Let us continue our qualitative discussion of the DC model of a large uniformly doped bulk MOSFET. In the previous lecture, we have discussed the distributions of electrons, holes, then the space charge, the electric field and the potential as a function of the y direction. That is, in terms of this diagram, the direction from the interface into the bulk. Now in this lecture, we continue this and move on to some other aspects of the spatial distributions of NP, JNJP, ESI and energy bands. Specifically let us consider the plot of JNX with y. Refer to the diagram here, JNX is the x component of JN, x component of electron density. X is the direction from source to drain as shown here and we want to plot this x component of electron density as a function of y. Now here is the section of the device rotated by 90 degrees so that the spatial axis is horizontal. So in this diagram, the x direction would be from this end to the other end, so vertically upwards. Now to plot the current density, we must look at the electron concentration and the distribution of the electron concentration is as follows, I am repeating a slide from the previous lecture. So this is the concentration of electrons in the inversion layer as a function of distance. Now the current can be because of drift or diffusion. The drift current would be proportional to the concentration of electrons shown here and the electric field in the x direction. On the other hand, the diffusion current would be proportional to the gradient of electron concentration in the x direction. This means that like you have this kind of a distribution for n as a function of y for some x, at another x you will have a similar distribution as a function of y but the values of the electron concentrations would be different. Now it is easily observed that the gradient of the electron concentration itself would also depend on the concentration shown here. So if the concentration itself goes to 0 or very small values, then the gradient of the electron concentration in the x direction that is this direction would also go to 0 by the same distance or at the same location. And therefore we can easily conclude that the Jnx which depends on the electron concentration and its gradient in the x direction would both have qualitatively the same shape as shown here for the electron distribution. And this is what is shown here that as the electron concentration falls to 0, the current density also falls to 0. The current density shown in the negative axis because the current is from drain to source whereas the x direction is from source to drain. So we conclude that the current of electrons from source to drain would be very close to the interface. At the interface the current density would be maximum and then it would decay rapidly and it would be restricted to the inversion layer. Now let us move on to the plot of energy bands with y. Now here are the various bias points where we could draw the band picture. We sketch the energy bands versus y at some x near the channel midpoint in saturation and sub threshold that is at bias points 2 and 3 that is this point and this point. I will be doing it for 2 and I will leave it as an exercise for you to do it at 3. You can also do it at 1. Let us consider the device in saturation. Now here is the structure of the section of the device that is relevant to us and here is the procedure for constructing an energy level versus y diagram. This procedure is repeated here from a previous module. Recall we have done a complete module on energy band diagrams and there we had listed out the sequence of steps in which any band diagram should be drawn. Now these are the steps. Let us look at the first step. We locate the junction, space charge and neutral regions. Now let us do that. Junctions in this device structure are the N plus gate silicon dioxide insulator junction that is shown here, the silicon dioxide insulator substrate junction that is shown here. Then you have the substrate and substrate electrode junction that is shown here. Then you have the space charge regions. This is one space charge region which is controlled by the gate and this is another space charge region because of the substrate-substrate electrode junction. YD represents the edge of the depletion layer due to the space charge layer controlled by the gate. The next step in this procedure is to draw EFN and EFP continuously everywhere from a knowledge of applied voltage across the junction V and current densities of electrons and holes. So you know that the gradient of EFN represents the current density Jn and the gradient of EFP represents the current density Jp. Now whenever we have options we should first draw the EFN and EFP in neutral region and among or between EFN and EFP we should first catch the majority carrier quasi-firmly level. Now this order is dictated by the ease in which we can draw the Fermi levels. So let us do that second step. So here we have drawn the Fermi level across the substrate-substrate electrode junction. This is easiest to do because as we have remarked earlier there is no voltage drop across this junction because the voltage drop across this junction that is any part of the applied voltage if it drops across this junction that would amount to a current flow across this junction and such a current flow is not possible because that current would have to flow through this insulator layer and this is not possible. Thus this PN plus junction is in equilibrium and therefore we can easily draw a constant EF across this junction. So EFN and EFP are both the same. Next let us extend this Fermi level over the neutral region of the substrate until the depletion edge. Now while doing so we have shown it as a dashed line because we want to first catch the majority carrier Fermi level and that is EFP. So the quasi-firmly level for holes will be shown with a dashed line. Evidently the quasi-firmly level for electrons in this region which is next to the depletion edge would not coincide with the quasi-firmly level for holes. Thus you can easily appreciate from your knowledge of a PN junction. So when you have a biased PN junction then in the region near the depletion edge in the neutral region near the depletion edge the quasi-firmly levels split. Now here this Svesta region corresponds to the inversion layer P substrate junction which can be regarded as an N plus P junction. The inversion layer is N plus and the V, the channel voltage V can be regarded as the reverse bias across the inversion layer P junction and therefore the EFN would not coincide with EFP. Now the EFP would be a constant line and it is easy to draw because the whole concentration would not be disturbed in spite of the bias. This is a reverse bias and therefore the injection level has to be low and therefore the majority carrier concentration is not disturbed. Therefore EFP remains a constant line okay which is the same as under equilibrium and that is why in this entire neutral P region the EFP would be constant. Next we sketch the EF in the gate region. Once again here we have sketched the EF in both the neutral as well as the space charge region of the gate and we have shown it as a constant line because the N plus region here is under equilibrium. No current flow can occur perpendicular to the silicon dioxide interface here and that is the reason why N plus region is in equilibrium therefore we can draw the EF as a constant line. EFN and EFP are the same. Now where do we draw this constant line? We know that there is an applied voltage VGB between this N plus region and the back N plus contact. Therefore the Fermi level in the back N plus contact and the Fermi level in this N plus region would be separated by Q times VGB. This VGB is positive and therefore the Fermi level in the gate is below the Fermi level in the substrate electrode because this is energy band diagram and any movement upwards would amount to moving to more negative energies or more negative potentials and therefore a positive potential will be realized by moving downwards. Next we extend EFP in the space charge region using the assumption of cos equilibrium according to which in the space charge region of a p-in junction the whole density, whole current density rather will be approximately equal to 0. Note that in the space charge region you have a large gradient of holes and as well as a large electric field and therefore you have large drift and diffusion currents which are however in opposition and the difference between the two currents is what results in the net current and this net current is what is represented by the gradient of the quasi Fermi level. So we are saying that individually the drift and diffusion are very high but they are in opposition and the difference between them is very small and that is what is implied by the statement that the gradient of EFP is very small or EFP is a constant line and therefore this is simply extended from the EFP outside the depletion region. Next we sketch EFN again as a constant line using the same cos equilibrium approximation for electrons. The distance between EFN and EFP is equal to the reverse bias across the junction that is Q times V so the channel voltage V is actually the reverse bias across the N plus inversion layer T substrate junction and since the inversion layer is positive with respect to substrate the quasi-familiar for electrons is below the quasi-familiar for holes. Now we connect the EFN from the depletion edge to the EFN deep inside the bulk as a continuous line. Now slope of this line represents the so called diffusion current of electrons which is flowing okay because of the reverse bias across the inversion layer P substrate junction. Now let us clean up the slide and show other aspects referring to the procedure for constructing the EY diagram. The third step is sketching of EC and EV in neutral regions from a knowledge of energy gap EFN, EFP and concentration of electrons and holes. Now that is what we do next so we first locate the EC in N plus region the distance between EC and EF is given by concentration of electrons in the N plus region. Then we sketch EV below EF in the P type substrate the difference between the Fermi level and the valence band edge in the P type substrate is given by the hole concentration in the substrate. So we are sketching the EV in the neutral region of the substrate. Then we sketch EC in the neutral region of the gate okay. So EC is a little above EF in practice for poly gates for heavily doped poly silicon gates EC and EF are assumed to be coincident however in our diagram we have shown a small difference between EC and EF both in the gate as well as here because we want to show the EC and EF clearly. Now using energy gap we place the other level we place the valence band edge in the substrate electrode and in the gate and we place the conduction band edge in the substrate in the neutral region of the substrate. Now the fourth step is to sketch a continuous E naught everywhere from a knowledge of potential and electron affinity so that is what we are doing here. So we place E naught in the neutral regions above EC at a distance given by Q times electron affinity. So we have done that in all the neutral regions okay. Now after sketching the E naught in the neutral regions we can now join these various E naught levels by continuous line as shown here. So we draw a continuous line for E naught in the space chart region joining the various ends. Now note that the slope of E naught is important and it represents the electric field therefore for example if you take E naught at the silicon silicon dioxide H in silicon dioxide region the variation of E naught has a constant slope because the electric field is constant. The gradient of E naught represents the electric field. In the substrate at the interface the electric field is different than the electric field in the oxide because of the dielectric constant differences between the oxide and the substrate and therefore you see the slope of E naught at the interface in substrate is different from the slope of E naught in the oxide. So specifically E naught in the oxide has a higher slope than the E naught at the interface in the substrate. In fact the slope difference would be a factor of 3 because of a factor of 3 difference in the dielectric constants of silicon dioxide and silicon. Now the next step is to sketch EC and EV in insulator and space chart regions based on electron affinity and energy gap. So let us begin with the insulator we place EC at a distance Q times the electron affinity of oxide below the E naught and we place EV at a distance EG below the EC. Now we place EC and EV in the space chart regions controlled by the gate. So both the space chart regions in the substrate as well as in the gate. Now this is done in a similar way as we have done for the EC and EV in the oxide. We are doing it from the knowledge of electron affinity. So this EC line here in the space chart region would be Q times chi s below the E naught line in the space chart region similarly for this EC line and the EV line here in the space chart region would be energy gap below the EC line and similarly here. Now let us repeat the same process for this space chart layer. Now we are not going to show any more levels such as a donor level or any trap level and so on. I leave that as an exercise to you. Let us look at some further aspects of this energy band diagram. Before that let us clean up and show only the energy levels. Let us show the voltage differences. Now Q times VGB is the difference between the Fermi level here and this line which represents the Fermi level in the bulk electrode. The same QVGB also appears as the difference between the E naught levels in the substrate electrode and the E naught level in the gate. As we have remarked earlier, the difference between the quasi Fermi level for holes and the quasi Fermi level for electrons in the space chart layer controlled by the gate is equal to Q times the channel voltage V, Q times the potential drop across the poly, Q times the potential drop across the oxide and Q times the