 discuss the Gelman and Lowe theorem, which I mentioned in the previous class. And essentially, let me quickly remind you what we are really dealing with, because our focus is on the treatment of the electron correlations, which we are very concerned about. We know that the Hartree Fock is not able to deal with these electron correlations. The Hartree Fock certainly took account of the statistical exchange correlations, but not the Coulomb correlations. So, we introduced quantum mechanics in various pictures. The Schrodinger picture is the one that we most often use in undergraduate and some other introductory graduate level courses. But then there are other pictures which are very useful in doing quantum theory of many particle systems, in particular the Heisenberg and the Dirac picture. So, this is what we have in the Schrodinger picture. And then what we did is to introduce the difficult term. This is what I have been calling as the unfriendly term. This is what makes the problem so cumbersome, the H 1. And you have the exact solution for H 0, which is the unperturbed Hamiltonian. So, this is solvable. This solution is known. We have the solution with us, but then we do not have the solution when we include the correlations and these are sitting in this term H 1. So, what we did was to insert a mathematical parameter. Now, this is the control parameter that is at the disposal of mathematicians. And it is not that there is anything in nature which lets us choose the presence or absence of correlations. But a physicist uses his tools, mathematical tools and we can choose appropriate values of alpha or t, which are useful to us. So, that we can develop a methodology which can be exploited to get the solution for the full Hamiltonian, which of course is H 0 plus H 1. The full Hamiltonian is H 0 plus H 1. H 1 is the difficult part of the Hamiltonian, the unfriendly part of the Hamiltonian, which makes it impossible to use a perturbative treatment. Like in the many electron system, the first order perturbation theory gives you the same result as the Hartree-Farck, but second order perturbation theory and higher order perturbation theory does not converge. So, this is the unfriendly part and we sort of scale this unfriendly part by this term e to the alpha t and the choice of the parameter alpha, when alpha is equal to 0, you have e to the power 0, which is 1 and you have got the full Hamiltonian. When t is equal to 0, which is the instant of time, you can say that it is the now moment for us. This is when we really want to deal with the full Hamiltonian. So, at t equal to 0, again you have the full Hamiltonian, but then if you look at the system evolution from t equal to or t going to minus infinity like this is the position vector r and this is the time which goes to minus infinity. Then as t goes to minus infinity, this e to the alpha t times H 1 goes to 0 and you have only the unperturbed Hamiltonian and that is the part of the problem for which we do have a solution and our effort is going to be always from the known to the unknown and what we do know is the solution to the at 0 and we have to figure out what is the solution to the full Hamiltonian, which is H. So, in the Heisenberg picture, we carry out these transformations. So, these are the transformation of the operators and this is how the Heisenberg state transforms, you get it from the Schrodinger picture wave function and the Heisenberg picture wave function of course, does not depend on t. All the time dependence is packed in the operators, the state functions themselves are independent of time in the Heisenberg picture. In the Dirac picture, which is also called as the interaction picture at times, here the transformation is again through a unitary operator, but whereas the unitary operator over here was the full Hamiltonian, in the Dirac picture it is only the unperturbed Hamiltonian. So, it is not the same, but it is similar, but not the same. Here the transformation is through the unperturbed Hamiltonian and the wave functions also transform like this and the result of this is that both the wave functions and the operators in the Dirac picture depend on time. So, at t equal to 0, which is the now moment as I call it, so at t equal to 0, all the operators correspond exactly equal to each other and so do the wave functions at t equal to 0. They all correspond to each other at this now moment, but otherwise at all other instance of time they are different. So, these are the primary relations that we work with and here notice that if this unfriendly part was simply not there, if the h 1 was 0, if the correlation part was 0, just in case wishful thinking, if it was 0, if h 1 was 0, then h would be at 0 and this is how the Schrodinger picture wave function would evolve with time from time t equal to 0 and the interaction picture wave function would be the same as the Schrodinger picture wave function at t equal to 0. So, that is a particular case and what it really does is that just in case h 1 were 0, which we know it is not, but in case if it were 0, then this wave function would