 So, we have been discussing about the density operator calculations, density matrix calculations. We introduced last time a specific formalism known as product operator formalism, product operator formalism which will facilitate the calculation of the density operator at any point in time in a pulse sequence. And this we said this is for weakly coupled spin systems, weakly coupled spin systems which is generally valid in most of the modern spectrometers which have very high fields. So, for this means condition was J by delta is much much less than 1. So, in this formalism the density operator is written as a summation of some basis set of operators which we wrote it in this form as BS. So, this BS set constitutes a complete basis set. This is a complete basis set. So, that any density operator can be expressed as a linear combination of the elements of this complete basis. So, then we started looking at what these individual basis operators mean. We said these are generally described in terms of the angular momentum operators because there is a natural thing to do. There are many ways of doing it and we considered the Cartesian angular momentum operators Ix, Iy and Iz for individual spins. Then you can make combinations of these to generate products of 2 spins, 3 spins and things like that which will generate a complete basis set using which we can express the density operator. Now, when we define the basis operators, we also must know what these individual basis operators mean. What does it encode? So, if you want to take the complete matrix, the row as a complete matrix as some huge matrix by n by n matrix or whatever that this thing is and which elements are occupied by the particular basis operator. See, if I want to take a basis operator BS, one particular basis operator which elements here are occupied by this. So, that actually gives us an information about what is the meaning of a particular basis operator BS. So, in this context, we actually looked at 1 spin, 2 spin, 3 spin, define the basis set of operators and we also looked at last time, particular operators like I kx and I lx for 1 spin system and also for a 2 spin system and what does it represent? What does I kx represent in this density of density matrix? So, we said this will represent in phase magnetization of k spin, in phase magnetization of k spin. Likewise, it is this if you look at the elements here. So, we had this 1 1 here, 1 1 here, this had 1 1 here and 1 1 here. So, this actually gives the in phase magnetization of the l spin. So, for the 2 spin system and this is the 4 by 4 matrix. In the individual case of 1 spin system, there will be 2 by 2 matrices. For a 2 spins case, it will be 4 by 4 matrices, but we can represent the individual k magnetization or the l magnetization in the 4 by 4 matrix itself to draw a meaning out of that one. And similarly, for a 3 spin system also we wrote what does a kx would mean in a 3 spin system of klm and this will be 8 by 8 matrix. And obviously, it will occupy 4 elements here. The kx if I have a klm which is the coupled like this k here, l here and m here. If all of them are coupled, then each one of them is a doublet of a doublet. So, there will be 4 lines for each one of those and this 4 lines will occupy 4 elements in this complete density operator. And we saw that they will all have the same signs and we will have for the each kx lx and mx, there will be 4 different positions occupied here, 4 different positions on this side and 4 positions on this side occupied to represent the in phase magnetizations of the kl and m spins. So, now we go forward from this so far we are considering only one spin elements in a multiple spin systems. Now we will look at the products, products of the operators. So, how do we do that and that is what we are going to do today. So, let us consider the particular element which we represent as 2 ikx ilz. So, what it means I have 2, this is now a 2 spin product so of ikx and ilz, k and l refer to the 2 spins. How do we calculate the matrix representation of this? So, we take the matrix of kx here and remember these are for the individual spins. So, therefore, these are since these are 2 independent and separate spins we have to take the direct product. If this is k spin and this is the l spin and we have the direct product of the k spin and the l spin, the x is 0 1 1 0 and the z operator for those single spin is 1 0 0 minus 1. So, and now to generate a 4 by 4 matrix we have to take the direct products of these 2 matrices. So, therefore, at the place where there is a 1 I get this entire matrix here 1 0 0 minus 1, here again where there is 1 I get 1 0 0 minus 1 and the other 4 elements will be 0 because you have 0 here and a 0 here you multiply 0 by 0 this entire matrix you get 0 here and a 0 there. Now you notice here these ones represent the single quantum coherences. So, if you recall the energy level diagram that will tell you we will see that in the next slide and we will exactly know how these ones mean single quantum coherences. So, these ones are for the 2 spin system we have 4 energy levels as we have mentioned and these ones must correspond to those 4 energy levels. This is indicated here explicitly. So, we have here the 2 i k x i l z the same matrix represented here and the 4 energy levels of the 2 spin system are alpha k alpha l alpha k beta l beta k alpha l and beta k beta l these are the 4 states and therefore, these 4 states are 1 2 3 4 on this side and 1 2 3 4 here and therefore, I can write these 4 energy states on on the top here this is alpha alpha alpha beta beta alpha and beta beta and here also alpha alpha alpha beta beta alpha and beta beta. So, now you see so what these elements are so we have this transition this is the 1 3 transition which means this is 1 2 3 this is the 1 is here and that is the 1 3 transition here and this is the k transition k is flipping from alpha to beta and l is remaining the same as alpha. So, this is the k transition right. So, therefore, we say it is one of the component of the x magnetization of k and this one is the other transition which was seen because it is the element which of the density operator which it has occupied is 2 4 and the 2 4 is this. So, and this again is alpha k to beta k and so we have here element minus 1 here, but notice one