 Good morning. We are now entering into the most fundamental and rigorous description of NMR. Previously we had used the vector notations for the magnetizations, magnetic moments, the fields, their interactions and so on. Although it is extremely useful, it becomes difficult to deal with those descriptions when we have to handle complex experiments involving multiple pulses and the responses difficult to describe. There will be many situations of equilibrium and non-equilibrium states. Various kinds of equilibria and non- equilibria can be present. There can be non-equilibria in the populations of the individual energy levels. There can be non-equilibria of the phase coherences between the spins. The phase coherence as you know results in transverse magnetizations. And the transverse magnetization evolved with a different time constants than the longitudinal magnetizations. And all of this has to be described in a more inclusive manner and an exhaustive manner and also more rigorous manner which obviously cannot be done by the vector descriptions. So the recourse is to go into the more basic description of the spin systems using quantum mechanics. So a certain amount of quantum mechanical knowledge is required here and I believe that they graduate students who are going to qualify in chemistry and some aspects of structural biology or things like that or even physics and they would have gone through such courses in quantum mechanics. And the description what we are talking about here is known as the density matrix description of NMR. Density matrix is not just a concept which is for NMR but it is a very general concept and therefore it deals with the description of any system in a very generalized manner. Let us begin by looking at the wave function of a spin system. In quantum mechanics you describe a system by a wave function which is of course a vector and if you want to represent it as a vector, you will represent it as a vector with an amplitude and a phase and almost invariably the wave functions are described in quantum mechanics as in some kind of a superposition of various eigenstates of a particular system. For example here the wave function psi of t is described with the superposition of various individual eigenstates of the spin system. Here these are the individual eigenstates and for the spin states if you have a nucleus with a spin i and it is as a mutual quantum number is M and we have already seen that it has 2i plus 1 eigenstates. So let us look at that in somewhat more detail. So if I have a nucleus with a spin value of i then I know that M takes 2i plus 1 values. For example if i is equal to half then we say it has the alpha state and the beta state represented like this. There are 2 states 2i plus 1 is equal to 2. If i is equal to 3 by 2 then I have 2i plus 1 states are 4 then I will have different 4 number of states. If I have 2 spins for example this is for 1 spin this is for a single spin. If I have 2 spins then each one of them can have alpha and the beta state. So this can have the alpha beta states and this other spin can also have the alpha beta states and therefore I will have a total of 4 states which we represent as alpha alpha, alpha beta, beta alpha and beta beta. This we have seen before when you are actually discussing analysis of spectra as discussed by Professor Ashutosh Kumar. Suppose I had 3 spins all these are spin half systems. If I have 3 spins then how many states I will have? I will have alpha beta for 1, alpha beta for the second, alpha beta for the third. So I will total of 8 states. So therefore we write it as alpha alpha, alpha beta, beta alpha, beta beta and then here I multiply this by alpha here, alpha here, alpha here and alpha here. Then in the same manner alpha beta here then I have here alpha beta, beta, beta alpha beta and beta beta beta. So total of 8 states. So like that we will have different wave functions. We represent the generalized wave function as a superposition of these individual eigenstates. These eigenstates are not time dependent these are stationary states. But the wave function itself evolves with the time. Therefore what we do? We represent this as a superposition of the states where the coefficients, these coefficients are time dependent. CMT this is the coefficient which is time the wave function psi of t is time dependent and these UMs are stationary states and therefore to get this time dependence here you write these coefficients as time dependent numbers. And since the wave function is a complex wave function because it is complex in nature as I said it is a vector it has an amplitude and it has a phase the coefficients themselves CMT these are also complex in nature. So these are complex numbers. And the various eigenstates the states UMI these constitute an orthonormal set orthonormal set what does that mean? So you have here for example for the alpha alpha state I have alpha, alpha this is equal to 1 and but alpha beta is equal to 0. So for the one spins case when you have the two states here alpha alpha is 1, alpha beta is 0. Likewise if you have the beta alpha it is 0, beta beta is 1. Similar thing applies to all other states in your spin system. For the two spins state you will have 4 states and all of them are orthogonal to each other and they are also normalized because individual matrix elements of these are equal to 1. Now so this is the generalized description of your spin system we