 Welcome to the lecture number 10 of quantum mechanics and molecular spectroscopy course. As usual, we will begin with a short recap of the previous lecture and carry on with the present lecture. In the previous lecture, we talked about the perturbation theory of the many states. Finally, using the first order approximation, we came about the equation a of t of a state f which is could be a final state, its coefficient will be equal to 1 over i h bar integral e to the power of minus i omega i f t integral f h prime t dt with some limit 0 to t prime. Now, what does this indicate? It indicates that if there is a two states, it indicates if there are two states a initial state i and a final state f ok. The transition from the initial state to final state and that coefficient is given by this equation. And you will also see that this transition from the initial state i to phi will be bought about this by this time dependent perturbation ok. Now, that is only the coefficient, but the probability is the square of the coefficients. Therefore, probability of transition to a state f will be equal to now 1 over i h will be minus 1, but then probability always has to be positive. So, this will be 1 by h bar square e to the power of minus i omega i f t integral f h prime of t i dt modulus squared. So, it has to be real number or a real positive number is what the p of t would be. So, this is the final equation. So, this is the probability and that depends on this integral and it turns out that this integral also governs so called selection rules. That means, under what condition you can start from initial state i and end up with the final state f. However, what we do not know as of now is this h prime t. Now, it is not very difficult to realize that h prime t or the perturbation Hamiltonian or the time dependent perturbation should come from the light. So, which means we need to look at light and that is the content of our lecture number 10. Now, light you know can propagate let us assume that it propagate along the x axis and then there is a vector or electric field that goes up and down like that. So, along y axis. So, your electric field vector is going along x and y axis and you know that electric field vector will go up and down something like that. Now, there is also a magnetic corresponding magnetic field which goes in the z axis something like that. So, this is my z axis. So, this along z you have this magnetic field y axis you have electric field this your x axis will be propagation direction. So, the propagation direction the magnetic field and electric field are mutually perpendicular to each other. Let us assume that there are 3 vectors that represent these quantities. Now, for propagation direction there is a vector called r which is nothing but the radial vector that means you are going away from some point and there is an electric field vector which I will call it as epsilon or epsilon like that and there is a magnetic field vector called b and these are unique vectors along that direction and you will see the electric field vector will keep changing the direction. Once it goes up then comes down and changes its sign and goes to other direction. Similarly, your magnetic field but the propagation direction is along the x axis in one direction that is not going to change its sign. Now, other thing that is important is something called wave vector. Now, wave vector is nothing but how many units of wavelength does the wave travel. Now, this is given by wave vector is also called as k and this is given by 2 pi by lambda k 4 equations. Those 4 equations are very well known called Maxwell's equations and this says del e or del dot e equals to 0. So, let us call equation number 1 then you have del dot b equal to 0 this is second equation. Third equation is del cross e equals to minus d b by dt and del cross b is equal to mu naught epsilon naught minus d by where mu naught is permeability of free space and epsilon naught is permittivity of free space such that c square that is speed of light is equal to 1 over mu naught times epsilon naught. Now, that is the general description of the light in terms of Maxwell's equation. Now, the Maxwell's equations are associated with electric field and magnetic field. Now, the electric field let us say e we will have 3 components e x, e y and e z this is in general you know light propagating in arbitrary direction. Similarly, one of them one of or more than one of them could be zeros say for example, if it is traveling along x direction then why it will either have y or z or y z it will be in y z plane. So, e x will be 0 something like that. So, you have b will also have b x, b y, b z but you see Maxwell's equations are only 4 equations involving e and b. So, we know Maxwell's equation is del dot e equals to 0 and del dot b equals to 0 and del cross e is proportional to d b by dt and del cross b is proportional to d e by dt ok I have just omitted the constant constants. But you will see they have there are 6 variables and there are 4 equations. So, there is some somewhat of redundancy in the number of variables ok, but that is not really the point that I would like to capture. The most important point that I want to say is that when you will treat Hamiltonian ok the Hamiltonian has kinetic and potential energy terms ok. So, Hamiltonian it has kinetic energy plus potential energy it has no concept of field and what the classical light is saying it has electric field and magnetic field, but what we want is kinetic energy and potential energy. So, we need to convert these electric fields and magnetic field into appropriate potentials that is what we need ok. So, now we make a transformation ok. So, in the transformation we start describing the electric field and magnetic field or the classical wave light in two other quantities called vector and scalar potentials because we need potentials and not fields ok. So, we make an alternate description of this and we get to what is known as scalar is noted and vector. Now, this will turn out to be phi is 1 that is a scalar potential. Scalar potential says just means that it is a number and there is a vector potential a and this will have 3 components A x, A y, A z. Totally we have 4 variables and now what you are going to do to move from electric and magnetic fields to scalar and vector potentials you are going from 6 variables to 4 variables that means once again you have redundancy. So, there will be infinite number of possible combination in which you can reduce 6 variables to 4 variables. Now, you can take some physically important fixations. So, I say I want to fix this quantity I want to fix that quantity and then you can