 Greetings. After doing the random phase approximation in unit 3, we will today start with the next unit and in this unit we will introduce Feynman diagram methods. I would like to suggest these two references which you see on your screen, Fetter and Wallacher's book, Quantum Theory of Many Particle Systems and the book by Reims, Many Electron Theory. These will be the primary sources for this unit. And before we get into the details of the Feynman diagram methods, today's class will be like warming up for this topic. We will revisit a little bit of what you would have done in your quantum mechanics course on the Schrodinger picture, the Heisenberg picture and the Dirac picture. So, we will spend some time just warming up and that is what we will do today and then build the topic from there. So, let me first remind you some results from the unit 3, from the previous unit. And I will refer to the Hartree-Fock model. And in the Hartree-Fock model which we discussed earlier in the course on atomic physics and later we dealt with it also in unit 3 of this course. And in particular, I would like to refer to the lecture number 19 and 20 of unit 3 and slide number 96 and 97 from that lecture. So, I will recollect one of the results from there and what we learnt over there that when you take into account the electron-electron interaction in the Hartree-Fock, you get an energy which is actually lower than that of the Sommerfeld gas which is somewhat surprising because you have the electron-electron interaction and you expect that it will increase the energy because it is repulsive. But then it has an attractive component which is the exchange energy and that is completely taken into account in the Hartree-Fock model. And what the exchange does is it is a result of the fact that the many electron wave function is an anti-symmetrized wave function in consistency with the Fermi-Dirac statistics, the electrons being half integers, spin particles. And what this does is to keep electrons with parallel spins away from each other and that is what reduces the energy as we have seen. And what we found is that the average Hartree-Fock energy per electron, so Hartree-Fock average energy per electron if capital N is the number of electrons, then this is given by a result which we have discussed in the previous unit and you will remember that result. Now what we also discussed is the fact that if you obtain this result using first order perturbation theory rather than the self-consistent field methodology of the Hartree-Fock scheme, you get essentially the same result. So we discussed this in the previous unit in the lectures 19 and 20. However, if you went to higher order perturbation theory, if you went to the second order or any order higher than the first, then you find that the methodology does not converge at all and you cannot use perturbative methods. So you have to look for methods which are non-perturbative and whereas the Hartree-Fock self-consistent field method is equivalent in a certain sense to the first order perturbation theory. Going beyond the perturbation method is important because Hartree-Fock takes into account this exchange correlation, the exchange interaction. So it takes into account all the correlations which are coming from the Fermi Dirac statistics. But then there are some correlations which are left out and the correlations which are left out are what we call as the Coulomb correlation. So correlations in a many electron system are of two kinds. One is the spin correlations. They are equivalently the Fermi Dirac correlations or exchange correlations, statistical correlations. They are all synonymous equivalent terms to describe this fact that the many electron system observes the Fermi Dirac statistics. But then there are other correlations and these are the Coulomb correlations and we are interested in taking account of these correlations and that is what many body theory is about. So these Coulomb correlations are ignored in this result and these can be addressed using a variety of different techniques and we are in the process of discussing some of them. And for this you need a formal many body theory. This is where quantum field theoretical methods or many body methods become necessary and they help us go beyond perturbation methods. So the technique that we discussed in the previous unit is the random phase approximation. It is due to Bohm and Pines which was developed in the mid 50s of the previous century. Very nicely described in the book by David Pines, elementary excitations and solids and it gives you a result beyond the first order perturbation theory. It gives you a correction and we have discussed some of these results in unit 3. So this is just a quick recapitulation of that. And the basic problem that we are trying to solve is just the many electron Schrodinger equation with this Hamiltonian. So this is the one electron part and this is the Coulomb part. But then all the statistical correlation is appearing in the Hartree-Fock formalism because the n electron wave function has to be an anti-symmetrized wave function. And as I pointed out earlier you do not have an exact solution for this problem. I had quoted Brown earlier also that if you are looking for exact analytical solutions then having no body at all is already too many and you cannot have exact solutions even for the vacuum state let alone for the n electron problem. So what do you do? You have non-perturbative methods one of which we discussed at length in the previous class the random phase approximation. Then there are methods which are alternative to the RPA some of which turn out to be equivalent to RPA in some sense. Some of them are different and you can do them using techniques which are based on configuration interactions or CI methods. And what they enable you to do is describe the n electron wave function not by a single slater determinant as we do in the Hartree-Fock but by a superposition of 2, 3, 4 or more or in principle it can be an infinite set of slater determinants. And these are the configuration interaction methods. So