represent it as a wave function as a superposition of various eigenstates of the spin system. Now when you actually make a measurement of an observable which is represented by an operator for example you may be measuring the X magnetization or the Y magnetization or something like that. Then we observe the time average of an ensemble. So here we have this is for individual spins. Individual spins are represented in that way. Each spin evolves in a particular manner. For example in the classical way we noted that the each spin is processing around in the cone surface of a cone. Each spin processes right when it processes it acquires a phase phase keeps changing that is what we say its time constant is I mean the Cm the coefficient is changing the phase is changing. When we say the complex numbers they have also an amplitude and a phase therefore their phase is changing. So each spin has a particular value of Cm of t. Now when you are making a measurement on an ensemble which consists of trillions and trillions of spins so what we will measure is a ensemble average or the time average. Same thing is called as the time average of a particular spin over a long period of time. So this is described by what is called as the expectation value of the operator. Every physical observable is described by an operator and you measure the when you make a measurement we say we are getting the expectation value of this operator this is basically representing a time average the ensemble average. So how does how is that calculated that is represented in this following manner. This is the standard notation for describing the expectation values. This is the standard notation here we have a cat here and a bra here generally we represent in this manner and if a is your observable on which you want to make a measurement you write this expectation value in this manner this is the psi wave function here and the psi wave function here this is the complex conjugate on this side and a is here. The same thing is represented in different notation if you want to use the integral because you want to take an average here the summation we take the integral psi star a psi we notice here we are assumed that the psi star we are normalized that means psi star this integral is equal to 1 otherwise you divide it by this integral here. So these are already normalized functions and therefore you can simply write it as psi star a psi d tau. Now d tau is a volume element which means it goes over all the variables in your spin system. For example if you want to calculate the expectation value of mx the operator for x component of the magnetization in terms of the functions umi and then you write it in this manner u mx phi is equal to summation. Now you see here you have one summation for this psi another summation for this psi and therefore and there I will therefore have two indices here one index for one wave function the other index for the other wave function therefore I will have summation cmt star cm because this is as I said it is a complex conjugate here therefore here we will have a complex conjugate and this will be corresponding from here this psi cmt and this is your umi corresponding to this and un i corresponding to this and in between we have this mx operator. Because since these are numbers we can take this out and keep only this eigen states and the operators here as in the vector notation in this matrix element representation. So these are numbers these numbers can be taken out these are numbers which can be taken out and then we write this umi this umi stays here mx stays here and un i stays here and we have double summation coming going over the indices m and n. Now we simplify this notation slightly instead of writing this umi and un i we simply drop the u and we will simply say m and n. So it is just to indicate that m is the actually the eigen state for a particular one particular this one and this is the n here this corresponds to this and this corresponds to this. So this is just a simplification of the notation. Now what we do now is we define an operator which is defined in this manner. This we define this p of t this is an operator now it is a time dependent remember this is a time dependent part right this is time dependent therefore the operator which we want to define is also time dependent and that is defined in this manner n p of t m e is given by c m star c n t this I did not this term which is present here we want to put it as a matrix of this form the n and m keep varying right for this therefore this will generate a matrix each one of them this generate as you keep varying n from all the states and m also for all the states you will generate that many matrix elements how many matrix elements will generate for this. So if n is equal to 4 m is equal to I mean the n runs from 1 to 4 m also runs from 1 to 4 you will get generate a 4 by 4 matrix. So like here so this will be you will generate a 4 by 4 matrix when you vary all the m's and the n's so therefore now the expectation value psi m x phi is now equal to the same thing this psi has come in between this is not required this is basically what we have here right. So we have put this n so star t c m star t n this one is put here okay this is represented by this and this remains the same as that okay this is not required here. So therefore and now this goes over this terms m and n okay. Now basis now this is again basic rule in quantum