become independent of time in the interaction picture. So, these are some special cases in which the Heisenberg picture and the interaction picture or the Dirac picture they correspond to each other. So, these are special situations. Now, what happens when you go to minus infinity as t goes to minus infinity, the Hamiltonian which is the full Hamiltonian, this t going to minus infinity kills any effect of this h 1 in this product term and then you have only the unperturbed Hamiltonian and the wave function at t equal to minus infinity in the interaction picture is nothing, but the solution to the Schrodinger equation in the Schrodinger picture corresponding to the unperturbed Hamiltonian and that is the known thing, it is this known from which we want to go to the unknown. So, phi 0 is known to us. Now, the interaction picture wave function at t equal to 0 is what I represent by this psi 0. So, this 0 subscript really corresponds to this time t equal to 0 and we are looking our interest is in the interaction picture. So, that is the one that we are going to be using in our analysis. So, this state can be obtained from the state as t goes to minus infinity by operating on this by the time evolution operator this is u. So, this is the unitary operator which tells you how the state would evolve from t at minus infinity and gives you the state at t equal to 0. So, the unitary operator this is the time evolution operator it actually tells you that if you know the wave function at t equal to minus infinity or t tending to minus infinity then how do you get the wave function at t equal to 0. So, this is the unitary time evolution operator and this one is nothing, but the Eigen state of the unperturbed Hamiltonian phi 0. So, if you know phi 0 and then you know this unitary operator then you can get the interaction picture state. So, that is the strategy that we are going to adopt. So, this is what we it really boils down to we will get the interaction picture solution state vector at t equal to 0 from the Eigen state of the unperturbed Hamiltonian through the time evolution operator u by fixing these two parameters for time which is t equal to minus infinity which is the start time and t equal to 0 which is the n time over which this evolution is observed. You can write it as vectors in the Hilbert space or a state functions. So, any which way so it is just a matter of notation. So, this is what we have got and there are these two limits which are of importance to us as t goes to minus infinity we know that we have the unperturbed problem. The alpha tending to 0 alpha going to 0 is when you will have the full Hamiltonian that that is the real problem of interest to us that is precisely the problem that we want to solve t going to minus infinity we have discussed how we get the solutions right because that is just the unperturbed part of the Schrodinger equation. So, that we know how to handle and the question is what is going to happen when you take the limit alpha going to 0 and this answer to this question what happens in the limit alpha going to 0 is provided by the Gelman and Loth theorem of quantum theory. So, this is a very fascinating theorem let me state what the theorem is and you can see the entire discussion in Fetter and Wallacher's book. So, you have got the unperturbed part of the Schrodinger equation this is solvable then you have the full Hamiltonian which reduces to the unperturbed Hamiltonian in the limit t going to minus infinity and we are addressing the question how do we get the Eigen state of the full Hamiltonian from phi 0. So, that is the question we are really interested in now we do know that the Eigen state in the interaction picture at t equal to 0 can be obtained from the unperturbed solution through the time evolution operator this is the one which of course, has the correlation setting in. So, that is how it is going to pick up the correlations and you will see it explicitly in the slides that follow. So, this is the statement of the theorem the theorem states that if the limit alpha going to 0 of this ratio exists this is the limit which is written as psi 0 over the projection of psi 0 on phi 0. So, this is the inner product. So, if this limit exists then the theorem states that it is an Eigen state of the full Hamiltonian and you can write it as an Eigen value equation H operating on this state is equal to E operating on that state. You essentially get an Eigen value equation for the full Hamiltonian provided the limit of that ratio exists. Now mind you you are interested in the limit of the ratio and you have got u alpha in the numerator you also have it in the denominator you have essentially the average value of the time evolution operator in the unperturbed ground state phi 0 that is what you have in the denominator and both the numerator and the denominator have the parameter alpha which is our mathematical construct and one could in principle ask what is the limit of the numerator in the limit alpha going to 0. One could also ask the question what is the limit of the denominator in the limit alpha going to 0. Now it