 time plus some constant. These two together we could write it as x mu going to x mu plus a mu, x 0 is like time with dimensions if you want to write, you have to write the velocity of light multiplied by time and x 1 is x, x 2 is y, x 3 is z. So, these are the compact way in writing in a space time notation. So, this is space translation, this is time translation, this is compactly a space time translation. So, this space time translation we elaborated so much there exists an operator right which when acts on x mu will give you psi prime of x mu and we determine what that operator was. We did this for simple one dimensional motion in x direction and then I generalize to arbitrary r vector and I also said what will happen for the time translation and what did we find? We found that t in general will be an exponential with a minus i and then you have an operator I am going to write it in a compact notation 11 operator associated with an operator there is a parameter divided by h cross and what is this operator going to be? O 1 will be O 0 will be Hamiltonian or the energy operator O 1 will be your momentum p x O 2 will be p y and O 3 will be p z is that clear? The corresponding parameters can take any value between it is not bounded you can take your time to be if you start the time as 0 then it is 0 to infinity is minus infinity then it is minus infinity in principle this is going to give you a parameters in three dimensional space and one time direction ok. But what you have to observe is that this operator is what we called it as a generator. So, what is the role of a generator is that if you do an infinite symbol transformation by infinite symbol I can write my m u to be delta m u times n you can do the finite translation in n steps each step having an infinite symbol step ok. So, these delta m u's are very small and n times this will be the total finite translation. So, if you do it this way you can write what happens. So, I should write that this T is dependent on m u dependent on m u and now I can write what is T as delta m u and this will have always the first term will be identity and then the next term will be some i by h cross the parameter times the generator which I can call it as some m u ok and instead of this delta m u you could replace it by m u by n ok. So, because this non trivial transformation takes you away from an identity state if you operate it on some state some vector the first one is like as if it does not do anything and this gives you a deviation or takes you away from its initial position and infinite symbol transformation is good enough to take you away from the initial position and that is why this is called generator for such a transformation. Equivalently you can write any finite transformation you could do this n number of times ok. So, T of delta m u you can do this n number of times this is for infinite symbol step and you are going to do n steps for every step you do a T of delta m u. So, this will be n number of times and this can be shown to be you can do that on this and show it to be an exponential and you have to take the fact that limit of n tending to infinity because the number of steps for a finite transformation will be infinite delta m u is really small this product will give you a finite value. So, this will give you an exponential form which I was telling you that minus i by h cross m u operator fact this plus this also can be in general you can use you can absorb these signs into your definition of your delta m u. So, since I have followed this let me just keep it like is this clear. So, this step is straightforward this is the definition of your exponential right you put delta m u as m u by n to the power of n n tending to infinity will give you a. So, that is why we call this operator as generator. Generator for translation in this particular case, but in general it could be some other transformation like rotation or other internal transformations which I could look at. So, right now let us just confine to some examples. So, translation is something which I have discussed you can even take it for rotations a deviation from identity which operator performs such a deviation that operator differential operator is what you call it as a generator of this such a transformation is that clear. So, coming back to the slide. So, we studied this under space time infinite simul translations you have a slight change away from identity as I said this plus or minus i is just a matter of definition you can absorb it into an epsilon. So, this gives us a epsilon mu is infinitesimal and x mu is the initial position and it gets shifted by an epsilon ok. So, infinite simul transformation will give parameters epsilon mu and the corresponding generators is for rotation in space you have to show that the generators are not linear momentum. So, we already saw for translation the generators were linear moment. We do this exercise for rotation what is rotation going to do you have a vector r and you do a rotation operation and gives you an r prime rotation will depend on an angle and a choice of an axis right will depend on some rotation angle let me call it as psi just for uniformity with my slides and a choice of a direction unit vector direction you have to choose z axis x axis y axis or arbitrary axis n hat denotes the arbitrary axis about which you rotate by an angle psi is that clear. You have an arbitrary axis usually a unit vector will be given it can it can denote your theta phi angle in your spherical polar core take a unit sphere the direction of it on the unit sphere will give you different direction corresponds to different theta and phi. So, unit vector on the sphere will generally depend on theta and phi if you take the z axis theta is 0. So, similarly you can take other axis also theta equal to pi by 2 will be your x axis and so on. So, you can choose your axis accordingly in general the rotation can be will be fixed by an axis of rotation and the amount by which you are going to do a rotation. So, this is the rotation operator and it is going to change your vector r to r prime. The next question