 Let us continue with the discussion of our density matrix. We wrote the equation for the density matrix M nth element of the density matrix as by the following equation delta Mn e to the minus En by kT where En is the eigenvalue of the state N and delta Mn is a Kronecker delta and for M is equal to N this will be the diagonal element. So diagonal element and this there then will represent the populations of diagonal elements are 0 at equilibrium at thermal equilibrium. This will be 0 for M is not equal to N at equilibrium okay and we also wrote in the previous class an operator form of the density for the density operator we derived an equation we wrote the density operator equation for the density operator rho is equal to 1 by z plus k by z i z where i z is the z magnetization and this z is the partition function and this represents the number of states okay. Now we also wrote earlier when you make a measurement let us say if you want to measure the x magnetization then we also derived earlier that we have to calculate the expectation value of this operator which represents the x magnetization and that is equal to is equal to trace of ix with the density operator rho we had derived this equation earlier. Let us take a simple example here for a one spin system i is equal to half we have we derived this expression matrix representation of the ix operator is equal to half 0 1 1 0 and the i z operator was 1 is equal to half 1 0 0 minus 1 okay and therefore the density operator for this rho is equal turns out to be this is equal to half into the unit matrix plus k by 4 1 0 0 minus 1 where k is gamma h cross h naught by k d okay. So now putting this together you calculate ix rho this product if you calculate it this turns out to be half into 0 half minus k by 4 and this will be half plus k by 4 and 0 okay. Therefore the trace of ix rho which is the sum of the diagonal elements of this product this is equal to 0. Therefore there is no x magnetization in the equilibrium state right this also we have said earlier when we defined at thermal equilibrium of diagonal elements are 0 and so this is mathematically shown here that the x magnetization is 0 in at equilibrium condition. Now suppose for example the you are dealing with a non-equilibrium state and that often comes when you do various kinds of manipulations you can create a density matrix the which is which is which has off diagonal elements which are non-zero. In this in the earlier cases you notice that the density matrix had off diagonal elements 0. So now we consider a matrix rho is equal to something like this P1 A here and P2 here. So we said these are the populations here the density matrix has off diagonal elements are the populations and we somehow consider a situation where the off diagonal elements here are non-zero but are equal to A. Now what will be the consequence of this now if I want to calculate the trace again of ix rho then this will be equal to half 0 1 1 0 multiplied by P1 P2 A A. So this will turn out to be equal to half A A P2 P1 and this is equal to A. Now you see there is a the trace is non-zero therefore there is a net x magnetization in this situation therefore the density matrix represents the complete situation about the system. You have the diagonal elements which have the populations and the off diagonal elements will represent transverse magnetization. Same thing will happen y as well. So you may create various kinds of situations and how does these things occur? How do these things occur? This will happen is the result of various kinds of manipulations you might do in your experimental sequence then it becomes necessary for us to actually calculate the density operator and then we can make the measurement. So the density operator calculation that will be dependent on time if you are making a pulse sequence where lots of pulses are applied and this go as a function of time along the time axis. So therefore we need to consider and calculate the evaluation evolution of the density operator with time. So how do we do this? This will require a framework where we can calculate the time evolution of the density operator. So this is what we are going to discuss in the next few minutes. So for understanding the performance of multiple experiments it becomes necessary to understand the time evolution of the spin system through the pulse sequence and this is best done by calculating the time evolution of the density operator. So for this we begin from the time dependence Schrodinger equation. See it is time dependence Schrodinger equation describe how the state evolves with time and that is the equation is given in this manner it is minus h cross by i d psi by dt is equal to h psi where h is the Hamiltonian describing the system containing all the interactions and psi is dependent on time so the evolution of psi tells you how the system is evolving with time. So to achieve this let us write the wave function psi as before as the superposition of various state functions Eigen states the C and T's are the coefficients and u n's are your Eigen states they constitute an orthonormal basis set as we defined before. Now let us substitute this in the Schrodinger equation we get minus h cross by i summation n d c n by dt into u n and that is equal to h and this is the psi again here the summation n c and t u n. Now we take the matrix elements of this with the state u k. So that means you multiply on the left you take this bra here k minus h cross by i k summation n d c n by dt and this is the ket here n because this summation goes over the n indices and that is equal to on the right hand side you have the h k here and the h c and t u n this is the wave function psi summation n. Now so therefore this is equal to now c and t is a constant and therefore we can take that out of this to the movie to the left and k will come inside so we will have k h n so c and t this k h and this will be the matrix element of the Hamiltonian and this is represented as h k n. So therefore this differential on the left hand side this is equal to summation c and t h k n. Now on the left hand side notice that there will be only one non-zero element this summation goes over all n but this n and k are orthonormal therefore this will survive only when k is equal to n therefore there will be only one element here d c k by dt therefore my summation will disappear minus h cross by i d c k by dt is