turns out and I will not have a chance to go through this in too many details I am not going to work out the detailed proof of the Gelman and Lowe theorem. I will refer you to just two or three pages from Fetcher and Wallacher's book the references provided here and it turns out that the limits of the numerator and the denominator if you take them separately you do get certain divergences and these limits do not even exist in the sense that they are not very well defined. But it does not matter because if whatever factor is unknown when you seek the limit for the numerator cancels the corresponding factor in the denominator exactly then the limit of this ratio is very well defined and that is the one that we are really concerned with because we are not using the limits of the numerator or the limit of the denominator separately. We are interested in limit of the ratio and that is well defined and this is the some of these things go into the details of the proof in the Gelman and Lowe theorem and essentially what this theorem tells you is that if this limit is defined and this is the limit the limiting value of what is in this box is written by this ratio here this turns out to be an eigenstate of the full Hamiltonian and this difference between E this is the eigenvalue of the full Hamiltonian E0 is the eigenvalue of the unperturbed part of the Hamiltonian the difference between them is the correlation energy which the unperturbed problem is not able to address. So, that is the one that we are trying to find out. So, the contention of this theorem is that this particular eigenstate which is which we met in the right hand side of the Gelman and Lowe theorem this is the same vector. So, this particular vector which is obtained in this limit it gives you an eigenstate and this eigenstate develops adiabatically from the eigenstate of the unperturbed Hamiltonian adiabatically in the sense that you are not adding any external field any perturbation external field and in that sense this is an adiabatic evolution of the state which is an eigenstate of the full Hamiltonian inclusive of the correlation from the solution that you have for the unperturbed Hamiltonian. So, this is our question what happens in the limit alpha going to 0. So, we are ok with this part and we are concerned with the limit alpha going to 0. So, mind you that as alpha goes to 0 you get the full Hamiltonian, but as t goes to minus infinity you get the unperturbed Hamiltonian and this Hamiltonian gives you the full Hamiltonian in the limit alpha going to 0 and also as t goes to 0 which is the now moment for this problem. So, also in the limit t going to 0 or alpha going to 0. So, our interest is in the solutions at t equal to 0. So, that is the instead of time in which we are interested. So, this is how the way functions in the different pictures are related. I mentioned that this limit need not be well defined it is not fundamental to our analysis it is the limit of the ratio which is of importance to us. So, these are some of the things that I really want to emphasize. What happens is that there is a phase factor and you get certain divergence in that in the numerator if you were to treat this separately and you would have a corresponding divergence in the denominator as well and they happily kill each other when you take the ratio. So, now let us begin with the statement of the Gelman and Lowe theorem and what we are going to do is to look at this whole thing as a vector because when this operator operates on a vector you get a new vector. The new vector is what I put in this larger ket which is colored in red. So, now let us look at these 2 vectors on the left side and on the right side. You have got a ket vector and I am going to construct an inner product of this vector which is to take its projection on another vector which is the adjoint of the vector corresponding to the ground state of the unperturbed Hamiltonian. So, the unperturbed eigenstate phi 0 is known to us. So, we have the adjoint of that vector and I take the projection of this vector which is in this larger ket the red colored ket and I take the projection of this on this. So, it is just the inner product that I am constructing. So, let us go ahead and write this projection. So, it is it will be this, it is the projection of this vector on phi 0 on the left hand side and the right hand side it is the projection of the right hand side on phi 0. Now, let us write these terms a little bit neatly because in the numerator you will have the matrix element of the full Hamiltonian in the states phi 0 and psi 0. So, let us rewrite this result a little bit neatly. So, you have got the matrix element of the Hamiltonian in these states. This is the eigenstate psi 0 which is here, this is the eigenstate in the interaction picture at t equal to 0. So, this is just the solution that we really want. So, this is what we are interested in and you have got the Hamiltonian operating on this psi 0 and we are taking its projection on phi 0. So, this is the result that we have got. So, let us write this result over here which we got