 interakciju. Now we remember how we wrote h, so h is equal to h0 plus v. Now h0 commutes with h0 and it simplifies this term. So we have that this is equal to e to the i h0t v e to the minus i h0t So we found the term evolution of the state in the interaction equation. So we have the term evolution. Now you see that it satisfied the same equation as before, so it is like a shodding equation, but now this operator here depends on the time, as you pointed out before. What happens now? Can we write formally the solution to this equation? You already know how to do it. I tell you. OK, first of all, let's write here, just to complete this scheme. So what is the solution? Interaction picture, we have that the state time involved is according to this equation. So we have the i time derivative of psi interaction picture so this e to the i h0t v e to the minus i h0t psi t interaction and the observables evolve according to this equation t is equal to o t interaction h0 h 3 standard pictures where we can study time evolution in quantum mechanics. OK, let's spend a few words on this, because maybe you are less familiar with the time evolution in the presence of Hamiltonian with time dependence. What is the solution to that equation? So you can, if you write psi t for example interaction picture as some time evolution operator applied to the standard initial time this is an i, this is an i interaction. OK. Then what you can prove is that you can write this time evolution operator as a sum that goes from zero to infinity can write a series this form, u v, u n that is my opti u opti u n opti where u 1 u 0 is just equal to the identity and u n of t is equal to minus i integral from zero to t of u n minus 1 opti this is a claim so you have this equation and now I am telling you that you can express always the time evolution this way how can you prove it you just take the derivative time derivative of this object and then realize when you take the time derivative of u n you find u n minus 1 every time the only exception is u 0 where you find 0 then if you then take all the all the derivative you realize that it satisfies this equation u i t is equal to that operator e to the i hzot v v e to the minus i hzot u so we found the the time evolution operator also for this case this is just okay because maybe you are less familiar with this so you know that you can try the solution also in this when there is a time dependence in the Hamiltonian the form of solution maybe you can find in books whatever you can find a compact rotation and we generally write this in the following form so we write that given a time evolution with a time dependent Hamiltonian we say that this solution the one that I wrote here and we have written this for as the time order operator of the exponential of minus i integral from 0 to t h of tau this is just a compact notation for this this is for a generic Hamiltonian here we consider this particular case but you can this is a general theorem you can prove it you can prove it it is not a very complicated theorem it is a lemma what is the meaning okay this is I don't want to do the details now let's just consider this is just a compact a compact notation to denote this one whatever I wrote below so it's not really important t is an operator that the operator is a super operator that orders the operator depending on the time so from the larger times to the shortest it's not really important just ui is the same okay this was just to present you different formalism to describe time evolution they are completely equivalent the problem is the same so we are interested in the expectation values of expectation values and we can use one of these or even others if you would like to for example you could prefer to consider some kind of strange interaction feature when you want to apply h0 to the state instead of the operator why not you can do the same and you describe your time evolution in a slightly different way with different equations okay so what about the general properties of time evolution in quantum mechanics just consider this so in particular let us assume now the Hamiltonian is independent of the time and okay and we are interested in the expectation values of some observable or maybe just okay let's consider just the state the time evolution of the state in the striding of pitch for example we have e to the minus i ht say 0 no, we can write the state in the basis that diagonalizes the Hamiltonian okay because we are aligned here the Hamiltonian to the state so it can be convenient here to insert some identity which is just the sum over all the state of the project on the other states of the Hamiltonian let's assume now that we have a discrete spectrum so we can do something like this so it is equal to the sum over all the other states of e to the minus i ent times the overlap between the state and the excited state corresponds to this energy say 0 and here you have I just inserted this identity this is 1, this is equal to 1 this is 1 in the space so what do we learn from this expression now let's assume that we have a finite number of states our our space is finite like for example spin chain with a fixed number of spins so this means that this sum is up to some capital N if you want to think about spin chains let's write 2 to the capital N so are there some general problems that we can infer just from this expression yes ok first of all we know this that if we just consider the contribution from a given idea state of this because the time appears only in this space so this means that for the single contribution we can immediately see that this contribution is periodic with a period which is equal to a period that period C which is 2 pi over N for given N because every time that you if you add this capital T to this term then you find that you are multiplying this space by e to the 2 pi i which is equal to 1 ok but here we have many many contributions but what happens instead if we