potential drop across the semiconductor or Q times the surface potential. These are all represented for the E naught variation from the gate into the bulk. Just for the information of the students psi p, psi ox and psi s are the potential drops occurring in these regions of the device and this is the polarity. Q times phi MS is actually the total variation of E naught across the space chart region of the substrate electrode junction. Now that is essentially the potential drop or the built-in potential drop across the substrate electrode substrate junction. Now let us summarize what have we achieved by drawing these band diagrams. We sketch the energy band versus y at some x near the channel midpoint in saturation to highlight the following. Now this is essentially the bias point at which we sketch the band diagram. The first point we highlighted was the applied voltage VGB and the channel voltage V. So go back and see this applied voltage VGB and channel voltage V. Then we wanted to show the potential components psi s, psi ox, psi p and phi MS of the applied voltage VGB. So psi p, psi ox, psi s and phi MS. These components of the applied voltage VGB which is the difference between E naught levels. So you see the various potential drops are shown on the variation of the E naught level okay whereas the applied voltages VGB and channel voltage V are shown as differences between quasi Fermi levels. Finally we wanted to show the behaviors of electron and hole concentrations, electron and hole current densities, electric field and psi all in a single diagram. So when you look at this energy band diagram, from this diagram you can conclude okay you can derive the information about electron concentration, hole concentration, electron current density, hole current density, electric field and psi. Now let us just explain how. Now you know that the difference between EC and EFN okay gives you the electron concentration at any point. Similarly the difference between EFP and EV gives you the hole concentration. The gradient of EFP gives you the hole current and the gradient of EFN gives you the electron current density right. Now the gradient of E naught gives you the electric field and similarly gradients of EC and EV also give you the electric field and the variation of E naught is actually nothing but the variation of the potential. So this is how NP, JN, JP, E and psi all are reflected at once okay in the energy band diagram. Now here is an assignment for you sketch the energy bands versus y near the channel midpoint from gate to bulk for sub threshold conditions that is for this particular bias point shown on a log ID versus VGS graph. Now let us turn our attention to plots of NP, JN, JP, E and psi as a function of X. Now so far we were plotting quantities as a function of Y that is from the interface into the bulk. Now let us plot along the interface from source to drain that is what we are doing now. Now we are not going to plot all of these quantities we are going to plot some of these relevant quantities I will leave it as an exercise to plot those quantities which I am not going to do here on the slides. First let us sketch psi S, EXS and EYS versus X in saturation that is at this bias point. Psi S is the potential of the interface okay silicon dioxide interface or surface, EXS is the X component of the electric field that is the electric field directed along this X direction from drain to source or source to drain whichever we look at it along the interface. So S here means it is along the interface and similarly EYS is electric field in the Y direction but plotted along the interface in the X direction. Let us look at psi S as a function of X this variation would look something like this so 0 is the source and X is equal to L is the drain so from source to drain the surface potential goes on increasing progressively okay the variation is slow towards the source but rapid near the drain. Now let us explain this with the help of this particular diagram so psi S is the potential of this interface okay with respect to the bulk. Now if I take the source end that is this so this is X is equal to 0 and this is the drain end this is X is equal to L now we are assuming strong inversion so the inversion charge concentration at the interface or the electron concentrate at the interface may even be more than the doping in this heavily doped source okay so we are ignoring any small potential variations that may occur okay across this region because of the difference between the inversion layer electron concentration and electron concentration in the source. So we will assume more or less both are at the same potential. Now psi S goes on increasing from source to drain this is evident because VDB is more than VSB so the psi S should increase. Now the psi S increases slowly near the source but rapidly near the drain. Now this is evident from the fact that the inversion charge is very strong near the source but weak near the drain because of which the electric field directed in this direction now this is your X direction. Now our electric field at this here it is like this okay you also have an electric field in this direction but we are concerned with the electric field in this direction right because we are interested in the surface potential variation in this direction. Now this electric field is high near the drain and low near the source because as we have remarked earlier the current has to remain constant from source to drain approximately constant at least there may be some contributions from thermal generation and so on which are small. If the same current has to be there when the inversion charge is more as well as when the inversion charge is less wherever the inversion charge is more the electric field in this direction should be small and wherever the inversion charge is less to get the same current I should have a higher electric field. Now that explains why the slope of psi S is small near the source end but high near the drain end. The value of psi S at the source is equal to phi t plus vsb where phi t is normally given by 2 times phi f where phi f is the difference between the Fermi level and the intrinsic level in the substrate. So this approximation is normally given in most textbooks however when you do numerical calculations then the accurate value is twice phi f plus 6 times the thermal voltage. This we have explained in our discussion of the MOS capacitor at the drain end the psi S is equal to phi t plus vdb so source end is here and drain end is here. So we are saying that at this end the potential of this point with respect to bulk is vsb plus phi t and at this end it is this vdb plus phi t so this potential with respect to this. So from psi S versus X we can easily plot ExS versus X. As we have already remarked ExS is nothing but this electric field okay this is the exaggerated electric field and at the interface. So this ExS is nothing but minus dou psi S by dou X and since the ExS is directed from drain to source it is negative and that is why we are plotting the modulus here. So this curve is nothing but the slope of this curve. Let us turn our attention to the Y component of the electric field. Now here is a plot of EYS as a function of X. You see the EYS is crossing the zero line and becoming negative for X is equal to L. Now this is because you recall we were considering a bias condition in which the vdb is greater than vgb which is same as saying vds is greater than vgs. Now for this bias condition we have sketched the equipotential lines field lines and so on right in one of the earlier modules and you recall when vdb is greater than vgb near the drain end the field is directed from substrate to gate whereas over most of the channel length near the source the field is directed from gate to substrate. At some point the field reverses direction. Now let us explain the shape of the EYS versus X curve. The EYS is high near the source and it progressively goes on decreasing. Why is it so? Now look at this diagram and this is your EYS that is the EY at the surface or interface. Now you know that the applied gate to bulk voltage falls partly across the polysilicon partly across the oxide and remaining across the depletion layer in the substrate. So we can write this as vgb is equal to psi p plus psi ox plus psi s where this is psi ox and this is psi s. Now let us ignore the psi p that is the potential drop in the poly. Now vgb is constant from source to drain but what happens to psi s? So you see the psi s is going on increasing from source to drain consequently psi ox goes on decreasing from source to drain so this potential drop decreases from source to drain. Now the total charge in silicon that is inversion charge plus depletion charge should decrease from source to drain because the psi ox is decreasing from source to drain. So total charge in semiconductor if you represent it as qs which is sum of qi the inversion charge and the depletion charge that is this charge and this depletion charge. Then the magnitude of this depends on psi ox so this is equal to psi ox into c ox this is by the parallel plate capacitor law. Now since psi ox decreases from source to drain the qs decreases from source to drain. Now if qs decreases from source to drain then by Gauss law the y component of the electric field which terminates on this charge which is actually the eys that eys also should decrease from source to drain. Now note that we have to argue in terms of psi ox because if you take the inversion charge and depletion charge individually inversion charge goes on decreasing from source to drain but the depletion charge increases from source to drain. So purely from this knowledge of variation of qi and qb we cannot conclude about qs because this is decreasing from source to drain this is increasing. So we cannot talk about the sum unless we talk in terms of psi ox. Now the next point of interest is the fact that the variation in eys is slow near the source and it is rapid near the drain. Now this is because of the same reason because of which the x component of the electric field that is the field directed from drain to source increases rapidly near the drain and the variation in this field is slow near the source. So the conditions in the channel change rapidly near the drain but they vary rather slowly near the source. Now let us see what can we do with this information. So we have sketched psi s, exs, eys versus x in saturation to highlight the mobility degradation because of the following factors. So what we are saying is from this information regarding the variation of the y and x component of the electric field we conclude something very important regarding how the mobility varies along the channel. The first point is the mobility degrades due to transfers field eys near the source where eys is much greater than exs. So here you can see near the source eys is much more than exs and therefore mobility degradation near the source would be because of eys that is it is because of the vertical field or the transfers field. On the other hand the mobility degradation is due to parallel field exs near the drain. So when I come near the drain the exs is much much larger than the eys. So near the drain the mobility degradation would be due to parallel electric field, electric field parallel to the interface okay and that is what causes the velocity saturation effects. Now here is an assignment for you sketch ns versus x in saturation that is at bias point 2 that is this point and ns, exs, eys and psi s versus x in sub threshold conditions that is at bias point 3. So at bias point 2 that is in saturation we have sketched exs, eys and psi s and therefore you are asked to sketch ns alone whereas for sub threshold conditions you sketch all these quantities with that we have come to the end of the lecture and so let us make a summary of the important points. Now in this lecture first we sketch the electron current density directed from drain to source in the y direction and showed that this current density is restricted to the inversion layer thickness. In other words the current in a MOSFET is restricted to the inversion layer thickness. Next we drew the energy band diagram in y direction near the channel midpoint. Now while drawing the energy band diagram we made the important point that the material associated with the electrode connected to the substrate and the electrode connected to the gate that has to be taken into account while drawing the energy band diagram in the y direction and therefore whenever you draw energy band diagram in the y direction you must explicitly show the gate electrode and the substrate electrode. For simplicity we assumed that the gate electrode had the same material as n plus gate and the substrate electrode was of the same material as the gate electrode and therefore that was also n plus. The band diagram was used to highlight the applied gate voltage VGB, the channel voltage V and the potential drops psi p, psi ox and psi s across poly oxide and the substrate as well as the built in potential or work function difference phi ms. Another thing we achieved by drawing the energy band diagram is that from the energy band diagram we are able to show the variation of all important quantities namely np, jn, jpe and psi. Then finally we sketched the potential variation at the interface from source to drain that is psi s and the electric field components directed parallel to the interface at the interface that is exs and the electric field component directed perpendicular to the interface Ey at the interface, variation of these quantities as a function of distance from source to drain. The field components were sketched to highlight the causes of mobility degradation and we remarked that near the source the mobility degradation occurs due to transverse electric field that is electric field from gate to bulk whereas the mobility degradation in the channel near the drain occurs because of the parallel electric field that is electric field along the silicon silicon dioxide interface. We shall continue this discussion in the next class.


