 I will just go back and introduce the creation operator first that if it acts on a vacuum state it gives you a state which has the ith spin orbit and then if you add a j dagger on this then you make a determinant which is j i which means essentially it is a determinant j is the first column so this is the rule of creating electrons and then we found that the a i dagger a j dagger plus a j dagger a i dagger is zero the anti-computator is zero so then we are introducing the reverse operator which is basically the annihilation operator so the annihilation operators do exactly opposite first to know that if it acts on a vacuum it simply gives you zero because it cannot annihilate nothing is there it however if it acts on a state or a determinant which has chi i on the left and then whatever is there chi i and then let us say chi one etc to chi n minus one then it generates a state which is chi one to chi n minus one so the idea is that these are annihilation operators which immediately remove a spin orbital exactly to the right immediately to the right so these are called as annihilation operators what we wanted to show is that these annihilation operators are adjoint of the creation operator so that is what we wanted to show so this is the definition of the annihilation operator why is it an annihilation operator I have written now in a i but of course this is a i dagger so it is deliberately written because I will now show that they are adjoints of this so we did that last time so let us take a determinant which is a i dagger a j dagger vacuum so exactly negative of this so let us take a two electron determinant a i dagger a j dagger vacuum so that is i j right by definition first j will be created then i will be created is it okay so I can write this as a i dagger then I can write this as a j dagger vacuum which is a i dagger j the same state k I am writing as a i dagger j because a j dagger vacuum is a state is a state where only one electron is there in chi j orbital correct so now let us take this conjugate of this so let us write this conjugate as j a i dagger dagger and I will call this j a i I will show that this a i is an annihilation operator right now this a i is just a conjugate of a i dagger do not interpret as an annihilation operator because I will show that they are actually adjoints of the creation operator so I just write this as j a i is it okay so then we do the normalization of k k what is the value of k k because k is a two electron determinant k k is one right because all these determinants are normalized they are all orthonormal orbitals so k k is one so I write for the conjugate k this j a i and on the right side I simply write k so this k is written as j a i and the right side I wrote just the k okay so now I am going to interpret j a i k is equal to one so what can be one unless this itself is j so I made this one in two ways one is this determinant normalization is also one two electron another is the orbitals are also normalized so now I propose that the a i k must be equal to j at this point a i is not an annihilation operator a i is simply adjoint of the creation operator right but now I am showing that this a i acting on k must give you j so what is my k k is i j so a i acting on i j must give me j hence I show that the a i is an annihilation operator and it annihilates exactly the next to it so both these rules okay so I initially of course yesterday I had discussed the annihilation operator but I have not showed I am not showing yesterday I was showing it actually but probably did not complete it that why are they adjoints of creation operator so to show this I just take a two electron determinant and then write this as a i dagger a j dagger vacuum which is j and write this the conjugate of k as j a i dagger dagger which is now defined as j a i but this a i need not be an annihilation operator then I show that the j a i k must be equal to one which means a i k must be equal to j which essentially means that a i annihilates on a spin orbital i from k is it clear I hope everybody gets the logic it is a little involved logic but it is easy you can this is a two electron but you can show it in a many many many other ways okay so essentially if you show this then it is easy to show the anti-computation a i a j equal to 0 just by taking the conjugate of this I have already proved I can prove it by showing that this annihilation operator in the same way yesterday I did that but a simpler proof is just to take a conjugate of this as soon as you understand that the annihilation operators are nothing but adjoints of the creation operator okay then we will when we ask the question what happens if I have a commutation between two operators a i and a j dagger so one of them is creation one of them is annihilation what happens to the anti-computation between these two operators okay these parts are clear any i and j it is 0 and of course specifically also for i equal to j i equal to j essentially shows that you cannot destroy twice because a i i a i a i becomes 0 so you cannot destroy it twice okay so remember the anti-computation here is very important if this was a commutation for i equal to j you would not have got anything because it means a i a i minus a i i is 0 that is a triviality right so you would not have got the new physics so here the physics is that if i equal to j a i a i plus a i a i is 0 the plus is important if it was a commutation then of course it is trivially 0 so I do not need I do not get a new physics you understand here because of the plus I can now show that a i a i must be 0 so null operator which means I cannot annihilate twice