 So, you are all here to study very advanced stuff, but I am going to start with some somewhat elementary stuff. So, that the language is uniformized. To this end I want to lay down what we consider to be the basics of quantum mechanics. So, I will try to be fast so that you do not get totally bored with the, but I think that it will give you some food for thought and for how the subject of quantum mechanics is organized. It looks like lots of formulae that you have, but effectively what it boils down to is the following. So, the first things are the novel features. One is the probabilistic outcomes right. Even if you have identically prepared systems or if you repeat exact same experiment the outcomes can be different. If you have an atom that is spontaneously decaying, it can emit a photon in this direction or in that direction, but if it is let us say s equal to 0, s transition if it was you would see isotropic photons only if you do enough experiments or if it was a P wave emission, you will see a P z if suppose you had oriented and spin correlated emissions. You will see the pattern of P wave only if you did lots of experiments. In any one experiment you will you only have a probabilistic outcome. Actually the related fact is that also, but definiteness of the value of the observable. So, probabilistic outcome about the observable, but the second point is definiteness of the value of the measurable in any one experiment. So, another way of keeping this mind are just the emblemic experiments that correspond to this are double slit experiment which in practice was some electron scattering experiment. The definiteness of value of the outcome is what is called Stern-Gerlach experiment right. The double slit experiment the photon could have gone through this or gone through that. So, there is its probabilistic which side it came from, but the definiteness of the value of the outcomes this Stern-Gerlach experiment which Feynman spends a lot of time discussing in his third volume is like this you have your magnets North Pole and South Pole and you send a beam of polarized particles through it. What happens here you have a screen here the and you are the because of the non uniform magnetic field this beam will split, but it will split precisely into two patches. The spin will be either half or minus half it will not give in between answers. Although which answer it would give would be probabilistic if you made the beam intensity sufficiently low. So, that only one particle was coming at in a given minimal measurable certain time you would find that you could get either this or that, but what you get is a precise value it is either plus half or minus half, but which of the two it will come is probabilistic. So, those are two very hallmark features of quantum mechanics and I like to point out a little historical thing that Einstein later in his life kept expressing surprise about quantum mechanics, but he was the guy who proposed photoelectric effect and if you think about it everyone in his time thought that for the light would come out like a wave which in when an atom emits they would imagine some kind of a spherical wave coming out. Whereas, Einstein was the guy was saying it goes out like a bullet if it does then it violates isotropy of the process it actually accords with these two things that were later to be confirmed by innumerable experiments, but Einstein's own first hypothesis is premised on the fact that it can come it can come out only in one direction in any one next emission process it can only go in one direction. So, it can only have a precise value which violates the isotropy of the process, but you recover the isotropy in the sense that the probability of going in any one direction is the same. So, these are the two slightly bizarre features of quantum mechanics which one has to digest one has to learn to live with them and the third thing is the mutual incompatibility of so, compatible and incompatible observables and this is what we popularly call as uncertainty principle because you have delta x delta p greater than something or the other right. So, this basically says that x and p observations do not commute. So, this is also a feature of quantum mechanics which probably is tied to this number 2 that any experiment that you do with x will make it collapse into a particular value of x. So, that so, maybe in some way the two are related, but not necessarily. So, these three are the general features. Next I want to do is write down for you what I call postulates of quantum mechanics which we all imbibe just by going through the two courses, but nobody somehow lays them out as simply and sequentially as I am trying to do now partly because it does require some mature watchability to do it, but let us put them down like this. The first is that and we can think of it as mathematical side and physics side ok. So, the states are vectors in a complex vector space in fact, in a Hilbert space. So, Hilbert space is when you go to the continuum of states, but otherwise it could be a finite dimensional vector space like spin, but they are essentially complex vectors in a complex vector space. What this means is that with a inner product. So, we have so, there we laid down the notation quickly. The second thing is that all observables are represented by Hermitian operators. So, the second thing is that so, again this is the physical side of it and the second half of the sentence is the mathematical side of it. It is a Hermitian operator. So, what do we mean by Hermitian operator is that yeah. So, the Hermitian conjugate is defined with respect to this with respect to this with respect to the inner product