the methodology which comes out of this when it is based on the Hartree-Fock is called as a multi-configuration Hartree-Fock. But you can have its relativistic analog in which the starting elements which go into the slater determinant are not the 2-component Hartree-Fock spinors the spinor wave functions the spin wave functions but these are the 4-component Dirac wave functions the Dirac bispiners as we call them right. So if the slater determinants are made up of these 4-component bispiners then you have what we call as a multi-configuration Dirac-Fock and more correctly it should be called as multi-configuration Dirac-Hartree-Fock and there is a whole scheme which has been developed pioneering work has been done by Frosch Fisher for the multi-configuration Hartree-Fock by A. N. Grant and some others on the multi-configuration Dirac-Hartree-Fock. So we have the multi-configuration Dirac-Hartree-Fock sometimes abbreviated as MCDF or MCDHF for the Dirac-Hartree-Fock as such and then there are other methods which are the Feynman diagram methods and this unit will focus on the Feynman diagram method. So all of these techniques are essentially they target the many-body problem and in particular they target the issue of how the Coulomb correlations must be dealt with in a many electron system. So let us have a look at this Hamiltonian now. Notice that this Hamiltonian is not an explicit function of time there is no time dependence in this. Now what we can do is the correlations are of course coming from the electron-electron interaction so you have got the 1 over r12 or 1 over rij electron-electron interaction term and if we can deal with this term as if it were time dependent then we could use methodology in which the operators are time dependent. This is an operator 1 over r12 or 1 over rij is one of the terms in the n electron Hamiltonian and if we can treat it as if it were a time dependent term then we can use those methods in quantum theory in which time dependent operators are made use of and in particular this is possible in what is called as the interaction picture or equivalently synonymously also called as the Dirac picture. So I would like to introduce these three pictures the Schrodinger picture the Heisenberg picture and the Dirac picture all of these essentially contributed to the development of quantum theory all of them got Nobel prize Heisenberg in 1932 and Schrodinger and Dirac shared it in 1933 and the formalism of quantum theory which we most often make use of and which is what we have used in our in our earlier courses. We have made use of the Schrodinger picture but there are equivalent you know formulations in the so-called Heisenberg picture and the Dirac picture and in today's class I will revisit these pictures. These are to be distinguished from what we often refer to as a representation for example you write the state of a system by a vector in the Hilbert space and then you can have a coordinate representation of this state vector which is the wave function you can also have a momentum representation. So these are different representations and you can go from one representation to one representation to another by carrying out appropriate transformations. A picture is somewhat different. So it is also you know different kind of representation in a certain sense but these are achieved through what are called as generalized rotations in the Hilbert space rather than usual rotations in the Hilbert space. So let me explain this. So let us begin with the Schrodinger equation h psi equal to i h cross time derivative of the wave function. What the Schrodinger equation does is to give you the time evolution of the system and the time evolution operator tells you how the wave function evolves from an initial time 0 to a later time t and the evolution is through this time evolution operator which includes the Hamiltonian. So this is what gives you the Schrodinger equation and this is the typical Schrodinger picture quantum mechanics that we are used to. So this is the description of a state of a vector at time t and we are essentially looking at stationary state solutions. So this exponential of an operator is understood as a power series expansion so that the Hamiltonian can operate as many times depending on whether it is the first term or the next term and so on. And the stationary state solution then is given by the solution e to the minus i omega t, omega is e over h cross and most often we write this solution that the time dependent part of the time dependent solution is given by e to the minus i omega t or minus i e over h cross t and the time independent Schrodinger equation has got this solution which is e is the eigenvalue and the eigenfunction is psi of r at time equal to 0. So this is the time independent Schrodinger equation for a stationary state. Now notice that the state functions are a time dependent and the entire time evolution is contained in this e to the minus i omega t term or e to the minus i e over h cross t term. This is the entire time dependence the state function is a time dependent function and the operators the operators are made up of the position operator the momentum operator right. The Hamiltonian operator is made up of the potential energy operator and the kinetic energy operator and all of these operators you write in terms of the q and p and these are completely independent of time. So in this picture it is the state function which is time dependent and the operators are completely independent of time and this is the signature of the Schrodinger picture. So this is the usual quantum mechanics that we do and that is the reason I have added a subscript s to emphasize that what we are looking at over here is the Schrodinger equation in the Schrodinger picture. But you can have alternative formulations in which the time dependence is not necessarily in the wave function you can actually transfer the time dependence to the operators rather than the wave function and still do equivalent quantum mechanics and let us see how this is done. This is done through certain transformations