 come to the lecture number 3 of quantum mechanics and molecular spectroscopy course. Before we get on with this lecture, I would like to have quick recap of what we have done in the previous lecture. In the previous lecture, we found that H of psi of x, r t is equal to ih bar T by dt of psi of x, r t. This is the Schrodinger equation. However, when the potential is independent of time that is V of x, r t can be written as just V of x. This is something we have discussed because many problems in chemistry or many systems in chemistry have potential that is independent of time, such as particle in a box or hydrogen atom or harmonic oscillator. In such scenario, we can get two different equations. One is ih bar d by dt of phi of t is equal to e phi of t and H psi of x is equal to e psi of x such that psi of x, r t is equal to psi of x multiplied by phi of t and we also showed that psi of x, r t is equal to psi of x into e to the power of minus i e t by H bar and this we called as the phase factor. Further, we also said psi of x, r t represents a stationary state because modulus of psi of x t square is equal to just psi of x square. So, this represents a stationary state. Apart from that, we also defined some integrals overlap integral. This was nothing but integral psi m star psi n d tau this is equal to delta m n also could be written as in bracket notation psi m psi n. Similarly, we have expectation value. This is nothing but integral psi m star operator a psi n d tau. Expectation value is always with respect to an operator psi m a. Similarly, we had defined nothing called action integral psi m. Okay, in the case of expectation value, this is I beg your pardon okay psi m star a psi n d tau this is nothing but psi m a psi n. Now expectation value is a special case of action integral. Then we also defined something called Hermitian operator and if an operator a is Hermitian, then average value of a should be equal to average value of a transpose star. In terms of integral one can write it as psi m a psi n equals to psi n a psi m. This one important thing that is about Hermitian operator is called turnover rule and turnover rule simply says that psi m star a psi n d tau equals to integral a psi m whole star psi n. Now this tells us that Hermitian operators can operate in both direction forward as well as backward. Now let us look at the some important aspect called time evolution. Now the Schrodinger equation is ih bar d by dt of psi of x comma t is equal to h psi of which means the psi of x and t will evolve in time. The time derivative simply says that how it will evolve in time and that will depend on the Hamiltonian. Now we also know that if you have a time independent potential, then your psi of x t is equal to psi of x multiplied by e to power of minus i e t by h bar and an average value of an operator a is defined as on expectation value of an operator a is defined as psi a psi this is nothing but psi of x a psi of you can see that if I take this integral this will be integral psi star of x a psi of x d tau. So, this is nothing but psi of x I can always multiply by e to the power of minus sorry, plus i e t by h bar a and psi of x multiplied by e to the power of minus i e t by h bar d tau because when I multiply this and this then of course, I am multiplying by e to the power of 0 that is equal to 1. So, this is nothing but integral psi of x comma t a psi of x comma t d tau which is equal to psi a. Now let us start with an operator a such that operator a does not have any time dependence which means t by dt of a equals to 0 this simply means operator a has no time dependence. Now, even though a may not have time dependence what will happen to the expectation value of the operator a. So, what I want to now evaluate is that I want to evaluate the expectation value of operator a its time dependence. Now this is given by d by dt of expectation value of a is nothing but psi a psi I have shown a minute back that this integral is equal to d by dt of psi a psi. So, let us expand this. So, d a by dt equals to d by dt of integral psi star a psi theta. Now, you have to differentiate because psi a and psi r 3 either operators or no function. So, one can write this as this is equal to integral of d by dt of psi star. So, I am using a product rule for differentiation a psi theta plus integral of psi star d a by dt psi plus integral of psi star a d by dt of psi theta. So, this I will write as 3 integrals. Now, we will see that the second integral this one this is equal to 0 because we said d to begin with we said that d a by dt is equal to 0. So, now I am left with only 2 integrals. So, d of a dt equals to integral of d by dt of psi star a psi theta plus integral of psi star a d by dt of psi. Now, here we are going to invoke the Schrodinger equation. What the Schrodinger equation says I h bar d by dt of psi equals to h psi. This implies d by dt of psi is equal to 1 over i h bar h psi. Now, if I take a complex conjugate of this d by dt of psi star equals to i h bar h psi star and with a negative sign because I will become minus i. Now, I am going to plug in so I am going to plug in d psi by dt here and d by dt of psi star here. Now, if I do that then I will get integral minus 1 by i h bar psi whole star multiplied by a psi d tau equals sorry plus integral 1 over i h bar psi star a h psi. Now, 1 over i equals to minus i. So, if I use that I can take this i numerator and this also I can take numerator. So, this will become i by h bar I can take a integral h psi star a psi d tau plus minus integral psi star a. Now, we have these two integrals. Now, if I use star now a rule here it says integral a psi star whole star psi m psi n equals to integral d tau equals to psi m star a psi n. If I use that turn over rule then I can write d and I can write d by dt of average value of a equals to i by h bar integral h psi whole star a psi d tau minus integral psi star a h psi. So, this will nothing but i h bar integral h a psi d tau minus integral psi star a h psi d tau. So, this is nothing but this is equal to I can think of it like this i by h bar. Now, this is nothing but integral so psi h a psi minus psi a h psi. This integral I have written like this and this integral I have written like this in terms of bracket notation. So, there is an h a operator that is going to operate over integrate over psi star psi. So, h a operator integrating over psi star psi similarly a h operator integrating over psi star psi. This is equal to i h bar. Now, these are two different integrals which I can combine into one integral that is nothing but psi h a minus a h psi. Now, you can see that h a minus a h I have two operators. Let us say you have an operator a and operator b. Then we can define what is known as a commutator which says commutator a comma b is equal to a b operator minus b a operator. Of course, this one thing that I want to tell you is that commutators have to be evaluated with respect to some function. But nonetheless, it is just the a b minus b a. Now, if you have a and b as numbers or functions, then commutator a b will always be 0, but that is not necessarily true for operators. So, a commutator may be equal to 0 or may not be equal to 0 of two operators a and b. So, this is equal to i by h bar psi commutator h comma a. This is one of the very interesting results in quantum mechanics and is very useful for spectroscopy. Now, I am going to rewrite that equation once more. T a dt is equal to i by h bar average value of psi h comma a commutator psi. So, this is in short can be written as i by h bar average value of h a commutator, T a by dt. Okay, what does it mean? This is a very important formula. It says that if a operator a commutes with h, of course, if it commutes with h, this will go to 0, h a commutator will go to 0, then d a by dt will be equal to 0. So, the average value of a, the time dependence of it will depend on whether a commutes with the Hamiltonian or not. If operator a commutes with the Hamiltonian, then its average value or expectation value does not have any time dependence. But if operator a does not commute with Hamiltonian, then its average value will have some time dependence. In spectroscopy, this is very interesting because if you start with the molecules and interact with the light and if you write an operator corresponding to light, it does not commute with Hamiltonian. If it does not commute with Hamiltonian, its expectation value will show time dependence. That means, for example, you start with the hydrogen atom and you apply this operator a corresponding to the light, interaction of light. Then what happens is that the average value will change with time because it has time dependence. So, something that started with 1 s will end up with some other expectation value corresponding to either 2 s or 2 p or 3 s or 3 p depending on the nature of your operator. So, it is very important that a state will move. So, this means that the state will move to a different value. You can move a state or an initial state from one expectation value to different expectation value only if you apply an operator that does not commute with the Hamiltonian. I will stop for this lecture. We will continue in the next lecture. Thank you.