discuss the Gelman and Lowe theorem, which I mentioned in the previous class. And essentially, let me quickly remind you what we are really dealing with, because our focus is on the treatment of the electron correlations, which we are very concerned about. We know that the Hartree Fock is not able to deal with these electron correlations. The Hartree Fock certainly took account of the statistical exchange correlations, but not the Coulomb correlations. So, we introduced quantum mechanics in various pictures. The Schrodinger picture is the one that we most often use in undergraduate and some other introductory graduate level courses. But then there are other pictures which are very useful in doing quantum theory of many particle systems, in particular the Heisenberg and the Dirac picture. So, this is what we have in the Schrodinger picture. And then what we did is to introduce the difficult term. This is what I have been calling as the unfriendly term. This is what makes the problem so cumbersome, the H 1. And you have the exact solution for H 0, which is the unperturbed Hamiltonian. So, this is solvable. This solution is known. We have the solution with us, but then we do not have the solution when we include the correlations and these are sitting in this term H 1.  Now, this is the control parameter that is at the disposal of mathematicians. And it is not that there is anything in nature which lets us choose the presence or absence of correlations. But a physicist uses his tools, mathematical tools and we can choose appropriate values of alpha or t, which are useful to us. So, that we can develop a methodology which can be exploited to get the solution for the full Hamiltonian, which of course is H 0 plus H 1. The full Hamiltonian is H 0 plus H 1. H 1 is the difficult part of the Hamiltonian, the unfriendly part of the Hamiltonian, which makes it impossible to use a perturbative treatment. Like in the many electron system, the first order perturbation theory gives you the same result as the Hartree-Farck, but second order perturbation theory and higher order perturbation theory does not converge. So, this is the unfriendly part and we sort of scale this unfriendly part by this term e to the alpha t and the choice of the parameter alpha, when alpha is equal to 0, you have e to the power 0, which is 1 and you have got the full Hamiltonian. When t is equal to 0, which is the instant of time, you can say that it is the now moment for us. This is when we really want to deal with the full Hamiltonian. So, at t equal to 0, again you have the full Hamiltonian, but then if you look at the system evolution from t equal to or t going to minus infinity like this is the position vector r and this is the time which goes to minus infinity. Then as t goes to minus infinity, this e to the alpha t times H 1 goes to 0 and you have only the unperturbed Hamiltonian and that is the part of the problem for which we do have a solution and our effort is going to be always from the known to the unknown and what we do know is the solution to the at 0 and we have to figure out what is the solution to the full Hamiltonian, which is H. So, in the Heisenberg picture, we carry out these transformations. So, these are the transformation of the operators and this is how the Heisenberg state transforms, you get it from the Schrodinger picture wave function and the Heisenberg picture wave function of course, does not depend on t. All the time dependence is packed in the operators, the state functions themselves are independent of time in the Heisenberg picture. In the Dirac picture, which is also called as the interaction picture at times, here the transformation is again through a unitary operator, but whereas the unitary operator over here was the full Hamiltonian, in the Dirac picture it is only the unperturbed Hamiltonian. So, it is not the same, but it is similar, but not the same. Here the transformation is through the unperturbed Hamiltonian and the wave functions also transform like this and the result of this is that both the wave functions and the operators in the Dirac picture depend on time. So, at t equal to 0, which is the now moment as I call it, so at t equal to 0, all the operators correspond exactly equal to each other and so do the wave functions at t equal to 0. They all correspond to each other at this now moment, but otherwise at all other instance of time they are different. So, these are the primary relations that we work with and here notice that if this unfriendly part was simply not there, if the h 1 was 0, if the correlation part was 0, just in case wishful thinking, if it was 0, if h 1 was 0, then h would be at 0 and this is how the Schrodinger picture wave function would evolve with time from time t equal to 0 and the interaction picture wave function would be the same as the Schrodinger picture wave function at t equal to 0. So, that is a particular case and what it really does is that just in case h 1 were 0, which we know it is not, but in case if it were 0, then this wave function would become independent of time in the interaction picture. So, these are some special cases in which the Heisenberg picture and the interaction picture or the Dirac picture they correspond to each other. So, these are special situations. Now, what happens when you go to minus infinity as t goes to minus infinity, the Hamiltonian which is the full Hamiltonian, this t going to minus infinity kills any effect of this h 1 in this product term and then you have only the unperturbed Hamiltonian and the wave function at t equal to minus infinity in the interaction picture is nothing, but the solution to the Schrodinger equation in the Schrodinger picture corresponding to the unperturbed Hamiltonian and that is the known thing, it is this known from which we want to go to the unknown. So, phi 0 is known to us. Now, the interaction picture wave function at t equal to 0 is what I represent by this psi 0. So, this 0 subscript really corresponds to this time t equal to 0 and we are looking our interest is in the interaction picture. So, that is the one that we are going to be using in our analysis. So, this state can be obtained from the state as t goes to minus infinity by operating on this by the time evolution operator this is u. So, this is the unitary operator which tells you how the state would evolve from t at minus infinity and gives you the state at t equal to 0. So, the unitary operator this is the time evolution operator it actually tells you that if you know the wave function at t equal to minus infinity or t tending to minus infinity then how do you get the wave function at t equal to 0. So, this is the unitary time evolution operator and this one is nothing, but the Eigen state of the unperturbed Hamiltonian phi 0.  So, that is the strategy that we are going to adopt. So, this is what we it really boils down to we will get the interaction picture solution state vector at t equal to 0 from the Eigen state of the unperturbed Hamiltonian through the time evolution operator u by fixing these two parameters for time which is t equal to minus infinity which is the start time and t equal to 0 which is the n time over which this evolution is observed. You can write it as vectors in the Hilbert space or a state functions. So, any which way so it is just a matter of notation. So, this is what we have got and there are these two limits which are of importance to us as t goes to minus infinity we know that we have the unperturbed problem. The alpha tending to 0 alpha going to 0 is when you will have the full Hamiltonian that that is the real problem of interest to us that is precisely the problem that we want to solve t going to minus infinity we have discussed how we get the solutions right because that is just the unperturbed part of the Schrodinger equation. So, that we know how to handle and the question is what is going to happen when you take the limit alpha going to 0 and this answer to this question what happens in the limit alpha going to 0 is provided by the Gelman and Loth theorem of quantum theory. So, this is a very fascinating theorem let me state what the theorem is and you can see the entire discussion in Fetter and Wallacher's book. So, you have got the unperturbed part of the Schrodinger equation this is solvable then you have the full Hamiltonian which reduces to the unperturbed Hamiltonian in the limit t going to minus infinity and we are addressing the question how do we get the Eigen state of the full Hamiltonian from phi 0. So, that is the question we are really interested in now we do know that the Eigen state in the interaction picture at t equal to 0 can be obtained from the unperturbed solution through the time evolution operator this is the one which of course, has the correlation setting in. So, that is how it is going to pick up the correlations and you will see it explicitly in the slides that follow. So, this is the statement of the theorem the theorem states that if the limit alpha going to 0 of this ratio exists this is the limit which is written as psi 0 over the projection of psi 0 on phi 0. So, this is the inner product. So, if this limit exists then the theorem states that it is an Eigen state of the full Hamiltonian and you can write it as an Eigen value equation H operating on this state is equal to E operating on that state. You essentially get an Eigen value equation for the full Hamiltonian provided the limit of that ratio exists. Now mind you you are interested in the limit of the ratio and you have got u alpha in the numerator you also have it in the denominator you have essentially the average value of the time evolution operator in the unperturbed ground state phi 0 that is what you have in the denominator and both the numerator and the denominator have the parameter alpha which is our mathematical construct and one could in principle ask what is the limit of the numerator in the limit alpha going to 0. One could also ask the question what is the limit of the denominator in the limit alpha going to 0. Now it turns out and I will not have a chance to go through this in too many details I am not going to work out the detailed proof of the Gelman and Lowe theorem. I will refer you to just two or three pages from Fetcher and Wallacher's book the references provided here and it turns out that the limits of the numerator and the denominator if you take them separately you do get certain divergences and these limits do not even exist in the sense that they are not very well defined. But it does not matter because if whatever factor is unknown when you seek the limit for the numerator cancels the corresponding factor in the denominator exactly then the limit of this ratio is very well defined and that is the one that we are really concerned with because we are not using the limits of the numerator or the limit of the denominator separately. We are interested in limit of the ratio and that is well defined and this is the some of these things go into the details of the proof in the Gelman and Lowe theorem and essentially what this theorem tells you is that if this limit is defined and this is the limit the limiting value of what is in this box is written by this ratio here this turns out to be an eigenstate of the full Hamiltonian and this difference between E this is the eigenvalue of the full Hamiltonian E0 is the eigenvalue of the unperturbed part of the Hamiltonian the difference between them is the correlation energy which the unperturbed problem is not able to address. So, that is the one that we are trying to find out. So, the contention of this theorem is that this particular eigenstate which is which we met in the right hand side of the Gelman and Lowe theorem this is the same vector. So, this