this result using the Gelman and Loh theorem. We just took the projection of the Gelman and Loh vector on the unperturbed state vector eigenstate of H 0. So, this is what we have got. Now, this is the full Hamiltonian which has got two pieces one is H 0 the other is H 1. So, I am going to separate out the contributions from these two terms. So, on the left hand side I will have two terms. So, this is the first term which is the matrix element of H 0 in these two states and then in the second term you have got the matrix element of the unfriendly Hamiltonian the perturbation H 1 in these two states. Now, here you have an eigenvalue equation because phi 0 is an eigenstate of H 0. So, phi 0 H 0 on the left over here will be E 0 phi 0 because it is just the eigenvalue equation over here. So, here you get E 0 eigenvalue pops out and then you get the projection of psi 0 on phi 0. Now, psi 0 and phi 0 of course are different. So, the projection of psi 0 and phi 0 is not equal to unity. If both are normalized then the projection of psi 0 and phi 0 would be 1 if they were the same eigenstates, but they are not the same and therefore, it is not unity it would be some non-zero value. But whatever it is you have the same factor in the denominator here. So, these two factors cancel likewise these two factors cancel and now you have got the eigenvalue of the full Hamiltonian on the right side on the left side you have got the eigenvalue of the unperturbed Hamiltonian plus this term. So, if you just rearrange the terms in this equation you get the difference between the eigenvalues of the full Hamiltonian and the eigenvalue of the unperturbed Hamiltonian. So, E minus E 0 which is like the correlation energy. So, you get this in terms of the ratio of these two quantities. So, let us figure out how we are going to evaluate this. This is the quantity of interest to us. Now, h 1. So, this psi 0 over here we know can be obtained from phi 0 through the evolution operator U by taking the evolution from the state minus infinity because this is the solution at minus infinity. This is the solution of the unperturbed Hamiltonian and that is what the full Hamiltonian collapses into as t goes to minus infinity. So, we know this and from this you can get the solution at time equal to 0 through this time evolution operator as we have just discussed. So, you have this psi 0 written over here you do the same in the denominator this psi 0 is also this then you have got h 1 over here. So, this is the result that you have got. However, our result cannot really depend on alpha which is only a mathematical construct, but that is not a worry because anyway we are taking the limit alpha tending to 0. So, that is the limit that we are going to discuss. So, this is our expression this is how you can get the states. Now, the state vectors which are involved over here you see a phi 0 over here and a phi 0 over here you see a phi 0 over here and a phi 0 over here. So, all the state vectors that you need to evaluate the right hand side are now known to us. So, we have succeeded in writing the difference e minus e 0 in terms of state vectors and the time evolution operator and the correlation term, but using only those vectors which are known to us because that is the eigenstate of h 0 that part of the problem has been solved. Now, this is the result the one of the top we just got this from the Gellman-Loev theorem. Using this result we will now show that this energy difference is actually equal to the limit alpha going to 0 and here on the right hand side I have got a new expression and we are going to see how this result shows up. It will turn out in the next few minutes that if you take the logarithm of the expectation value of the time evolution operator in the state phi 0, then take the derivative of this log with respect to time the partial derivative, then evaluate the value of this partial derivative at t equal to 0 and then take the limit alpha going to 0. So, you go in that order first take the log then the derivative then the value of the derivative at t equal to 0 and then finally, the limit alpha going to 0. So, do not do anything in any arbitrary order because that is the sequence in which things have to be done. So, let us look at this right hand side let us look at this limit alpha going to 0 of this expression on the right which is i h cross del by del t of this log at t equal to 0. So, let us consider what is inside the square bracket this is just the time derivative of this log. So, it will be this 1 over this expectation value times the time derivative of this expectation value and I have pulled in this factor i h cross over here inside the bracket and then of course, I retain the t going to 0 constraint because that is the one which is of importance. So, this is what we have got. So, the limit alpha going to 0 i h cross del over del t. So, this is the left hand side and now I just write this a little bit neatly. So, that I have got this factor in the numerator which is this and this factor in the denominator comes below it. So, I have just read written this expression. So, that