an easy way for me to express the time evolution operator so I have my time evolution operator again let's set t0 to 0 and let's suppose we have this time independent Hamiltonian I can express this in the basis of energy eigenstates let's take those energies just for the moment to be discrete energies okay we'll get it back in a little bit so the set this form of basis for Hilbert space that is to say they form a resolution of the identity again I'll just for the moment take these to be a discrete set but this might be interval so now I want to express this in this basis what is a representation of you in this basis remember these are eigenstates of the Hamiltonian there are also ways you can get at this you can put the energy on both sides oops excuse me but because the Hamiltonian is diagonal in this basis right that's to say the Hamiltonian is diagonal in this basis with those eigenvalues any function of the Hamiltonian is also diagonal in that basis with the eigenvalues just given by those functions is that clear so this is a representation this thing is sometimes called the propagator because it propagates the system in time so people would like spontaneous measure this not this but it's a good question which is if I'm in an excited state of an atom which I think about as a energy eigenstate it's not a stationary state it decays to the ground state well what that's telling us the dynamics of the electron say the proton is not just described by that binding force of the electron to the proton in the hydrogen atom for example because if it were it would be a stationary state and the reason is that it's also coupled to other quantum degrees of freedom and those other quantum degrees of freedom are in a electromagnetic fashion and so if I had a stationary state of the vacuum and the atom it wouldn't do anything but it's not a stationary it's not an eigenstate of the coupling to the vacuum and it's because of that that it evolves and the fact that it does so irreversibly in that unitary is related to what we talked about last time when you had a large number of degrees of freedom and all that stuff of complication and sophistication very good so I want to finish with an example let's go back to the good work I've ever talked about this semester a stick we'll talk about something else so let's talk about a spin what has particle in a magnetic field and we'll take this to be a constant magnet so we've discussed this before the energy of the spin in the field is related to the interaction between the magnetic moment dipole moment and the magnetic field and that energy is minus magnetic moment not the magnetic field now in this case the magnetic moment is an operator we take the magnetic field to be classical there's my Hamiltonian and the magnetic dipole moment is proportional to the spin and that proportionality concept we typically call gamma this is what's called the gyro magnetic ratio no that's the g fact it's got this has got the word magnet on another stock so this is equal to the following so let's write this as minus so this is minus gamma b dot s and s is equal to s is equal to this is an energy if this is an energy and this is h bar what does this have to have to have the units of h bar something is energy h bar omega right h bar omega is energy so this is a frequency energy divided by h bar is frequency better than that and so we'll call it capital omega so my Hamiltonian here I can write as minus h bar omega over 2 dot s and let's take the magnetic field to be in the z direction we'll call the direction of the magnetic field the z direction that is tradition and so that tells me that the Hamiltonian is minus h bar omega over 2 sigma z so now I ask you what are the eigenvectors and eigenvalues of the Hamiltonian what are the stationary states and the energy levels of this Hamiltonian that's the eigenvalues and what are the eigenvectors yeah they're up and down the z axis so the eigenvectors the eigenvectors of the Hamiltonian are plus and minus along the z axis and the eigenvalues are minus plus so if I were to draw an energy level diagram of this system it says that spin up along the z is the ground state and spin down along the z is the excited state and the splitting between them is that this one goes down by h bar omega over 2 and this one goes up h bar omega over 2 this corresponds to the case where the spin or mu v are aligned and this corresponds to the case where mu and v are anti-aligned that's the excitement the system wants to align the spin for the magnetic moment now I made an explicit assumption just a little aside this is will at some point that gamma was positive a positive number right um if gamma is a negative number then it splits it's always the case that the magnetic moment when it's aligned with the magnetic field is the lower energy and when the magnetic moment is the higher energy but it's not always the case that the spin in the magnetic moment are in the same direction because if it's an electron they're in the opposite direction because of the charge because of the band-band fragment but you want to change the actual edge of the electron what would happen is spin down would be the lower energy because spin down would correspond to see in this case the spin and the magnetic moment are in the same direction if it's a bit of a collapse then gamma becomes important to know if it's a bit of a