ok. So, with respect to the inner product the operators A its Hermitian conjugate is the Hermitian is defined by whatever that is that if it had acted first on the left on the bra side then it would be it would give the same answer. So, I hope everyone knows that in this complex vector spaces the multiplication is linear with respect to the ket side and anti-linear with respect to bra side right. If you multiply scalar here and try to take it out you will get complex conjugate of that. Whereas, if you multiply this side by a scalar and take it out it is a it remains real. So, this defines Hermitian conjugate and if it is a self conjugate if it is self conjugate then it is Hermitian and that is what observables are and of course, observables have specific eigenvalues. So, now, this thing leads to the interesting fact because from the mathematics side we know that if they are Hermitian then their eigenvalues are all real and their eigenvectors are complete are a complete set. So, now here I have been a little vague they means you actually have to find all the mutually commuting Hermitian operators and that nails down the system completely, but we are coming to commutativity later, but let me just say that by there we mean one should identify all the mutually commuting Hermitian operators right. So, that goes back to this part compatible and incompatible, but anyway. So, there are some overlaps between these they are not necessarily logically completely sequential, but together they make a consistent set of statements. The third thing is that symmetries are implemented by unitary operators change of physical basis or change in choice of complete set of observables right. So, if you change the set of observables right. So, if you change the set of observables through which you are going to specify the system like instead of angular momentum and energy you say linear momentum and energy something like this right. So, if you do a change of basis then that is implemented by unitary operators in particular. So, actually unitary or anti unitary operators that is the term Wigner invented. So, we are going to use it and in particular symmetry operations are also realized through unitary operation. So, this is a more general statement the operation you are carrying out need not be a symmetry operation, but you are doing some operation on the system then there will be a unitary operator that will connect the two ways of mapping the system, but further symmetry operations are all fall in this category. So, the anti unitary is like the time reversal it involves a complex conjugation as well I am not getting into the detail of it right now, but that then the third fourth is of course, the probabilistic interpretation or the this part that the definition. So, these two things together in a way are stated by saying that the outcome of any observation. So, you can say this postulate as collapse of the wave function to the possibility of a particular outcome of Eigen values you know there corresponds a projection operator. So, we talked about Hermitian operators unitary operators and projection operators. So, there is a projection operator corresponding to every possible outcome of a set of observables. And so, for example, if you have a if you have an observable a with Eigen values a or the observable values a it can be written as a sum over this spectrum of Eigen values a this is how one writes it. You can think of it as sum over a times the outcome a and the projection on to the. So, times p of a belonging to sigma of a I will write down what is sigma of a. And here we say projection operator on to the state with Eigen value a and because p is a generic thing if you want to be sure what you are talking about you give you tell which operator we are dealing with. And where sigma a is the spectrum. So, when you go to advance notions in quantum mechanics that is that is when you go to the continuum and the more subtle issues of the Hilbert space. This is called a spectral decomposition that an operator can be written out like this has to do with the fact that. So, p a has a representation p a a is like this it can be written like this it should probably add that it has to do with a, but it is ok. So, p essentially is a projection operator that this representation for an operator works has to do with the fact that of course, its Eigen vectors are complete. And it is watertight in the case of finite dimensional vector spaces, but when you go to continuum then the availability of this kind of spectral decomposition is not always guaranteed. So, this is called spectral decomposition. The point is that if you have a operator like x or p one dimension then they are unbounded operators that Eigen value list is going straddling infinity to infinity. So, then what this kind of sum means etcetera all becomes complicated, but roughly speaking this is what it is. And so, the fact that this projection operators are available we can then write the statement about the probability we can write familiar things probability of outcomes for Eigen value a to b. So, what it says is the this is the mu is the probability and mu is borrowed from measure theory. So, it is a measure of. So, the probability for on trying to observe observer a in state psi returning the value a is the projection operator for that particular Eigen value of that particular projection operator of that and that observable acting on psi. So, that is what the outcome probability of outcome of a particular Eigen value is and in general where we do not get then we get a average value psi a psi and in this language it will become equal to. So, this symbol is just help us to nail down what we