 Greetings, we will begin unit 2 today and this will be on many body theory. Our focus will be on the study of electron correlations and techniques which make use of diagrammatic perturbation theory, the Feynman-Golstone diagrams and so on. So, we will develop these techniques which are of importance in studying atomic collisions and spectroscopy. And I will spend a few minutes at the start of this unit, recapitulating a few points that we did earlier in the previous course on the special topics in atomic physics. In particular, I will invoke the discussion on the Hartree-Fock formalism which is also a many body theory in a certain sense. It is a many electron theory, but then we are going to go well beyond the Hartree-Fock in the discussion in unit 2. So, let me recapitulate a few points from unit 4 of the previous course on atomic physics. And our interest in studying atoms molecules condensed matter of any kind is in studying electron interactions and electron correlations and of course, how the many body system responds to various probes like electromagnetic radiation or other particles like projectiles, electrons, positrons, composite particles and so on. So, I have underlined the words electron interactions and electron correlations to draw attention to the fact that these two terms will have different connotations in our context. And in particular, when I refer to correlations, I will be referring to two kinds of correlations. One are the exchange correlations and the other are Coulomb correlations. So, obviously Coulomb correlations and Coulomb interactions are two different things. Coulomb interaction is the usual 1 over r 1 2, Coulomb correlations is what many body theory is about. So, that is the focus of unit 2. Now, the exchange correlations come from statistics. Exchange correlations they are because of the fact that electrons are Fermi particles, Fermi Dirac particles and the statistics they must observe is the Fermi Dirac statistics and that requires that a many electron wave function must be anti-symmetric. So, there is a certain correlation coming in from statistics. In addition to that, there is a correlation which is the Coulomb correlation which is not included in the Hartree-Fock formalism and that is the one that we are going to talk about. That is a result of what we will refer to as many body correlations or many electron correlations and let me quickly remind you that statistics enters classical mechanics and quantum mechanics in different ways. In classical mechanics it enters just because you are dealing with a large number of particles and there is too much of information not all of it all the details are interesting. So, you do some averaging like in principle in classical mechanics since every single particle is distinguishable and it obeys dynamical laws in which you can predict the motion using the equation of motion. In principle you can follow the trajectory of every particle, but all this is not interesting. What is interesting is what is the average kinetic energy and what it generates is a thermodynamic parameter like the temperature. So, these are average properties and this is how statistics enters classical mechanics. In quantum mechanics statistics enters because of the uncertainty principle. So, laws of nature are intrinsically statistical there is a probabilistic interpretation that quantum mechanics demands even when you are dealing with a single particle or even when you are dealing with vacuum. So, it is nothing to do with the number of particles. Number of particles only makes it more complicated, but having fewer particles having a single particle does not eliminate statistics. And then you have the spin which is an intrinsic angular momentum for elementary particles and this also leads you to a statistical considerations because there are two kinds of particles in nature fermions and bosons and they observe different statistics and they have different spins. So, if you have two identical particles and the signature of quantum mechanics. So, far as many body formalisms are concerned is the fact that these particles are indistinguishable. So, if you have two identical particles one at coordinate q 1 and the other at coordinate q 2. If you interchange these two particles and I is the interchange operator and on the result you carry out the interchange yet again. So, you are carrying out an interchange twice you naturally expect that you will recover the original state. Now, this is the geminal wave function which is a wave function for the twin particles they are completely identical in all respects. And you will recover the same function which means that if you interchange only once you can at the most change the phase of this wave function by e to the i alpha. And e to the 2 i alpha must be unity which is what you will get over here which means that alpha is either 0 or pi and depending on it is 0 or pi you have either both particles or Fermi particles. Now, all particles which have half integer spin are Fermi particles or Fermi Dirac particles and all particles with integer spins are both particles or both Einstein particles. And typically the wave function would be written in terms of the spin quantum number as well and then you will have a one electron state written as n l m l m s for a particular state which is alpha. So, you have got a set of commuting operators. So, this gives you a complete description a complete set of commuting operators or a complete set of compatible observables as Dirac calls it CSCO and these give you the complete description and this complete description must include the spin. Now, it is very interesting that you have this connection between statistics the spin of the particles and the sign of the wave function on interchange because on interchange of both particles the wave function remains invariant whereas, on interchange of Fermi particles the wave function