 Hello, welcome to lecture number 12 of the course Quantum Mechanics and Molecular Spectroscopy. We will have a quick recap of the previous lecture before we start with this lecture. In the previous lecture, we talked about the time dependent perturbation, okay. So when the total Hamiltonian is H is equal to H naught plus H prime of t, we evaluated H naught prime of t by putting the charged particle in the electromagnetic radiation and evaluating the Lorentz force from that. In the Lorentz force, we got the corresponding momentum and that momentum was used in the kinetic energy operator and from there we did some mathematics or simple algebra and we caught the final momentum kinetic energy operator in terms of H prime t, okay. So the H prime t that came out in the last class was ih bar by m q a into del which could be written as minus q by m a. So this is the H prime of t. Now we know the total probability or the probability of a transition p of f time t is nothing but 1 over h bar square modulus square of integral 0 to t some t prime e to the power of minus i omega i f t integral f prime of t i whole square and we told this integral that is nothing but f H prime of t i will give you selection rules, okay. However, to evaluate this integral one needed H prime t, okay and now by doing this we got the H prime t which we can plug it in this integral, okay. So your integral f H prime of t i will be nothing but integral of f minus q by m a. So this is what we need to evaluate. So let us go back to H prime of t and look at it in a more detailed way. So this is nothing but minus q by m a, okay but a in plane wave notation you can I can always write in plane wave notation. So that is nothing but minus q by m, plane wave notation of a will be a naught so long, okay e to the power of minus sorry i k dot r minus omega t plus e to the power of minus i k dot r minus omega t, okay. So this is nothing but is equal to I am just going to write is equal to minus q a naught by m into e to the power of i k dot r minus omega t plus e to the power of i k dot r minus i minus omega t epsilon dot, okay. So this is the equation that we are looking at. Now I am going to make one approximation, okay. What is the approximation? Approximation is let us suppose you have a molecule or an atom, okay. What is the size of atom? Size of the atom is typically let us say a one nanometer or a molecule is one nanometer. Atoms are much smaller, okay because radius of hydrogen atom or the bore radius of hydrogen atom is 0.529 angstroms, okay. H atom bore radius 0.529 this is equal to 0.0529 nanometers. Now there is one problem is that let us say even if you take the let us look at the electronic or the electronic absorption or the infrared absorption or the micro absorption in the rotation. So what is the wavelength of the light? Even if you take go to you know vacuum ultraviolet it will be about 100 nanometers of the order of 100 nanometers in vacuum ultraviolet. But in general UVV spectroscopy starts from 200 to 800 nanometers. This is UV or electronic absorption, okay. So this is about this much. Let us suppose which means by just let us suppose this is one full wavelength, isn't it? So there does say this is about 200 nanometers, okay this distance. That is you know one wavelength of light. Now if I put one nanometer particle in it, so I have to make 200 divisions of this, isn't it? One tooth like that. I have to make 200 divisions of one nanometer. And if my size of the molecule is say one nanometer, so this will be very small something like that. So the size of the molecule is so small in comparison to the wavelength. That means effectively you do not see the wave going up and down because for a small portion of the light the wave is not changing at all, okay. So that means 1 over lambda, that means lambda tends to infinity, okay. That is because it is very long because at least 200 or you could have even more, okay. At least 200, so you get you know the change in the profile of the electric field or magnetic field can hardly be felt. So as lambda tends to be infinity 1 over lambda tends to 0 and your k modulus of k is given by 2 pi by lambda. That means k will tend to 0. So in your interaction Hamiltonian h prime of t is equal to minus q by m A naught 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 k is this k is tending to 0. That means k dot r term can be approximated to 0, okay or we neglect k dot r term, okay. So this is something called you know size of the molecule limit of the size, limit of the wavelength in the size of the molecule regime you know it can be neglected. That means your h prime t will now become minus q by A naught by m e to the power of minus i omega t plus e to the power of i omega t. That means this is what is this? This is nothing but minus q A naught by m this is cos omega t into 2. So this is equal to minus 2 A naught q by m cos omega t, okay. That is the equation that you have. So your h prime t now if you go back and look at the plane wave description, okay. Then you had E naught equals to minus 2 A naught omega, okay. So I am going to make substitutions in this. So minus 2 A naught omega is equal to E naught. Then if you go back and look at you know plane rotation about couple of classes before. So this will come out to be minus, okay. This is minus 2 A naught omega. So this is minus. So what I will do is I will multiply the numerator and denominator with omega. So it will get 2 A naught omega q by m omega cos omega t, but this value minus 2 omega naught is nothing but E naught. So your h prime of t is equal to E naught q by m omega cos omega t to epsilon dot p. So let me rewrite this equation in the next slide, okay. So what you have is h prime of t equals to E naught q by m omega cos omega t. What is this epsilon? This epsilon is nothing but unit vector of the electric field. And what is E naught? E naught is the maximum amplitude. Now there is something very interesting that has come. We converted electric field into scalar potential and vector potentials, okay. And eventually by doing enough mathematical transformation we are back with electric fields because E naught is an electric field that is the maximum amplitude of the electric field. It represents electric field and epsilon is a unit vector electric field. But we have to do this because we wanted the potent h prime t that is your time dependent perturbation, okay. Now the time dependent perturbation that you have is this h prime t, okay E naught q m omega cos omega t epsilon dot p. That means your h prime t that I would like to write is equal to E naught q by m omega cos is nothing but e to the power of i omega t plus e to the power of minus i omega t of course divided by 2. So, I will write it 2 here epsilon. So, that is my electric dipole Hamiltonian. And why do we need this electric dipole Hamiltonian? Because if you look at the selection rule if h prime of t i you need to plug in this. Now there is some small modification I want to do to h prime of t before we finish this lecture. Now what is happening? You are taking only q that means only one charge particle one charge particle of charge q. But in general atoms and molecules have multiple charge particles. So, your h prime t. So, for multiple charge particles you have h prime t is equal to E naught cos omega t sigma over n q n pi m n, okay. So, q n is the charge particle. So, what are we doing? You are looking at the charge collection of charge particles not just not one because atoms and molecules are charge particles collection, okay. So, you have to run this over all charge particles and this Hamiltonian is the electric dipole Hamiltonian for many charge particles. So, we will stop here in this lecture and continue in the next class.