it is easy to read. So, that is what we have here in the top of this slide. Now, what is this? This is nothing but the Schrodinger equation for the time evolution operator in the interaction picture. So, in the interaction picture the time evolution operator satisfies the differential equation which we often call as the Schrodinger equation itself right. This is the Schrodinger equation for the time evolution operator. This is in the interaction picture and this is nothing but h u. So, this is very similar to how we write the time evolution operators Schrodinger equation in the Schrodinger picture except for the fact that here this is the interaction picture Hamiltonian right. So, this is just the transformation of the interaction part or of the correlation part. So, that is the only one which is being focused upon in the Dirac picture or the interaction picture. Now, here you have the time derivative of the time evolution operator the partial derivative with respect to time. This partial derivative is given by h u. So, that is the h u I write here instead of the derivative del u by del t. So, instead of del u by del t I put in h u I have included the i h cross which was sitting here. So, that has been included and the right hand side on the top is now rewritten as this. Now, put the limit t equal to 0. We have already taken the time derivative that is what gives us h u right. Now, put t equal to 0. So, this t will go to 0 this t will also go to 0. So, let us do that. So, now you have this t equal to 0 this t the second parameter is the start time which is minus infinity and you do the same in the term in the denominator. So, this is now taken care of and this right hand side is nothing but e minus e 0 as we saw from the Galman-Low theorem. In other words e minus e 0 can be written as the limit alpha going to 0 of this expression. So, we have just written a consequence of the Galman and Low theorem in a form which we will find extremely useful to deal with these correlation terms. Now, it is going to get messy before it becomes better, but you will see how that happens. So, this is what we have got from the Galman-Low theorem. This is the energy difference in which we are interested. We have got an expression for this difference which we got in the previous slide which is given by this result, but how did we get this result? We got it from the Galman and Low theorem. We got it from the adiabatic hypothesis. We got it from the mathematical construct of the parameter alpha. What do we expect it to correspond to? We expect it to give us information about whatever was missing from the unperturbed Schrodinger equation from the edge 0 because that is the problem that we solved. We expect this energy difference to correspond to that. So, in some sense, we expect it to correspond in the perturbative sense to what perturbation theory would give us if we were to take into account the correlations which are missing in the term edge 0 and edge 0 problem is solved. It is the H Eigen value equation for the full Hamiltonian which is not solved. So, we expect it to correspond to what we get from the usual perturbation theory which is what I shall refer to as the Rayleigh Schrodinger perturbation theory because that is the common form of the perturbation theory that we work with. So, that is the usual Rayleigh Schrodinger perturbation theory and we expect some sort of a correspondence between the energy correction delta E which we have got from the adiabatic hypothesis using the Galman and Low theorem and we will like to ask what exactly is this correspondence and how do we relate these terms in the perturbative sense. It does not mean that we are using perturbation theory, we are not. If we were to use the Rayleigh Schrodinger perturbation theory in its raw form the way it is constructed in introductory courses in quantum theory that will not give us any result which will converge. That is the reason you cannot approach you cannot use that approach you are not using the Rayleigh Schrodinger perturbation theory, but in the perturbative sense the results that you are going to get will correspond to how you would get information about this part of the solution to the problem which is missing in the problem which has been solved which is the problem for the unperturbed Hamiltonian H0. So, our correspondence will become clear when we analyze the time evolution operator because this is the most important creature here we have to get its expectation value in the unperturbed eigenstate in the eigenvector of the unperturbed Hamiltonian and the form of the time evolution operator has now to be studied in some details because that is where all the correlation is sitting. Now, we did this in not the previous class, but in the class before that so that was in lecture number 25 and I am borrowing a result from that lecture which is on slide 45 of that lecture that the time evolution operator was written as infinite terms if you remember, but we rewrote those infinite terms as an infinite series n going from 0 through infinity of this and the order in which these