 Electrons have spin, so whenever we talk of electron eigen function, we have to talk in terms of spin as well. So let us say that the eigen function of this H of 1, although H need not contain spin, H may be only a function of space coordinates, electron still have to be attached on that space wave function with a spin alpha orbiter. So we will call these orbitals or spin orbitals. So let us say this is the eigen value equation of the one particle operator. So now please note a symbol that I am using kai. So these kais are what I now call spin orbital. So we will try to make sure that we use this symbol whenever we have spin orbital. What is a spin orbital? Again I repeat it is a one electron function containing the space coordinates and the spin coordinates. So let me write it very specifically. So H again depends on only r1, three dimensional r1 which can be r1, theta1, phi1 or x1, y1, z1 whatever, whatever coordinate system. However kai now depends on r1 of course but also on the spin coordinates which I now call omega. So my spin coordinates are r and omega. So this will be energy of kai r1 omega1. So this is the long form of this equation. When I am writing the short form one, remember one refers to the coordinates of the electron one and whatever coordinates are required. So this is space only, this can be spin. This can also have a spin in a many particle problem or in general also even for hydrogen atom when there are other effects like relativistic effects and so on. We are not worrying about those effects in the Hamiltonian. So our entire discussion is actually a non relativistic quantum mechanics here. So the operator itself does not have a spin. However the wave function of the electron must have a spin and we are calling this spin coordinate as omega1. Remember the orbital spin, this orbital, the spin part is not omega1. The coordinates are only omega1 just like the coordinates here are r. The functions which depend on omega will be called the spin functions and these two, there are two such functions. One is an alpha function with a half up spin, another is a beta function with a down spin. If I can solve the regular space part, all I have to do is to attach the alpha part or the beta part to generate the chi. So let us say what are the chi's? So chi's are these chi's which are called spin orbitals. So the spin orbitals chi, i are now product of the space part and spin part. We have to be very, very careful about these symbols as much as possible from now on. So let me write down a chi of r omega. Note that this was r1 omega1, I am writing generally r omega. So it can be r1, r2, whatever. So this has to be a product of a space part alone and I am calling this phi. Let us say from phi k r and a spin part which is either alpha or beta. So it will make it very clear to you that what I mean by the spin coordinate omega. The functions are actually alpha and beta as you have learned, up spin and down spin. But those functions also depend on a coordinate just like your 1s orbital in hydrogen atom depends on r theta phi. So those coordinates are what I am calling omega. So omega is not the actual function, the function is either alpha or beta. So for every space part, how many spin orbitals? This is an orbital, this is a spin orbital. So for every orbital, I can generate two spin orbitals, trivially by attaching two spin functions. And there are only two spin functions, alpha omega, beta omega. There are many space parts but spin functions are only two. So for every space function, I can generate two spin orbitals. So these are called spin orbitals just because they are one electron functions. But functions of space and spin coordinates and these justify k of r, we will call orbital. So please make sure that the symbols you understand. When I say orbital, that means it is just the space part. When I say spin orbital, it is the space part multiplied by the spin part and this is just the spin function. So you do not have to bother about this. This space and spin, there is no coupling in my Hamiltonian. We cannot do that if the spin orbital coupling is there. There is a coupling, what is called spin orbit coupling, which is not there in Hamiltonian. So that is what I know, I first said that my Hamiltonian depends only on the space part. So spin is just attached. There is no coupling. It has to be in the wave function with electrons as spin. But those two spins are uncoupled with the space part. So they can just like a non-interacting Hamiltonian, we are attaching the Eigen function as a product. So here we simply attach as a product because there is no coupling. So otherwise you are right, they actually have a separate basis called the spinor basis, which are more complicated basis for those kinds of couplings. And those are dealt in a different level and quantum mechanics. But our quantum mechanics is non-relativistic. So the spin simply gets attached. The spin is very simple to a one particle part. But when you generate many particles, you will see that there are complications even here. But as of now, for one particle it is just trivial. You simply attach it. So that is the reason I said orbital is a three-dimensional quantity, spin orbital is a four-dimensional quantity. Because this r has three and omega adds to a one-dimension, so you have total four-dimension for the spin orbitals. So let us assume that we know how to solve this one particle problem for a non-interacting Hamiltonian. Then we said that the Eigen function of these must be a product of these spin orbitals. But now we know that this product must be anti-symmetrized product. This must be an anti-symmetrized product. So we will see how to do that because we have already written that they must be anti-symmetrized product. So let us assume that we