 The quantum mechanics, formal paper of quantum mechanics actually came in 1926, I think, when Schrodinger wrote a letter, paper in physical review. So, this was a very, very important paper in physical review, which actually signaled most historians think this is the first paper of quantum mechanics, proper quantum mechanics. And many of these postulates and all that evolved. And very next year, actually the chemistry got a big boost. 1927, when Heitler and London produced a paper based on quantum mechanics and what is called the valence bond theory. And that was the genesis of valence bond theory in 1927, very interesting to read the history. And Nils Bohr actually started talking about it long back, but later on people realized that the Bohr's theory is not correct. Bohr's theory is only right for hydrogen and hydrogen like atoms, it could not explain. So, the angular momentum quantization that Bohr brought in was only a very restricted part of quantum mechanics and it does not explain. So, then later on of course, in the Schrodinger's paper along with Heisenberg, he actually enumerated the whole quantum mechanics and that is today known as the cornerstone of what is called the postulates of quantum mechanics. So, what I will now do is as I move forward, I will quickly remind you of the postulates. Again, I will not go into any detailed discussion which we have done. Again, those who have attended the 4 to 5 class, we have done very detailed discussion of the postulates, but I will remind you of the postulates and maybe I will ask some people to tell now. So, can somebody state the first postulate of quantum mechanics? I mean, that is historically and traditionally accepted as the first postulate. You can say no, no, that is the second postulate, but that is traditionally accepted as the first postulate which we have taught. So, can somebody explain or not explain, can somebody state? Explanation is more complicated. Who will be able to state? So, let me see how much you remember. Come on, I can see some of the faces, we have got good marks also in 4 to 5. Can I start? Every state of a system is described by, so what happened? That is the first thing. The first thing is to remember the first postulate, otherwise you cannot start. So, the first postulate shows that, so let me write down very cryptically. Again, I will not write in details that every state of a system, now system, when I say system, of course it is a quantum system. That is obvious because we are looking at postulates or quantum mechanics. So, quantum system, the system can be a single electron, system can be a collection of atoms, single collection of molecules and so on, whatever we defined as quantum. Every state of a quantum system is described by a wave function. It was called actually a wave function by Schrodinger, by wave function and the standard nomenclature was psi. Of course, you are allowed to give any other nomenclature. The psi is usually a function of either coordinates or moment of the systems. Remember, these are not measurement of the coordinates. As I told you, delta x delta p is greater than equal to h cross. So, you cannot actually measure the position, but this is only a variable of the functions. So, this psi could be the position, so x1, x2, etc., up to xn. This could also be momentum. We have not discussed that the standard quantum mechanics actually is based on the position representation or coordinate representation and that is why we call it the coordinate representation of quantum mechanics. Of course, there is a momentum representation of quantum mechanics, which is a Fourier transform of this representation, which will not go into this, but this is the most standard representation. Then the interpretation is that the mod psi square is now says the probability density of finding this system at x1, x2, xn. So, now we can see how this connects to the uncertainty principle. We are merely saying that there is a wave function for every state, but now we are saying that if you take a mod psi square, that will be also a function of x1, x2, xn. That will basically be the probability density of finding one particle at x1, another particle at x2, another particle at x3 and so on, nth particle at xn. So, now that we have said the probability density, this is a very important term. It essentially attests to the uncertainty principle. That means we are acknowledging that there is an uncertainty. We cannot find the electron where we can only say what is the probability of finding the electron at a particular point. So, although there is an uncertainty, there is a probability of finding the electron at a particular point. So, that is what we are acknowledging. So, you can see that the very first postulate actually acknowledges the uncertainty principle. It does not say that the first electron it x1, second at x2, third at x3. The interpretation of this is very, very important that the mod psi square, so this essentially is a function which is a probability function of getting an electron at x1, x2, xn. Again remember since electrons are indistinguishable, in fact all quantum particles are indistinguishable. I hope you know that the quantum mechanics and classical mechanics, one major distinction is apart from the size that the quantum particles are indistinguishable. So, since it is, we cannot say that the particle 1 is at x1, particle 2 is at x2. We can only say there are n particles whose positions are x1 