of the geminal changes sign. So, this connection is rather interesting and it is nevertheless not so very easy to understand as Tomonaga tells us in the statement that the relation between spin and statistics is apparent, but hard to understand and I would like to quote Feynman one more time over here that you have a rule which can be stated very simply, but the explanation is complicated it lies deep down in relativistic quantum mechanics. So, I will not go into those details because our focus in unit 2 is the discussion on how many body correlations are to be treated and what they are in the first place. So, let us deal with electrons on interchange the electron geminal the 2 electron wave function changes at sign and you consider the wave function which is a function of 2 electrons q 1 and q 2 one at q 1 the other at q 2 to be separable in one particle coordinates. So, the 2 electron wave function is written as a product of 2 1 electron functions, but the product must be an anti symmetric product as we have discussed in details in our previous course anatomic physics on the in the discussion on the Hartree-Fock. So, you have to reconcile 2 properties 2 attributes which can appear to be contradictory the particles are indistinguishable in the sense that you really cannot separate one from the other yet you talk of them as individual elementary particles in which each has an identity of its own as an elementary particle the elementary particle of nature is the electron not the twin it is not the pair of electrons. So, the elementary particle is still the electron, but the 2 electrons are indistinguishable and this is indicated over here that the 2 electron wave function is written as a product of one electron functions which respects the elementary nature of the 2 electrons, but then it is a superposition of these 2 states electron at q 1 in the quantum state 1 and electron at q 2 in the quantum state 2 and the other possibility that the electron at q 1 is in the quantum state 2 and the electron at q 2 is in quantum state 1. So, these 2 situations are not distinguishable. So, indistinguishability demands that you construct a linear superposition of this and this superposition must be an anti symmetric superposition. So, you can see that you can write this as a determinant this is just an equivalent form and this is the 2 by 2 determinant for the 2 electron system this is called as the Slater determinant you can automatically see that the poly exclusion principle and the anti symmetry of the wave function is automatically built into it, because if you interchange q 1 and q 2 the determinant would change its sign and if you have 2 rows to be the same the determinant will vanish that is the poly exclusion principle. Now, basically the single electron wave functions which are called as spin orbitals these are made up of a spin part and the orbital part. So, there is a separation between spin and orbit part over here, but the spin orbit coupling of course will give you a quantum number which is different, because l and s will combine to give you a j and then j m j will be good quantum numbers and not m l m s. So, basically you deal with these spin orbitals in the Slater determinants for an n electron system now we go quickly from the 2 electron system to an n electron system where n is any number it could be 2 or 3 or 4 or whatever and you have an n by n Slater determinant in which the columns are labeled by the q's which are the coordinates and the rows are labeled by the quantum numbers these are the complete set of commuting operators and their Eigen values. So, the quantum number is obtained from the Eigen value of the observable which is measurable and once you have a complete set of commuting observations you get a set and this set is what I call as alpha. So, alpha is not just 1 quantum number in our case it is 4 quantum numbers and together this gives us 1 set. So, each row in this Slater determinant corresponds to a particular set of alpha where alpha contains all the 4 quantum numbers. So, these are the 4 quantum numbers in alpha n, l, m l and m s and you can write this determinant also as you can take all the diagonal elements construct a product of this and then carry out the permutations of these this is how the determinant yes you have a question. The other e subscript is a combination different combinations of n, l, m, m. Exactly right. So, either one or more quantum numbers will be different when you go from one row to the other no two rows will have the same set of quantum numbers that will violate the Pauli exclusion principle and the determinant will vanish, but if even one of them is different then you are ok right. So, that difference is manifested inside the q the position I mean in each row in each row or each. The q's go from one column to the next. So, the first column is q 1, the second is q 2 and the last column is q n. So, in a particular up to q n. Yeah. So, look at all other quantum numbers Yes. So, basically what you are doing is interchanging the positions that is where the indistinguishability of the particles comes in. So, typically this is a measure of the probability amplitude that an electron at coordinate q i is in the quantum state alpha, but in the next row belonging to the same column you will have the probability amplitude that an electron at coordinate q i will be in the quantum state alpha plus 1, the next one right and that cannot be rolled out because these particles are completely indistinguishable. So, you must construct a superposition of these and this must be an anti symmetric superposition. So, that is what this later determinant guarantees. So, you can write this basically you are carrying out these permutations between these coordinates and then you must have the parity of permutation. You will have factorial n permutations possible and then 1 over root factorial n is the normalization as you