particular vector which is obtained in this limit it gives you an eigenstate and this eigenstate develops adiabatically from the eigenstate of the unperturbed Hamiltonian adiabatically in the sense that you are not adding any external field any perturbation external field and in that sense this is an adiabatic evolution of the state which is an eigenstate of the full Hamiltonian inclusive of the correlation from the solution that you have for the unperturbed Hamiltonian. So, this is our question what happens in the limit alpha going to 0. So, we are ok with this part and we are concerned with the limit alpha going to 0. So, mind you that as alpha goes to 0 you get the full Hamiltonian, but as t goes to minus infinity you get the unperturbed Hamiltonian and this Hamiltonian gives you the full Hamiltonian in the limit alpha going to 0 and also as t goes to 0 which is the now moment for this problem. So, also in the limit t going to 0 or alpha going to 0. So, our interest is in the solutions at t equal to 0. So, that is the instead of time in which we are interested. So, this is how the way functions in the different pictures are related. I mentioned that this limit need not be well defined it is not fundamental to our analysis it is the limit of the ratio which is of importance to us. So, these are some of the things that I really want to emphasize. What happens is that there is a phase factor and you get certain divergence in that in the numerator if you were to treat this separately and you would have a corresponding divergence in the denominator as well and they happily kill each other when you take the ratio. So, now let us begin with the statement of the Gelman and Lowe theorem and what we are going to do is to look at this whole thing as a vector because when this operator operates on a vector you get a new vector. The new vector is what I put in this larger ket which is colored in red. So, now let us look at these 2 vectors on the left side and on the right side. You have got a ket vector and I am going to construct an inner product of this vector which is to take its projection on another vector which is the adjoint of the vector corresponding to the ground state of the unperturbed Hamiltonian. So, the unperturbed eigenstate phi 0 is known to us. So, we have the adjoint of that vector and I take the projection of this vector which is in this larger ket the red colored ket and I take the projection of this on this. So, it is just the inner product that I am constructing. So, let us go ahead and write this projection. So, it is it will be this, it is the projection of this vector on phi 0 on the left hand side and the right hand side it is the projection of the right hand side on phi 0. Now, let us write these terms a little bit neatly because in the numerator you will have the matrix element of the full Hamiltonian in the states phi 0 and psi 0. So, let us rewrite this result a little bit neatly. So, you have got the matrix element of the Hamiltonian in these states. This is the eigenstate psi 0 which is here, this is the eigenstate in the interaction picture at t equal to 0. So, this is just the solution that we really want. So, this is what we are interested in and you have got the Hamiltonian operating on this psi 0 and we are taking its projection on phi 0. So, this is the result that we have got. So, let us write this result over here which we got this result using the Gelman and Loh theorem. We just took the projection of the Gelman and Loh vector on the unperturbed state vector eigenstate of H 0. So, this is what we have got. Now, this is the full Hamiltonian which has got two pieces one is H 0 the other is H 1. So, I am going to separate out the contributions from these two terms. So, on the left hand side I will have two terms. So, this is the first term which is the matrix element of H 0 in these two states and then in the second term you have got the matrix element of the unfriendly Hamiltonian the perturbation H 1 in these two states. Now, here you have an eigenvalue equation because phi 0 is an eigenstate of H 0. So, phi 0 H 0 on the left over here will be E 0 phi 0 because it is just the eigenvalue equation over here. So, here you get E 0 eigenvalue pops out and then you get the projection of psi 0 on phi 0. Now, psi 0 and phi 0 of course are different. So, the projection of psi 0 and phi 0 is not equal to unity. If both are normalized then the projection of psi 0 and phi 0 would be 1 if they were the same eigenstates, but they are not the same and therefore, it is not unity it would be some non-zero value. But whatever it is you have the same factor in the denominator here. So, these two factors cancel likewise these two factors cancel and now you have got the eigenvalue of the full Hamiltonian on the right side on the left side you have got the eigenvalue of the unperturbed Hamiltonian plus this term. So, if you just rearrange the terms in this equation you get the difference between the eigenvalues of the full Hamiltonian and the eigenvalue of the unperturbed Hamiltonian. So, E minus E 0 which is like the correlation energy. So, you get this in terms of the ratio of these two quantities. So, let us figure out how we are going to evaluate this. This is the quantity of interest to us. Now, h 1. So, this psi 0 over here we know can be obtained from phi 0 through the evolution operator U by taking the evolution from the state minus infinity because this is the solution at minus infinity. This is the solution of the unperturbed Hamiltonian and that is what the full Hamiltonian collapses into as t goes to minus infinity. So, we know this and from this you can get the solution at time equal to 0 through this time evolution operator as we have just discussed. So, you have this psi 0 written over here you do the same in the denominator this psi 0 is also this then you have got h 1 over here. So, this is the result that you have got. However, our result cannot really depend on alpha which is only a mathematical construct, but that is not a worry because anyway we are taking the limit alpha tending to 0. So, that is the limit that we are going to discuss. So, this is our expression this is how you can get the states. Now, the state vectors which are involved over here you see a phi 0 over here and a phi 0 over here you see a phi 0 over here and a phi 0 over here. So, all the state vectors that you need to evaluate the right hand side are now known to us. So, we have succeeded in writing the difference e minus e 0 in terms of state vectors and the time evolution operator and the correlation term, but using only those vectors which are known to us because that is the eigenstate of h 0 that part of the problem has been solved. Now, this is the result the one of the top we just got this from the Gellman-Loev theorem. Using this result we will now show that this energy difference is actually equal to the limit alpha going to 0 and here on the right hand side I have got a new expression and we are going to see how this result shows up. It will turn out in the next few minutes that if you take the logarithm of the expectation value of the time evolution operator in the state phi 0, then take the derivative of this log with respect to time the partial derivative, then evaluate the value of this partial derivative at t equal to 0 and then take the limit alpha going to 0. So, you go in that order first take the log then the derivative then the value of the derivative at t equal to 0 and then finally, the limit alpha going to 0. So, do not do anything in any arbitrary order because that is the sequence in which things have to be done. So, let us look at this right hand side let us look at this limit alpha going to 0 of this expression on the right which is i h cross del by del t of this log at t equal to 0. So, let us consider what is inside the square bracket this is just the time derivative of this log. So, it will be this 1 over this expectation value times the time derivative of this expectation value and I have pulled in this factor i h cross over here inside the bracket and then of course, I retain the t going to 0 constraint because that is the one which is of importance. So, this is what we have got. So, the limit alpha going to 0 i h cross del over del t. So, this is the left hand side and now I just write this a little bit neatly. So, that I have got this factor in the numerator which is this and this factor in the denominator comes below it. So, I have just read written this expression. So, that it is easy to read. So, that is what we have here in the top of this slide. Now, what is this? This is nothing but the Schrodinger equation for the time evolution operator in the interaction picture. So, in the interaction picture the time evolution operator satisfies the differential equation which we often call as the Schrodinger equation itself right. This is the Schrodinger equation for the time evolution operator. This is in the interaction picture and this is nothing but h u. So, this is very similar to how we write the time evolution operators Schrodinger equation in the Schrodinger picture except for the fact that here this is the interaction picture Hamiltonian right. So, this is just the transformation of the interaction part or of the correlation part. So, that is the only one which is being focused upon in the Dirac picture or the interaction picture. Now, here you have the time derivative of the time evolution operator the partial derivative with respect to time. This partial derivative is given by h u. So, that is the h u I write here instead of the derivative del u by del t. So, instead of del u by del t I put in h u I have included the i h cross which was sitting here. So, that has been included and the right hand side on the top is now rewritten as this. Now, put the limit t equal to 0. We have already taken the time derivative that is what gives us h u right. Now, put t equal to 0. So, this t will go to 0 this t will also go to 0. So, let us do that. So, now you have this t equal to 0 this t the second parameter is the start time which is minus infinity and you do the same in the term in the denominator. So, this is now taken care of and this right hand side is nothing but e minus e 0 as we saw from the Galman-Low theorem. In other words e minus e 0 can be written as the limit alpha going to 0 of this expression. So, we have just written a consequence of the Galman and Low theorem in a form which we will find extremely useful to deal with these correlation terms. Now, it is going to get messy before it becomes better, but you will see how that happens. So, this is what we have got from the Galman-Low theorem. This is the energy difference in which we are interested. We have got an expression for this difference which we got in the previous slide which is given by this result, but how did we get this result? We got it from the Galman and Low theorem. We got it from the adiabatic hypothesis. We got it from the mathematical construct of the parameter alpha. What do we expect it to correspond to? We expect it to give us information about whatever was missing from the unperturbed Schrodinger equation from the edge 0 because that is the problem that we solved. We expect this energy difference to correspond to that. So, in some sense, we expect it to correspond in the perturbative sense to what perturbation theory would give us if we were to take into account the correlations which are missing in the term edge 0 and edge 0 problem is solved. It is the H Eigen value equation for the full Hamiltonian which is not solved. So, we expect it to correspond to what we get from the usual perturbation theory which is what I shall refer to as the Rayleigh Schrodinger perturbation theory because that is the common