and to xn and the probability of finding that is mod psi square. We are not saying that they are actually there. Again I repeat where many people get confused. We are merely saying that the probability of finding them at x1, x2, xn is mod psi square. So, if it is a one particle problem, so let us take a single particle problem and that is a very nice model problem, single particle in 1b. So, then of course, I can write that as a psi x square and I can say mod psi square which is nothing but psi x of star psi x is the probability density of finding that quantum particle. It could be electron, it could be boson, it could be something else of finding the particle at x. So, some mod psi square would then be defined as a probability density. Remember each time I am writing probability density and I will define it what the density means of finding the particle at x if it is one particle. So, that is mod psi square which can be written as psi star A x i x. Note also that when you are talking of a quantum mechanics postulates we have not said specifically these are electrons. Postulates are of quantum particles. So, quantum particles are small particles in that region which follow quantum mechanics which are indistinguishable. That is important. At this point we are not distinguish electrons or fermions or bosons. So, some of these postulates are identical for that. So, what does the density means? So, density is essentially means that this is the probability per unit volume, volume or length. So, when I say the mod psi square let us say this is P of x. So, the P of x is the probability of finding in a unit length around x. How do I do that? Because as x changes of course P of x will change. So, what do you do? You use the convenient way that we do in calculus that you take at let us say x equal to x 0 the value is P of x 0 the probability density. Then if I write this from x 0 to let us say plus delta x which is an infinitesimally small quantity or I let me write this as the infinitesimal delta delta x then I can assume that within this delta x the probability density is constant because this is a very small amount. Then I will say that the total probability between x 0 to x 0 plus delta x is nothing but P of x 0 into delta x. Is it clear? So, P of x 0 is defined as the probability density at x 0. So, this is my probability density. So, at x equal to x 0 I define P of x 0 which is psi of x 0 square. So, then I say that the total probability of finding the electron or the quantum particle between x 0 and x 0 plus delta x would be the probability at x 0 times the delta x because this is now the length. So, this is total probability divided by the length. So, that is how the concept of density comes in. So, if you look at now the why it is called density that P of x 0 is nothing but total probability divided by the length. So, of course, you can define that as a density and then we can write in the same manner the probability of finding the particle between two finite distances. So, for example, if I want to find the particle P between A to B where A and B are of course large separated not like x 0 to x 0 plus delta x then you have to do this as a mod psi of x square d x sorry mod psi x square d x between A to B just like you do in calculus. So, essentially what you are doing is that we are starting from A dividing into infinitesimal thing A to A plus delta x etcetera etcetera and then integrate between A to B. So, just like you do in calculus the whole idea of calculus integration we actually bringing here when A and B are now separated by a finite distance. So, it could be between here to a larger distance if it is infinitesimal you do not need to do because it you can consider it uniform. So, you simply multiply by the volume and that is where the density concept or density comes. So, I hope everybody clear because this is a very important concept many people confuses the mod psi square as a probability it is actually probability density. So, you must understand the subtle difference between probability and probability density. Probability can be defined as a total probability between certain regions probability density is at a point. So, you cannot define at a point what is the probability there is no meaning. So, if you give a space if space is infinitesimal then you simply multiply if space is finite then you have to integrate. So, that is very important to understand that how do I calculate probability from probability density. So, the bond the interpretation that came here and this was actually originally given by max bond another celebrated physicist the interpretation of psi was the mod psi square as probability density. Now, again we are writing these for one single particle you can easily generalize this for the n particle n particle in one dimension n particle each of them in three dimension whatever the generalization is very easy integrations will have many many dimensions that is all. So, you can you can keep doing that generally. So, this is actually the postulate one which is very important and it actually seems with the uncertainty principle because we are already bringing in the concept of probability density. So, it is already bringing in the concept and we are essentially saying that the quantum particles cannot be located. So, it is a very important part that comes in. Now, can somebody tell me the next postulate? Yes, good good. So, so every dynamical observable which are observables