can get if these individual spin orbitals which are elements of the determinant are individually normalized. So, these things we discussed at some length in our previous course on atomic physics and you may want to refer back to some of those details and I just wanted a very quick recapitulation of that and the many electron Schrodinger equation is h psi equal to e psi for the n number of particles and this is what we called as the catch 22 situation. The reason is it is catch 22 a phrase which is which comes from the novel by Joseph Heller and the main idea in this is the following that if you write the n electron Hamiltonian as a sum of these single particle operators this is the single particle of kinetic energy. This is the kinetic energy of each electron in the field of the nucleus, but this is the electron-electron repulsion and the 1 over r i j which is the electron-electron repulsion. So, this requires for your consideration the distance between these two electrons and this distance can be specified only in terms of distance between the two probability densities because charges these are not classical point charges. So, you need the probability densities which can come only from the wave function. So, you need the wave function even to construct the Hamiltonian and unless you have the Hamiltonian you cannot set up the differential equation at all. This is the catch 22 situation and to break it you follow Hartree's procedure and have approximate numerical solutions in which you obtain self consistent field solutions. So, some of these details we have discussed in our previous course I will not go through it now. This is just a quick reminder that these are the techniques which are used and then of course, you go beyond Hartree you must include the spin and that is the Hartree-Fock formalism. This is a nuclear part this is a single atomic system. So, there is a single nucleus for molecules of course, you have the nuclear-nuclear interaction this is a single nucleus. So, I am developing the formalism in the context of a single atom, but of course, in condensed matter where you have a molecule or a cluster then you have the nuclear-nuclear term as well. So, the contribution from the nucleus is not neglected it is z over r i over here for in molecule where there are more than one nucleus, you can carry out the Born-Oppenheimer approximation or you can do even better that, but that is a matter of detail. So, essentially it is hopeless to look for an exact solution of a many body system. There is no chance that you can get an exact solution, it is not that it is difficult. So, it is absolutely hopeless I think you can understand it only in those terms. So, it is impossible to get an exact solution and this is best stated in remark by Brown in his book where he says that having no body at all is already too many if you are looking for an exact solution. So, you have to look for approximate solutions and even to write down these solutions you have a very huge task because just how much of storage space would you need to write handle these slater determinants. How many terms do you have in the slater determinant? If you just take a small atom like every electron has 3 degrees of freedom. So, it has got 3 n variables and if you write the wave function just on a 10 point grid actually it is more than 3 because you know that there is a spin also. So, just to simplify this I have taken 3 if you take a 10 point grid then you have 10 to the power 3 n numbers to tabulate and you will get a very bad wave function on a grid which has got only 10 points. But even that you have 10 to the power 3 n numbers and what kind of a number is it for n equal to 1 it is already 1000 and then for neon for mercury atom these are huge numbers. So, you need techniques and the occupation number formalism the second quantization methods that we are going to discuss in this unit will tell you very elegant ways of doing this. So, you have to deal with spin as well and we know that self consistent field solutions are possible in the Hartree-Fock formalism and the main strategy of Hartree-Fock is to get an extremum of the expectation value of the n electron Hamiltonian in the slater determinant. This is the single slater determinant mind you and in this you obtain this extremum subject to the condition of constraints of the orthogonality of the single particle wave functions and their normalization as well. So, you can determine the expectation value of the single particle and two particle operators and proceed and this is what we discussed in our previous course anatomic physics. The two electron geminal state which is always nice to have some kind of handle on because all the two electron interactions are expressed in terms of these functions. Now, you can have the anti symmetry of the wave function that when you interchange q 1 and q 2 q 1 is here and q 2 is here when you interchange the first one is q 2 and the second one is q 1 the wave function must change its sign, but this sign must be attributed either to the spin part or to the orbital part and either of the two possibilities will give you an anti symmetric wave function. So, there are two possibilities here that the orbital part is anti symmetric and the spin part is symmetric or the orbital part is symmetric and the spin part is anti symmetric. So, you have two kinds of you know solutions and you must ask which part is symmetric and which part is anti symmetric and depending on whether you are dealing with a symmetric orbital part or a symmetric spin part you get the triplet or the singlet states at this again I will not spend any time discussing this we have had fairly extensive discussion in our previous course in atomic physics. So, you can certainly refer back to that and I will just like to remind you that if you consider the diagonalization of the coulomb interaction in this you know that the triplet state