for example the strange situation where all these excertation energies are in a rational ratio so let's assume that e to the N over em is a rational number rational number means it can be written that we belong to integers p and q where p and q are integer let's assume that we are in this condition then is there something that we can infer from this expression why quasi in this case it's really periodic it's periodic because if you now consider all this ratio and then you consider the you can always find the time which is proportional to the product of all the denominators here and these numbers after which you reconstruct the initial state so in this condition there is actually there are infinite in many times times t tilde such that psi of t tilde is exactly equal to psi 0 the system is periodic here for any if you let's assume that the entire spectrum of this property you pick two generic energies and you always find for the final spectrum and let's assume that we have this kind of strange symmetry in our system and then we find that this is periodic we already know this kind of system if you can see a single oscillator is it true that we have infinitely many I guess states but if you remove the one half omega the energy of the ground state and then you realize that the all the ratio of the excited states are rational numbers so for these states you see that there is always this time which is very large because you should consider that here is equal to the capital N states so these dominators can be very, very large and so there is a very, very large time after which you reconstruct the initial state clearly you cannot expect to have this kind of relation in general system also in the easy model so if you consider ratio of two energies you find something which is not rational nevertheless you can approximate real numbers with rational numbers so what happens is as long as the number of states is finite you can always say that the system is quasi periodic so the idea is that you can fix a distance between the initial state and your a distance and you can always find a time such that the initial state is close to one another within the error that you chose that you chose and this is called quasi periodicity here the condition we should have a finite generally a finite number of eigenstates the spec should be discrete then we can relax some of these conditions in particular cases for example for the harmonic state but generally if you have these two conditions enough to state that the state is quasi periodic so exactly periodic where you need to generate this here you can always consider shift you can shift all the energy by some closely if you want here the important want this one so as a traffic all the energy difference makes sense so we have to subtract the for example the energy of the ground state so we learn here that as long as we can see a finite number of degrees of freedom this system is quasi periodic and so maybe after this you could you could start thinking that we cannot describe for example statistical physics but just by starting the time evolution of your states because there is a big difference between what we see in statistical physics so that there is some time direction so there is the entropy for example increases so you have something the state time evolves it doesn't come back to the original state instead here I'm telling you that there is always this time where the state is exactly equal to the initial state how can you how can you go against this this paradox if you want and the way is to consider thermodynamic limit so you have to you have to break some hypothesis, some assumption here in order to have some time evolution that doesn't come back to the initial state if you want to see if you want to describe some some non-tria I'll say some some time evolution resembles our idea of statistical problems we'll see that in the thermodynamic limit all these kind of theorems don't hold anymore and we can have interesting, more interesting dynamics so this was just to give you an idea of time evolution and now let's come back to the first lecture for a moment about the quantum measurements ok so let's start think what happens when you perform measurements at different times so what I mean the idea is to prepare our initial state have a system in some initial state say zero ok, this state is pure for example and so we can describe the state for using this density matrix let's now assume that we we measure some observable now you remember what I told you that after measurement if I don't tell you the outcome then the density matrix becomes equal to the sum let's assume that we measure this observable which is spectral decomposition is this one so you have some eigenvalues and here you have a project on these on these states ok, so this is our observable so the density matrix is projecting from this operator to N, N you remember this so these are the projectors on the eigenspace so this is our measurement in particular outcome and the density matrix after the measurement is written in this form so you have the projectors on the left and on the right and so on so let's assume that we we take this measurement and then we consider time evolution under some Hamiltonian so if you consider time evolution of this density matrix for h independent of time I show you before what is the did I did I show you how the density matrix time evolves now I'm not sure maybe I didn't ok, maybe I forgot so, just a moment so we have seen before that the state time evolves in this way we consider now the density matrix the density matrix of time t is equal to psi t psi t which is equal to e to the minus i ht psi 0 psi 0 e to the i ht so this is how the density is time evolves if you are interested in the question I mean I satisfy by the density matrix of this one so h ok so it's the same equation satisfied by the operators in the same picture but with the opposite sign ok, this is just because