unitary transformations we know that they leave the physics invariant. The physics as such remains invariant you can have unitary transformations even in time independent Schrodinger picture formalism and essentially by large what it amounts to is that you can expand the wave function in any linearly independent basis. You can have you can orthogonalize that basis and then you have got an orthonormal basis and you can have an orthonormal basis made up of you know like in three dimensional space you have this you can twist it turn it move it right and any basis which is equivalent will work and you can carry out transformation of the wave function from one basis to another through unitary transformations the coefficients are then the cosine functions which connect the two orthogonal basis. Now that is something that you know very well and they leave the physics invariant. We will introduce what are called as generalized rotation. So a rotation from one basis to another this is the usual rotation that we talk about when we talk about unitary transformations. Now what I am introducing are generalized rotations which are not transformations of this kind but they are transformations which are very similar but with a difference and this difference tells you how you carry out a transformation from the Schrodinger picture to the Heisenberg picture or to the Dirac picture. So this is how you carry out the transformation of an operator from the Schrodinger picture. So if omega with subscript s is an operator in the Schrodinger picture then the corresponding operator in the Heisenberg picture is given by this transformation. This is the transformation rule which tells you how you carry out a transformation from the Schrodinger picture to the Heisenberg picture and this transformation is effected through these time dependent operators in which the Hamiltonian plays a big role. So the Hamiltonian has to appear in this transformation. There is a time dependence which is explicitly as you can see and this gives you a new operator. So this operator it is what you call as the Heisenberg picture operator and this is the Schrodinger picture operator. This is the relation between Heisenberg picture and Schrodinger picture. So the operator which is effecting this transformation is this e to the i h t over h cross. This is the operator. So you have got o omega s omega dagger which gives you the Heisenberg picture operator. Correspondingly when you carry out a transformation the states will also be transformed and you get new states which are then called as the Heisenberg picture states and these pictures in the Heisenberg picture these states the Heisenberg states will become independent of time. So the Schrodinger picture the wave functions are time dependent. The operators are not in the Heisenberg picture it is the other way around. The operators become time dependent but the wave functions become independent of time and you will see how they become independent of time. So this is the main difference between the Schrodinger picture and the Heisenberg picture. So you still need a time dependent formulation but the time dependence is transferred from the wave function to the operators. To do quantum mechanics there are two things that you do. One is represent the state of the system by the state vector and then deal with operators instead of the classical dynamical variables and these operators we are looking at the properties of these operators and now we will introduce time dependent operators which we did not use in the Schrodinger picture. So you have a generalized rotation effected through the transformation operator O which is e to the i h over h cross t and the corresponding wave function when you carry out transformation of the operators omega s to omega h there is a transformation of the wave functions psi s to psi h essentially through the same transformation which is operator O and because we are dealing with the operator O which effects the transformation to the Heisenberg picture I have now added a subscript h over here. So this operator transforms the Schrodinger picture wave function to a Heisenberg picture wave function and this Heisenberg picture wave function is now e to the i h over h cross t operating on psi s. So this is an operator which is made up of the Hamiltonian and when this operates on the Schrodinger picture wave function you get a new wave function which is a wave function in the Heisenberg picture. And notice that you can write the Schrodinger picture wave function is an eigen function of the Hamiltonian belonging to an eigen value e. So this e to the i h over h cross t is equivalently replaced by the corresponding eigen value which is e to the minus i e over h cross t. So this is the stationary state solution of the time dependent Schrodinger equation and if you now put this back in the Heisenberg picture wave function. So this is the Heisenberg picture wave function. How do we get it? By operating by this transformation for generalized rotation operating on the Schrodinger picture time dependent wave function which is now e to the minus i omega t times the time independent wave function at t equal to 0. But then e to the minus i omega t can be pulled out and then you have got this operator operating on the Schrodinger picture wave function at t equal to 0 and that will give you e to the plus i e omega t and these two terms will then cancel each other and you get psi at r 0. So you have got a time dependence, a formal time dependence over here but it turns out that it really does not depend on time because this is the wave function at t equal to 0. So essentially what we see is that the time dependence is lost. So all the time dependence is in the operator here. This is where you have got the time dependence. So far as the wave functions are concerned in the Heisenberg picture there is no time dependence. But the physics remains the same and that is the reason this is called as the generalized rotation from one picture to another. However remember that if the operator which is being transformed this is any