is less punished as Landau and Liffchitz call it has got a lower energy than the singlet. So, now you can of course write the single particle quantum numbers either as n l m l m s or as n l j m j and you can write one in terms of the other by simply carrying out this rotation of the base vectors in the coupled angular momentum space. You have got two sources of angular momentum for the electron one is orbital angular momentum the other is spin angular momentum and you can have their uncoupled spaces or you can have the coupled spaces and you can write both are equally valid you can write the wave functions either in terms of one basis or the other any basis is good all you need is a complete set of basis and some of these details again we have discussed at length in our previous course in atomic physics and some of these video lectures are also available on the internet and you can refer to that if required, but I just want to mention that you can use either of the two designations and depending on the coupling scheme if you are coupling you know if in the j m basis you have these spherical harmonic spiners which are you know which have two rows and one column. So, you can write these wave functions in terms of these spherical harmonic spiners so instead of the m l m s basis. So, here again the details can be found in unit 3 of the previous course in atomic physics, but I want to quickly go over to the many electron formalism which really requires us to go in for second quantization methods. Now, let us take a particular case of many electron system let us take the case of magnesium. Now, magnesium has got 12 electrons the usual configuration is 1 s 2 2 s 2 2 p 6 3 s 2 the 2 p is a spin orbit split because of the relativistic effects in the non relativistic quantum mechanics you will not have the spin at all. In relativistic quantum mechanics or in non relativistic quantum mechanics in which spin is plugged in on an ad hoc basis then also you can do some quantum mechanics with spin then you have j becomes a good quantum number and j is either half or 3 half for l is equal to 1 for 2 p l is equal to 1 s for the electron is half. So, you have two possibilities l plus half and l minus half from angular momentum coupling and the l plus half gives you the j equal to 3 half state in which you can have 4 electrons because this has got a degeneracy which is 4 fold for each j m j can take 4 values. So, for j equal to 3 half m j can take 3 half 1 half minus half and minus 3 half. So, it has got a 4 fold degeneracy and for j equal to half you have got a 2 fold degeneracy for j equal to half m j can be plus half or minus half. So, this is your configuration the superscript S D stands for the slater determinant and this is the slater determinant which I call a slater determinant 1 which I write as a subscript which is an anticipation of the fact that we will have a slater determinant 2, 3, 4, how far can you count? Many right. So, you can have a number of slater determinants let us consider this one. Now, instead of 3 s 2 I have got the first 10 electrons go in the same states, but the 11th and the 12th electron instead of going in 3 s go into 3 p. Now, this is also magnesium atom you have got the same number of neutrons the same number of protons the same nucleus the same number of electrons, but the configuration is different and if you are to write a slater determinant for the first configuration compare it with the slater determinant for the second configuration you obviously, recognize the fact that the 2 slater determinants must be different from each other. In other words each slater determinant corresponds to a particular configuration there is a 1 to 1 correspondence between the given configuration and the slater determinant. There is a 1 to 1 correspondence and you can have many different slater determinants you can have 2 electrons in 3 p 3 half you can have 2 electrons in 3 d 1 half 3 d 3 half 3 d of course, does not have 1 half 3 d has got either 5 half or 3 half. So, you can have what else 4 s 5 p anything you have really infinite number of possible slater determinants and strictly speaking to have a complete basis. You must include all of these and according to the fundamental consideration that we have in the expansion of an arbitrary wave function in a linearly independent complete set of basis. You must have expand the magnesium wave function in terms of this complete set of basis. So, you really have infinite slater determinants yes the coefficients of most of them may go to 0, but the coefficient of many may not go to 0 and how many really you really do need to include depends on the nature of the correlation that is there in the system in the 12 electron system. So, you have got a many body system which contains 12 electrons this is your many body system this is your many electron system and it is a correlated system. Now, if you have more slater determinant what happens to your Hartree-Fock in the Hartree-Fock you did an extremum of the Hamiltonian the expectation value of the Hamiltonian in a single slater determinant right, but now you recognize the fact that you have more than one slater determinant 2, 3, 4 may be more a certain number of slater determinants and all of these slater determinants need to be considered this is what is meant by saying that you must go beyond the Hartree-Fock and what requires you to go beyond the Hartree-Fock is the Coulomb correlation. So, this is the definition of Coulomb correlation Coulomb correlation is what is left out of the Hartree-Fock in the single configuration Hartree-Fock you of course, have the Coulomb interaction all those 2 electron integrals that you did in your previous course they are matrix elements of 1 over R i j that is the Coulomb interaction right. So, you had the 1 Coulomb interaction you also had the exchange how did you have the exchange because you had anti symmetric wave