I forgot I do this ok, so well, this assume that we are considering the density matrix of the entire system ok, so let's warn it let's now assume that we are interested in the time evolution of reduced density matrix in this way is it clear why it has a more complicated time evolution the time evolution here is simple because we are considering an isolated system when you consider a subsystem then the system is open ok, so you cannot expect to describe the time evolution in this simple way with such an equation and you can immediately see this if you want let's see this so, let's assume that the our Hamiltonian can be written as the sum of three terms ok, there is one terms which acts only on the on the subsystem a and an x like the identity on the rest of the system then we have another term that instead x like the identity on the subsystem and like some non-trivial operator on the subsystem b but then you can also have some terms which acts on both a and b so you can always write the Hamiltonian in this way for example if we consider just three spins let's consider two spins and with the Hamiltonian sigma 1x sigma 2x plus sigma 1z plus sigma 2z then this term x only on the first spin so belongs to this this term x non-trivial only on b so belongs to this but then you have this term a couple of the two spins so this is part of this you can always decompose the Hamiltonian in this way then ok we know the resolution of the total state which is this one so we we can take the trace of this equation over the subsystem b for example what happens when we take this trace so here the trace of the subsystem b is just the density of the subsystem a so we have the the high derivative respect to the time reduced density matrix now is equal to the trace over b of what? the commutator between h a plus h a commutator now when you consider this term here you have the commutator between h a and b and rob and then we have the trace over b does not act on b so you can move the operator outside the trace and so this here you find that is equal to h a comma rob of t and this is the first term then there is this term and now you have that h acts on b but there is a commutator between h b and rob now because you are tracing out this is equal to zero the trace of the you can convince yourself that is equal to zero the trace of the commutator so this does not give any contribution but you have also this term that we cannot simplify in any way so trace over b of h i b so what is important here you see that when you consider reduced density matrix you cannot write any question using just rob a you have also some other contribution so the term which is island on trail for reduced density matrix and this is related to the fact that the system the subsystem is not closed there is an interaction between a and b and this is the interaction I am talking about between the subsystem and the rest ok questions about this so as long as the system is isolating the amilton is concerned about simple equation simple equations when we consider this term let's see so the second term is written as the trace over b now the trace over b what is it is the sum trace over b of we are here b times rob minus hb this is what we have now trace over b is written as the sum over complete basis of states in the subsystem b so we have a sum over all the state of psi and b of the same here yes it's a but there is rob here rob t is here and then you have ia hb because that is a commutator so we write this minus hb for example we could just to see immediately result as a base this is an arbitrary basis ok we can choose a basis that diagonalizes the hb why not when you see that when you apply hb to this you find the energy but you find the same here so this term simplify now the exact expression is not important just know that it's not simple it's not as simple as in the in the isolated case so now let's let's go back to what we were doing here so the idea was to prepare the state the system in the state of psi not which is described by this density metric which is the projector on the state then we consider a particular observable generic observable and then here is to is to take a measurement of this observable at a time zero so now the density matrix is described by this after the measurement hb equals zero in exam hab equals zero is if also in the case another nice example I doubt if there are cases they are actually related to the particular state that you are considering so so you can write this kind of general equation for every state maybe there are particular cases in which you find we were here so we carry out this measurement this is our state after the measurement for example we are considering this chain computing for example the Zz here in position and we don't know the outcome just we have these apparatus that carry out this project measurements then we follow the time evolution so the density matrix at the time t is equal to e to the minus i ht and this operator evolution after the measurement then let's assume that we we want to measure another observable after this time so the new observable would be this was a1 let's say let's put some one this will be one this will be one one and then we measure the observable o2 and these are the other states of the other observable so here we measure o1 and here we measure o2 so the new density matrix we can even buy the sum where would this space m of m2 e to the i minus a i ht n1 sum of m m m ok, so here we measured o1 then we time evolved until the time t and then we measure another observable, o2 and we are computing density matrix after the second measurement ok so what we have to do is to project on the on the agis space of the second of the second operator and here e to the i ht m2 so just to give you idea of what we are doing so let's assume that this chain extends up to the I don't know some unknown planet and the first measurement actually was done by this alien here they did this