 by a composition law that allows us to translate from one time to the next. And this is a lead group, it's a continuous group. It is a differential group and we can talk about the elements of the group that are near identity. And a small translation away from identity is therefore generated by the object that is the conserved quantity and that is energy. And that's defined by the Hamiltonian operator. And from that we thus arrived the differential equation that is sometimes known as a form of the time dependent order. Or alternatively we can look at this as an equation for the time evolution of the state. So for example if I have a pure state at time two zero then in a system that for which we gain nor lose any information we know that we have unitary evolution of the system and that unitary time evolution is defined by this equation we can then write a differential equation for the state evolving according to the Schrodinger equation. And the general solution we can write down. So the solution for the time evolution operator can be written down formally in terms of expedition and we allow ourselves for this Hamiltonian operator to be explicitly time dependent. If we have some time dependent parameters like a magnetic field that's changing in time then the general form we had is actually kind of complicated. But if it is the case that this operator commutes with itself at different times then we can write down the solution explicitly. And what we'll typically be dealing with unless we say otherwise is the case where the Hamiltonian is independent of time. It's a constant operator. And in which case the time evolution operator is just the exponentiation of the Hamiltonian times the time propagation. A particular interest as you know are the set of states that are the eigenstates of the Hamiltonian. And the solution to this equation or this equation itself is called the Schrodinger equation. Also, it's the time independent Schrodinger equation and it plays as you know a central role in quite a period of time. In fact, if you're a chemist this is all you really care about or a material scientist at the time you just solve it for the end of the notes. So if we have this solution then we can write down a simple representation of the time evolution operator in terms of the energy eigenvectors and eigenvalues because you know a function of an operator can always be written in the basis of the eigenvectors of that operator in which case this is diagonal operator with these eigenvalues, right? And if we can write that down easily then we have a nice way of writing down the time propagation, the solution to the initial value problem. So if I'm given the state at the initial time let's say t equals zero then we can find the state at any later time by just propagating it according to the time evolution operator and if I insert this form of the time evolution operator here then I solved it because the solution at some later time is just this superposition with these now amplitudes that are given by this. So if I know what the initial decomposition of the state is in terms of the energy eigenstates then the decomposition in a later time is just I put this phase factor on each of the probability amplitudes and resum. Okay? So I think to, of course, note about this is that the probability to space in the energy eigenstates e lambda at a later time if the system evolves according to this is what? It's the same at any time, right? Because this is the square. It doesn't mean that nothing's happening as a function of time. It would be true if there was just if the state itself were an energy eigenstate then there would just be one coefficient and that would be then the same state at all times up to an overall phase which is irrelevant and that's why they're called stationary states but if the state is initially at the initial time a superposition of energy eigenstates with different energy eigenvalues then the state will evolve as a function of time because the relative phases between the terms and the superposition are different at different times and it's the relative phase between phases states that determines what the state is, right? And we saw this in an example where we looked at a lot more per session where we imagine we have some magnetic moment in a magnetic field and in that case we wrote down the Hamiltonian minus mu dot b take this to be a constant magnetic field this is the magnetic moment, dipole moment and that moment is proportional to of course a particle with a total angle momentum that is its spin it's proportional to the so mu here is proportional to the spin and that proportionality concept we call the gyromagnetic ratio and this is the Marmor frequency it's called the Marmor frequency in classical dynamics so the spin in a magnetic field that's what you get this is the frequency by which a magnetic moment will process in a magnetic field this is an operator and in that case we could simply write this as if you like and I do like h bar over 2 dot sigma and if this is in the z direction if b is in the z direction and this is the Hamiltonian and then we could just simply if we had a situation where say at the initial time the spin is say pointing along the x direction then according to this we just wrote down this, that's the kind of evolved state with this Hamiltonian we plug that in and what we found when we did all that was this is cosine the Marmor frequency over 2 x minus i sine we get b over 2 spin down and so the probability to be spin up along back to the function of time what this is equal to is cosine squared square of this amplitude which is 1 minus cos or 1 plus cos omega over 2 and the probability to be spin down along back to the function of time is 1 minus or sine squared omega t over 2 which is 1 minus cos and so as we wanted as a function of time what we see is an oscillation and this is over pi