operators come is determined by this chronological order chronological operator T. So, these operators are all time ordered which means that operators containing the latest time stand to the left. So, that is the chronological ordering that the T operator guarantees. So, that is the result that I am now going to use and we have obtained this result in some detail in the 25th lecture of this course. So, let us look at these terms there are infinite terms I separate the first term corresponding to n equal to 0 which is nothing but the unit operator and then I have the remaining terms which so what is being summed over is the same except that the summation is now going from n equal to 1 through infinity because n equal to 0 term has been separated out. So, I first separate out the n equal to 0 term. So, now I write this time evolution operator as a sum of the unit operator plus again an infinite term, but n going from 1 through infinity and what is being summed over is this un. Now, this is where all the correlations are setting it and this is where they will show up and if we can evaluate this term then we will get the term delta e that is the quantity of interest. So, you have the time evolution operator u alpha which is a sum of this 1 plus these remaining infinite operators when you remove 1 from infinity you still have infinite terms there is nothing I can do to help you with that. So, one would think that when you remove something you will have to deal with less, but that is not the case always and I am really sorry I cannot help you with that. So, you have got this full Hamiltonian and these interaction picture Hamiltonian of the correlation part has got these e to the alpha t term. So, this mathematical constructs e to the alpha t they are all sitting there in this. So, in h i t 1 which is here you will have an e to the alpha t 1 here you have h i t 2. So, in h i t 2 you will have an e to the alpha t 2. So, you will have this e to the alpha t term several places actually n places in each one of these from h i t 1 right up to h i t n set all of these n places you will have these alpha parameter exponents and then of course, that is not being enough you of course, have a summation over n going from 1 through infinity. So, there is a lot to deal with. Now, let us look at this expression now subsequently we will have to take the logarithm and so on, but let us just deal with this expression for the time being piecewise. So, let us look at this expression you separate the first term the first term is nothing, but the inner product of phi 0 with itself it is the norm of this state and let us say that this state is normalized. So, you get unity from here and then a sum of these expectation values of u n's and each u n is this. So, this matrix element is now what I call as a n. Now, you will you might wonder why are we introducing new symbols we already have those expressions and if you did not do that you are going to have to work with terms which are already cumbersome and you can only make them more cumbersome by expanding your notation. So, this is just some strategy to save the ink that you would use to write in your notebooks. So, these are compact terms so and you will see this very soon we are heading toward it. So, you have this a n which is the expectation value of u n. Now, you have to take the logarithm of this this is just the expectation value of the time evolution operator what about the log. Now, this is the logarithm of 1 plus another quantity, but this is a small quantity what is it made up of it is made up of these u's what are these u's made up of it is made up of the correlation which was left out in our earlier problem, but our contention was that we have done most of the problem and it is a tiny thing which is left over right and we are worried about this tiny thing. So, the only thing we know about it is that it is a tiny thing. So, it is nothing, but an expression similar to logarithm of 1 plus x where x is a small quantity at this power series expansion is known so we can use that result. So, let us plug it in and now you see how many terms you have infinite terms you already have various other kinds of infinities in our analysis and you have some more because now you have got infinite terms in the expansion of the logarithm term. Now, I am sure you did not want to write the right hand side over here instead of a n here you have the square you have got the quadratic terms and then the cube and the fourth power and the fifth power and the sixth. So, you have infinite terms over here. So, let us look at this result now. So, you have got these infinite terms n going from 1 through infinity each a n is a very compact creature it is nothing but the expectation value of the operator u n but the operator u n itself has got a very complex structure. So, this is a lot to worry about. So, now this is what we get for delta e delta e is the result that you will get after you take the limit alpha going to 0 of i h cross then the time derivative of this logarithm function. So, you have got the time derivative operator here and for this