form of the perturbation theory that we work with. So, that is the usual Rayleigh Schrodinger perturbation theory and we expect some sort of a correspondence between the energy correction delta E which we have got from the adiabatic hypothesis using the Galman and Low theorem and we will like to ask what exactly is this correspondence and how do we relate these terms in the perturbative sense. It does not mean that we are using perturbation theory, we are not. If we were to use the Rayleigh Schrodinger perturbation theory in its raw form the way it is constructed in introductory courses in quantum theory that will not give us any result which will converge. That is the reason you cannot approach you cannot use that approach you are not using the Rayleigh Schrodinger perturbation theory, but in the perturbative sense the results that you are going to get will correspond to how you would get information about this part of the solution to the problem which is missing in the problem which has been solved which is the problem for the unperturbed Hamiltonian H0. So, our correspondence will become clear when we analyze the time evolution operator because this is the most important creature here we have to get its expectation value in the unperturbed eigenstate in the eigenvector of the unperturbed Hamiltonian and the form of the time evolution operator has now to be studied in some details because that is where all the correlation is sitting. Now, we did this in not the previous class, but in the class before that so that was in lecture number 25 and I am borrowing a result from that lecture which is on slide 45 of that lecture that the time evolution operator was written as infinite terms if you remember, but we rewrote those infinite terms as an infinite series n going from 0 through infinity of this and the order in which these operators come is determined by this chronological order chronological operator T. So, these operators are all time ordered which means that operators containing the latest time stand to the left. So, that is the chronological ordering that the T operator guarantees. So, that is the result that I am now going to use and we have obtained this result in some detail in the 25th lecture of this course. So, let us look at these terms there are infinite terms I separate the first term corresponding to n equal to 0 which is nothing but the unit operator and then I have the remaining terms which so what is being summed over is the same except that the summation is now going from n equal to 1 through infinity because n equal to 0 term has been separated out. So, I first separate out the n equal to 0 term. So, now I write this time evolution operator as a sum of the unit operator plus again an infinite term, but n going from 1 through infinity and what is being summed over is this un. Now, this is where all the correlations are setting it and this is where they will show up and if we can evaluate this term then we will get the term delta e that is the quantity of interest. So, you have the time evolution operator u alpha which is a sum of this 1 plus these remaining infinite operators when you remove 1 from infinity you still have infinite terms there is nothing I can do to help you with that. So, one would think that when you remove something you will have to deal with less, but that is not the case always and I am really sorry I cannot help you with that. So, you have got this full Hamiltonian and these interaction picture Hamiltonian of the correlation part has got these e to the alpha t term. So, this mathematical constructs e to the alpha t they are all sitting there in this. So, in h i t 1 which is here you will have an e to the alpha t 1 here you have h i t 2. So, in h i t 2 you will have an e to the alpha t 2. So, you will have this e to the alpha t term several places actually n places in each one of these from h i t 1 right up to h i t n set all of these n places you will have these alpha parameter exponents and then of course, that is not being enough you of course, have a summation over n going from 1 through infinity. So, there is a lot to deal with. Now, let us look at this expression now subsequently we will have to take the logarithm and so on, but let us just deal with this expression for the time being piecewise. So, let us look at this expression you separate the first term the first term is nothing, but the inner product of phi 0 with itself it is the norm of this state and let us say that this state is normalized. So, you get unity from here and then a sum of these expectation values of u n's and each u n is this. So, this matrix element is now what I call as a n. Now, you will you might wonder why are we introducing new symbols we already have those expressions and if you did not do that you are going to have to work with terms which are already cumbersome and you can only make them more cumbersome by expanding your notation. So, this is just some strategy to save the ink that you would use to write in your notebooks. So, these are compact terms so and you will see this very soon we are heading toward it. So, you have this a n which is the expectation value of u n. Now, you have to take the logarithm of this this is just the expectation value of the time evolution operator what about the log. Now, this is the logarithm of 1 plus another quantity, but this is a small quantity what is it made up of it is made up of these u's what are these u's made up of it is made up of the correlation which was left out in our earlier problem, but our contention was that we have done most of the problem and it is a tiny thing which is left over right and we are worried about this tiny thing. So, the only thing we know about it is that it is a tiny thing. So, it is nothing, but an expression similar to logarithm of 1 plus x where x is a small quantity at this power series expansion is known so we can use that result. So, let us plug it in and now you see how many terms you have infinite terms you already have various other kinds of infinities in our analysis and you have some more because now you have got infinite terms in the expansion of the logarithm term. Now, I am sure you did not want to write the right hand side over here instead of a n here you have the square you have got the quadratic terms and then the cube and the fourth power and the fifth power and the sixth. So, you have infinite terms over here. So, let us look at this result now. So, you have got these infinite terms n going from 1 through infinity each a n is a very compact creature it is nothing but the expectation value of the operator u n but the operator u n itself has got a very complex structure. So, this is a lot to worry about. So, now this is what we get for delta e delta e is the result that you will get after you take the limit alpha going to 0 of i h cross then the time derivative of this logarithm function. So, you have got the time derivative operator here and for this logarithm expression you now have these infinite terms which are in this beautiful bracket you put take the time derivative of everything that is there in this beautiful bracket put that in a square bracket like a box bracket and then take the value of the box bracket the rectangular bracket at t equal to 0 and then you take the limit alpha going to 0. So, that is what we are going to do now. So, these are the terms that we have to worry about now let us write these summations infinite terms a n. So, a n summed over n going from 1 through infinity. So, you have got a n plus a 2 plus a 3 and so on now this is the square of a sum of these terms.  Because it becomes senseless to take the quadratic terms in a 1 square and ignore a 1 a 2 because they are of the same order it makes no sense to take account of a 1 a 2 but not a 2 a 1 it is of the same order it becomes even trickier when you deal with the cube. So, here you have got a 1 cube a 2 cube a 3 cube but then you have got a third order term coming from a 1 a 2 a 3 but you also have a third order term coming from a 1 and a 2 square and also from a 2 and a 1 square. So, there are all kinds of permutations which are possible and we have written terms only up to the third order and only sum of the terms and then you have to take the time derivative and once you take the time derivative then take the value of the derivative at t equal to 0 and then you take if you are still alive by then take the limit alpha going to 0. Now, you can write the expression delta e when you take all of these terms carefully as a sum of contributions from first order terms second order terms and the order is what I indicate by this superscript and these are in various powers of h 1 which is the correlation term and these correlation terms become progressively small but that does it mean that you have convergence and that is the essential reason why we are developing these alternative techniques. So, notice that various order terms come from different terms like second order terms I showed you from a 1 square but also from a 1 a 2 and also from a 2 a 1 third order terms from a 1 cube a 2 cube but also from a 1 a 2 and a 3 also from a 1 a 4 square right. So, the third order terms from you know different combinations of a linear term and a quadratic term or a product of three linear terms and so on. So, all of this has to be kept track of. So, this is your energy correction now this is the sum and substance of what we have got. So, it is a fairly complex structure and it is a lot of mass is it it is a lot of mass and let us see if we can look at just the first order terms to begin with. So, always from the simplest to the more complex from the known to the unknown I got it right this time Blake. So, from the known to the unknown and from the simplest to the more complex terms. So, we will begin with the first order terms and the first order term you have limit alpha going to 0 i h cross del over del t of let us say let us deal with just the a 1 term just to look at that. So, let us see how we are going to deal with this term. So, this is what your a n in general is therefore, we know what a 1 is a 1 is just out of these n terms you have only one which is this h i at t 1 there is only one time the to worry about in the n th term you have got n different instance of time to worry about t 1 t 2 up to t n what you have over here is a chronological time ordered product of these operators for n equal to 1 there is only one instance of time that you are working with which is this t 1 and this is the dummy variable which gets integrated out from the start which is minus infinity up to the current time t whatever it is and after doing this integration you can then take the derivative of this term with respect to time and then take the value of the derivative at t equal to 0. Now, this is the relationship between the interaction picture operators and the Schrodinger picture operators the evolution that the transformation is through the unperturbed Hamiltonian at 0. So, that tells us that this h i at t 1 this is the interaction picture term is the transformation of h 1 using this evolution operator the transformation operator which contains the unperturbed Hamiltonian at 0 this is the term which will go in over here and what is this h 1 now that is something we do know in terms of the creation and destruction operators. In the earlier unit in which we did second quantization we wrote the electron-electron interaction term in terms of the creation and destruction operators. So, you had one particle terms which are over here and I am sure you remember this form of the Hamiltonian which we discussed in our earlier discussion on second quantization. So, in the second quantized form we have written the full Hamiltonian now you have got the one