equal to c n t h k n. So now we also know the basic definition of the density matrix right so d by dt k rho m is equal to d by dt c k c m star this is also the basic definition of course the ensemble average is implicit here so ensemble average for the coefficients is implicit in this equation. Now on the right hand side we differentiate this explicitly you get c k d c m star by dt plus c m star d c k by dt. Now these are remember these are coefficients these are time dependent of course and therefore we can move them around. Now looking at this we can calculate what d m star by dt is we derive this equation for d c k by dt so therefore d c k by dt will be equal to minus i by h cross summation c and t h k n. Now I want for the complex conjugate so if I want to take the complex conjugate here I want to take the complex conjugate on the right hand side as well so when I do that it becomes i by h cross because of the minus i by h cross will become i h cross and this coefficient c n star will be now c n will be now c n star and this element will be h you have to represent because you have to turn them around this way right. So therefore this will become h and m that is instead of k here m here it will come the other side d m star will be m will come on this side n will come on this side because that is what happens when you take the complex conjugate on this of this matrix element. Substituting this then we get d by dt k rho m is equal to i by h cross c k c n star h n m minus i by h cross c m star c n h k n that was the substitution of those equations here. So this will now be equal to i by h cross summation n now for this once again this summation k rho n we write this again for the density operator c k n c n star is k rho n this is this matrix element here and h and m will retain as it is h and m is n h m and minus i by h cross here once again in the summation we have c m star c n is written as n rho m we turn put it around because these are coefficients we can move them around here we can move this here and this there and we keep this h k n here h k n here and this one we will write it here as n rho m. So now the summation is taken out here put it together for the entire equation we have put the whole thing in a curly bracket here and you have a summation n here. Now we notice an interesting that has happened here with the summation n we have got these element here summation n this is the ket and this is the bra this runs over all the n's similarly here also we got this ket n and bra n so this summation runs over all the n's and you remember from previous classes that summation n this side of a operator this is the projection operator sum over all the projection operator is equal to 1. Therefore this element vanishes similarly this element vanishes and the summation also vanishes therefore I will have i h cross k rho and this h Hamiltonian because this has gone and similarly on this side I have here k is of course there and the k appears here h this one has vanished h rho and this is m this will survive as it is. So therefore I have here i by h cross this is a bra k and inside here rho h minus h rho and m state here on this. Now what is this here this is called as the commutator between the two operators rho and h it is typically written in this manner rho h m. So therefore if I combine this if you compare these two equations then I will have d rho by dt is equal to i by h cross rho comma h and this is the commutator of rho h k this is called as the Livial von Neumann equation this is the most important equation for calculation of all the density matrix elements of any experiment. So this is the fundamental equation where the time evolution of the system is described and you calculate the elements of the density operator as you carry out various manipulations with your spin system and in the end when you actually are ready to measure you take the expectation value with regard to the ix or the iy operator and that gives you the measurement. So therefore this is the crucial equation here and we have to see how this equation can be solved and this there are standard methods of this we will now go into the details of the solution how it is solved but we will take the solutions of this as they appear and we can use this solution to calculate the evolution of the density operator as a function of time. Now if the Hamiltonian is explicitly independent of time in that Livial equation then the solution of this is easily written in this manner we will we are not deriving this explicitly from that point but this equation can be verified by explicit differentiation. Rho of t is given by e to the minus i by h cross h t rho of 0 e to the i by h cross h t. So the rho of 0 means this is at time t is equal to 0 and these two operators on the left and the right these describe the evolution of the density operator as a function of time. So this can be easily verified by explicit differentiation of this. This is independent of time these are the two things which actually depend on time therefore you can differentiate it and reorganize the elements and then you will find that it satisfies that von Neumann equation. Now using this definition let us try and calculate the off diagonal elements of the density matrix. So what we do we take the matrix elements here rho in the middle and then m here and m here this is basically rho m n this is rho m n. So therefore this is m here now for rho of t we put this solution here e to the minus i by h cross h t rho of 0 e to the i by h cross h t and then you have n here. Now how do we calculate these elements? Now we notice earlier we defined with exponential operators what is the result when you operate it on a particular state. If you have an eigenvalue equation then e to the i by h cross h t n gives you e to the by h cross e and t where e n is the eigenvalue of the state n. So for the Hamiltonian h for the operator h then you get this eigenvalue e n for the state n. Therefore e to the i by h cross h t e to the by h cross e and t n. Similarly e to the i by h cross Hamiltonian t is operating on m gives you e to the i by h cross e m t operating on ket m. Now if I take the complex conjugate of this if it is the complex conjugate then what I get on the left hand side I get here instead of the ket I get the bra this here and this one will now get the minus sign e to the minus i by h cross h t essentially this will be