logarithm expression you now have these infinite terms which are in this beautiful bracket you put take the time derivative of everything that is there in this beautiful bracket put that in a square bracket like a box bracket and then take the value of the box bracket the rectangular bracket at t equal to 0 and then you take the limit alpha going to 0. So, that is what we are going to do now. So, these are the terms that we have to worry about now let us write these summations infinite terms a n. So, a n summed over n going from 1 through infinity. So, you have got a n plus a 2 plus a 3 and so on now this is the square of a sum of these terms. So, what will you have you will have quadratic terms you will have even square a 2 square a 3 square and so on you will also have a 1 a 2 1 a 1 a 3 and a 1 a 4 you will also have a 2 a 1 a 2 a 3 a 2 a 4 and so on and you really have to keep track of each one of these terms. Because it becomes senseless to take the quadratic terms in a 1 square and ignore a 1 a 2 because they are of the same order it makes no sense to take account of a 1 a 2 but not a 2 a 1 it is of the same order it becomes even trickier when you deal with the cube. So, here you have got a 1 cube a 2 cube a 3 cube but then you have got a third order term coming from a 1 a 2 a 3 but you also have a third order term coming from a 1 and a 2 square and also from a 2 and a 1 square. So, there are all kinds of permutations which are possible and we have written terms only up to the third order and only sum of the terms and then you have to take the time derivative and once you take the time derivative then take the value of the derivative at t equal to 0 and then you take if you are still alive by then take the limit alpha going to 0. Now, you can write the expression delta e when you take all of these terms carefully as a sum of contributions from first order terms second order terms and the order is what I indicate by this superscript and these are in various powers of h 1 which is the correlation term and these correlation terms become progressively small but that does it mean that you have convergence and that is the essential reason why we are developing these alternative techniques. So, notice that various order terms come from different terms like second order terms I showed you from a 1 square but also from a 1 a 2 and also from a 2 a 1 third order terms from a 1 cube a 2 cube but also from a 1 a 2 and a 3 also from a 1 a 4 square right. So, the third order terms from you know different combinations of a linear term and a quadratic term or a product of three linear terms and so on. So, all of this has to be kept track of. So, this is your energy correction now this is the sum and substance of what we have got. So, it is a fairly complex structure and it is a lot of mass is it it is a lot of mass and let us see if we can look at just the first order terms to begin with. So, always from the simplest to the more complex from the known to the unknown I got it right this time Blake. So, from the known to the unknown and from the simplest to the more complex terms. So, we will begin with the first order terms and the first order term you have limit alpha going to 0 i h cross del over del t of let us say let us deal with just the a 1 term just to look at that. So, let us see how we are going to deal with this term. So, this is what your a n in general is therefore, we know what a 1 is a 1 is just out of these n terms you have only one which is this h i at t 1 there is only one time the to worry about in the n th term you have got n different instance of time to worry about t 1 t 2 up to t n what you have over here is a chronological time ordered product of these operators for n equal to 1 there is only one instance of time that you are working with which is this t 1 and this is the dummy variable which gets integrated out from the start which is minus infinity up to the current time t whatever it is and after doing this integration you can then take the derivative of this term with respect to time and then take the value of the derivative at t equal to 0. Now, this is the relationship between the interaction picture operators and the Schrodinger picture operators the evolution that the transformation is through the unperturbed Hamiltonian at 0. So, that tells us that this h i at t 1 this is the interaction picture term is the transformation of h 1 using this evolution operator the transformation operator which contains the unperturbed Hamiltonian at 0 this is the term which will go in over here and what is this h 1 now that is something we do know in terms of the creation and destruction operators. In the earlier unit in which we did second quantization we wrote the electron-electron interaction term in terms of the creation and destruction operators. So, you had one particle terms which are over here and I am sure you remember this form of the Hamiltonian which we discussed in our earlier discussion on second quantization. So, in the second quantized form we have written the full Hamiltonian now you have got the one electron terms and then you