functions. So, you had the exchange you also had the Coulomb interaction, but you restricted yourself to a single slater determinant. So, the confinement to a single slater determinant is the single particle model when you go beyond that you begin to include correlations it does not mean that there is no correlation in the Hartree-Fock there is the exchange correlation, but there is no Coulomb correlation sorry because it is coming once again from the 1 over R 1 2, but the 1 over R 1 2 is coupling not only states from a single configuration, but also from this there are so many other configurations and all of these correlations they are coming because of the Coulomb interaction, but then coming from different configurations in the last 2, but they could refer in many others. Now, you have different corpical angular form right that is the I mean I thought that is why we have called Coulomb. No that is just another possible state of the 12 electron system at the reason you have this alternative state as an accessible state for the 12 electron system is because of the Coulomb correlation. They should all have the same angular momentum. So, finally, they must couple to the same j that is a matter of detail, but I am not getting into that at this point due to the anti symmetry of the wave function. It is there in each slated determinant. So, this is one slated determinant which is completely anti symmetric. So, if you interchange any 2 rows or any 2 columns the sign of the determinant will change. So, the sign of the determinant will change under any interchange. So, the exchange correlation which is the anti symmetry of the wave function or the poly exclusion principle that is already built into this and this is a different slated determinant, but the system wave function is a linear superposition of these 2. So, you are then lead to what is called as a multi configuration harsh form. So, now you cannot deal with just a single slated determinant, but a linear superposition of slated determinant. What is it coming from? It is coming because you have to consider the interaction between the first configuration and the second configuration. This is what is called as C i or the configuration interaction. So, this is the configuration interaction and this requires you to go beyond the harshly fork and that is what many body theory is about. So, even the harshly fork in a certain sense is a many body theory. In a certain sense it also has the correlation, but the correlation it has is only the exchange correlation, but the name many body theory is typically reserved for those formalisms in which you go beyond the harshly fork consider the configuration interaction and include the coulomb correlations not just the fermi correlations not just exchange correlations. So, these are what are typically called as many body correlations and I will describe it once again for the magnesium atom. So, if it has this 2 p 6 3 s 2 this is the usual configuration that one talks about this is the default that one that comes to your mind and the slated determinant for this will consist of 12 columns. So, q 1 through q 12. So, as you go from left to right you are going from first column to the second to third until you get to the 12th column and when you go through the rows 1 to the next and so on you go from different quantum states this is n l j m j quantum numbers are what I have used. So, this is the 1 s up state and then the last one will be the 3 s down with m j equal to minus 1. So, that is the 12th row. So, you have got a 12 by 12 determinant this is the determinant wave function and this is corresponding to different single particle states each row is labeled by a set of 4 quantum numbers. All the 4 quantum numbers are the same in a given row they change only from 1 row to the next in at least 1 quantum number. They may change in more than 1 quantum number like if you look at the first row and the last row the first row has got m j equal to half the last row has got m j equal to minus half but that is not the only difference the first row has got n equal to 1 the last row has got n equal to 3. So, they may change in 1 or more quantum numbers but there must be at least 1 quantum number which is different if all of them are the same then you have 2 rows which are equal in a determinant the determinant would vanish. So, this is where you have the slated determinant corresponding to the 2 p 6 3 s 2 configuration but then we agreed that you have not only this configuration but you also have the possibility of the 3 p 2 configuration and not just this there are many more but let us take this as an example. So, if you are to write a slated determinant for this then you will have the first 10 rows will be the same but the 11th and 12th row will be different and the 12th row will now have l equal to 1 here mind you because this is the 3 p state. So, you have got a 3 p number and not a 3 s anymore. So, what you are going to do is you have these different single particle states and you can arrange them in some order you can say that n equal to 1 l equal to 0 j equal to half m j equal to half is my state number 1 n equal to 1 l equal to 0 j equal to half m j equal to minus half is my state number 2 and you start giving numbers to each row. So, the first 10 you have label then 11 and 12 you label according to the 3 s 2. Now, these 2 rows you cannot label any more as 11 and 12 because the label number 11 corresponds to the 3 s up. So, this will be your 13 and 14 what is happening in the slated determinant is that 11 and 12 are empty and 13 and 14 are occupied the first 10 are the same but what was 11 and 12 with the previous configuration is now vacant and you have an occupancy of 2 in 3 p. So, in other words these different slated determinants which correspond to different configurations you can refer to them in terms of occupation numbers of single particle states. What is