just as I cannot create twice the same orbital so the so this is why the anti-computation is very important the fact that I have an anti-computation relation and not a commutation relation because commutation relation would not have given this because that would be an identity for i i equal to j okay so let us ask this question now for a i a j dagger let us do it in two steps so let first I take i equal to j so it means I want to define what is a i a i dagger plus a i dagger a i right so I am asking this question of a i a j dagger or a j a i dagger whatever for i equal to j so I have the anti-computator which is now a i a i dagger plus a i dagger a i correct so quite clearly let us let it act on a general determinant so I will find out what happens if it acts on a general determinant so there are two three possibilities let us say the determinant contains chi i okay then of course this will act but this will not act okay if it does not contain chi i then this will act if it contains chi i then this will act this will not act so only one of them will act okay so let us take this a i a i dagger when the determinant does not include chi i so what will it do it is a very trivial thing it will first create and then it will annihilate so I will simply get back the determinant and of course if a i dagger a i acts it will be 0 so if chi i is present in the determinant then I can say that the anti-computator a i a i dagger acting on the determinant is equal to the same determinant if i is present in the determinant so that is a simple way of writing it is a simple group notation that i is included in the determinant okay i is not included in the determinant that is what I prove okay so then of course a i a i dagger will first create and annihilate the next is i is included in the determinant if i is included in the determinant then of course the first one will not act that will give you 0 so what will act is just a i dagger a i so a i dagger a i acting on the determinant and let us say now the determinant is something like this chi i some chi n so chi i is included somewhere I do not know where somewhere it is included however to let this act on chi i I must bring chi i here so a simple way to do that is to flip k and a i so I get minus a i dagger a i chi i chi k k chi n so simply replace k by i so two two columns of a determinant just changed then of course it just a negative sign so then I can allow a i to act on chi i and then make a i dagger here so I will get back minus chi i chi k chi n correct because I will I will first annihilate and then create and then I push it back again here so I will get chi k here chi i here chi n here with the plus sign so exactly as it was right so I first push it but let's say that let's say my first column was k some k and i arbitrarily lies somewhere so I first change this to first column then I annihilate and create and then push it back so the two negative signs will again give me positive and the other one will no longer act a i dagger a i the a i a i dagger so essentially I can again say that the commutator of a i a i dagger of this determinant is also the same so this two is one i is included in the determinant another i not included in the determinant in either case if the commu anti commutator works on the determinant it gives you back the same determinant right and of course these two sets of determinants span the entire set because either in a determinant i is present or not present that's only two options right so then I can say for any general function if this anti commutator works it will give you back the same function so in one case one part of the anti commutator it is working in another case the other part of the anti commutator is working among the two so I can now write that a relation that a i a dagger is equal to one when i is equal to j then we come to the next case if i is not equal to j so explicitly now we are talking of case b where i and j are different remember that they are different now so we again look at a i a j dagger now plus a j dagger a i acting on a general determinant and let us say when this will survive when this will not survive if you look at it very carefully if either way j is present and i is absent it will not survive because if j is present and i is absent this cannot act neither can this act because j is present so I can't create right and in fact if either of them is either of them is true this will not act so the only time this will give a non zero value possibly is when i is present and j is absent is it clear so this determinant can exist or give a non zero value this action can give a non zero value only if i is present and j is absent in the starting determinant otherwise it will become zero is it clear to everybody because then either a i will not act or a j dagger will not act and now remember a i and a j dagger are different i is not equal to j so you can't say that no no i first create and then annihilate that is already done they are now explicitly different okay so so if if j is not present if j is present then of course this will automatically give you zero if i is absent then this will give you zero and this will also give you zero because even if I create j I can't annihilate i because i cannot be equal to j so each of the strings will give you zero each of the strings so there is an unlike in that case that either this will act or this will act each of the string is going to be zero unless both are both are true which means i must be present and j must be absent both must be true is it clear because you look at this string it is a i a j dagger a j dagger a i correct so so if if if