have solved this problem and we have got set of spin orbitals, chi-1, chi-2, chi-3 etc. Which are basically nothing but set of space orbitals with spin attached. And each of them has an eigenvalue epsilon-1, epsilon-2, epsilon-3 and so on. Quite trivially these space parts can be solved separately, in which case of course the eigenvalues will be equal. Like if I have these two have same space part, the eigenvalues of E1 and E2 will be equal and so on. But in general I am writing this as E1, E2, E3. Right now you do not have to bother which are equal and not equal. Many of them may be equal. So I have a set of spin orbitals for this one particle problem and a set of eigenvalues of this one particle problem. Then I want to construct the wave function for the two particle problem. So this is the wave function. Remember this wave function is no longer orbital because orbital has to be one particle. Spin orbital or orbital has to be one particle. So it is a two particle function and just calling it wave function. Which should be the eigenfunction of this Hamiltonian. What does our theorem says? What does our theorem says? The theorem says that this psi of 1, 2 must be an anti-symmetrized product of the eigenfunctions of the one particle problem. So for example psi of 2, 1, 2 can be a product of any sets of chi 1. Now the only catch is that they must be anti-symmetrized product. So let us assume what we can. Let us say I put these both the electrons in chi 1 to start with. Let us say. So let us analyze this function. Where I write psi of 1, 2 as a product of two one particle functions and both of them are in chi 1, spin orbital. Now let us analyze this. Is it anti-symmetric? What is the answer? No. All of you are convinced? That is if I interchange 1 and 2, they become identical. So it is not anti-symmetric. It is actually a symmetric function. And this can be a valid non-interacting Bose eigenfunctions for Bose particles. But not for, so not for the electron. So what can I do now? So let us analyze another function. I put chi 1, 1 and chi 2, 2 since I cannot put in the same. So let us write this as a chi 1, 1 into chi 2, 2 and I ask the same question now. Now I know that already that one spin orbital cannot have two electrons because I cannot anti-symmetrize. In fact, you already know that that is Pauli-exclusion principle. So what I just now showed is basically the Pauli-exclusion principle because of the anti-symmetry Pauli-exclusion principle is valid. But now what about putting one electron in one spin orbital and another in one spin orbital? So for example, chi 1 can be space part phi 1 into alpha, chi 2 can be another same space part phi 1 times beta or different space part with a different spin part, does not matter. Now this is clearly possible and we know that for example Helium atom, we have 1s alpha, 1s beta, so this is clearly possible. So I can obviously put two electrons, one in chi 1, one in chi 2, but is it anti-symmetric? Answer again is no. So even though it is possible and we know that we can do it, clearly it is not anti-symmetric. So then the question comes how do I anti-symmetrize? If you note the next wave function, so let us analyze this wave function now where I put chi 1, 1 and chi 2, 2 and then subtract from it chi 1 of 2, chi 2 of 1, which means I have put the two electrons one in chi 1 and one in chi 2 except that the first electron here was in chi 1, the second electron here was chi 2. I am now mixing it up the second electron in chi 1, first electron in chi 2 and I have put deliberately a minus sign. Is it anti-symmetric? The answer clearly is yes because if I make interchange of 1, 2, this part becomes this part, this part becomes this part and because of the sign, one negative sign, they become anti-symmetric. So now in this case, this is a valid wave function. Right now I am not worried about the normalization. Remember it may not be normalized unity. So that part I am not worried, that can be trivially done but it is a valid wave function. So it is possible to construct a valid wave function of a two electrons by using at least two spin orbitals, only two spin orbitals I should say, not even at least. Two spin orbitals, I can construct a product. If I have more spin orbitals, I can construct more products, many more than one product but at least two are required. One is not enough because the one I cannot anti-symmetric. Note that you might have said why do not you do the same exercise here, write chi 1, 1, chi 2, chi 1, 2 and then write it again minus but you will see they are identical, so it will become 0. Had I written this chi 1, 1, chi 1, 2 and interchange 1 and 2 subtracted, it would have become 0. So there is no way and that is the Pauli principle. For at least two different spin orbitals, I can anti-symmetrize by properly combining with a negative sign, I interchange 1 and 2 with a negative sign and that becomes an anti-symmetric. Anti-symmetric, otherwise you will not get anti-symmetric. If it is plus, it is not anti-symmetric. It will become symmetric function, again Bose function because you have already said that the electronic functions must be anti-symmetric. So it is true to anti-symmetric. So there is a tremendous significance. In fact, if you put plus then it will go to a Bose function, it will not be electrons. You understand because then if I put 2 1, then this will become this, this will become this, they are they are plus, so it is a same function. So minus is extremely important. So let us analyze how do I write this function in a manner which is generalizable to a larger n particle problem. Right now we are discussing only two particle problem. Eventually you have to write for n particle problem for any pair. Before you do that, let us analyze these two's dysfunction.