the occupation number of state number 1? What is the occupation number of state number 2? What is the occupation number of state number 11? What is the occupation number of state number 13? What is the occupation number of state number 20, because you have got an infinite set of slated determinants. So, you can write all of this in terms of these occupation numbers. Now, that is the formalism which corresponds to the occupation number space, and this is where we use second quantization methods, which is what we are going to discuss. Yes, Ankur, you have a question. You will always have 12 by 12, because you are describing a 12 electron system. The 11th label is empty. The 11th label is empty. The 11th row is occupied by what would be the 13th label. The 12th row will be occupied by what is the 14th label. You will always have each determinant for an n electron system will be an n by n determinant. So, it has to have 12 rows and 12 columns, but which 12 rows that set is what we are talking about. And in this set, the labels each label corresponds to a particular one electron state. And the last two labels in this are different from the last two of the previous. So, if you look at this, you have got the 3 p half is the last one. In the previous one, you had this is you had the 3 s. So, the 12th row was occupied by the 3 s quantum numbers, but now the 12th row is occupied by 3 p 1 half. So, you have 12 rows and 12 columns, but which 12 rows is different in different slated determinants. In the previous case, we are denoting by this 2 and the position is by q 11th and q 12th. And in this case, whether we should use some q 13 and q 13. No, no, you are talking about 12 electrons. You are talking about 12 electrons. What is the probability amplitude that a particular electron at coordinate q y is in the state alpha. So, here this is the probability amplitude that the electron at q 1 is in the state. This probability amplitude is different from the probability amplitude that the electron at q 1 is in the 3 s state. So, you have got 12 electrons, they are where they are in the spin orbital space, but the probability amplitude that they are in different quantum states, single particle quantum states is different, which is why the rows are different. The coordinates of course, will be from q 1 to q 12, they are only 12 electrons. It is certainly possible. So, you have the levels 11 and 13. You could have that as a possible configuration. So, you will have infinite configurations. Of course, there are some other constraints because the total angular momentum must be the same and so on. So, those are the details that we can talk about, but basically you will have many different kinds of possibilities. You can have two electron going from first configuration to the other. You can have one going from one state, the second going to a different state. So, here you have what we have done is we have elevated 3 s 2 to 3 p 2. You can elevate 3 s 2 to 3 d 2. You can elevate 3 s 2, one going to n l and the other going to n prime l prime. So, there are all kinds of possibilities and that is what makes many body theory very challenging. And all this is happening because of the electron correlations. So, there are these possibilities and the configuration interaction wave function is now a linear superposition of all these later determinants. And depending on how many configurations need to be considered in this, n can be either a small number like 2 or 3 or 4 or it can be quite large and people do calculations with hundreds, sometimes even with thousands of configurations. Now, it is a mess because each determinant has got factorial 12 terms. We already felt like quitting when we were dealing with a slingle determinant when we thought about a number like factorial 10 just for the neon atom, which is a small atom. Now, you have factorial n terms and so many of them. So, there is a good way of handling this, which is what makes use of the occupation number formalism. This is where the second quantization methods come into the picture. So, the occupation number formalism is sometimes called as the second quantization. Why second? Because the first quantization was doing away with classical dynamical variables q and p position and momentum, which were simultaneously measurable accurately in classical mechanics, recognizing the fact that such a simultaneous accurate measurement is not possible and therefore, replacing them by operators that was your quantization. Now, what we are going to do is that the wave functions that you are talking about will also be treated as operators. In the first quantization methods, the wave function psi was like a scalar function, it was not treated like an operator. Now, in the second quantization method, so it is a technique and because it is quantization of the scalar field, it is sometimes also called as field quantization, but these are different expressions to talk about the same technique. The technique is essentially that of occupation number formalism, also called as second quantization or field quantization, field theory and all of these are related terms depending on different context. They do have different emphasis contained in these alternate expressions and they are inspired by the fact in the context of the fermions that each slater determinant is decided by a particular configuration, which means that which single particle states are occupied. In the first configuration, other than the first 10, these two are occupied and these two are empty, but in this case the 3 s are empty and the 3 p are occupied. So, the occupation numbers are different and you can talk about going from one to the other by destroying two electrons in the first and creating two electrons in the other. So, you can begin to make use of operators, which are called as creation and destruction operators or creation and annihilation operators and the second