mechanics that when you have a complete orthonormal set this summation of this type over all the states m this summation is equal to 1 this is the ket here and this is the bra here you take a sum over all these states then this is equal to 1 okay this is the identity and we can simply use this identity for simplifying this equation you notice here this identity appears here in the middle right. So from here to here this identity appears right therefore if I take this sum summation m and take this portion of this equation then this actually goes to 1 therefore this simplifies my equation okay. Now one more correction here once I remove this one of the summation I do not need this so I will have only one summation and that is going over n okay so I will have n p t m x n now therefore there is only one summation. Now what does this tell me this tells me that I have if I have a matrix of p t m x this is one operator this is another operator the product of them is also an operator and I can have a matrix representation of this operator if I want to write it like that so this is the matrix representation of p t m x and all these n n n n these are the diagonal elements of this matrix then I am taking a summation over all these matrix elements n n n is equal to 1 2 3 4 5 so like that so depending upon whatever number of states are there so you will have the summation over all those states so that is called as the trace so this is the trace of this matrix p of t m x so this is the sum of the diagonal elements of the matrix p m x. Now in an ensemble the different individuals will have different coefficients as I said before the different prints will have different coefficients and therefore we will have to take an ensemble average in order to represent the expectation value correctly so therefore when you take the ensemble average of this operation then we will have this same equation coming back c m t and this is summation over the two indices m and n the average will be taken over this because this is not time dependent if m x is not time dependent this is not time dependent because m and n states are not evolving in time what evolves in time are the coefficients and m x is my operator m and m states are not dependent on time therefore when I take the ensemble average I take the ensemble average over these time dependent quantities which are the c's. So now this I represent by a new operator to represent the ensemble average I call that as rho of t this operator now instead of the p of t when I take the ensemble average without the ensemble average this bar I had called it as p of t now with the bar I simply call it as rho of t and this is called as the density operator and density operator has a matrix representation depending upon the various states what we have for instance if I have a single spin I will have two states and therefore I will have a 2 by 2 matrix and the two states are alpha beta here and alpha beta here because in each of this side because n goes from alpha to beta m also goes from alpha to beta therefore I will have 2 by 2 matrix which is so there are 4 elements so if I have a 2 spin system that will have 4 states so 4 states how will I write that here so I will have 4 states alpha alpha alpha beta beta alpha and beta beta and similarly alpha alpha alpha beta beta alpha beta beta beta so this will be 4 by 4 matrix that is 16 elements therefore the density matrix this is called as the density matrix and so the operator which is it represents is the density operator so similarly if I have 3 spins then we will have 8 by 8 matrix and so on so when you go to larger and larger spin system for the density matrix also becomes larger and larger. So now what we have to do is we have to understand the density matrix more clearly. So let us try and calculate the elements of this density matrix you put this definition once more a particular element of the density matrix see is represented in this manner right rho mn is cn t cm star t this is one particular element of the density matrix. Now the coefficient cn as I said are complex quantity and hence ensemble average can also be written as follows. Now what I do is I simply represent this as complex quantities so it has an amplitude and a phase. So for the cn t I write it as some modulus of cm and e to the alpha m and for cm t this is complex conjugate here therefore it has a minus if it is minus i this is the complex conjugate will be the plus sign. So therefore cn cm star t is the modulus of cm and cn e to the minus alpha n minus alpha m. Now this is the phase and this is the total amplitude with alphas represent the phases and the cm represent the amplitudes this particular matrix element therefore has an amplitude and a phase. Now let us see what is the consequence of this as you look at the individual elements here is we have to understand this very clearly every sentence is quite important we can go slow and to try and understand this very clearly. At thermal equilibrium say by the hypothesis of random phases all phases of alpha in the range 0 to 360 are equally probable and hence ensemble average vanishes for m not equal to n. When you take an average over this alpha minus alpha n all possible orientations in the space are possible because every phase is possible when I say phase it is an angle in the transfer plane. So therefore all possible orientations are possible that means all the off diagonal elements vanish