measurement and then after a while we are here and we are measuring this observable here after a time t this o2 and this o1 ok, this is a measure of zero time the alien measures this observable and now I consider time evolution and after a time t we are here and we measure something and this is the density matrix so after the alien measures the observable then we got this we evolved for some time then we measure some observable and we obtain this how can you rewrite this this can also be written as e to the minus i ht times sum over m and n of e to the i ht e to the minus i ht and you have here n 2 then we have the same e to the i ht e to the i ht clear what I am doing all the steps at least so first measurement o1 then time evolution for a time t then second measurement of o2 and this is what we find ok this is kind of complicated no, rather complicated because it depends on the high states of the observable of the first observable the high states of the second observable with the time evolution but now we recognize here that this is the time evolution of an operator in the Isenberg picture right so what I mean is that if we consider the time evolution of the second operator in the Isenberg picture this is just given by sum over m of lambda n2 e to the i ht m definition of the Isenberg the time evolution of the operator in the Isenberg picture and we see this is exactly the same there are no eigenvalues because we don't know the outcome of the measurements especially because we don't know from by alien very far away from here we cannot communicate with aliens so there was just this measurement we don't know what the outcome and so this is the data in schematics so now let's assume this is now assumption assumption that this operator here commutes with o1 so what happens so what's the consequence what can you find you can find a common basis that diagonalizes both operators so this means that we can actually choose here which is the same for this and this other operator in other words if you want we can choose n 1 to be exactly equal to e to the ok let me why there is this something that I don't let's change this is correct and one like e to the I check these signs I don't want to check but they are correct and two we can do this we can impose this because this is just the basis that diagonalizes the operators that they commute we can actually identify these two eigenvectors why not but then what happens here this was an orthonormal basis this is an orthonormal so this means that this scalar product is either equal to 0 or 1 depending on whether n is equal to m or n is different from m so this means that finally your density matrix becomes i ht and you have sum over m of e to the i ht m because this is equal to m2 e to the i ht ht n2 e to the minus i ht e to the i ht so under this assumption this is a chronicle delta e to be equal to this e to the i ht n and helps in this this simplify what is this this is the density matrix after the measurement of o2 after the time t so if you measure to at the time t you find this density matrix no information anymore about the operator o1 you have lost everything so this means that unfortunately we cannot infer from our measurements whether aliens exist or not so you see that if these operators commute then independently of what I can see of the measure of the fact that you measure some observable o then the density matrix is the same after the second measurement if they commute commute without t questions? so you see that it is useful to consider this kind of quantity because well if there are situations when we can if there is some theory that tells us that two observables commute or almost commute then we can infer that the two that measurement the two quantities are completely independent you are unaffected by the measurements of the first observable if you wait for time t if this condition is satisfied so what do you so what I propose is to consider the models that we studied in the last week and see what happens just see what happens if we compute the commutator between in observables in a given position at a time zero like for example in the harmonic chain let's consider quantum quantum harmonic chain so we could compute something like the commutator between the position of the j-pions the displacement of the j-pions and for example if I compute this the time t and the momentum of the of of the held ion so what does it mean here so this means that we know that if these two commutes it means that if I measure the momentum of the held ion and then after time t I measure the position of the j-pado then I'm not affected by the first measurement if this is equal to zero so that is if you see, well just compute it try to see what we find and how can we compute this quantity in our system so let's let's write the definition of the termolution of x in the azimov picture so you have the x j j of t is equal to e to the i which t x this is the Hamiltonian of the harmonic oscillado and let's write this Hamiltonian in terms of the ladder operator it will be defined here so we have that this is equal to the exponential of i then we have a term like p squared divided by 2 mn plus sum over k of epsilon of k a dot k in k clearly the constant is irrelevant because it's simplified as we should so we have this times x j and then the same with the minus sign now the idea is to rewrite x in terms of the of this ladder operator you can create phonons or the strat phonons and we compute the expression last time so we wrote that let's write it again that everybody is we have last week we found this expression x j is equal to the sum from 1 to capital N minus 1 of e to the minus 2 pi n j over capital N 1 over square root of 2 m capital N omega sign of pi n over capital N i a capital N minus n minus a dot and now we have to compute the scan operator the time evolution of this operator so we have to consider term by term no in definition of x there is capital X otherwise