quantization methods make use of these operators and that is the formalism that we are going to learn in this unit. So, your slater determinant in the first configuration was made up of these states 1, 2, 3 and 4. Then you had the 2 p 1 half, you had the 2 p 3 half and then you had two electrons in the 3 s states. So, these are the 12 single particle states. When you talk about the second configuration, the 11th and 12 those two states electrons in those two states are effectively destroyed and you create two in 3 p 1 half. So, in the second configuration you have instead of the 11 and 12, you have 13 and 14 that is what I was referring to. And you can refer to these in terms of single particle operators, creation and destruction operators and these are the ones, which are used to describe the n particle system, essentially you are describing them in terms of a complete set of compatible observables or Eigen values of complete set of commuting operators from Dirac's CSCO and you have n number of identical particles. So, the individuality of the particle that idea is carried over, but so is the indistinguishability. So, is the statistics and so are now also the many body correlations. So, you use the single particle labels, even in the presence of not just interactions, but also in the presence of correlations and I have made a very clear distinction between the term interaction and correlation. So, correlation has got a specific connotation in the context in which we are discussing this. So, you describe a particular slater determinant, a particular configuration by spelling out how many particles are there in state alpha 1, each alpha is a set of 4 quantum numbers, how many particles are there in state alpha 2. Now, this number is either 1 or 0 for Fermi particles, but it can be anything for both particles, because there is no poly exclusion principle for bosons. You can put any number of particles in a one both state, as a matter of fact you can put all of them and get both bosons and condensation. So, there is no such restriction in that, but the basic property of any configuration is of any many electron system is how many particles reside in these single particle states and that number is the occupation number of that state. This will give you one configuration, but then of course, the system wave function will contain many other configurations and you will have a linear superposition of such states. So, this is the idea behind indistinguishability, you take into account the correlations, you recognize the elementary nature of these particles, although they are indistinguishable. So, you reconcile with that and essentially you can get these numbers as Eigen values of the occupation number operators, which are complete set of commuting Hermitian operators. You define these occupation number operators and of course, the description will be different in terms for both particles and for Fermi particles, because if you have created a particle in one vacuum state and if that particle is a Fermion, you cannot create any more in that state, but if it is a Bose particle, the creation operator can operate twice or 10 times or have a Bose Einstein condensation and pump everything into that. So, the properties, the commutation properties of the creation and destruction operators for Fermi and Bose particles will be different. So, the statistics will now be incorporated in the operators. So, you have some sort of an arrangement, this alpha 1, alpha 2, alpha 3 and is some predetermined sequence, you decide what is your state number 11, you decide what is your state number 12, decide what is your state number 13, 14 and then you say that state number 11 and 12 is occupied in this and 13 and 14 is vacant and in the next configuration 11 and 12 are vacant and 13 and 14, but these numbers have to be pre-assigned. So, you have some sort of a predetermined sequence and in this sequence, you give the occupation numbers and then you have a vacuum state in which all of these particles, all of these states are vacant. You can have a single particle state, in which there is only one particle in the state i, so in the i th state n i is equal to 1 and whenever j is not equal to i, this occupation number only for the i th state the occupation number is equal to 1, whenever j is not equal to i, the occupation number is 0. So, this is how you would describe a single particle state and depending on the number of particles, you can write different occupation number states. So, I will conclude my discussion today, I will be happy to take a few questions and we will go from here in the next class. In the meantime, there are some references which I will like to draw your attention to, for Hartree-Fock, we have already given these references in the previous course, which is either Betty and Jackie or Branstad and Joshain, these are the primary sources and for second quantization and occupation number formalism, I will primarily really use Fetran-Valekas book or the book by Reims called many electron theory. Questions? In the magnesium 12 electron system, the alphas will be the set of alphas, you are referring to each configuration will be having 12 ones and all of them. So, each of that is a configuration. Yes, exactly. So, out of the infinite numbers, 12 will be occupied, all the others will be 0, but which 12 is going to be different for each configuration, a particular choice of 12 defines once later determinant. So, there is a one to one correspondence between a configuration and a slated determinant and an occupation number state vector. So, these are state vectors in what is called as the occupation number space, sometimes also called as a Fox space. Maybe a range annihilation and… Right, right. So, in our next class, we will get into these second quantization operators, the creation and destruction operators and the number operator and so on. Alright, thank you.