electron terms and then you have the two electron terms. So, this is your operator h 1, but we are going to have to deal with this put this in our expression for a 1 and then make a further we have to progress further and then see how we are going to work with the second order and higher order term. So, you see what a mess it really is and this is where the Feynman diagrammatic methods become handy those are the ones which I am going to introduce in the next few classes. So, now I think we have the basic machinery ready. So, we have to we are our interest is in getting delta e not just the first order, but to higher orders as well we could not get it using the Rayleigh Schrodinger perturbation theory. So, I am going to stop here for today's class and we will proceed from this point in the next class we have the electron electron term. So, this is the difficult part of the Hamiltonian this is the unfriendly part as we have been referring to this is the one which is responsible for correlations. The static average gives us the Hartree-Fock which is the time independent Hartree-Fock right. So, that we can get from which is equivalent to first order Rayleigh Schrodinger perturbation theory which converges only for the first order, but for higher orders it does not. So, this is the H 1 part this is the difficult part of the Hamiltonian with the difference that we have now scaled it by this factor e to the alpha t which is our mathematical device. So, our Hamiltonian is not just this term plus what is in this purple blocks block this is the western color is it not. So, this is in the purple block this is the interaction term H 1 it is more than that it has been scaled by this mathematical construct e to the alpha t which is the adiabatic hypothesis which goes into the government and low expression which we have used. So, I will take the discussion from this point in subsequent classes yes. So, in the last slide we have the explicit expression for A 1 in the red box. So, there the alpha dependence is did we miss it or no in the same slide 75 yeah. So, the H 1 there is an alpha dependence right. So, this H i t this H i t 1 this H i t 1 is what you will get by simply transforming from the Schrodinger picture to the interaction picture. So, we are doing two things over here one is carrying out a transformation from the Schrodinger picture to the interaction picture, but there is something more that we are doing and that is the adiabatic hypothesis which is why H 1 which is in this first purple box is to be replaced by that term scale by this e to the alpha t. So, when we are going to work with these correction terms in A 1 A 2 and higher order terms we will bring in the alpha with every time there is a correlation term which is to be addressed. In fact, that is the that is the point I was anticipating in one of the previous slides. Let me go back to the just to emphasize this let if I can quickly find it here yeah. I alerted you to the fact that you have got in this u n in the nth term n number of different instance of time parameter. You have got a t 1 here and a t 2 here and a t n here, but associated with each of these there will be an alpha parameter. So, you will have an alpha t 1 you will have an e alpha t 2 and so on. So, all of these terms will have to be plugged in when we evaluate these terms carefully.  So, we are doing you know more than just the Dirac picture. The Dirac picture is the basic platform that we use and over and above this Dirac picture in which we can do quantum theory. We are now doing the adiabatic hypothesis which is the insert which is the insertion of this mathematical device which is the e to the alpha t which we can use to let the Hamiltonian the full Hamiltonian at t equal to 0 evolve from the unperturbed Hamiltonian at t going to minus infinity. So, that solution is known to us at t going to minus infinity from that eigenstate we want to get the eigenstate of the full Hamiltonian at t equal to 0 which is the now moment. So, that is where the alpha parameter comes in and it will of course, be inserted not just for a 1, but for every order term and there will be several terms like with each of these time parameters t 1 t 2 and t n there will be an e to the alpha t 1 e to the alpha t 2 and so on. So, save this consideration at the back of your mind we are going to need it in one of the next classes. I do not know if it will be the very next class or the one after, but we are going to need it soon. And you see what a mess all this is and this is where the Feynman diagrams come in handy because instead of dealing with so many terms there you have to do the integration there are these Volterra type of integrations then you have the chronological operator you have to take the logarithms you have infinite power expansions. Now, instead of all of this if you were to work with just pictures would it not be nice some fun diagrams and that is what we will be discussing in the next few classes. So, thank you all very much and we will take it from here in the next class.


Welcome to the Global Symposium for Regulators 2019 here in Port Vila, Vanuatu where I'm very pleased to be joining the studio today by Dorian Bogdan-Martin who is the director of the Telecommunication Development Bureau for ITU. Dorian, welcome to the studio. Thank you, Max. Great to be here. Now, I'd like to ask you a little bit about GSR. Why is GSR a unique gathering? So the GSR is a unique gathering because it's the only global platform that brings together the world's regulators. Today we have 168 separate regulators and it's the only event that brings them all together annually. It's an event that was started in 2000 and since then the regulatory population has grown to 168. Now this year's theme for GSR is inclusive connectivity. It's focusing on the future of regulation. Why is this important, you think? It's important because we have succeeded in getting half the world online but half the world's not online and we know that getting the other half of the world online is not going to be so simple. We know that the regulatory approaches that were used to get the first half connected which were largely based on competition and market liberalization are not going to be the right practices to get the other half of the world online. So we're gathered here to look at what we need to do, what are the challenges that regulators and the private sector are facing and how can we face those challenges together? Okay, well it's going to a little bit more depth then. Countries worldwide are going through rapid digital transformation as you know, technologies such as Internet of Things, Artificial Intelligence, 5G already here or certainly on their way. What can regulators do to keep pace with these changes? Regulators need to find a balance. They don't need to be heavy-handed and risk stifling innovation and growth. They need to strike a balance between public and private interest and really try to find ways that they can really be connecting the last mile, the last billion. So I think it's all about balance and finding sort of the best approach to fostering and enabling the needed investment and helping to grow their connectivity numbers. You mentioned before that half the world is still not connected. I wanted to ask you, what more could countries do to fast-track connectivity? I think countries need to be more collaborative and that's very much the theme of this year's GSR. It's about collaboration, ways that we can cooperate, the ways that we can better communicate with each other, with different stakeholders. We see that getting the other half online is not going to be something totally left to the private sector, nor is it going to be something that's left entirely to the public sector. So it's really about collaboration. How can we work together through partnerships and how can everybody take responsibility to ensure that we get the other half of the world connected? Finally, this year's GSR is in Port Vila in Vanuatu. It's given an opportunity for a lot of people to come, sometimes great distances, to get a view of what the world is like over here and what the challenges might be. I just wanted to ask you, finally, what do you hope that people will take away with them from this year's symposium? I think everyone has been, let's say, delighted by the really warm welcome that we have received here in Vanuatu. The government has gone above and beyond in making us all feel at home. Vanuatu, as you may know, is ranked number one in terms of risk of natural disasters. So a country very much at risk that has suffered tremendous cyclones in the past. And they've done a lot to use technology to prepare their people and to respond to limit the damage. Yet at the same time, Vanuatu has also ranked one of the happiest countries. And so I think colleagues will leave here feeling optimistic about the future, feeling happy, and feeling better informed that we've all learned something from each other and that together we can really make a difference. To our book to modern, thank you very much indeed. Thank you.


Dear viewers brothers and sisters in Islam, welcome to another episode of Life from Kirbala with me or host Ahmed Ali. For the dear viewers who didn't get the chance to view the previous episodes, the previous two episodes, we've been discussing and examining as well as comparing between the justice and the genuine respect of Alimin Bi Talab to the citizens and the declaration of the rights of men and of the citizens of 1789. Over the past few nights and two nights, we examined the sings of Amir al-Mu'mineen and compared those sings to the French declaration. As a matter of fact, yesterday we examined the way of ruling of Amir al-Mu'mineen Ali Bi Talab peace and blessings be upon him and compared his ruling with the rest of the rulers and the caliphs who ruled the Muslim world after Prophet Muhammad and Alimin Bi Talab peace be upon him. However tonight, insha'Allah, we will discuss, we will further discuss the topic of human rights but first let's welcome our very special guest who has joined us over the past two nights, Sayyid Muzaffar al-Qazmini. Assalamu alaikum Sayyidina. Assalamu alaikum. How are you? Alhamdulillah. We're approaching aid, Sayyidina. Insha'Allah. Have you bought aid clothes? Insha'Allah. It's very busy in Karbala, especially between the Haramain, between the two hushrines of Muzaffar al-Qazmini and Fadal al-Abbas. Sayyidina, yesterday we examined the ruling of Alimin Bi Talab and it's, it wouldn't be, it is safe to say that we came to the conclusion but it's really clear that Alimin Bi Talab's ruling was based on justice and equity. But the fourth article that insha'Allah we're going to examine tonight states that liberty consists of doing anything which does not harm others, thus the exercise of natural rights of each man has only those borders which assure other members of the society the enjoyment of such rights. These borders can be determined only by the law. Yesterday we saw that Alimin Bi Talab, peace be upon him, was the only ruler who gave freedom and liberty to his citizens. We examined the story of Talha and Zubair yesterday and over the past nights when they entered to Alimin Bi Talab and desiring to acquire political position in the ruling of Alimin Bi Talab. So when he saw them he extinguished one candle and lit the other. So they were you know kind of shocked. They came and asked him can we speak to you, told them what you have to speak about, does it concern the matter of the Muslims or is it a personal question, they said no it's a personal question. So the Imam turned off the candle and when they asked him why he turned off the candle he said because it's bought from the public treasury from Bayt al-Mal. If you're not speaking to me about the interests of the Muslims then this needs to turn off and actually they sat in darkness in some and he lit another candle that belonged to him not to Bayt al-Mal. And he said you know I lit this candle because it's bought from my own pocket. When they saw this they just left without saying any other word because they saw how strict he was. They seen the equity of Alimin Bi Talab, peace be upon him. But Alimin Bi Talab as we mentioned he knew that Tarhan Zubair were leaving to Basra to gather troops and mobilize troops against him. Definitely. But liberty in this article how does it compare with the liberty of Alimin Bi Talab, peace be upon him and the freedom of Alimin Bi Talab. Well first of all you know every human being is born free. By nature every single creation is born free. Every human feels freely, thinks freely, speaks freely, acts freely, unless he or she is forced otherwise, correct? By a force that eradicates his or her freedom. And Amir al-Mu'mineen, peace be upon him, understood this. He understood it fully thus he gave them the ability to roam freely. InshaAllah we will compare the words of Amir al-Mu'mineen today to the fourth article and see how Amir al-Mu'mineen's words compare. And how Amir al-Mu'mineen gave liberty and freedom to his citizens and the people he ruled over. You know not just human beings, even the sun when it beams if there is no barrier freely it will beam over anything it sets on, correct?  