Schrodinger picture operator if this is the Hamiltonian itself. Then this Hamiltonian operator of course commutes with e to the minus i e h t over h cross. It will commute with every term in that infinite series. And you can then swap the positions of this operator h with this and then essentially you find that if the operator which is being transformed happens to be the Hamiltonian itself then it remains the same. So the Hamiltonian in the Heisenberg picture is the same as the Hamiltonian in the Schrodinger picture and this is not a function of time. Otherwise there is a time dependence for all of the other operators. So this is let me summarize the essence of this. So you have got a Heisenberg picture operator which is time dependent and therefore its time evolution can be studied because it is time dependent and its time evolution is given by the rate at which it changes with time. So now you have got the rate at which omega h changes with time and the time dependence will come from this term and also from this term and you can treat it just as if it is a function of three product of three functions two of which the first and the third are time dependent the middle one is time independent. So you get the time derivative of this and then you have got omega s e to the minus i h over h cross t plus you have got the this term omega s is not dependent on time. So it stays as it is and then you have got the partial derivative of the third operator which is e to the minus i h over h cross t. So now let us take these derivatives. So you will get i h over h cross here and you will get minus i h over h cross here. So now you have got these two terms the third term is missing because omega s is independent of time. Now you have got these two terms and let us write them in a slightly different way because the operator products are associative. So you can look at these terms in a slightly different way by recognizing that this Hamiltonian here and this function of the Hamiltonian which is e to the minus i h over h cross t these two operators can be swapped. Their positions can be swapped because they obviously commute one is made up of the other. So naturally the two operators swap. So if you interchange the positions of these two you bring e to the minus i h over h cross t to the left of this omega s and to the left of this operator h. So this h now is written at the end rather than the penultimate position that it had in the previous step. So the Hamiltonian as I pointed out is the same in the Heisenberg picture as in the Schrodinger picture. The time dependence of an arbitrary operator in the Heisenberg picture is given by these two terms and what is this? This is nothing but the transformed operator omega in the Heisenberg picture. So you have got a commutator over here. So if you factor out i h cross is common you find that the time derivative of the Heisenberg picture operator is given by i over h cross times the commutator of h the Hamiltonian with this operator h. So obviously this gives you the time evolution and this is sometimes called as the Heisenberg equation of motion for the operator omega. So this is how the time derivative of the operator is expressed in the Heisenberg picture. It is the operators which are time dependent the wave functions are independent of time. Now this is contrasted with the Schrodinger picture in which the operator is not a function of time whereas the wave function was and the relationship was affected through these generalized transformations which are unitary transformations but generalized rotations in the Hilbert space through operator this is the transformation operator e to the i h over h cross t. So this gives you the transformation from the Schrodinger picture to the Heisenberg picture. Now there is another picture that is the Dirac picture it is also called as the interaction picture and in this picture quantum mechanics of those problems which cannot be solved perturbatively using perturbation methods this is particularly useful for such problems like the Coulomb correlations in our problem. In the many electron problem we recognized that the statistical correlations the spin correlations could be handled using the anti-symmetrize wave function which is a Hartree-Fock theory which we learnt is equivalent to the first order perturbation theory result but there is a residual correlation in the many electron system that is the Coulomb correlation and that is something we cannot handle using perturbative methods because whereas the first order perturbation theory gives you a result which is the same as the Hartree-Fock theory the second order and higher order perturbative methods do not give any converged result. So now we are looking of ways to deal with this term the electron-electron interaction which is responsible for the Coulomb correlations and for these problems for which you cannot solve using perturbative methods the interaction picture provides you with very powerful tools to solve the problem. So how do you do that? You again carry out generalized rotations in the Hilbert space but this time the transformation operator is e to the i h over h cross t but mind you this is the h 0 which is the soluble part of the Hamiltonian. So this Hamiltonian consists of two parts one is h 0 for which you can get exact solutions then there is a residual part which is the culprit in a certain sense which gives you these complications that you cannot handle using perturbative methods you can get approximate solutions Hartree-Fock is a solution to this problem at 0 plus h 1 but it is only an approximate solution it is approximation to the extent that it has taken into account the statistical correlations but not the Coulomb correlations and it is the Coulomb correlations that we are now interested in handling. So this is left out in the transformation Hamiltonian. So notice the difference between this transformation operator which is the full Hamiltonian here in the Heisenberg picture but over here it is only the unperturbed soluble part of the Hamiltonian which is used to effect the generalized