 So, just to show you that experimentalists have seen these frequencies which are non-degenerate a 1 symmetric a 1 bending then b 1 asymmetric stretch and then there are these linear molecules which have 3 n minus 5 degrees of freedom vibrational degrees of freedom and then you also have similar to your water molecule there are other molecules ok and methane of course, I just briefly mentioned the TD symmetry and you will have these 3 n minus 6 or I have written some things here or it is already there it is not that the thing is some of these modes all these frequencies for example, if you saw these numbers they are all in the infrared regime ok, centimeter. So, they are all whatever you going to measure associated with the vibrational mode of a molecule has opened up a new field called infrared spectroscopy where they do these infrared spectroscopy to find the spectra of the molecules and so on. So, in that sense what we are trying to only say is using the group symmetry these are the kind of things have to be measured we do not know those frequency values. If you want to find these frequency values you have to go through that long method find the basis vectors and then find the frequencies by solving those basis vectors which are not done here ok. You can do it, but it is a little more work to be done, but experimentless can give you that these frequencies are seen in the infrared spectrum these are seen in the Raman spectrum and so on ok. Suppose I say a irrep is allowed for both infrared and Raman I will just explain what they are as far as the group theory person is concerned and if they see the same frequency in both we are actually validating it ok. So, let me give you some flavor of what exactly is happening ok. So, whatever we did so far is first you need to know the point group of the molecule and then you obtain either you can obtain a complete motion of the molecule which has 3 and degrees of freedom or you can subtract out the translation and rotation and write the vibrational representations also. Those are things which we can do, but essentially you have to remember that the way we write those characters are which of the atoms are in moved by the symmetry operation that is encoded in the NC. NC is the number of atoms which is not moved by the operation and then reduce it by the conventional decomposition. Eliminate translation and rotation if you started with 3 and degrees of freedom and the remaining what you get are the vibrational modes ok. So, just to give you a flavor of what these experimentalists look at the molecular spectra is the spectroscopy. So, you do have various vibrational rotational transitions and also you can have electronic transitions. Let me not get into too much of a detail here. Transition rules we have already seen for you to have a transition from one energy to another energy such a transition which is triggered by an interaction is possible if they satisfy the rule that you know the tensor product of that interaction operator, the irreducible representations of the states if everything adds up to give you a unit representations multiplied tensor product then you know that such a transition is allowed that we have seen in the two lectures before this ok. So, what I am trying to say is that here experimentalists have tried to look at these vibrational transitions and they as I said the energy or frequencies are in the infrared. Rotations of course, are in the microwave which I am not going to get into it. So, for a molecular vibration to be seen in the infrared spectrum they call it IR active, you should have in the vibrational modes the irrep of these vibrational modes should be same as the irrep which I associate with a dipole moment ok. So, this is what is the approach of looking at basically the molecules which as a net dipole moment they are supposed to be infrared active. So, what happens in the water molecule is that you do have A 1 irreps, A 1 irreps is like the z component basis right and they are like the dipole moment z component. So, in that sense you can say the two frequencies associated with the two. So, for the water molecule we found that the vibration is 2 times A 1 plus B 1. A 1 irreps the basis was z basis equivalently I can write the electric dipole moment which is the z basis. So, since A 1 is transforming like the z component of the electric dipole moment I can say that the vibrational mode associated with the irrep A 1 which is seen in the water molecule should be IR active ok. So, the A 1 the two A 1s must be IR active. So, that is what I am trying to say here if you see the basis the two vibrations with A 1 symmetry as z as a basis function they will be seen in the infrared spectrum of water. This will result in two peaks you have to get two peaks at different frequencies you should not see them to be very close should not be degenerate because the C 2 v symmetry is abelian ok. So, that is why it is the water molecule is sometimes called as a intra reductive spectrum. The other Raman process have you all done this Raman have you how many raise your hands ok. So, let me just give you what exactly I am I am trying to say in this case is just that you can send in you know you can have an initial level with some frequency omega it gets excited, but it gets de-excited with a lower energy. So, omega naught is the initial energy and then omega vibration is that it is the final. So, this is the difference ok. You start seeing a peak this difference is you can see that omega naught minus omega vibration is positive. So, these are called Stokes lines usually when we have been doing all our exercises if you excite an atom take it to an excited state it de-excites back and you see that that energy is equal to the energy when it de-excites back. This is something which we always do which is the Rayleigh scattering Rayleigh line