this is not true each of them will become zero because if i is not present a i cannot act if j is present then a j dagger can attack yes that's what i and j then then a j dagger will not act a i will act so what eventually whole thing has to give you non-zero you take take a determinant a j dagger a i both i and j are present so let's take chi i chi j okay so this will act this will give me a j dagger chi j this is zero see eventually the string has to act the product operator has to act so it is not a question of that this acts and gives you something after that what happens okay so this is the only condition in which this will survive and let us let us analyze what happens then is it clear because I have already eliminated i equal to j k so I have shown that this is equal to 1 a i dagger a i equal to 1 so now you cannot say that I will create an annihilate the same orbital because they are two different orbital now explicitly i is not equal to j okay is it clear alright so let me now take that determinant so my determinant is i is present j is absent so I have a i a j dagger plus a j dagger a i on a determinant which is now let's say generally chi k etc somewhere chi i is exactly like the way I write chi l or chi n whatever so I am just marking chi i chi i is an arbitrary determinant the orbital which is now present and of course very importantly j is absent so j is absent so let us see now how how do I do this before I even start this problem let me bring chi i here because I know I have to do that so let me write this as a minus a i a j dagger plus a j dagger a i with chi i here and chi k here so all I have done is to interchange chi i and chi k so that is you don't have to bother about how many operations are there just interchange so my whole impetus is to bring chi i right in the beginning so now let us allow each of these operators to act and then sum and and don't forget the negative sign okay so when a j dagger a i acts what happens you create you'd annihilate i and immediately create j right here so i is replaced by j correct and when a i a j dagger comes again j comes but then I have to first interchange chi i to annihilate okay so let's do this first so a i a j dagger chi i chi k chi n so what is the result of this I first have a i then I I have one electron extra now because I am allowing a j dagger to act so chi j chi i chi k chi n correct then I interchange so I make it minus of a i then chi i chi j chi k chi n now I can annihilate chi i so this is gone so the result is minus chi j chi k chi n so what did I do I said if a i a j dagger acts on this determinant chi i chi k chi n it will give you minus chi j chi k chi n because of the fact that I am first creating j and then annihilating i I had to do another interchange I have a negative sign here so don't forget it so that I will do at the end okay so the next is a j dagger a i so a j dagger a i acting on the same determinant now chi i chi k chi n remember I had an arbitrary determinant I have first this negative sign is to bring chi i here okay with chi interchanging with chi k now this is easy because simply i is annihilated j is great this is very easy if you first annihilate and create all you do is to create in that place okay but by interchanging you can easily show this so this now becomes chi j chi k chi n okay exactly like this determinant but with a without the negative sign now it is very clear either I take the negative sign or doesn't matter it is 0 the sum of these two is 0 because it is the same determinant with opposite sign so of course I can then conclude that a i a j dagger plus a j dagger a i acting on this determinant where what is this determinant this determinant is where I present j absent remember explicitly is also equal to 0 and we have already noted in all other cases the anti this is anyway 0 the anti commentary 0 that I have already noted that it will survive only for this case but even in this case it is 0 right so I can then say that if i not equal to 0 by again the same argument that any given state can be expanded in terms of all these sets of determinants I can argue that the a i a j dagger is equal to 0 if i not equal to j for the case of a i dagger a i it is fairly simple okay that is if there is i you annihilate you bring it annihilate and bring it back so it is same if i is that is if i is present if i is absent the this a i dagger a i will become 0 but then the a i a i dagger will act so either one of these two terms will be there depending on chi is there chi is there or not there and each in each case if you will get back the determinant so basically the sum of this is equal to 1 so either 0 plus 1 or 1 plus 0 so if chi is present then the if chi is present then this will act a i dagger a i first it will annihilate then it will create you can't create first because chi is already present so the question is that if chi is absent this will act and the other one you will not question is what you do first that is very important i am a i dagger a i of course means i am creating an annihilating the same orbital but what you do first is very important so depending on this will survive one of these two terms will survive but since you are talking of plus and a anti-combatator anyway one of the terms will survive to give you one whereas for i not equal to j i showed everything anyway becomes 0 only if i is present j is absent it can survive but even in this case i showed that this is equal to 0 okay so first we had shown remember a i dagger a j dagger equal to 0 anti-combatator then i argued that the annihilation operator that we defined are the adjoints of the creation