in classical mechanics see I can see that if I give you problem you are very good if I solve ask you to solve immediately particle in a boxing but this is not a problem this is an articulation and that is where there is a weakness that if I have to articulate what is it because that is very very important problem solution you can all do if you are reasonably good in maths, but maths I again repeat excellent maths does not make you a good physicist I repeat it a good physicist can be a poor mathematician because the good physicist has to have the concepts now somebody can bring the language mathematician remember mathematics is language. So, it is like you know something you are not able to explain many of you are actually good in mathematics. Now, I am realizing what you have to be good in physics because quantum mechanics is a subject of physics in fact I do not mind if you say no no no I know exactly what to do you have to solve the difference I cannot solve the difference integration I do not actually mind provided your entire framework is very clear you know that is something you can learn, but I can see that after doing the full quantum mechanics course you are still struggling to tell me the postulates I am not saying that you are not telling, but you are struggling to tell that is that is the what I will use. So, every dynamical observable which are defined in classical mechanics actually is represented by an operator which I now call it quantum mechanical operator which I just call quantum mechanical operator, but actually it is an operator and and for all all measurable observable these operators are Hermitian. So, that is another important thing that all physical observables which I can measure correspond to what I now call Hermitian operator. Now, here comes a little bit of mathematics to know what is an operator, what is an operator? An operator acts on a function to make another function. So, that is an operator and those are Hermitian operators. So, Hermitian operators have some special properties can somebody tell me what is one property of Hermitian operator? Self-adjoint very good. So, Hermitian operators are self-adjoint, what is adjoint? So, let us say an operator A I again repeat acts on a phi to create psi, I am using what is called the Dirac notation. Again I hope all of you are familiar with this, this is basically a function of whatever variable this is also a function of whatever variables and the function is represented as a notation k which again is something that we are going to use. So, I again repeat that people who have not done Dirac notation, we have done it in the last class please get yourself familiar. So, this is this is the normal notation it will be written A operator acting on phi of whatever is the coordinate, let us say x to give you a psi of x. So, this is actually written in this language. So, the so the coordinates x, y, z or x 1, x 2, x n whatever is actually silent. So, whatever coordinates are there here the same coordinates are here we do not need to write. So, this is called the ket vector. So, this is the Dirac notation. So, we said that if A of phi gives you psi then the adjoints are defined in the following manner that the conjugate of phi if you act adjoint of this operator from the right you give ket conjugate of psi. So, the question is if I have a function phi the operator acts on the left to give you a function psi if I have a conjugate of phi then the definition is that the adjoint of this operator if it acts on the right it gives you the conjugate of psi. So, this is called the adjoint and of this operator A and the Hermitian operator this A is equal to a dagger. So, I do not have to write a dagger. So, that is why they are called self adjoint. So, Hermitian operator so for Hermitian operator A equal to a dagger. So, this is how the quantum mechanics platform is defined because so far we have only defined a state of a system and how to define the probability density. The question that now you will beg is how do I make measurements in the system what happens? So, for that this postulate 2 actually prepares you that for every observable there is an operator then the question is what do I do with this operator how do I still measure it still not answers the question. So, that will come in the next postulate which is the postulate 3 and that is a little difficult postulate but it just says that can somebody state this postulate anybody can state how do you make a measurement after that. So, if I have a state of the system psi and if I have an observable to measure for which there is an operator A then what do I do? So, let us so first of all what does the postulate says? It says that if I make a measurement of A of that observable let us call it A whose quantum operator is A hat then this measurement always leads to and that is a very important statement the Eigen values of the operator. Remember every Hermitian operator has an Eigen values every operator has an Eigen value every Hermitian operator also has an Eigen value. What is an Eigen value? Who does not know what is an Eigen value? Everybody knows what is an Eigen value engineers must be knowing of course the physicists the chemists fine then I need do not need to worry about it the Eigen values for Hermitian operators happen to be real. So, that is a very important definition. So, all the measurements always lead to real values but I am saying that it lead to any one of the