over 2 where this is the probability to be spin up along the x as a function of time and this is the probability to be spin down along back so the spin is processing around the z axis the frequency of omega it starts out at time people 0 spin up along x later time it's been up along some other direction at a time this is upside down at a time omega t is pi over 2 it's a equal superposition when omega t is pi over 2 what is the state of pi over 2 this is cosine pi over 4 which is and this is sine over pi over 4 which is 1 over 2 so this is spin up x minus sine is actually equal to spin up along y so it's processing and this is an example of what we call coherent evolution of the quantum state it is coherent because we are maintaining the coherence of reposition it's not like a quantum jump that you learn about you know software physics where you say it's in this energy level and then it goes to that at your level and there's a quantum jump that's not described simply by the time dependent Schrodinger equation because that's not a reversible evolution this is completely reversible it's coherent it's a quantum jumps which are much more complicated than I understand alright any questions? alright then let's continue so the way that we're described the time evolution of the system by how the state evolves as a function of time is known as the Schrodinger picture the Schrodinger picture what we say is that the state evolves as a function of time observable goals are constant operators unless there is a explicit time dependent classical parameter that unless part we're going to ignore for the most part let's see okay now so there's this kind of asymmetry here the states evolve and the operators are constant now if we think about what the quantum theory does what we demand of the quantum theory is that we use quantum theory to predict the outcomes of measurements that's what it's supposed to do at least with some probability that's my word so the physical content and those measurement outcomes for example we have the boredom which for example we say the probability of finding a particular say eigenvalue of an observable is equal to say for a projected metric we could write this more generally it could be a p-o-v-n element and it says this could be a density matrix in which case we'd have a sum over all the states in my system to make sure but that would be it's generically of this form or we might have expectation values which are sums of this weighted by the eigenvalues say the expected value of some observable function of time is equal to this as you notice physical content that we are extracting from here isn't about the state itself but about matrix elements it's the matrix elements that have the physical quantity it's not the states alone okay so the physics depends on the matrix elements which means that if this is the thing that I want or this is the thing that I want I kind of have a choice of how I'm going to calculate it let's write this thing out let's say I'm looking to be expected value of this observable pure state I could write this in terms of this density matrix as well time evolution is plugging in for the time of all state so in principle I can counter think about this in a different way I can think about that there is a uniterally transformed operator or matrix that's evolved as a function of time this is my I can define this as a of t in which case it's perfectly equivalent to say that this expectation value as a function of time is equal to whatever the initial state was but this is this expectation value it doesn't matter what the t is no I just here I'm just emphasizing this is looking at the expectation value it's a some function of time so in this picture whatever we want to calculate the state is constant what's evolving is the operators and I can do exactly the same thing for probabilities the probability to find a certain outcome here would be equal to this and I can equally write it as this now to keep this notation straight what we often do is say that there's two pictures in this picture we call the Schrodinger picture in the Schrodinger picture I say the state also has a function of time and this is a constant so in order to denote that we'll put a little label which often we don't do once we know what picture we're working getting but I'll denote this with that superscript s which means the Schrodinger picture alternatively in this picture the state is constant with a function of time alright and so this is this picture is known as the Heisenberg picture in the Heisenberg picture we take the in the Heisenberg picture the state is constant and it's equal to whatever the Schrodinger picture state was at times equals zero okay and we take it to the same state all times the state is constant in contrast the observables evolve and again they are taking to agree at times equals zero so this operator at times equals zero is whatever the Schrodinger picture operator was so they agree at the initial time and then they propagate forward okay so we have two different ways of doing our time evolution when do we use one and when do we use the other that's the point well we could as I said this they're absolutely formally equivalent they have some pluses and minuses in terms of when you might consider looking at the Schrodinger picture in the Schrodinger picture one thing to notice is that given the state at all times you can calculate experiments like predict all experiments at time two meaning that if I have the state the total state then I can use this to calculate the probability of any measurement outcome because that probability is given by the appropriate point right whereas in the Heisenberg picture we must consider the observable above for each experiment so if I want to know the probability of measuring outcome A about observable A I have to find the evolution of this observable if I want to