operators i showed this today so then of course the anti-combatation relation of the annihilation operator comes straight forward from the adjoint to the creation operator and then i showed the anti-combatation between one creation one annihilation operator which essentially takes care of everything so this particular thing can now be rewritten as of course you know how to rewrite with a chronica delta right so don't forget your chronica delta so whenever this is 1 and 0 so i want to avoid so eventually of course it is just delta ij and then delta will take care is it okay let me ask the question can you show that two spin orbitals kai and kai j of course you know equal to delta ij and this is the cornerstone of our anti-combatation relation but can you show this now by second quantization you understand the question if i just show using the second quantization anti-combatation relation show that because obviously the anti-combatation relations are based on orthonormality so now orthonormality you should be able to prove by anti-combatation so i am just reversing the problem remember it's a chronica delta so if i equal to j it is 1 i not equal to j it is 0 so now you know we just now did i hope all of you can do it but let me just solve it so what is kai i adjoint it is vacuum ai because remember what is kai i ket kai i ket is ai dagger vacuum correct so what will be the conjugate kai i vacuum ai dagger dagger and i have showed that this is annihilation operator so it is vacuum ai so the annihilation operator acts on the left of the vacuum creation operator acts on the right of the vacuum annihilation operator cannot act on the right of the vacuum because there is nothing to annihilate but left it will act because of an adjoint then i write kai j it is a j dagger vacuum so i am writing the left hand so vacuum ai now what will you do you will actually go back and use this anti-combatation so how do i write the anti-combatation ai a j dagger remember the anti-combatation ai a j dagger equal to delta i j minus a j dagger ai correct remember how to use this ai a j dagger plus a j dagger ai equal to delta i j so ai a j dagger is delta i j minus a j dagger ai so let me put this here vacuum i am just writing v to simplify v delta i j minus a j dagger ai v correct now look at the first half delta i j is a number chronicle delta is a number either one or zero does not matter so it comes out and vacuum vacuum expectation value is i norm is one correct so vacuum also is normalized state so you have just delta i j minus vacuum a j dagger ai vacuum correct is it is it okay and now you see a j dagger ai acting on vacuum is zero because ai cannot annihilate neither can a j dagger can dagger can act on the left so this is zero so the result is delta i j note what we are doing is that based on the orthogonality we already did this anti-combatation relation now i am reversing the problem i am saying the anti-combatations are god once i have anti-combatation relations i don't need to know anything else that is the whole idea of second quantization can you now construct everything from the anti-combatation so including the fact that the spin orbitals are orthonormal because if my anti-combatation is right the spin orbitals must be orthonormal so the creation of the electron in those spin orbitals those spin orbitals must be orthonormal so i am just trying to prove that chi i chi j equal to delta i j so i hope you see how the proof goes okay let's take another problem which i wanted to take first remember among the most used anti-combatation you will actually need to know this this will be the most used anti-combatation most useful anti-combatation and this is written in this form ai a j dagger is delta i j minus a j dagger a of course this could have been a j dagger ai this could have been ai a j dagger it doesn't matter they are plus so it doesn't matter so now let me take another problem yeah so let's say given a determinant k so given a determinant k which is chi 1 chi 2 chi n that's like a Hartree-Fock determinant right it's a Hartree-Fock determinant or whatever determinant doesn't matter how do i write this i write this as a 1 dagger a 2 dagger a n dagger correct remember this is how i have to write it the first one will come actually last because i am building the first column with a n dagger then a n minus 1 dagger etcetera etcetera a 1 dagger so just please learn how to write this just looking at the determinant the order is very very important because if you interchange then there will be sign change so this determinant can be written this in terms of v is just a short form of vacuum so that's a given now we want to show that k ai dagger a j k equal to 1 if i is equal to j and that orbital i is included in the subset of 1 to n okay i hope you understand is 0 otherwise otherwise means otherwise so i hope specific things you should be able to understand if i is equal to j i am annihilating ai so ai must be included here otherwise i can annihilate right okay i am now saying that if if otherwise what happens if i is not equal to j then what what are the other possibilities that i can annihilate j here okay and then i if i annihilate i have to i have to create but i cannot create if i and j are in the subspace so the only trick only thing to understand is that both i and j i should also write both i and j are in the subspace of 1 to n so of course in that case i cannot create so you can physically understand so that is my i and j i and j are basically among the 1 to n orbitals so in such a case it is equal to 1 if i equal to j and 0 otherwise that means if i is not equal to j it will become 0