rotation. So that is the big difference and this is what gives you the Dirac picture operators omega i and the corresponding interaction picture wave functions are obtained from the Schrodinger picture wave functions by operating these Schrodinger picture wave functions by the same transformation operator which is e to the i h 0 over h cross t. So this is the Dirac picture and in this picture there is time dependence in the wave functions the interaction picture wave function is time dependent. There is an explicit time dependence over here as well as over here and then there is an explicit time dependence over here. So both the operators and the wave functions depend on time. In the Schrodinger picture only the wave functions depend on time. The operators do not in the Heisenberg picture it is the operators which depend on time but the wave functions do not and in the interaction picture both depend on time but the physics remains invariant. So the important difference in the Heisenberg picture in the Schrodinger picture is that the transformation operator consists only of the soluble part of the Hamiltonian. So here we are this is what we get for the interaction picture operator and the wave function and if you now operate on the interaction picture wave function by the adjoint of this operator which will be e to the minus i h 0 over h cross t then you get the Schrodinger picture wave function and the Schrodinger picture wave function you know what we can do is rewrite this wave function this Schrodinger equation because this h is nothing but h 0 plus h 1. So that is what we have over here then you have got the Schrodinger picture wave function which is psi of s then you have got i h cross del over del t of the Schrodinger picture time dependent wave function but the Schrodinger picture time dependent wave function is related to the interaction picture wave function by operating upon this by e to the minus i h 0 over h cross t. So this is an equation which looks very similar to the Schrodinger equation as such but then there are these subtle differences that you have to keep track of and get used to. So this is the physics is the same as we had in the Schrodinger equation in the Schrodinger picture. What are the consequences? Let us take the time derivative of the right hand side. So there is time dependence over here and the time dependence over here. So you have two terms on the left hand side one coming from the operation on psi s by h 0 and the other which comes from the operation on psi s by h 1 which contains the complicated terms leading to electron-electron correlations. So you have got two terms on the left hand side on the right hand again you expect two terms because there is a time derivative which has to be taken of this term as well as this term. So the time derivative of the operator here gives you minus i h 0 over h cross and then this function and then you take the time derivative of the interaction picture wave function which we also know is time dependent. So you get a time dependent term over here. So there are two terms now however this term psi i r t is related to the Schrodinger picture wave function through this generalized transformation operator e to the i h 0 over h cross. So you plug it in and using this you find this e to the minus i h 0 t over h cross operating on psi i gives you the Schrodinger picture wave function. Now notice that both sides have got this term the left hand side has got this coming from one of the two terms which come from the time derivative and the left hand side has got this term because the left hand side Hamiltonian was written in two pieces one the soluble part is 0 and the other is the correlation part. So these two terms cancel each other. So these two terms are essentially the same they cancel each other and you are left with the remaining relationship which again looks like the Schrodinger equation but there are subtle differences and important differences. So this Schrodinger picture wave function is now given by this this is the transformation. What it does is if you now write this only in terms of the interaction picture terms you write this wave function in terms of the interaction picture wave function but to do that you must operate upon this by e to the minus i h 0 over h cross t the right hand side is the same at this stage and now on this result you operate on both sides by e to the plus i h 0 over h cross t this is the transformation operator O which affects the generalized rotation to the Dirac picture. So you take this relation in this blue box and operate on both sides by this e to the i h 0 over h cross. So this term together with this term on the right hand side will give you the unit operator and you are left only with i h cross times del over del t of psi i on the right side and on the left side you have got this transformation of h 1 and what is this transformation? This is the transformation of the h 1 operator from the Schrodinger to the Dirac picture because that transformation is affected through the operator O which is O omega O dagger and you have essentially the interaction picture Hamiltonian but now even if this equation looks just like the Schrodinger equation there are important differences the physics is the same when we looked at a similar relation in the Schrodinger picture we had the wave functions which had the entire time dependence we had the operators which had no time dependence but now you have this interaction picture Hamiltonian but what is this Hamiltonian? It has got information coming in from both the soluble part as well as the part which is not soluble the soluble part is coming over here h 0 is the part of the Hamiltonian which is the soluble part h 1 is the one which we are not able to deal with so you have got a relation which looks very similar to the Schrodinger equation it has got the same physics but here the focus the central term is h 1 it doesn't mean that it has nothing to do with the soluble part it is of course implicit because after all it is this product of these three operators in that particular