find the probability of measurement outcome B associated with the rule B I have to find that differential equation separately now that comes out of tradeoff because this is typically easier for a given observable and this is typically hard and that kind of makes sense because in some sense this contains all the information about all observables whereas this only contains the information about that particular observable so if you want the information about all possible measurements you can make out of all possible observables well then you want to solve the Schrodinger time dependent Schrodinger equation but if there is only some particular observables that you care about then typically you would use the Heisenberg picture in terms of solving the time evolution of course if the picture is equivalent if you have the whole time evolution operator if you have the propagator if you have the whole thing which we were able to do first in that particle well then you have both pictures all right because it's just related but typically you can't solve that what else can I say about this Schrodinger picture this is the Heisenberg picture in this I would say to cross with that dynamics is a skewer meaning if we have the state of the volume in time it's kind of hard to connect the particle if I want to think about this what's the equivalent classical trajectories that things would do is typically a skewer it's not easy to see in the Schrodinger picture where as we'll see the Heisenberg picture connection is exposed that if we want to consider multi-time correlations are not, whereas in this case they're simple what do I mean by that suppose I want to know what is the correlation between the spin at one time and the spin at a later time no in Heisenberg there's time dependence the time dependence is in the observables observables the thing is that you could say so I could look at what I mean by two-time correlation function and say what is the I have say two spins and I want to know what is the correlation between the spin at time one and the spin at time two I can calculate that easily in the Schrodinger picture because they're time dependent whereas what would that mean in the Schrodinger picture there's only one state and it has one time so we'll come to that all right so if we have the Heisenberg picture we want to the state is fixed and we have to find the observable at a later time and if we have the time evolution operator then we just do this right but generally we don't have the full time evolution operator and so as in the Schrodinger picture we want to write a differential equation and typically we would solve differential equations in evolution so what we are going to now are the Heisenberg equations of motion so what is that so what we said is that the Heisenberg picture operator as a function of time is given by the general is this so this is how I get from the Schrodinger picture to the Heisenberg picture operator at the initial time is the Schrodinger picture operator alright so let's see what is the differential equation that this thing satisfies this is equal to plus and we know how this evolves is general and that is equal to what and this is the aggregate which is 5 over h bar times 1 you dagger h h is from mission and when you take the dagger switch the order so the difference for inventing the adjoint is that what that means or if I look at that equation it doesn't make a difference I can write the adjoint so putting that all together what do we see now h and u commute right because u is a function of h so I can put it on the other side no problem so this is equal to i over h bar h u dagger t h Schrodinger minus i over h bar h u dagger h Schrodinger t right and this is a minor and so is this what did I do wrong I did something wrong this is on this side commute with a here absolutely but it doesn't necessarily it's a commutator indeed so what we find is that the Heisenberg equation of motion says that the time derivative of some observable in the Heisenberg picture is given by i over h bar plus i over h bar the Hamiltonian now I want to make one small point which uh is a subtle very subtle point you come back over here a very small type here it's something that looks like that equation but it's not that equation it's not in one important way it's got this minus sign this is a Schrodinger picture evolution it's not the operator it's the state the state is evolving okay because in this when I have a general mixed state the state is an operator okay that's a Schrodinger equation that's not the Heisenberg picture this is the Heisenberg picture and in this case rho is constant so in the Heisenberg picture the state is constant in the Schrodinger picture the state evolves the operators are constant in the Heisenberg picture the operators evolve and the state is constant the size we should note that this looks in classical mechanics we're going to talk about this in more detail in coming lectures let's say I have a Hamiltonian h which is a function of momentum and suppose I have some observable physical quantity which is some function of position of momentum it might be position it might be momentum it might be angular momentum it's some observable and I want to know how does this evolve as a function of time do you know how to get that if I have the Hamiltonian you've done that in classical mechanics there's something called Hamilton's equations of motion right so Hamilton's equations of motion that's why it's called Hamiltonian have a general form and that form could be written explicitly as this where this is known as the Watson bracket and it's equal to the partial of the first thing to x times the partial of the second thing to p in the other way around those are the Hamilton's equations of motion in a very generic form that you may not have seen if you get to this point in classical mechanics these are