thing that these 2 have opposite signs this one is plus 1 and this one is minus 1. So, to represent that we put it arrows in this manner if I put 1 2 3 as a positive arrow here to represent positive signal here then the 2 4 I represent by a negative sign this is only a representation to convey the meaning as to what we are trying to say when we actually measure the magnetizations of these coherences in NMR experiment when I get the spectrum I will get these 2 lines in this manner 1 3 line will be positive because it is the plus 1 here and the 2 4 line will be negative and that is because of the minus sign here and this is going down. So, similarly for the ky Lz I have here the same 4 states but again once again it is the k spin Lz L spin in the z direction and ky transfer magnetization is of the k spin previously but transfer magnetization was of the k spin but it was a x component. So, here we have the y component this is the product is ky Lz. Now what we see here we see minus i and i and this one is represented in the same manner here. Now once again you see this you have opposite signs what is the meaning of opposite signs first of all these are imaginary numbers the imaginary numbers meaning we have a dispersive component this is a 90 degree is out of phase. So, if I call x to y it is a 90 degree is out of phase and this is represented here in the by an imaginary number here. So, we get an i here. So, therefore if I have 1 3 as a particular way if it is in this manner then the 2 4 will be opposite to that and therefore this goes in this manner here. So, you notice here if this is positive going like this then this has to be negative which is going in this manner. So, accordingly you could have chosen either this way you could have chosen this way as a particular sign and this is a particular as an opposite sign it does not matter. So, you will have the opposite signs for these 2 signals but they will be dispersive line shapes. Y magnetization means these are dispersive line shapes. Now let us look at a product which is kx Ly. Now both these components are transverse components of the k and the l spins. This is x component of the k spin and the y component of the l spin. So, now what we do? So, we have to add here this again I will put k and this is l. So, for k I have the operate the matrix for the x and that is 0 1 1 0 and for the y I have 0 minus i i 0 this is for the l spin. Now I have to take the direct product of this and then I get here 0 0 0 minus i 0 0 i 0 0 minus i 0 0 i 0 0 0. It is very interesting you see which elements are occupied here remember this was the beta beta state and this side is the alpha alpha state. So, alpha alpha to beta beta that is actually double quantum coherence. So, therefore this element represents a double quantum coherence therefore this is the 1 4 element of the energy level diagram and this will be a double quantum coherence and this has a minus sign and it has it is i here. Now you also have a nonzero element here and this is i and this represents 2 3 state 2 3 coherence right this is energy level 2 and this side is energy level 3 and therefore I have a 2 3 here and the 2 3 state is actually 0 quantum coherence because they are both alpha beta and beta alpha their m values are 0 and therefore we have a 0 quantum coherence here. Therefore and similarly and these are the corresponding complex conjugates in this in this area. So, therefore this matrix represents a combination of double quantum plus 0 quantum coherences this was k x l y and now if we look at k y l x let us look at k y l x. So, what I have to do is simply I have to interchange these 2 for k I will put 0 minus i i 0 and for l I put here 0 1 1 0 ok. So, now if I take this deduct product so I get here 0 0 0 minus i 0 0 minus i 0 0 i 0 0 i 0 0 0. Now it is the same 4 elements which are populated which are nonzero here the difference however is that these are minus i minus i here and these are correspondingly i i. So, therefore there is a sign difference between these 2 if this way to represent d q plus z q then this will represent one of the elements is changed to minus sign. So, we call it as d q minus z q. So, double quantum minus 0 quantum coherence for this element. So, 2 k y l x is represents d q minus z q both of them are of course are combinations of double quantum and zero quantum coherence but they have different signs and therefore in this case I have d q plus z q here I will have d q minus z q. So, now what I do? So, I have here the same elements represented here 2 i k x l y put the same matrices here. So, I have 0 0 minus i 0 0 i 0 0 this is a d q plus z q and I put here d q minus z q. Now I take an addition of these 2 I take a sum of this because in a density operator when we are doing it may be that you will have combinations of basis operators it cannot be that you will only have one of them as your density operator. So, your density operator remember is a summation of the various basis operators. So, if I take a combination of these 2 k x l y and k y l x added over here then what do I get? I get here minus i I get here i and all the others are 0 that means I get pure double quantum coherence. So, this is the unique way of getting pure double quantum coherence and this is because it is minus i here we call it as y although you cannot represent a double quantum coherence as x as along the any of the Cartesian coordinates as x y z we simply say by convention for the sake of ease of representation we call this as the y component and this is represented by i here this is basically imaginary and one could have called them as real and imaginary as well but for convenience or some convention which has been used so we call it as pure double quantum y. Now, suppose I do a subtraction of the same 2 elements i k x i l y here i k y i l x here and take a subtraction if we do a subtraction I will get i and minus i here and all other elements are 0. So, what that means I get pure 0 quantum y see the y coherence pure 0 quantum and this remains as minus i i and therefore it is called as y. So, now let us look at the other product here to i k x i l x now in this case I have the x components for both k and l spins. So, let me write here once more the these are for the individual spins this is for k and this is for l. So, once again it is for k