are talking about there are not any mathematics you can start computing from, but it is just a more precise way of stating what we are trying to say or stating it in symbols. So, we have this thing of the states there are Hermitian operators there are unitary operators there are projection operators which capture the essence of observation process and then then we come to the interesting parts more interesting parts. So, this is just the setup. So, now, we come to postulate number where did we go force of 5 which is quantum kinematics. In first year physics you know what kinematics means it is when nothing much is happening you define things like velocity and acceleration and this is it is all kinematic if nothing is no energy is being added to it and free particles are essentially is what is described by kinematics. In the case of quantum mechanics there are more subtleties because we are living in some Hilbert space and this is where we come to the mutual incompatible observables as well, but the point is that we have operators for every observer for observable. So, the quantum kinematics part essentially is the free commutators is what I want to talk about, but what I want to be specific about is the fact that we do some simplification here a very big simplification. So, operators corresponding to observables need not commute it would then mean that for every pair of incompatible observables you will have to specify what their commutator is. The grand simplification is that we assume that it is enough to specify the commutators of q and p and that all other observables are expressible algebraically in terms of them and that their commutators therefore, can be worked out knowing those of p and q. So, the p q the canonical variables. So, we work under two assumptions and two that the canonical observables. So, we assume the availability of such canonical set in terms of which all other observables can be expressed and then the all other commutators. So, this as you remember why we call it classical analogy is that this is same as Poisson bracket the classic corresponding classical Poisson bracket. So, commutators of all other observables are derived from this. Now, this itself is a slightly big assumption because firstly the ability to represent. So, there are ambiguities in this process because of the operator ordering. So, if you have some like angular momentum if you which involves r cross p, but we know that the r and p is do not commute if the indices are same. So, there is always a slight doubt as to how you define such quantities and that is called operator ordering ambiguity. So, results in, but what I want to emphasize is that technically, technically all you had to do was gather enough experience about the commutation relations of all the individual operators and then you it is the ambiguity arises not because of some problem with quantum mechanics, but it is because of the grand convenience you try to derive from the availability of observables that in classical limit will become the classical canonical variables. So, we are now approaching the end of the list of the postulates. The next is the dynamics which is either Schrodinger or Heisenberg picture and the last part is the most important part. So, number 6 and this works in analogy with classical mechanics, but in classical mechanics we have p dot equal to minus d d h by d q, right which is equal to p h Poisson bracket. So, we have to do the same thing here except that we replace the p h Poisson bracket by the commutator bracket and divide by i h cross. So, i h cross p dot is therefore, equal to p h commutator. So, this is Heisenberg picture as you know and there is the Schrodinger picture in which we write psi t. So, I am using the notation that Dirac uses in his book. So, psi is a generic state and you show its time dependence by putting a t. So, d by dt of i h cross d by dt psi is equal to h acting on psi t. So, this is a Schrodinger picture state and recall. So, just to explain you the notation recall that wave function psi x is the projection of psi x t. Is to take the generic state psi t and then project it on to the x space. So, this is this is how Dirac delineates what this mysterious thing is, it is actually just we it is just that we are looking at it in the x basis. Often you have seen also phi p t momentum space that is simply this. If the simplest system of point particle in one dimensions there are only two observables position and momentum of course, you can construct energy operator which will be p squared. So, no ambiguity is there, but and we know the that x and p are mutually incompatible. And we know that if you list all the momentum Eigen states then you can list the system completely because energy is just dependent variable. So, you can have either x basis or p basis for a point particle in one dimensions and these are the corresponding wave functions. And incidentally this also reminds me that instead of writing this q p we can also write x p sorry now x p equal to e raise to i p x over h cross. So, this relation is equivalent to this canonical commutator. What in Heisenberg picture is this statement in the kinematic statement of in terms of commutator is exactly in terms of states it turns out to be this because you can see that if I wedge up p here in between take a x I should of course, get p time sorry for the bad notation, but repeating p in various ways, but we know that this is what you should get, but this is what you will get if you take. So, you recover this relation from setting the x p overlap to be this. And this has this will reproduce the x p commutator correctly if you apply right. Let it act on some function