reduce the number of possibilities ok. And in electrodynamics this is called gauge fixation ok and for this course we employ what is known as Coulomb gauge C O U L O M V Coulomb gauge. Now, before that we also need a relationship before we do the gauge fixation we also need a relationship between the electric field magnetic field and the scalar and vector potential. So, your electric field E is given as minus del phi minus dA by dt and your magnetic field B is given by del cross A ok. Now, then we have I told you there are 4 variables phi A x, A y, A z and these have to be fixed from 6 variables of. Now, then we use something called Coulomb gauge in which it is a constraint Coulomb gauge is a constraint in which we say phi equals to 0 and del dot A equals to 0. So, this is the Coulomb gauge Coulomb gauge fixes these two quantities ok. Now, when I fix these quantities then your A E becomes minus dA by dt just A is just time derivative and B will be equal to del cross A. Now, you can see very simply the following ok. Now, the vector potential is time derivative of E and B which is the magnetic field is nothing but the curl of A ok. That means, that B will have B will be in perpendicular direction with respect to E ok. Now, we have 3 vectors by the way E let us say we have a unit vector we had unit vector epsilon for E and you had unit vector B for magnetic field ok and turns out that A will also be along the unit vector epsilon because it is just the time derivative. So, the direction is not going to change, but del cross A is the curl. So, it is going to change ok. So, A will be perpendicular to B that means vector field A will be perpendicular to magnetic field B. So, what is perpendicular to magnetic field B the electric field. So, the direction of the electric field and the vector potential will be the same. However, they are going to be even the direction is same they are not going to be same because A that is the vector potential is a time derivative of E ok. Then you have the wave vector k which tells you the propagation direction. So, which means totally what you have is that your k vector is perpendicular to E vector which is perpendicular to B vector. So, all the 3 the dodging of propagation or the wave vector the epsilon that is the electric field vector and B that is the magnetic field all 3 will be perpendicular to each other ok. Now, very generally let us take A should be equal to A naught some maximum into epsilon that is going to be the unit vector ok e to the power of minus e to the power of i k dot r minus omega t plus e to the power of minus i k dot r minus omega t. Now, k dot r is nothing but the dot product of the wave vector and the propagation direction which simply means that the wave the propagation vector and the wave vector all along the same direction. So, which simply means that this term is now called phase factor. Now, if you remember the classical wave equation this is nothing but your alpha remember alpha was 2 pi by x by lambda minus mu t. Now, if I convert this 2 pi into x pi lambda minus 2 pi mu into t. So, this is nothing but 2 pi by lambda is k k vector x let us say the direction of propagation is x, but it could be r. So, r is generally minus 2 pi mu is omega. So, that is exactly what you have here ok. Now, if I want to write A ok let me write once more A naught epsilon e to the power of i k dot r minus omega t plus e to the power of minus i k dot r minus omega t. Now, this is nothing but e to the power of i theta plus e to the power of minus i theta this is cos theta. So, this can be equal to 2 A naught epsilon cos of k dot r minus omega t ok. Now, where you know I only told omega is angular frequency r is propagation direction and k is wave vector. Now, in the Coulomb gauge or in general e was equal to del phi minus d A by dt and my A is equal to A naught into e to the power of minus i k dot r minus omega t plus e to the power of minus i k dot r minus omega t by the way this is also called plane wave representation. Now, this is nothing but if I take e ok. So, e will be the first derivative of this with respect to time ok. Now, you will see k dot r will be constant and minus omega r will be just be the this one. So, this will be A naught omega into epsilon divided by e to the power of i k dot r minus omega t minus e to the power of minus i k dot r minus omega t ok. Now, this is nothing but this is if you see this is cos theta. So, this will be equal to sin theta ok proportional ok. So, this will be 2 A naught omega epsilon dot r minus omega t ok and this also can be written as there is a negative sign I am sorry I am sorry because when you take a derivative this will minus omega t will have negative sign. So, there is a negative sign. So, this I can write it as E naught epsilon sin k dot r minus omega t where E naught is equal to minus 2 A naught omega. I have now started looking at the thing as electric field and vector potential ok. So, the vector potential and the electric field are time derivative with respect to each other. So, you can see that the vector potential is cos theta function and the electric field is sin theta function and you know sin theta and cos theta have a phase shift of pi by 2. So, therefore, the vector potential A and the electric field E are out of phase with respect to each other by with respect to each other by pi by 2 that is 90 degrees. So, whenever the electric field goes up the vector potential comes down and whenever the vector field sorry electric field goes down the vector potential comes up and the other way around ok. So, the electric field and the vector potential are time derivatives with respect to each other and one can give off classical analogy of the you know harmonic oscillator or pendulum. Now, when you take the pendulum and go to the top. So, let us suppose there is a pendulum and it will move like this in a harmonic path and let me redraw it think of a pendulum ok which has movement like that this is the maximum ok. So, when you go to the top we have maximum potential energy because the potential energy is away you are moving it away. So, the potential energy will be maximum. However, when you reach here its momentum is 0 because it is going to if it went up but it has to come down ok. Similarly, at the bottom it will have maximum maximum momentum because it has to go through this but it has the minimum potential. So, the momentum and the potential they are out of sync with respect to each other in the case of harmonic oscillator or a pendulum. This is the same in the case of the electric field and the vector potential when one is maximum other is minimum ok. We will stop here and continue in the next lecture.