this is not the right operator so this is the the position of the central mass we consider when we consider the correlation we subtract it but therefore the variable position variable so we have this and now we are interested in the time evolution how do we do it we use the algebra just the algebra of the operator as always so we remember this commutation relations ak a dot q is equal to delta kq ak aq and we have that x b is equal to i and we have that all the other commutation are equal to 0 so we will just use this so let's consider for example this contribution from capital X that we are here so the time evolution of x in the Isenberg pitch x of t in the Isenberg pitch so this means let's put this operator here ak a commut with this with this x written there this ak a commut with p so this means that all this there can be move the right of the operator and simplify so you end up just with this part e to the i d squared over 2m capital N e to the minus i d squared over 2m capital N x and p are complete the couple from the rest operator so you can just consider time evolution under the free time evolution how do you compute this any idea? and how do you see there is a commutator here why? it's correct just asking ok and so you are suggesting to use the particular identity and she is the following so if you have to consider the exponential e to the i h t is generic 0 e to the minus i ht how can you write this? just expanding the exponential what you find is that this is equal to o plus i h o commutator plus i i h plus i h and so on so you see this is like the expansion of the exponential so you have one over in factorial and here you have to compute n commutator in the Hamiltonian operators yes? everywhere this is correct I guess ah the time ok yes we can use some some notation how can you write this if you understand what I mean so you write the expansion of this commutator just that some notation to indicate this kind of series so we can do the same p we know the commutator between p and x so what we find ok the first commutator you commutator between i v square over 2 mn and x we have the constant i over 2 mn times the commutator between p and x using the trick that I told you last week times the derivative of p so you have times 2 p which is equal to one over m capital N times x this is the first term of this series then we should consider the other term when we have i v square over 2 mn oh sorry there is p here no p then we have to consider the other term so the commutator between this again and this term and this commutator here but now this commutator is proportional to p and p is equal to zero so we find zero and then if we consider all the other all the other elements of the series expansion are equal to zero because of this so in other words we can just we can we can tronkate the series expansion up to just the first order we have all the terms so this means that x which is the zero order when we expand the exponential at the zero order time one and then you have plus what we wrote there p, oh there is a time here sorry, I forgot the time plus p t over m n are we surprised about this result? we are just saying that a free particle moves according to shouldn't be so surprised about this result so but it's ok, we are playing this kind of trivial result using quantum mechanics it should be happy ok so we see that the term of this term is simple just a free particle moving with momentum p over capital N and then what about the other terms we have to compute the time evolution of this of this operator but it should be simple as well because we otherwise we didn't introduce this operator it wasn't simple the Hamiltonian is diagonal in this basis so we have to compute the exponential of i epsilon k sum over k of epsilon k d sorry a a dot k a k t then there is f the generic a n for example that and then here we have the exponential of minus i t sum over k epsilon k a dot k ak ok, I drop this part because p squared commutes with a so as before it simplifies we have to compute this again we have to write the commutator between this term and the a dot n what is the commutator? so we have to compute the commutator between i t sum over k of epsilon k a dot k ak and a dot n this is the first term first of all all these terms but k equal n commute with a dot n so this is equal to i t epsilon of n and then we have here a commutator between a dot n a n and a dot n what is this? this is equal to i t epsilon of n a dot n commutator between a n and a dot n which is equal to i t epsilon of n a dot n here what I did here? I used that a, b and c this is equal to a, b, c plus a, c, b I used this event so this is what we found ok, the next step would be to compute again the commutator between this and the result but now you see that the result is proportional to the original one so in other words if I call this w we show that w with a dot n is equal to i t epsilon n a dot n so this means now if we compute the commutator between w and this we find i t epsilon squared applied to a dot n and so on every commutator will be applied we just have to multiply this number by i t epsilon so in the end what we find is that e to d I call this w so e to the w n e to the minus w is equal to e to the i i t epsilon n a dot n so the time evolution of this in this ladder operator is very simple they just time evolve with a phase now if you if you consider the conjugate of this you also find the time evolution of a n and what we find is that e to the w a n e to the minus w is equal to e to the minus i t epsilon a dot n so now we have all the ingredients to compute the time evolution of of x and I can raise it therefore the right one so what we find is that e to the in the isomer picture is equal to we rewrite this let's start here and now we replace the operators with a time evolving with a time evolution in the isomer picture so we replace x by this and we replace this a by themselves times the phases so x to the t is