of x then this represented this is correct representation for p in x space and that is why this overlap basic over. So, this is as fundamental and statement as saying that the canonical commutators are these they are equivalent ok. So, all this is well known stuff. So, I am just going fast and the last thing which I personally believe should be always taught along with all quantum mechanics that is somehow postponed indefinitely to tail end of quantum two courses is the Bose or Fermi statistics ok. So, I state it as yeah. And the correct thing to say is not the word statistics strikes fear in the mind of most people who like precise science. So, suddenly the thing there is statistics it is there is nothing statistical about it is correct enumeration ok. So, the so, Bose Einstein and Fermi Dirac it is more an enumeration process than any kind of statistics for same species of. Now, I personally think that unless this is done upfront from the beginning it does not bring out quantum mechanics correctly. You will find that most discussions of quantum mechanics only deal with 1 to 6 ok, especially all the mathematicians spend a lot of time worrying much more about the how you go to the continuum and how the unbounded operators make sense and all that, but I think the heart of real quantum theory lies here. It is this bizarre enumeration of states that one does with changed our perspective completely. And the reason why the word statistics comes is because it was done in the context of thermodynamics first. So, we know and this is completely elementary statement, but excuse me for repeating it, but it is important enough that we said. If you toss coins you have 2 outcomes you have all these outcomes 4 outcomes suppose, you toss 2 identical coins then you have 4 outcomes. In classical mechanics we have the probability for this is a quarter probability of this is quarter and this together is a half because although they are indistinguishable that number of times this will turn up is half of the times, but if you have both statistics then it is exactly one third one third and one third right. And this is what is the strange thing about really the strange thing about quantum mechanics that the 2 are quote indistinguishable. They are so indistinguishable that actually the word indistinguishable is a misnomer. If there is in principle possibility of distinguishing something then you can later say they are indistinguishable, but that negative putting a negative indistinguishable is actually wrong. What it says is that they always have only completely symmetrized states and that it is impossible to tell them apart tell one photon apart from another ok. So, it is. So, identity identical particles is in some sense better than saying they are indistinguishable because indistinguishable means that their identical billiard balls, but my manufacturer was so clever that he I cannot observe them, but this is a matter of principle. It is not that somebody really polish them very well so that they look the same. They actually are such that they produce only one state together and for me on is even more bizarre because it is 0 and 0 and 1. Right only the they cannot be in identical. So, assuming this is the only quantum number. So, the two of them cannot have the same quantum number and they will have to be in this state and they will obviously, be in the anti-symmetrized state. So, it is this counting this the way to enumerate the states that makes quantum mechanics so completely different and I think that most quantum mechanics is lost upon you if you keep worrying about this and all the general public in the general world worry about. Of course, they have got to this by through the so, called Einstein, Podolski, Rosen, Paradox which I think is daily life thing for a quantum system, but it has to do with this as you know if you the system remains correlated even over a space like distance because there is only one the problem with that is that they think that there are two distinct photons there which have become indistinguishably entangled, but there is no such thing from the Hilbert space point of you there is one state. The number two is a psychological thing, the number one is the correct enumeration of state, but since you think that there are two now you can play many mind games and say I observed this one and then this collapsed and this all this is really good amusement it is a practical way of thinking, but I think that there is nothing fundamentally paradoxical about it that is how the that is how the enumeration rules of quantum mechanics are, but it has become a very big issue because you do have states that are spread over space. So, the point is in quantum mechanics if you think in terms of quantum physics in space and time a quantum state is intrinsically and always delocalized. In classical mechanics somebody says there is a ball or there is a particle it is in some point in space time. Quantum mechanics is not like that ever that is the main lesson. You have a system it will be spread over the space over the basis states enumerated by its set of observables here the observable being location. Generic state will always be a linear combination of a lot of special Eigen values Eigen vectors and therefore, it looks like it is a delocalized system, but whatever that non-locality is valid it is actually physically correct it does not violate the relativity principle and is the integral part of it is the postulate number 7 of quantum mechanics. So, you can either call it a postulate or you can call it a paradox the second one will provide more amusement than the first. So, that is the end of my summary of quantum postulates.