Even the wind, the wind, the waves of the ocean unless you put a barrier, birds will fly freely, animals will roam freely unless they are put in a cage or taken in place in a zoo or they are hunted down. Same thing with human beings. And Amir al-Mu'mineen, peace be upon him, understood this. That you can't cage people, you have to give them their freedom unless it goes against the law. And that law in the government of Ali ibn Abi Talib was if you're harming others, if you harm another person that's when you have broken a law. If you harm society that's when you have broken a law and authority will stand against you. Or else no, you're doing something in your own house, far from everyone, authority will be far away from you. If you're not spreading mischief in society or in the government then you're a free man. And this is what Amir al-Mu'mineen, peace be upon him, practiced. So when they told him we're going to Basra even though he knew that they were committing treason but he gave them the freedom to do so until they stood against the law. They stood against the law and wanted to overthrow the power of Amir al-Mu'mineen, peace be upon him, and to lead a war against him and which they did. They led a war against him and that war was the battle of Jammel, the battle of the camel. In an article between Amir al-Mu'mineen, peace be upon him, with Al Mughayra ibn Shoaba. Al Mughayra was one of the workers of Umar in his era of ruling and he also worked for Mu'awiya as well. So in the period of Umar he ruled over Kufa and at the time of Mu'awiya I think after the Shahada of Amir al-Mu'mineen, peace be upon him, he ruled over Basra at the time of Imam al-Hassan. But when Amir al-Mu'mineen, peace be upon him, was in power he spoke to Al Mughayra ibn Shoaba.  So Amir al-Mu'mineen, peace be upon him, tells him will you be with me or Mu'awiya? Right now there is a battle going between us. You're either with me, I'm the Khalifa or you're against me. So Al Mughayra ibn Shoaba says well I will lay down my sword, I will neither fight for you or against you.  The hands of justice will come and get you.   When the khawaraj, when Mu'awiya put the Quran on spears, that's when they split the army of Ali al-Mu'talib, they became the khawaraj and Manik al-Ashir, his famous words, he said Ali and Quran al-Nataqah, heard Prophet Muhammad say that Ali is the Quran, but they still stood against him, but Ali al-Mu'talib did not punish them, he let them free. And that's how Mu'awiya was not defeated in the battle, they were very close to being defeated and destroyed, but with this conniving move, they put the battle to an end. Another segment from Amir al-Mu'mineen, alaihi wa sallam, to Malik al-Ashtar, when Malik al-Ashtar was the Wali of Amir al-Mu'mineen in Egypt, in Egypt. Amir al-Mu'mineen, alaihi wa sallam, sends one of the most beautiful articles written ever in history to Malik al-Ashtar. I have a meta person who has read this piece and was not amazed by the words of Amir al-Mu'mineen, by the justice of Amir al-Mu'mineen, by the eloquence of Amir al-Mu'mineen, by the laws of Amir al-Mu'mineen, by the description of Amir al-Mu'mineen, how to govern, how to rule the rich class, the middle class, and the poor class, how a ruler should carry himself, how he should speak, how he should treat his citizens, very beautiful piece by Amir al-Mu'mineen, alaihi wa sallam, Amir al-Mu'mineen tells Malik al-Ashtar regarding the fourth law of the doctrine of the man and citizens in comparison. Amir al-Mu'mineen tells Malik al-Ashtar, in this matter you should be enduring and watchful, even though it may involve your relations and favorites. You should watch your relations, and you shouldn't give someone a leverage because he is related to you or he is your favorites, from your favorites. Everyone in your government should be equal, so if you're looking for a judge, it shouldn't be your cousin, it should be who actually deserves to be a judge. The head of police, it shouldn't be someone who is related to you or someone who has bribed you or someone that you favor, it should be based upon equality, doesn't matter whether he's black or white, he's ugly or handsome, fat or skinny, no, it just depends upon whether he could carry the task or not. But today, even today in governments, what do we see? When a minister enters a ministry, his whole family goes into the ministry, correct? Yeah. If someone in this government has a relative and a crucial position, then everyone in his family is benefiting. Yeah, definitely. And overnight, they're the richest families. The rich? Or they have power. Yeah. They have power. They could do whatever they want. When they abuse it. They abuse it. You know, they become power, authority, nobody can speak to them. Police can't speak to them. They're above the law. They're above the law. And this is what we see even today in our governments. Yeah. Ameer al-Mu'mineen, a.s. says, you have to give justice to everyone. There is no favorites or relations when it comes to position of power or governance upon your citizens. Also, a little segment from the letter, he states, there are two kinds of people in this world, either a brother to you in religion or a reflection to you in humanity. And this statement is a guideline for the United Nations. And it's a guideline for the United Nations right now. But it's unfortunate to see that the West is actually pondering upon the saying, I have a student, but yet, I mean, the Muslim world keeps on declining. Of course. That's because out of hate, the West doesn't hate Ali ibn ibn al-Aliyya. The West is willing to learn from any great mind. But within the Muslims, there is hatred towards Ali ibn al-Aliyya. It's unfortunate to see that. I mean, why? Because he's greater than his greater than their leaders. This is a different topic, inshallah, we'll get into this topic another time. But even today, if you look at the time of King Abdullah in Saudi Arabia, who did most of his work, his brothers and cousins, there's rivalry between them, correct? Because each one wants to be the king. So who becomes the king's aides? His mother's brothers and children. And same thing. As soon as Salman came into power, he removed all the aides of Abdullah, which were the brothers of the mother of Abdullah and their sons, the Khals of Abdullah, and he brought the brothers and the sons of the brothers of his mother. And that's how basically that's his cabinet. That's his everything of his government. That's what runs his government. So is it based on your credentials? Is it based on where you've studied? You have a PhD? You have a master's? No. It's whose son you are. Yeah, even in this country, I mean. Even in this country. Even in this country. What's even worse is that they plagiarize or they just change their the resumes to, you know, he has a PhD in psychology. He's a doctor.  And that's basically governments of today, all around the world, all around the world. It's based on, you know, whose son you are or who you're related to. Is there equality and governance?  It doesn't matter whether they're Christian, Jew, Muslim, whether they wear hijab, they don't wear hijab. But today, it's not. You remember what happened to the girl in the, about a month ago in the American Airlines? The lady didn't even give her a closed can, because she wore a hijab. Or how many people do we know from our sisters or cousins that wear a hijab? They're not given opportunity to work in Western societies. The Bill of Human Rights was passed after or during the 1950s, yet after 9-11. Yeah, of course. All your rights are gone. All the rights are gone.  Sorry to say that. Then Amir al-Mu'mineen, alaihi wa sallatu wa sallam says, and keep in view the reward of that which appears burdensome on you because its rewards are handsome. Yes, it might be difficult for you to give someone simple a job because he deserves it, but the rewards from Allah subhanahu wa ta'ala are handsome. They're great rewards because you have ruled an equality. You have given all your subjects the right to gain an occupation and employment in your government. Now we will move to the fifth law, and between the fourth and the fifth, there is a much difference because it's all based upon whether you're harming others or not. If you're not harming anyone, then you have been given freedom, or if you're not breaking a government's law. So yeah, the fifth article says that the law has the right to forbid only actions harmful to society, similar to the previous article. Anything which is not forbidden by the law cannot be embedded, and no one can be constrained to what to do on other words. Yes, very similar. They're very similar, but this adds that anything that's not forbidden, and by law you can do, and Ali al-Mu'talib implemented this years, 12 centuries before this, and it's significant to see how they learned from Ali al-Mu'talib. This was our introduction to the program. Where did they come up with these laws? You know they have read the words of Amir al-Mu'mineen, if it's in their constitutions. Yeah, they have scholars that read every single book and every single publication, from Adam until now, and we build our laws on favors. So Amir al-Mu'mineen, we just have to see how he implemented these rules and guidelines in his government. If it harms someone, if it harms someone, then there is authority to stand before you. If it does a harm society or your peers, then you're a free man. Thus we see sometimes Amir al-Mu'mineen would punish based upon the person. Would this person, before punishment, was he sincere about forgiveness or not? Would he go back to the same crime or he wouldn't? Sometimes people repent. They commit a crime, but they repent, and they will no longer be harmed to society.   