Eigen values I do not know which one. So, this brings a very important question that how can I say which one the this question was actually answered it says that in general you cannot say which which Eigen values you get but you will get some Eigen values unless the state psi happens to be an Eigen function of the operator A. So, if the state psi which I am measuring on which I am measuring the A happens to be an Eigen value Eigen function of the operator A with an Eigen value small A then of course the measurement will always lead to Eigen value. This will always lead to Eigen value if it is not then it will lead to some other Eigen values and that is a very complicated state but these states are fairly simple states because their behavior is very much classical that at least when I measure I get a same Eigen value and these states are called the stationary states. So, these are called the stationary states. So, I can have also a stationary state of energy which means if energy operator is Hamiltonian then the stationary states are defined by states which are Eigen function of the Hamiltonian operator is it okay. So, this is for example this stationary state of energy alright. So, if I can define these states which are Eigen function of the Hamiltonian I define what are called the energy Eigen states where measurement will always lead to EI for a given psi I and this is very famously known as the Strodinger equation okay but this is actually a stationary state of energy and we are more interested in stationary state of energy because energy plays a very pivotal role in physics and chemistry all of you know why energy plays that is because of the thermodynamics. Thermodynamics is governed by free energy and so on that is the reason we are biased by the stationary state of energy but note that every observable has similar stationary states where that particular observable will be measured with a certainty that this is the value that I will always get. Why I am talking of this certainty because the very important part of quantum mechanics actually leads to the states which are non stationary. So, for example I look at a state psi which is not an Eigen function let us say psi is not an Eigen function of Hamiltonian to start with okay. So, let us assume this. So, this is the concept of what I call the non stationary states correct because it is not an Eigen function of Hamiltonian and I want to measure energy. So, I want to measure energy in this state. What will still happen let me just for the sake of symbol let me use these as some other symbol phi because I have already used psi as Eigen states of energy. So, let us say phi which is not an Eigen function of energy and I may want to measure energy then what will happen is that I will still get one of the EI because the first thing that I wrote it leads to always Eigen values of the operator. Eigen values of this Hamiltonian is of the EI. So, I am going to get some EI but which one I cannot tell. If it was a stationary state I can definitely say I will get that particular Eigen value but here I can get E1, E2, E3 any of them with some probability. Now note that these non stationary states sorry non stationary states evoked a serious concern of quantum mechanics where Einstein had serious disagreement. It says that I have a state which is not an Eigen function of Hamiltonian but so what I have a state and I make a measurement of energy I do not know which value I am going to get I mean that will never happen in classical mechanics. So, this was a very nice experiment what is called the Gedanke experiment which was actually done. It says that if I make an identical systems hundreds of identical systems thousands of identical systems and if I make a measurement by measurement is done by robots who do not make any error in measurement they will still get different values which is the one of the most I would say disruptive thoughts in quantum mechanics that how can this happen because if it is identical system robots will always get the same value and remember this is not because of experimental error this is inherent in the theory. This actually the non stationary state actually cause serious pain and serious discussion in quantum mechanics till of course Schrodinger said although they will get different values what I can predict is the probability with these values will come. How will the probability come in the following manner if I can always expand this phi as a linear combination of the actual Eigen states of the Hamiltonian is always possible in mathematics that a function can be expanded in terms of a complete set and remember Hermitian operator Eigen functions form a complete set. So, I can write this as a linear combination of psi and then what was related is that the mod C i square is the probability of obtaining a particular Eigen value E i. So, this is the concept which was known as the superposition of states and this is very very important concept which actually I just called it postulate for it does not matter what is the number you give superposition of states. So, superposition of states can actually explain many of the non stationary states but these coefficients are now square of the coefficients are now related to the probability just like in the first postulate mod psi square is the probability density. So, that is the reason this square is very important mod C i square. In fact, we showed in detail how it can be done and in fact