ok. Anti-Stokes is the opposite of Stokes you excite it with lower energy, but it de-excites with a higher energy. So, this is something which is not very normally seen in molecules, but it could be generated is what is the these are the three lines and they call this as Raman active if you find such frequencies in these vibrational modes of the molecule ok. So, one of the quantities which one should look at is the dipole moment time dependent dipole moment related to the electric field and the object which it connects is the polarizability. So, polarizability is what tensor this is a rank one which one will be it will be binary basis or primary basis no. So, it will be a binary basis it is another like moment of inertia tensor this will also be a tensor and basically you can see that the frequency with resin changes Rayleigh frequency with changes Stokes and. So, these are the three terms which turns out out of this time dependent interaction term and what is going to be seen is look at the binary basis for the group symmetry. If the irreps has any of these basis then you say it is also Raman active that is all we do ok. So, for a molecular vibration to be seen in the Raman spectrum its irrep the vibrational irreps must be same as the polarizability of the molecule. The polarizability has the same symmetry properties as a binary basis or quadratic functions ok. So, you can look at here that the two vibrations with a one symmetry also has these quadratic basis. So, you can say that it should be the Raman spectrum as well and they do see in the Raman spectrum those specific frequencies with they see in the infrared spectrum. So, that is kind of validating that you get the same frequencies and these are the three vibrational modes of the water molecule ok. What about b 1? b 1 has x and x is that as basis again the vibration will be both infrared active and Raman active. So, you do see this as a peak in both the spectrum Raman spectra as well as in the infrared spectrum. So, that is the summary and end of my discrete groups. Now, I will start with continuous groups that does not mean you can forget whatever I have taught in discrete groups. You will see a lot of parallel in understanding how there could be connections between what I have taught in the discrete groups, how it happens in the continuous groups. Of course, continuous groups are having a set of modified you will have some modification because continuous groups are not finite order elements right. The group will have infinite elements and we need to handle it in a different way by introducing generators. In the discrete groups also we introduce generators. Even if we introduce generators the number of generators and the number of elements are not significantly large that you need to worry. You could also write all the elements of the group and work with us, but in the context of continuous groups we have to work with generators and that will be the theme of how to see this in the rest of the semester ok. Any questions on this? I am going to get on to the board to give you a flavor today and next we converts we will go more on SU 2, SU 3 and so on ok. So, the plan will be. So, we will do a warm up on translation now and then a similar thing can be done for rotations. So, translation where you can translate by a vector a any position where a could be any value. What do I mean by that? This can lie in R 3 is what I would say technically. What I mean is that A x, A y, A z can lie between minus infinity and plus infinity that is the meaning of this. This is like a three dimensional space unbounded. A x translation which you want to do on an object can be any value between minus infinity and plus infinity. Similarly, rotations will involve theta x, theta y, theta z where the values are at the rotation about x axis, y axis and z axis. The values can lie between 0 to x axis and 0 to 2 pi. Then I will slowly introduce groups which are leak groups. First I will do a warm up to get a clarification on the notations. So, the leak groups specially I will confine to this S denotes I will tell you what it is. S denotes that you write 2 cross 2 unitary matrices with determinant plus 1. So, this denotes this one denotes unitary matrices with the lowest dimension as 2 cross 2, lowest non-trivial dimension as 2 cross 2 and this S denotes determinant to be equal to plus 1. Next thing I will do is SU 3 that will be again the same notation the lowest non-trivial dimension is 3 cross 3 unitary matrices with determinant plus 1. Then we will get on to Lorentz transformations Lorentz transformation is on three space and one time right. So, let me denote it by 3 comma 1. I have to remember that it is like R 3 space and then you have one time coordinate which is having a slightly a different meaning ok. So, the Lorentz transformation will also involve various parameters. What are the parameters you will have? You will have rotations in so let me write it space time which we call it as 3 comma 1. So, this will include rotations in R 3. So, it will include rotations in R 3. You will also have boosts by boosts I mean that you translate one inertial frame with respect to another inertial frame by introducing a velocity along the x direction. So, these three rotations are theta x, theta y and theta z. The boosts are translating or moving one inertial frame with respect to another inertial frame with velocity v x, v y and v z. You all know this I suppose how many of you have not done basic relativity, special relativity, special theory of relativity. You have done this right. So, what are the ranges? You can see that theta x, theta y, theta z the ranges will be 0 to 2 pi, v x, v y, v z will be minus minus c to plus c. That is because of special theory of relativity which says that no particle can, no frame can move with velocity greater than velocity of v.