whenever m is not equal to n when you take an ensemble average all the elements will vanish m is not equal to n we call it as off diagonal element when m is equal to n we call it as a diagonal element. Now so all off diagonal elements vanish non vanishing of off diagonal elements means there is a kind of a phase coherence between the individual spins in the ensemble that is why there is no cancellation complete cancellation does not happen so some amount of phase coherence is present among the spins and that leads to non vanishing of the off diagonal elements. What do the diagonal elements represent? The diagonal elements when alpha m is equal to alpha n that means I mean m is equal to n then numerator part actually goes to 0. Here m is equal to n this one goes to 0 then I have a modulus of c n square or c m square. Now what does this modulus of c n square represent? It represents the probability when it took the summation when you wrote the wave function as summation c m t u m i then the coefficient actually represents the square of that coefficient represents the probability that the particular spin is in that eigen state and therefore we are taking the superposition of the individual eigen states to represent the wave function. Therefore this modulus of c n square essentially represents the probability that the spin system is in that eigen state. So therefore we write that in this manner and what is the probability? Probability tells me the population of the individual state. We have discussed this before the what is the probability that the spin is in the particular eigen state and that is essentially asking what is the population and the population is given by the Boltzmann distribution this also we have considered before. So therefore the rho m n represents the delta when it is the diagonal element it is e to the minus en by k t divided by z. If it is off diagonal element then it is 0. So that is why we put this as delta m n delta m n is a chronicle delta. So if m is not equal to n delta m n means is a chronicle delta is equal to 0 for m not equal to n and it is equal to 1 for m is equal to n. So therefore if m is equal to n this is the diagonal element and the diagonal element is simply e to the minus en by k t divided by z and where z is the partition function. This is what we have seen this is the result of the Boltzmann statistics. What happens next? Boltzmann statistics is given by this the partition function definition we had given earlier here and this is the summation over all the energy values all the probabilities for the individual states where n is the total number of states. So n what we have here is the total number of states on your spin system that depends upon the value of i for any i you have the number of states is 2 i plus 1. So let me write that here n is equal to 2 i plus 1 n is equal to 2 i plus 1 for particular if there is a single spin or if it is multiple spins then it is a product of the two spins. If I have things like a and x then I will have 2 i plus 1 of a multiplied by 2 i plus 1 of x. Therefore for two spins two spin half systems I will have 2 here and 2 here I get 4 by 4 matrix. So that n depends upon the number of spins and i values. So this once e to the minus en by k t runs over all the individual states. Now we expand this as we did before as 1 to n 1 minus en by k t plus 1 by 2 factorial en by k t square and so forth as an exponential expansion under high temperature approximation we have seen that en by k t is much much smaller than 1 and then z can be approximated ignoring the higher order terms in this equation. So when we do that we see all the higher order terms are gone I keep only the first one here z is equal to 1 minus en by k t. This automatically leads to a further simplification. Now I separate these two I write summation 1 minus 1 by k t summation en this is simplification. Now what is the summation en? In order to understand this let us look at the energy level diagram once more here. Let us take i is equal to 3 by 2 for a single spin the same thing applies to other spins as well. So in the absence of any magnetic field we have the degenerate states all the 4 states are degenerate here i is equal to 3 by 2 means these 2 i plus 1 is 4. So and these states now split when you apply the magnetic field and they two of them go up and two of them come down by similar amounts to the extent this goes up this comes down this goes up this comes down to the same amounts therefore you have a negative energy here and a positive energy there and the total of this is 0. So similarly if you take 5 by 2 I will have 6 states here 3 states up here and 3 states down here and the sum of all of this is 0. So in general therefore the summation en over all the states is actually 0 therefore this portion becomes 0. So what happens therefore z is simply equal to the number of states. Now here is a little bit of a digression to be able to follow the mathematics little bit better afterwards suppose we have the Hamiltonian represented as h and if it has eigenvalues lambda i and we often write this as h i is the eigenstate gives you eigenvalue lambda i and the state i. So this is the usual eigenvalue equation and if this is the case then it follows bus that e to the power h this is an operator exponential operator of the same Hamiltonian and this if this is operating on the same state i it gives me e to the lambda i. So here lambda i is the eigenvalue