collapse then you need to know what gamma is if you have higher spins it depends that's a much more complicated question that's what the LFR higher spins I just wanted to get across the point that when we have to be careful about what this gyromagnetic ratio is when we look at real physical problems ok but with that aside this is correct ok so so I'm just going to hear say gamma is greater than 0 so now the question is let's look at the dynamics ok let's suppose at time t equals 0 the state of the system is spin up along x what is the state of the later time yeah well so how do you solve the initial value problem there's lots of ways to do it but the simplest way is to follow you decompose the initial state in terms of the eigenstates of the Hamiltonian that's the first thing you do so the first thing to do, step one decompose the initial state as a superposition of energy eigenstates so generally that involves putting in a complete set and doing all that but you can just read that off because we know that it's better than this what is this so this is a superposition of spin up along z and spin down along z ok and so initially the probability of spin up along z is equal to the probability of spin down along z is a half ok what is the state now at the later time we just did that so we just apply the propagator to the state so what is that well this guy picks up the energy of the plus eigenstate and this guy picks up the phase factor associated with the spin down state each one picks up that phase is here I mean given by those energy levels so we plug that in what is the probability to be plus z at some later time the same and because these are stationary states the probability to be in the eigenstates doesn't change the time the spectrum of the state relative to its the eigenvalues of the Hamiltonian are not changing the time but that doesn't mean nothing is changing the time what about relative to the x basis which doesn't commute with z sigma x and sigma z don't commute so at a later time something won't happen well there's lots of ways to do that what we want to know is let's express the state of the later time in basis up and down along x we could do that so I'm just going to I can put in a complete set of states or I can just remember that up along z is a superposition of up along x and down along x and spin down along z is a superposition but the other sign I just substituted it that's one way of doing it not too late to do it so this then is equal to e to the i plus e to the minus i over 2 spin up along x plus e to the i over 2 minus e to the minus i over 2 down along z I mean x which is sine and cosine what we have here thus is that with respect to the x basis we have the cosine over 2 spin up along x plus i and down along x so what is the probability to be spin up along x at some later time cosine squared excellent, good job and similarly for down it's sine squared or 1 minus cos omega over 2 plus and this is the minus for using some trigonometric ideas so if I were to plot this what I see is that as a function of time the probability as a function of time starts out with the probability to be up along x and it's the probability to be down along x it started out with x and then it becomes spin up I mean it comes from down along x where this should be all of the same amplitude and this is i over 2 it's in a linear superposition of the 2 it's not the parable lecture of both it is, it's exactly what's happening so what it says is let's just complete this discussion saying the following thing what is the expected value of say the spin along the x direction as a function of time 0? well let's know at any fixed time on that average over time this is equal to by definition the sum over the energy the spin eigenvalues times this is equal to hr over 2 times the probability to be spin up along x at time t minus hr over 2 is the probability to be spin down along x there's lots of ways to do this problem I mean this is this I could write it out more explicitly to this operator this matrix that's the expectation value this is 1 plus cos this is 1 minus cos so this is equal to cos maybe t the mean value of the spin is cos if you did the y we times hr over 2 and the y component if you did the same thing you would find sin so what's happening in this problem is exactly what z suggested here's the z this is the magnetic field in the z direction we started the spin along the x axis right so here this is the x axis this is the y axis we started the spin here at time t equal to 0 the expected value was here at later times it's here and it's possessive so far more perception that is exactly what you would see classically if you put a magnetic moment in a magnetic field you put it in a direction other than it it will possess around that if I put it like this it would possess like that like a gyroscope and the quantum evolution is exactly the same as far as the mean value goes it's not a classical spin that's in a certain direction because if I measure it I'd still have this I mean if I measure this bit I have to find it here and I have to finally find it there but the point of the matter is the mean value the expected value rotates now of course at all times there's a 50-50 chance of finding it here and here because all the states in this equator present that alright if you give up your homework I have you left here you can use the tail