order which gives you the h i t so this is the subscript i for the interaction picture correlation part of the term okay so this is a very similar relation but it has important differences and it is a result of these transformations affected through the generalized transformation generalized rotation in the Hilbert space affected through the soluble part h 0 of the Hamiltonian now this is a transformation only of the difficult part or the trouble monger if you might call it right but then note that h 0 the soluble part also plays a role because o and o dagger involve the transformation terms okay so this is the interaction picture term interaction picture Hamiltonian this is the transformation of the difficult the trouble part but it is affected through the simple nice yes the interaction picture term we are using the interaction picture only because of the see the other term the wave function the act of the h 0 on psi only gives you a phase that is the only reason why yeah using the interaction picture right yeah the interaction picture is going to render the operators time dependent the Schrodinger picture operators are not dependent on time okay the Hamiltonian in the Schrodinger picture is completely independent of time we are developing methods so that we can use a formulation in which the operators become time dependent and the operators become time dependent in the Heisenberg picture they also become time dependent in the interaction picture and to develop the powerful techniques of Feynman diagrams the most convenient framework is that of the interaction picture so I am laying down the groundwork for that you are doing essentially the same physics so as of this stage we have not introduced any methodology to deal with the residual correlation but that is something that we are going to develop in the next few classes in this unit so the first task in today's class is to show how you affect these transformations and these are necessary because our essential problem is that we are not able to deal with the electron-electron correlations in the Schrodinger picture we tried using perturbation theory we succeeded with first order perturbation theory but then when we used the second order or higher order perturbation theory we fail so perturbative methods are not useful to deal with that one way of dealing with it was the RPA which we discussed in the previous unit but now we are using we are developing other methods which make use of Feynman diagram methods and these will make use of the interaction picture but to be able to use the interaction picture we need the operators to be time dependent so what we have done through this transformation the transformation is affected through this transformation operator e to the i h0 over h cross t so this is the transformation operator O it provides a generalized rotation of the state vectors so this is the wave function and this is transformed to a new function which is on the left hand side and the new wave function is obtained from the previous old wave function in the Schrodinger picture by a transformation operator which is made up of this exponential function operator but it has only the unperturbed part of the Hamiltonian only the h0 and this makes both the wave function psi i time dependent and the operators omega also time dependent now when you carry out this transformation on the electron electron term and that is where the correlations are coming from that is where the Coulomb correlations are coming from the statistical correlations we know how to handle but the Coulomb correlations have to be handled and through this transformation this operator h1 which is otherwise independent of time this operator h1 which is independent of time now becomes time dependent the time dependence is coming here so we have succeeded in expressing the electron electron interaction term this is the difficult term which was otherwise independent of time but we have not succeeded in expressing it as a time dependent operator but correspondingly the wave functions are also transformed and then we will do physics in the interaction picture rather than the Schrodinger picture so the subscript i over here looks similar to the number 1 over here so the two fonts are very similar but keep track of the fact that h1 is the electron electron interaction term hi is the corresponding interaction picture term which is a transformation of h1 through the generalized rotation affected by the operator which carry out the transformation from the Schrodinger picture to the interaction picture so this is the interaction picture term and this is a transformation of what only the h1 part not the h0 part this is a transformation only of the h1 part but the transformation is by the operator by the operator e to the i h0 over h cross t and this is where the soluble part of the Hamiltonian shows up so you have got time dependence which is contained in the wave function psi i the interaction picture wave functions are time dependent and the interaction picture operators are also time dependent both are time dependent and the time dependence is governed both by h0 as well as by h1 as it ought to be because the physics is the same so this is again a unitary transformation and notice that if there were no interaction the electron electron interaction were 0 then this h1 would be 0 hi would be 0 and what if hi is 0 then the time derivative of the interaction picture would also be 0 and then the interaction picture becomes independent of time so it is particularly tailored to deal with the electron electron term which is the one which we really want to focus on the soluble part we already know how to do it in quantum mechanics you know it in your first course in quantum theory so it is the difficult part and that is what is the focus in the Dirac picture so if there were no electron electron interaction then the interaction picture wave function becomes time independent and you get a result similar to the Heisenberg picture but that is a special case when there is no electron electron interaction all right now let us study