equal to sum 1 e to the minus 2 pi j n over capital N 1 over square root of 2n not 2m omega omega sine pi times i then we have e to the minus i t then we should consider the energy of capital N minus n which is equal to sine of pi n over capital N minus n minus e to the i t sine of pi n over capital N x las ok, the structure of this of this equation is general so every time that you have some linear combination of the ladder operator and all the fermionic operator in the asian chain and the Hamiltonian's quadratic in these operators then you will find that the time evolution of these operators is still a linear combination of the original Pozonic or fermionic operators like in this case in other words the time evolution under quadratic Hamiltonian preserves the number of particles so if you have here two particles, two a together with two a's it doesn't increase the number of particles this is how you see that the time evolution is not interacting because it doesn't increase the number of particles you are not allowing decay decay or creation of particles ok, so we found this expression and well, let's also write the expression for pL the momentum this is just the expression that we computed last time which is equal which is given by it was equal to sum plus a capital N minus N plus and this was what we computed last time so we have both the operator at the time t and the momentum and zero and what we want now what to do is to compute the commutator between x and p I give you let me proset that I didn't give you why so we started half past two so it's a quarter past no, it's no, it started from half past two oh, it's not two hours ok it's a dense solution ok yes ok now if you compute this commutator it's kind of simple now because you know how to compute the commutator between these operators and the other ones you know that the only term different from zero is when the indices are the same and one is a creation operator and the other is the distraction operator so what do you find if, ok I assume that there are no typos here so what I found is that xj in the Isenberg pitch at the time t commutator pL isenberg pL is equal to I over capital N sum over N from zero to capital N minus one cosine at two pi N j minus L divided by capital N cosine of omega sine of pi N over capital N t and now if you take the limit N goes to infinity in this expression you find that this is equal to this approach is I times the best function to j minus L here we have no idea what is the best function over this it's just a special function but I I just plot the behavior the approximate behavior of this function if I remember it ok, I hope so so it's something like this now I plot j to N of omega t as a function of omega t I'm not sure ok, this is zero start from zero it's very close to zero and it becomes larger and this behaves like maybe I wrote it here let me check yes, this behaves like like omega t over the behavior here is like omega t over 2 to the 2N 1 over 2N factorial so you see that this is extremely small for large N and moreover this is a very large power of t it's almost flat you see that it becomes different from zero when it is close to close to N I guess and then it starts like this and it approaches zero like 1 over square root of omega t what do we learn from this so we learn that there is if you consider small times short times we respect the distance the two operators practically the two operators commute yeah, commute as long as they the time is sufficiently sufficiently small then when the time becomes of the order of the distance then where the commuter becomes different from zero so you start feeling the two operators in other words there is what we call a light cone behavior light cone and it means that we and I assume this is our chain and this is the time we started measuring in this case for example the operator pL here and then at the given time we measure this time we measure the operator xj this is j and this is l that if the time is smaller than the distance then you don't see the two observable commutes so you don't see the effect of the app so from here you can when you consider the time evolution you can actually draw a kind of light cone starting from the operator and you can be sure that if you are outside the light cone this operator commutes with this if you are outside the light cone likely relative what is surprising here is that we are considering non-relativistik models and ok we find this for the harmonic oscillator and tomorrow we will see that we find the same behavior for the quantumism model and then I will tell you that this is actually general in at least as long as we consider spin chains or what this is kind of general this is well I am using here only the thermal and the thermal limit I don't need it why large time you will have this also at large time but also at short time but then ok this means that your time is very very large so it decreases as a power law instead here you have a power law moreover you have a dependence from distance it was like 1 over n factorial faster than exponential it goes to 0 faster than exponential with a distance this slide this slide this slide comes from the bosons that diagonalize the model which have a finite velocity maximum velocity and the information between one point and the other is carried by these particles that move but if you are outside the light it means that no particle has reached that point and so you I mean if I want to make the slope of this hole if you want to make the slope of this hole yes the ratio of the over and what you have to do is to see the velocity of the excitations which is the derivative of epsilon with respect to the momentum and you consider the maximum value so this is the fastest particle because you cannot well if you are outside the space that can be reached by the fastest particle means that the correlation should be very very small if you are not able to this is your theory Thank you