Hamilton's equations of motion now there is something that looks pretty darn similar between the classical Hamilton's equations and the quantum Heidenberg equations we will see in the way in which in some sense this is some appropriate limit of this but we see this parallel and Dirac noticed this first and said oh yeah we quantized there's some way in which the Watson bracket becomes the commutator but this looks very similar in classical dynamics okay believe this is true let's just do a quick example okay thank you so let's say a of x and p is position and I want to know how does the position change the function of time well this I don't mean to get it backwards greater change in this respect to time well the partial of h respect to p over f and the partial of h respect to x is 1 that's the velocity and the rate of change of momentum with respect to time is 0 because momentum is conserved generally this would be which is the gradient of potential and there's no potential integrating the Heidenberg equations of motion is you know done typically in a case by case basis but we can write down a formal solution which is informative and sometimes calculationally useful so let's formally integrate this so if I just look at the integral of that that says that the Heidenberg operator at time t is given by well there's an initial condition and then plus i over h bar the integral from 0 to t of that that's just writing down the integral I didn't really solve anything here because I have the time dependent operator on this side of the equation and on that so not really a solution yet but I can iterate now now I can plug in this solution again and I get a formal series here so now I plug in so what I did here is I integrated once wrote down the formal solution and now I plugged in for this at t prime and this at t prime is equal to this from the integral from 0 to t prime from the integral over t double prime yeah why isn't it integral from 0 to t for both of the integrals because I'm plugging in here t double prime right so when I plug in here I said this at time t prime I plugged in for that is equal to this at time 0 plus the integral to t prime because that's the integral and then integrate t double prime minus over h bar h the Heisenberg picture operator t double prime so I just plug that in I mean this is just changing this to t prime so this is what's called a time 6 so just real quick is that on the second time that's an e t prime yes that's an e t prime now this would be a way of getting a power series solution for short t but we can look at it as a general power series solution let's so if I look at this this thing is actually independent of t it's equal to t times that commutator and I'll just write it down that's what that equals because this is independent of time and then I have this term well this I can't do anything but I can plug this one in again it'll be a 0 a times people 0 which is the initial a shorting of picture a plus a commutator of a commutator right so this guy when you do that out it's going to be equal to one half t squared if you do that integral because you get t and then the integral from 0 to t is one half t squared times the commutator of h and h and a plus order t cubed terms so what we have not thus is a way of expressing the solution as a Dyson power series which says that the in the heithberg picture the solution at time t which is equal to this is given by the sum over powers one over n factorial s i over or plus i over h r to the power n a commutator of a where this is a shorthand for h and this is often a useful thing to use I'd say once in a while this is really a result from Lee group theory close I have I look at the following so remember if we thought about Lee this say B as a anti commission operator this is like a generator so this is like a u and u to happen right? so if I have this kind of thing and I do this I write each one of these as power series one together and I do all the commutators I get the following result this is equal to the sum over n one over h factorial a commutator of B with a if we have u dagger of t sum a ut and this is equal to e to the plus I h t over h r a u to the minus side h u over h r well this result is this where now B is equal to I times the Hamiltonian times time over h r and you plug that in and you get back okay I should say there's a related result that I want to just note here suppose I have two lead rebeloments e to the a and e to the b and I want to combine them together there's a composition line now this generally does not equal this if a and b don't commute then you can do that but if they don't commute you can't do that however what is true is that e to the a times e to the b equals e to the a plus e to the b plus yeah because they have plus higher order terms okay if the commutator we'll use this a lot when we talk about the harmonic oscillator if the commutator of this with either a or b is zero so if this thing is some operator that can reach with both of these guys well then you can factor it out because that e to the a plus b is equal to e to the a times e to the b times e to the minus the half the commutator of a and b if this is true this is a form of what's known as baker camel house let's do an example let's do the same example but now in the Heisenberg equation let's look at a longer perception look how convenient is this awesome black word for number one okay so that's what we're going to do we're going to look at this problem but we're going to look at it now in the Schrodinger equation I mean in the Heisenberg equation so we want to find the Heisenberg equations of motion for the operators so this is equal to so this is my Hamiltonian so my Hamiltonian is this or I could write it equivalently so I want to find what and this is something to keep in mind when we write it this way this is the Hamiltonian and this Hamiltonian is