you will ask is how do we write the infinite simul transmission. So, just for simplicity let us take the rotation about z axis. So, let me take it to be rotation by some infinite simul angle which let me call it as theta or if you are getting confused let me put it as psi about z axis. So, this is z cap direction you all know what this is that is first one you can write this as cos delta psi sin delta psi minus sin delta psi cos delta psi. Of course, I have written it in the x y z basis. So, which means I have to make it into a I am looking at rotation of vectors in three dimensions. So, this is going to be and because delta psi is small you can rewrite this as identity plus a delta psi. Now you tell me what is the generator up to some normalization and sign. The generator let me call it as O subscript z for rotation will be proportional to 0 1 0 minus 1 0 0 0 0 0. This is the matrix representation for the generator executing a rotation about infinite simul rotation about z axis. Just like the way I did here I would like to write this also as r vector plus some infinite simul quantity delta r was a delta a mu here I want to find what this delta r is. Can you help me out with this using this. So, just check that delta r for rotation about z axis can be written as delta psi z cross can you check this is that right. So, if you use this you can show that x prime is x plus delta psi y using this r r matrix x prime the components x prime y prime and z prime will be I could compactly write that the change in vector for a rotation about z axis is this, but in general I could write this as if it is arbitrary axis by putting a vector rotation here or of course is a this is clear because this is a z component cross product gives you only either x or y the z component cross z component will be 0. So, that brings me to writing this delta r under rotation about arbitrary axis can be written as delta psi this is now an angle, angle which is a vector in a sense that you could resolve it into components about z axis x axis and y axis. So, this is this psi will be on an arbitrary hat n axis cross orbit this is clear. I just given you explicitly an operation where it is done with respect to z hat. So, this has the this is just a z component other components are all 0, but you can choose it to be an arbitrary n hat where it will have z x y z components and then you can write the change in the vector in three dimension under the infinite symbol rotation by an angle psi. The psi is I am using the psi to be the angle of rotation about arbitrary axis, arbitrary axis is given by the unit vector or unit vector will give you theta and phi coordinate is that clear ok. So, now once I give this what is the next step I need to find an operator a with respect to rotation which acts on the wave function sorry I am using the same thing, but I am putting a capital psi here psi prime of r wave just like what we did for translation. We want to find an operator where you can also use the additional fact what is the additional fact psi of r vector is same as phi prime of r of r, r is the rotation operation on this vector and these two are one and the same ok. I am just doing whatever we did for translation in the context of rotation. You rotate the wave function state as well as the space nothing changes and you want to find an operator at the same point how the functional form of psi changes. Any questions on this? So, tell me now you will use this fact to write the r which I have. So, you can do. So, do this and tell me what will happen. So, you can write this as r inverse of r prime right to write it and then do the transformation. So, let me write it for you this one is same as this is right this one is r prime r prime is r plus delta r. So, same as this for infinitesimal whatever I call it as r plus delta r let me call it as r then the delta r comes here with an opposite sign right. This implies you all with me I do not see anybody nodding their head sleepy tired fine good. So, so this is the expression which we get a Taylor series expansion. So, do a Taylor series expansion for this. So, the first term is psi of r minus delta r times sorry this psi should be a capital psi. So, in fact, I should write technically as a dot product with the gradient just a Taylor series expansion and what is delta r? Delta r is delta psi cross r ok. So, this is putting a capital psi for the wave function what is that one that is a scalar triple product. I can play around the scalar triple product and write it as tell me. So, this let me call it as p with a i by h cross introduced here ok. So, this is equivalent to is that right if you have made a mistake on the signs, but you understand what I am saying ok. So, this is a scalar triple product and it can be rewritten as delta psi dotted with r cross p given you now the generator. So, a r of psi of r identity minus i by h cross delta psi dotted with r cross p are you all with me. Now, tell me what is the generator for rotation now? Infinitesimal rotation which I am doing this r is for the rotation this is the operator which takes you away from identity the one which takes you away from identity for an infinitesimal rotation is this piece and this piece is what we call it as a generator for the corresponding transformation which is a rotation cross p which you all know what it is is angular momentum orbital angular momentum orbital angular momentum is the generator for rotations in 3 dimensional space 3D space. I am confining myself to rotations in 3 dimensions. So, how many components are there? It is r cross p is a vector. So, there are you can have L x L y L z like p x p by p z there are 3 generators and the corresponding parameters are your delta psi z delta psi x delta psi y depending upon your fundamental axis of rotation which is like choosing x axis y axis z axis. So, what have I told you now always you will find number of generators will be equal to the number of parameters that is why you can do this doting dot product of parameter with generator otherwise you will not be able to determine what is the generator for simple cases like this.