So Amir al-Mu'mineen, alaihi salatu was-salam tells him to bring this man to me. So when Amir al-Mu'mineen, alaihi salatu was-salam brings this man, he asks him, did you commit a robbery? And at first he declines. He says no. Amir al-Mu'mineen asks him again, did you commit the robbery? He says yes, what was my first time? Amir al-Mu'mineen, alaihi salatu was-salam says no, no by Allah, this wasn't your first time, because Allah SWT, his justice would not allow him and his mercy would not allow him to reveal someone on his first sin. We see that in our own lives. This is your 40th time that you have committed the sin and Allah SWT has revealed you. This man falls into tears. He says yes, Amir al-Mu'mineen, you're right. This is my 40th time. At this time, this man must be punished. It's not his first time. If he's not punished, people will learn from him in society. Crime will spread in society and he would never learn. So Amir al-Mu'mineen, alaihi salatu was-salam punishes him and his hand is amputated. This man carries his amputated arm and he goes, sits in one of the streets, one of the alleyways and one of the enemies of Amir al-Mu'mineen in Kufa overhears this man praising Amir al-Mu'mineen alaihi salatu was-salam. He's shocked.  He says yes, why shouldn't I praise a just ruler? I deserved what I deserved. Even with the punishment. I deserved it. This is reality. Amir al-Mu'mineen is the most just ruler to rule after Rasulullah sallallahu alaihi wa alaihi. He was happy about that.  He says the lion of the Arabian peninsula cut off my hand. It wasn't a normal man. This is not a normal man. I wasn't under punishment by some normal man. This is Ali ibn Abi Talib. The person who stood against our arrogance and those who worshipped idols, this person came from a pure lineage unlike all of you that I know. This man came from a pure lineage. He's the father of Al-Hasan and Hussein. He was the first person to become Muslim. His message Ali was the first man to believe and the first man to pray to Allah subhanahu wa ta'ala after Rasulullah sallallahu alaihi wa alaihi. And the last disciple from Al-Yasin the household of Rasulullah sallallahu alaihi wa alaihi. The person who was aided by Jabra'il, the person who is second aid is Mikael. This is not a normal man. This man is the vice gerent of Allah subhanahu wa ta'ala upon humans and the sons of Adam. What happened to me was a divine law. It was a divine ruling and Ali ibn Abi Talib set that rule upon me. And that is the prince of the believers. Amir al-Mu'mineen overheard the words of this man. Imam Hassan alaihi sallam was walking through the alley and he heard the voice of this man, the words of this man. He went to his father Amir al-Mu'mineen alaihi wa sallallahu alaihi wa sallam and he told him, Oh my father, this man is praising you even though you punished him. This man truly loves you even though he had committed a mistake. He's a criminal but his heart is pure. He's filled with love for you. And my father, your enemies have surrounded him and they're telling him you were punished by the person you love. Oh my father, we are not a family that punishes those who love them. Amir al-Mu'mineen told Imam Hassan to go call upon this man. He went and called him.  Address him by uncle.  My father is summoning you to his Masjid, Masjid al-Kufa. This man carries his hand and he walks towards Amir al-Mu'mineen.  And he recites a prayer, two unit prayer and asks Allah subhanahu wa ta'ala to give this man his hand. He removes the cloak and Allah subhanahu wa ta'ala shows the Muslims a miracle conducted by Amir al-Mu'mineen alaihi wa sallam. But it shows you justice. Punishment was placed when it needed to be placed. Punishment was placed by Amir al-Mu'mineen upon this thief. And in another instance, a woman comes to Amir al-Mu'mineen alaihi wa sallam. You know, occasions like this occurred many times in the lifetime of Amir al-Mu'mineen, especially during his time of leadership and Khilafah, where someone would come to Amir al-Mu'mineen alaihi wa sallam and he or she would tell the Imam to purify them from their sins if they had committed adultery or fornication. And Amir al-Mu'mineen alaihi wa sallam would tell them to go. Why? Because the ruling in Islam for the punishment to occur for just witnesses have to witness the act. So if a person comes and says, I have committed this act for the first time, Islamic ruling cannot be placed. So Amir al-Mu'mineen knows that this person is coming because he has repented or wants repentance. There is remorse in his heart. Amir al-Mu'mineen would usually tell them to go, to go and ask Allah SWT for forgiveness and to never go back to the sin. This woman comes to Amir al-Mu'mineen alaihi wa sallam and she tells him, Ya Amir al-Mu'mineen, I want to be purified from my sin. I have committed adultery and I'm pregnant at the moment from that sin. Right now, if it was Boka Haram or Daesh, they would throw her from the top of a cliff without even seeing where their witnesses, their were witnesses. It's different with them. They just blame even if they don't know. So Amir al-Mu'mineen alaihi wa sallam tells her go until you give birth. Go, there's a child in your stomach. Go give birth and then we'll see what happens. Months pass, days pass. This woman comes back to Amir al-Mu'mineen alaihi wa sallat wa sallam and she tells him, Ya Amir al-Mu'mineen, I've given birth. Purify me. Amir al-Mu'mineen alaihi wa sallat wa sallam turns to her. He says, go, breastfeed him for two years. Breastfeed him for two years and then come and we'll see what happens. You know, he knows this lady will never harm anyone else or spread mischief in society ever again. She's coming over and over again because she fears the punishment of Allah SWT for those who have committed the sin on the day of judgment. Two years pass. She comes back to Amir al-Mu'mineen alaihi wa sallat wa sallam and she tells him, Ya Amir al-Mu'mineen, two years have passed. Purify me. He tells her, go until your son is old enough to take care of himself. Basically, he's giving her time for repentance, giving her time for repentance. And he knows she will never harm anyone in society ever again. So if she had committed a mistake or a crime that didn't harm anyone, then it's between her and Allah SWT. It's between her and her Lord and her favor towards repentance. I mean, when we hear such stories, we find that Ali bin Abi Talib peace be upon him was not just a regular person who ruled a regular government. I mean, until now scholars and even scientists, they use Ali bin Abi Talib's government as a role, as a role model for them to guide their nations. I mean, we see that the United Nations also use the words of Ali bin Abi Talib to, you know, to act justly.  A man committed adultery and came to Rasulullah. He said, Ya Rasulullah, I've committed adultery through the young man. So he turned away from him. Told him, don't tell me. Go. Just go. And he turned away. He turned to Prophet Muhammad peace be upon him again. And he just allotted the same thing.  Yeah, I mean, it would have been better for you to go and ask Allah SWT for forgiveness and repent to Allah SWT instead of, you know, broadcasting or telling me, I mean sins are not there for us to, you know, to share with others. The only person or that could forgive us is Allah SWT. And the best way, I mean, you know, we're going off topic. But the best way, in short, the best way to ask forgiveness from Allah SWT is through Ahl al Bayt. Insha'Allah, may Allah forgive us. And the last part I want to mention in tonight's program is going back to segment four. And also, it also has to do with segment five, giving the opportunity to all your subjects to employment. This was known about Uthman bin Affan that as soon as he came into power, all of Bani Umayyah came into power, whether they were righteous men, but I don't know one from Bani Umayyah that was a righteous man. He brought all of them, all his cousins, his family members. And that's how Muawiyah came into power. In his time of ruling, it became so bad that his cousins and family members took advantage of the Muslims and the wealth of the Muslims that ended up in his killing. For example, Uthman brought one of his cousins. His name was Al-Walid bin Aqbah. Al-Walid bin Aqbah was a known alcoholic. And he had placed him as the Khalifa or the ruler of Kufa. This man would drink fornicate and steal the Muslims' money, openly and publicly. Amir al-Mu'mineen sent Malik Al-Ashtar to Uthman, told him you either change your cousin or we'll have to do something about it. Because a leader of the Muslims cannot carry such traits. This man is an alcoholic and everybody in Kufa knows. He takes advantage of the Muslims, the wealth of the Muslims. And there's nothing about him that says I am Muslim. He drank publicly. He drank publicly. So Amir al-Mu'mineen, alaihi wa sallatu wa sallam, kept repeating to Uthman, changed him. And Uthman was taking it into consideration until another Mal'oon came to Uthman and brainwashed him, Marwan ibn al-Hakam. He came and he told Uthman, no, no, no, you don't have to worry. Keep him. Keep Al-Walid, Al-Walid bin Aqbah. There's no problem with it. Why? Because there were cousins, there were family, and they were all benefiting. And Uthman changed his decision from moving such a man who was so far away from Allah subhanahu wa ta'ala and the message of Rasulullah sallallahu alaihi wa ala to rule over the Muslims. But this is what we see in Islamic heritage, Islamic history, and the history of the Muslims. And history repeats itself till today. It's unfortunate to see that. It's very unfortunate because when we examine, I mean, the lives of Ahl al-Bait and compare them, because when Prophet Muhammad was on his deathbed, he told the ummah, if you follow the two, my Ahl al-Bait and the Qur'an, you will not be led astray after me. But yet, their wills for this materialistic world, their love for this world, made them blind for no reason. I mean, now, where are they right now? I mean, that great Christian poet who went to visit Ayunabi Talib in Najaf, and he went to Syria to see where Muawiyah was, and it's a washroom. And he told his famous poetry lines. You see the imams, they're alive through the pilgrims of Imam Hussain, through the shrine of Imam Hussain, the shrine of Abbas, millions of people visit them annually. And you see Muawiyah, Yazid, and people of their likes embedded till today in whom? Isis. You don't see anyone else praising Muawiyah or Yazid besides Isis members, or being proud to support them or to follow them? It's weird to see that. I mean, we see, not to mention anyone, you know, but a small example is the scholar in Syria, what's his name? Laar Oor? I mean, one time on TV, he said, why when Al-Mahdi, he doesn't say, he says Al-Mahdi, why when he reappears, he kills Bani Umayyah. We are Bani Umayyah. What have we done to him?  Their lineage is still apparent. Definitely, but you know, Sufiani will be from the lineage of Bani Umayyah. Subhanallah. So it's the fruitful tree, the pure tree, and the tree of monkeys, and the tree of evil. Yeah. Monkeys that were playing on the pulpit of Rasulullah. So Sayyidina, we're coming to a conclusion of the episode, if you'd like to conclude. Insha'Allah. Insha'Allah, we'll continue the message of Amir al-Mu'mineen, for the nights to come. If our viewers have any questions, I'm sure they can send to your email or to the channel's email. What is the email address for the channel, if people want to inquire questions or anything? Inshallah. Our viewers, we're looking forward to your questions and comments. May Allah bless you all, and insha'Allah, we send you peace from the shrine of Imam. Insha'Allah, thank you very much Sayyidina for joining us tonight, and thank you very much respected viewers for tuning in for tonight. I hope, insha'Allah, we can learn from the justice of Arabic we thought of, to not oppress anyone, and to abide by the rules of Islam and the rules of Ahlul Bayt. As Sayyid mentioned, if you have any questions, you can also inbox us on Facebook at imohasayn3tv, also on Twitter and YouTube. Leave a comment or review the previous episodes. Stay tuned for the next episode. Salam alaykum Sayyidina, alaykum.