operating on the left here. And that is equal to m and you take the complex conjugate here so this state will come here so this is e to the minus i by h cross e m t. So you put that definition here because now we want to operate this one on this side and this one on this side. So this operating on this side gives me e to the by h cross e n t because this now is a number this one operating on this side gives me this number m e to the minus i by h cross e m t. So therefore I take out these two numbers out then I get e to the i by h cross e n minus e m e n comes from here and minus e m comes from here t and the state m and rho of 0 stays here and n stays here. So therefore the off diagonal element of the 10 theory matrix rho m n is equal to e to the i by h cross e n minus e m t m rho 0 n. Now you notice this is the energy difference between the two states n e n n between the states n and m. So now if I write e m is equal to h nu m this is the frequency now and this is a Planck's constant and e n is equal to h nu n and I define w e n or omega m n is equal to 2 pi nu m minus nu n. So now this is in radians these are in Hertz frequencies. So therefore if I put that in here now rho m n is equal to e to the i omega m n t and this matrix element m rho 0 n. Now we also know that at rho m n this density matrix is simple this definition c m c n star the ensemble average. Now going back to the earlier definition of this coefficients that they have an amplitude and a phase. So we write it in this manner and assemble average is written this way c modulus of c m modulus of c n these are the amplitudes of these two functions and the phases are written in this manner here e to the i m minus alpha n alpha m minus alpha n these are some phases some numbers which represent the phases. Now if these are random by the hypothesis of random phases so these elements this average will go to 0 in the case of equilibrium state. However if this is non-zero then non vanishing of rho m n implies existence of phase coherence between the spins in the states m and n in the ensemble. At thermal equilibrium all phases occur with equal probability which implies that c m star is equal to 0. So therefore at equilibrium these off diagonal elements of the density matrix are 0 and at if there is a deviation from equilibrium then you will have off diagonal elements non-zero and we showed in the very beginning that if there is an off diagonal element of the density matrix which is non-zero that amounts to transverse magnetization i x or i y. The appearance of i x or i y therefore implies a phase coherence here between the spins in the states m and n. So therefore transverse magnetization therefore gets connected to the phase coherence between the spins. So anytime you have a transverse magnetization we say there is a phase coherence between the spins and decay of that phase coherence amounts to decay of the transverse magnetization. So now putting that thing here at equilibrium this equation has to go to 0 and now this is never 0 sorry this one is never so this is an oscillatory function. So if this function has to be 0 then it is this one which has to go to 0. So that means this element is 0 at equilibrium this element is 0. So if you create a non-equilibrium state wherein this element is non-zero then we have created an off diagonal element which is non-zero. So any non vanishing off diagonal element implies a non-equilibrium state. So therefore extending to multiple spin systems and a variety of situations a very generalized density matrix can be written in this manner. In the most general case you have rho of t which is written as we have the populations on of the n states here. Suppose you have total of n states here you have the populations along the diagonal p1, p2, p3, p4 and so on and so forth and on the off diagonal elements you have these elements coefficients this and then you have the oscillatory function i omega 1, 2t between this represents the energy difference between the states 1 and 2 and this is that coefficient which we saw in the previous slide and likewise this coefficient keeps varying from element to element here and this oscillation frequency also is changing with element to element because the energy levels are changing. So if you have so many energy levels there will be oscillations of different types and one can create all these kinds of phase coherences which means that if I have an element which is non-zero here it would mean that I have created a phase coherence between the states 1 and 3. Here there is a phase coherence created between the states 1 and 2 this that is by the c12 term c13 term yes and this is by the 1n element this is the but the first state and the n state and this is the oscillation this time evolution is given by this and this actually depends the non-equilibrium situations where you have created a phase coherence between the states 1 and n. So this will be the complex conjugates here so you have c21 p2 so and it will be written in the same manner c23 and so on so forth c2n and here we get written as omega n1 if you notice here so therefore if I want to write it as omega 1n to represent this then I will put a minus sign here and then it will be consistent with what we said before. So when I say n1 n2 n3 this remains the same then they have the populations here therefore the density matrix has complete information about the spin system it carries all the information through the pulse sequence if you generated a pulse sequence spin echo for example we consider or just a Fourier transform experiment with single pulse what all happened how many if there are many frequencies how they all become magnetizations which is observed in the FID all of this can be calculated and you have to take the trace of the observable with this density operator and it will tell you what is all the information present in the spin system and which can be used for measurement and interpretation of your results. So I think this is the crucial stage and we can stop here and evolution of the density operator calculation will be required for interpretation of any experimental sequence we may design. So we will stop here and we continue with this density operator density matrix calculations in the future classes we may also think of some simplifications in the calculations how to arrive at these results in a much simpler form those things will come in due course.