and here e to the power h is the operator if I take then I will have e to the lambda i therefore if I take this matrix element between the two states j and i of the exponential operator e to the h then I will have j e to the lambda i i and that is equal to now this is the number e to the lambda i is the number therefore I can take this out and put this as delta ij that means if i is equal to j then this is equal to 1 or if it is not equal to i is not equal to j then this is 0 this is the chronicle delta. Therefore I write it as e to the lambda i I take it as this out since this is the number and then I represent this j i as delta ij chronicle delta multiplied by e to the lambda i and we are going to use that here. So therefore we wrote rho mn is now written as delta mn e to the minus en by kt by z. So and now I recall again that rho mn is 0 if m is not equal to n then delta mn goes to 0 and if m is equal to n then I have simply e to the minus en by kt divided by z this is the partition function this is the partition function and this is the probability and this represents the populations. So together I write this as rho mn is equal to delta mn e to the minus en by kt divided by z. Now let us write this in the in the matrix for like this rho mn is equal to now we have seen that z is equal to the number of states so it is 1 by n and then you have this ket here m and e to the minus en by kt n. So therefore essentially this if I were to take this out this I have m and n here and that is delta mn. So delta mn multiplied by e to the minus en by kt. So this is the same thing written in this manner. Now what I do for the en now I know that the en is the eigenvalue of the operator H or when Hamiltonian operates on the state n it gives me the eigenvalue en therefore here I will put this as the result of the exponential operator operating on the state n. So instead of en by kt of multiplying on the state n I write this as now the coming out as the result of the operation of the exponential operator operating on n. So therefore now we have an operator term introduced here. So the rho mn is now a matrix element here between the states m and n with the operator in the middle as e to the minus H by kt. So therefore if I write this also as in the in the operator form so this also I write it as m rho n and this is equal then what happens I can take this mn out ignoring this m and n I will have rho is equal to 1 by n e to the minus H by kt. So simply I take this out this states out and rho is equated to 1 by n e to the minus H by kt. So this gives me a different representation of the density operator this is the density matrix and this is the density operator. So once again I will do the same trick expand this density operator as a power series rho is equal to 1 by n is 1 by 1 minus H by kt plus 1 by 2 factorial H by kt whole square and so on and so forth. Once again under high temperature approximation the hydrogen terms can be neglected and I will have simply 1 by n 1 minus H by kt. So H is now we know what is a Hamiltonian H for a single spin I can simply write it as minus gamma H cross H naught I z this is the Z 1 interaction Z 1 interaction between the single spin with the field H naught and that gives you minus gamma H cross H naught I z remember this is mu dot H naught and the mu dot H naught gives me mu z H naught and mu z H naught is minus gamma H cross H naught. So this is the if you simply recall the previous lectures in the previous classes what is the Hamiltonian and you will simply see that this is for a single spin minus gamma H cross H naught I z for spin half. So this will be number of states is 2 therefore rho is equal to 1 by 2 1 plus gamma H naught H cross H naught I z divided by kt. Now if you calculate the matrix elements of this operator the matrix elements of rho will be rho alpha alpha you can calculate this explicitly here rho alpha alpha is equal to 1 by 2 this is the result what we will get but I will show you the explicit calculations and when you do this the first element of the density operator matrix is equal to half of this much and the this is the fourth element for the two spin system you know this is a single spin for the single spin there are this is a 2 by 2 matrix and there are four elements alpha alpha is one element beta beta is one element alpha beta is one element and beta alpha is one element and it turns out that when you actually calculate putting this I z operator here they calculate the matrix elements alpha alpha then this alpha beta and beta alpha go to 0 and alpha alpha gives you half 1 plus gamma H cross H naught by 2 kt and beta beta gives you half 1 minus gamma H cross H naught by 2 kt and now if I put this all together I will have rho is equal to half this two matrices 1 0 0 1 plus gamma H cross H naught by 4 kt 1 0 0 minus 1. So this one is 1 plus if I put it together of course the same thing is put in this manner this one is here this one is here and these ones have entered into this term where these ones and these zeros are coming these zeros are coming because alpha beta and beta alpha are 0 because you see remember this is alpha and this is beta here and alpha here and beta here so this alpha bit the cross terms are 0 and it is only the diagonal terms which are nonzero. So I think this is a good point to stop and we will continue with the discussion of the elements of the density operator in further detail in the next classes.