and this is for l. So, here this direct product now will have real numbers you see this product gives me 0 0 0 1 0 0 1 0 0 0 1 0 0 0 once again this is a mixture of double quantum and 0 quantum coherence is this is the double quantum this is 0 quantum this is the mixture of the 2 and they have the same sign. So, I represent this mixture of double quantum plus 0 quantum here. Now, if I take k y l y then now I have 0 minus i i 0 direct product is 0 minus i i 0 and this gives me once again real numbers here in the 4 by 4 matrix I get here minus 1 1 1 minus 1 and all other elements are 0. So, therefore because of the opposite signs of these 2. So, this will be represented as double quantum minus 0 quantum. So, now what do we do we take we do the same trick as we did before. So, we take a summation of these 2 elements i k x i l x i k y i l y remember this 2 elements is always as the normalization factor. So, I get d q plus z q addition d q minus z q I get here 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0. So, therefore take this addition I have only these elements non-zero therefore this will be pure 0 quantum and I call it as x because this is the real number this is not imaginary. Let me now take the difference if I take the difference then what do I get the same matrices here take the subtraction here I get here 1 I get here 1 and all other elements are 0. So, what do I get here therefore I get a pure double quantum x coherence. So, therefore we have seen from these matrices for single quantum coherences and double quantum coherences how we can get the representation in the density operator. So, these different elements occupy different places in the total density operator and whenever they are present we know we have created these elements. And often when we write the total density operator there will be mixtures of such elements and they will all be present in. So, by looking at what elements are present in the density operator you can say which coherences you have created and which coherences are observable and all of that will become helpful in analyzing the results of your NMR experiments. Now, there is one more here and this is 2 ikz ilz can of both are z here z component. So, this is k l and z and this is 1 minus 1 here and for each of this once more I have to put here k and l k and l. This represents a different kind of a situation we have 1 0 0 0 0 minus 1 0 0 0 0 minus 1 0 0 0 0 1. Notice that all the diagonal elements are occupied here. All the diagonal elements are occupied all off diagonal elements are 0 which means this operator does not represent transverse magnetization or it does not represent coherences between the spins in the individual states. What does it represent? So, this represents what is called as zz order. Notice while this has to do with the populations this is not the z magnetization of the any of the particular spins. If you remember for the z magnetization for a two spin system we if you were to write ikz plus ilz this was the representation for the total z magnetization of the two spin system and this was simply 1 and minus 1 here and all other elements were 0 and so this was ikz plus ilz representing the total magnetization total population difference between the two for the two spins and that was contributing to the total magnetization. But here there is some sort of a correlation between the populations of the k and l spins and therefore we call it as zz order. This must be distinguished from the z magnetization of the two spin system. So, now to summarize all of this so what we have is thus the basis operators give a physical insight into the spin system because we have seen what the individual operators are how they are represented in the density operator which elements do they represent and that is gives us an indication into what this the individual basis operators represent and which is easy to calculate. Iz operator represents the populations and the z magnetization, ix and iy operators in a multi spin system represent in phase single quantum coherences along the x and y axis respectively. And now we saw here today 2ikx ilz 2iky ilz represent single quantum coherences of k spin anti phase with respect to l along the x and y axis. Notice once again the kx lz and kylz. So, therefore they are both k magnetizations anti phase with respect to the spin l because that is in the z but these represent the real and imaginary component or the x and the y axis respectively. Similar interpretations hold good for the l spin single quantum coherences. For instance if I had ilx ikz and ily ikz then they would represent the l spin single quantum coherences. Then we looked at 2ikx ily 2 spin products with both transverse components and 2iky ilx 2ikx ilx 2iky ily all of these represent mixtures of double quantum and zero quantum coherences and suitable combinations of these represent pure double quantum and single quantum coherences. And we have the x components and the y components represented here as well but that is a kind of a convention you cannot actually draw a double quantum and zero quantum coherence on the Cartesian axis. And then we said if we take a 2ikx ilx plus 2iky ily represents x component of zero quantum coherence 2ikx ily minus 2iky ilx represents y component of zero quantum coherence and then 2ikx ilx minus 2iky ily represents x component of double quantum coherence 2ikx ily plus 2iky ilx represents y component of double quantum coherence. Then, the 2ikz ilz represents 2 spin zz order. This has to be distinguished from the z magnetization as I indicated before. So, we have now seen the various components of the basis operators, products, what do they physically mean and the interpretation they can give, the insight they can give in the density operator will be useful for understanding the experiments which we will discuss in greater detail for varieties of experiments that we will discuss in greater detail in later classes, but this forms the basis for understanding all of those experiments. So, in the next class we have to see how these evolutions, how these basis operators evolve with time because that is what is required to be calculated when you require the response of your spin system through the pulse sequence and that we will stop here and that will be taken care in the future classes.