have the two electron terms. So, this is your operator h 1, but we are going to have to deal with this put this in our expression for a 1 and then make a further we have to progress further and then see how we are going to work with the second order and higher order term. So, you see what a mess it really is and this is where the Feynman diagrammatic methods become handy those are the ones which I am going to introduce in the next few classes. So, now I think we have the basic machinery ready. So, we have to we are our interest is in getting delta e not just the first order, but to higher orders as well we could not get it using the Rayleigh Schrodinger perturbation theory. So, I am going to stop here for today's class and we will proceed from this point in the next class we have the electron electron term. So, this is the difficult part of the Hamiltonian this is the unfriendly part as we have been referring to this is the one which is responsible for correlations. The static average gives us the Hartree-Fock which is the time independent Hartree-Fock right. So, that we can get from which is equivalent to first order Rayleigh Schrodinger perturbation theory which converges only for the first order, but for higher orders it does not. So, this is the H 1 part this is the difficult part of the Hamiltonian with the difference that we have now scaled it by this factor e to the alpha t which is our mathematical device. So, our Hamiltonian is not just this term plus what is in this purple blocks block this is the western color is it not. So, this is in the purple block this is the interaction term H 1 it is more than that it has been scaled by this mathematical construct e to the alpha t which is the adiabatic hypothesis which goes into the government and low expression which we have used. So, I will take the discussion from this point in subsequent classes yes. So, in the last slide we have the explicit expression for A 1 in the red box. So, there the alpha dependence is did we miss it or no in the same slide 75 yeah. So, the H 1 there is an alpha dependence right. So, this H i t this H i t 1 this H i t 1 is what you will get by simply transforming from the Schrodinger picture to the interaction picture. So, we are doing two things over here one is carrying out a transformation from the Schrodinger picture to the interaction picture, but there is something more that we are doing and that is the adiabatic hypothesis which is why H 1 which is in this first purple box is to be replaced by that term scale by this e to the alpha t. So, when we are going to work with these correction terms in A 1 A 2 and higher order terms we will bring in the alpha with every time there is a correlation term which is to be addressed. In fact, that is the that is the point I was anticipating in one of the previous slides. Let me go back to the just to emphasize this let if I can quickly find it here yeah. I alerted you to the fact that you have got in this u n in the nth term n number of different instance of time parameter. You have got a t 1 here and a t 2 here and a t n here, but associated with each of these there will be an alpha parameter. So, you will have an alpha t 1 you will have an e alpha t 2 and so on. So, all of these terms will have to be plugged in when we evaluate these terms carefully. So, just keep this at the back of your mind and then we will put in all these terms together. So, we are doing you know more than just the Dirac picture. The Dirac picture is the basic platform that we use and over and above this Dirac picture in which we can do quantum theory. We are now doing the adiabatic hypothesis which is the insert which is the insertion of this mathematical device which is the e to the alpha t which we can use to let the Hamiltonian the full Hamiltonian at t equal to 0 evolve from the unperturbed Hamiltonian at t going to minus infinity. So, that solution is known to us at t going to minus infinity from that eigenstate we want to get the eigenstate of the full Hamiltonian at t equal to 0 which is the now moment. So, that is where the alpha parameter comes in and it will of course, be inserted not just for a 1, but for every order term and there will be several terms like with each of these time parameters t 1 t 2 and t n there will be an e to the alpha t 1 e to the alpha t 2 and so on. So, save this consideration at the back of your mind we are going to need it in one of the next classes. I do not know if it will be the very next class or the one after, but we are going to need it soon. And you see what a mess all this is and this is where the Feynman diagrams come in handy because instead of dealing with so many terms there you have to do the integration there are these Volterra type of integrations then you have the chronological operator you have to take the logarithms you have infinite power expansions. Now, instead of all of this if you were to work with just pictures would it not be nice some fun diagrams and that is what we will be discussing in the next few classes. So, thank you all very much and we will take it from here in the next class.