always constant in time because the Hamiltonian is with itself so it doesn't evolve so I can think about this as a Hamiltonian in the Schrodinger picture or in the Heisenberg picture it's the same bar right so now I want to find the equations of motion let's just emphasize this this operator everything in the Heisenberg picture so we're not going to write the little h's everywhere or in the Heisenberg picture which is the h bar the commutator of the Hamiltonian with Sz and what is that 0 this is Sz, Sz can be put this out right that's what we expect so Sz as a function of time is Sz at 0 which is the Schrodinger picture which was 0 in the basis and that's as we expect the z component of the spin is a constant of the motion because it commutes with the Hamiltonian okay what about the other components well let's look at the x component according to Heisenberg equation this is equal to plug it in minus i times the law more frequency over h bar the commutator Sz with Sz which is y so there's a it's an i times Sz plus or minus how do we know because it's a significant permutation so this is i times Sz equal to h bar so this is equal to omega and similarly Sz so that's those are the Heisenberg equations of motion and we can solve these simply by now if I take a second derivative well that's omega times the first derivative of that which is minus Sz and similarly the second derivative of Sz minus omega the first derivative of Sz which is minus omega squared so both of these x and y satisfy this differential equation do you recognize the differential equation that we do that's a simple harmonic oscillator ODE and we know the solution the solution is that the solution to this equation is well there's an initial condition for both the first what it was at times 0 and derivative the general solution and similarly for y so what this says then is that the x component at time t is equal to the x component at time 0 which we just call Sz and then the first derivative at time t equals 0 is equal to omega times Sz but that's Sz so this is plus Sz Sz at time t is equal to Sz minus this minus sign Sz the absolve behind the word now here's something that is often confusing these are the Sz and Sz operators as a function they're not constant operators so what are they what are those operators we can write a matrix representation of those so Sz as a function of time so in the basis let's say the standard basis of spin up and spin down along those boundaries what are these operators well let's write down the matrix representation what does Sz look like in this basis well it's h far over 2 times Sz and what does Sz look like well it looks like that that's the Sz operator so when Sz with no time dependence is the initial one which is the shorter picture one which is the one you know that's what it is times cosine and then we have Sz so that's equal to h far over 2 minus i in this basis it's not a constant matrix in fact when omega t is equal to pi over 2 what does this matrix look like if this what is Sz t equals pi over 2 we've just got to plug in here omega t is pi over 2 e to the minus i pi over 2 negative i in here it's plus i so what does this matrix look like this one is sigma y so sigma x at this later time became what sigma y this is what's confusing about the Heisenberg the matrix is not constant with a function of 5 but you have to just and that's exactly what you see right when omega t is pi over 2 cosine of pi over 2 is 0 and sine of pi over 2 is 1 sigma x at time t becomes sigma y which is Heisenberg's so with that said today this is used to calculate things for example if I wanted to know what is the mean value of the spin along say the x, y and z directions of the function of time and what is it well this is what it is at time people because that's your problem and this is equal to whatever the initial value was that's exactly what we saw in the Schrodinger picture though it wasn't as obvious what the mean values were because we calculated the probability amplitudes and then we have to translate those probability amplitudes into expectation values and the Heisenberg picture which is obvious what's going on the mean value of the spin is processing around the z axis with a lot more procession so it's exactly the picture we do before so this is the spin along the z direction sorry that's the magnetic field along the z direction we had some initial magnetic moment it's long more processes frequency omega around the z axis it's component along the z axis is constant and it's component in the x, y plane is the mean value is doing that so in some sense it looks exactly like the classical if you solve this classically for a magnetic moment it would do exactly the same thing the mean value in this case follows exactly the classical trajectory now that's not always true that's a very special thing about this kind of Hamiltonian which is the linear we will see but in this case here the mean value of the magnetic moment as a function of time exactly follows the classical shape we can just see that we will see the distinction under what kinds of Hamiltonians because the thing follows exactly the classical thing what does it mean to you when do the mean values function even though the mean value is following the classical trajectory it doesn't mean that this thing is classical if I think about this as a spin if I measure it it's not like I'm going to find that value if I measure one of these y-axis I'll find it this way or that way what's really processing here is the probability amplitudes and the predictions of what outcome I will see although we often think about it as if you're doing for an NMR a guy working somewhere you think that this thing is processing but it's not quite right it's not a classical vector that's doing that it's really telling you about what the probability of the outcome is how it's changing in the sense I have your