the time development operator now now what the time development operator does is it gives you the wave function at a later time t from an earlier time t0 so this is the time evolution from the time t0 to the time t and this is the result of the time evolution operator and its properties are quite familiar to you because you can carry out a transformation from t0 to t through an intermediate instant of time t1 so first carry out look at the evolution from t0 to t1 and then the evolution from t1 to t so you can have a cascading effect like this so this is an essential property of the time evolution operator you can also see that if you look at the evolution from an instant of time t to the same instant of time t then of course you have the unit operator right so if you break this into any intermediate step you have u t comma t equal to 1 from t1 evolution from t3 to t1 can be done in two stages t3 to t2 and then t2 to t1 you can do it you can break the time interval into two pieces three pieces or even infinite pieces it is a continuous time interval you can break it into infinite pieces so these are the properties of the time evolution operator we have already seen that if you break it from if you look at the time evolution from t to t0 and then from t0 to t then you find that the inverse of t0 to t evolution is the same as t t0 essentially if you look at these properties you find that the inverse exists there is a closure property the unit operator exists and these are the properties of operators which constitute a group because you have the closure you have got the existence of the unit operator and you also have the inverse so they these time evolution operators they constitute a group so this is the unitary transformation the reason it is unitary is because it preserves the norm you look at the wave function you construct its norm so this is the norm of the wave function in the interaction picture and you obtain it by looking at the evolution from t0 to t on the this adjoint vector is given over here at this operator being unitary you have udagger u equal to 1 and what you find is that the time evolution operator is essentially a unitary operator because it preserves the norm so these properties will be utilized in our analysis so you have got the interaction picture Schrodinger equation now this is the time evolution from time t0 to t of the interaction picture wave function this is for an arbitrary time t0 and essentially you get from this because this result holds good for any arbitrary interaction picture wave function so there is a corresponding operator equivalence of the operator which is operating on this so that operator equivalence is h i u on the left hand side is equal to i h cross del over del t of the time evolution operator so this is the equation of motion of the time development operator so we have got in this the initial state was recognized at t equal to 0 but if the initials time is some arbitrary time t0 then here instead of this minus i over h cross h t instead of this t you will have t minus t0 so it is just an offset of the 0 of your timescale so if your 0 of the timescale is at t0 you have this more general form and for this form you have the t minus t0 coming over here keep track of the fact that there are two exponential operators one containing the unperturbed part of the Hamiltonian which is at 0 that is the soluble part here also you have got a similar term but this is the full Hamiltonian so you have to keep track of these details okay so that is where what we have got this is the Schrodinger picture wave function which is related to the interaction picture wave function so you can write this result for the interaction picture in terms of the interaction picture wave functions so now this term which is equivalently written by this term and now you have this left hand side psi i r t coming over here and in the right hand side you have got this term which comes here you have got the middle term which has got the full Hamiltonian which comes over here with the t minus t0 and then the last term is given over here so this becomes your time evolution operator for the interaction picture this is what describes the time evolution from t0 to time t for the interaction picture time evolution operator so this is the time evolution operator in the interaction picture and you can explore its properties also so you must remember that you have got the exponential operators in which you have got the Hamiltonian but you have got the unperturbed Hamiltonian here you have got the unperturbed Hamiltonian edge 0 over here you have got the full Hamiltonian over here so we know the usual property when you take the joint of an operator of a joint of the product of an operator you get this relation so if you take the joint of this operator you get the adjoint of this then the adjoint of this and the adjoint of this comes over here or essentially you find that this is also a unitary operator so in the interaction picture also you have got the time evolution described by a unitary operator so these are you know more or less obvious properties but then you have to demonstrate their properties using explicit transformations and the transformations have to be done carefully because you have got the Hamiltonian in the exponential terms but the unperturbed Hamiltonian here and here but the full Hamiltonian over here so this is the equation of motion for the time development operator now and it has got a formal solution which we have just seen which is given by this product and this is where I will take a break and stop for the day and we will continue from here in the next class remember that in general the full Hamiltonian does not commute with the unperturbed part so the order in which you carry out this product is critical that is absolutely important and that is where I will stop and with this platform I will introduce the chronological operator which is known as the Dyson's chronological operator in the next class if there is any question I will be happy to take otherwise we break here and resume in the next class.