in all such cases for a non stationary state the way to obtain a value is through what is called the average value and that brings in the concept of average value or sometimes we call it expectation value. So, it says that the average energy in a state is integral psi star h psi volume element divided by psi star psi if psi is not a stationary state because then we have a probability of values E 1, E 2, E n with mod C i square is the probability and one can show that the average E average is nothing but integral psi star h psi in a direct notation sometimes it is written as h average equal to psi h psi by psi psi. The denominator comes because I am assuming that it may not be normalized if it is normalized unity then of course the denominator need not be written. So, this is a very very important concept the average value concept by which a non stationary state could be written note again in our previous class I had shown in detail how this comes by expanding psi in terms of the C i phi i, C i psi i and then showing that the psi i is an eigen state of h with eigen value E i one can easily show that this is nothing but this h average is nothing but sum over i mod C i square E i which is what my probability should be. Note because I am getting all the values of E i with a coefficient with the probability mod C i square. So, what would be the arithmetic average is like a weighted average sum over i mod C i square E i which I showed in the previous course that this is nothing but this I hope all of you will be able to do this just stick in the psi and psi star. So, that I could ask in some exam how to get this value. So, that is an average value which I had actually shown in the class please remember all that I am doing is just a rehash an overview of what I did before and I think it is important to do this in the very first class at least and so then it almost puts the framework of stationary as well as non stationary states of quantum mechanics yes well let us not let us not worry about it but to give you an answer if the same measurement is repeated then they will always lead to the same value because this is a very very difficult interpretation of Dirac which says that after you make the first measurement each state gets converted to its corresponding eigen function. See for example if I somebody may get E 1 somebody may get E 2 the person who gets E 1 for whom the state will be converted to psi 1 and once it is psi 1 now it is a stationary state. So, it gets stuck. So, this was an absolutely mind blowing interpretation which Einstein said this is absolutely stupid because depending on the observer some state is going to psi 1 some state is going to psi 2 you do not know which observer will get what also because it is a probability this is an absolutely mad kind of theory. So, he never accepted the theory that is one of the reasons he never accepted the theory but you know that is how the Dirac interpreted that once I make a measurement the person who will get E 1 for him the system will be locked to psi 1. So, essentially they it brought in a new concept of what is called the subject-object interaction depending on the subject who is measuring and the object the interaction throws it to another state which is always an eigen state of that operator and then it stays there once it is an eigen state no problem. So, this part is very beautiful the concept of stationary state because this is like classical mechanics once you have locked a state into a stationary state then you have no problem. The question that is also very important is that can I have a stationary state for multiple operators that is I calculate energy I calculate angular momentum I calculate dipole moment many many things the answer is yes if those operators commute mutually commute. So, that is why there is a very interesting theorem which says that if two operators commute they have simultaneous Eigen functions again again we have discussed this theorem in detail if two operators which is basically Hermitian operator of course they have simultaneous Eigen functions. Now very clearly you can see the the simultaneity essentially means that simultaneous Eigen functions essentially this means that if I have one state which is an Eigen function of an one operator then another operator which commutes with it also can be measured with certainty because it is an Eigen function of both the operators. So, that is a very important concept that we bring in the simultaneity of measurement and in fact this is very very important part of quantum mechanics, but do not forget the non stationary quantum mechanics this is something that is very important and it is resolved by what is called the super position of states. So, this is basically super position of states it is very important whole of most of the quantum chemistry I now repeat that we will be doing we will be interested only in the stationary states of Hamiltonian. So, this is what is important, but I must say this because this is a very very important part of quantum mechanics, but we will actually be interested more on this how do you solve this for a given Hamiltonian. So, this will be our quantum mechanics for many particle problems this is what this course will contain we have already solved this for various one dimensional problems exactly solvable even two dimensions three dimensions I will come to that when can I solve it exactly all right. So, I think today I will stop here.