 I will probably skip several details, nevertheless I will certainly like to introduce some ideas. The reason is that they are really important and sometimes these topics are considered to be advanced topics, which is really unfortunate because Relativistic Quantum theory means it's Dirac formulated this in 1928. So nobody over here was born at that time, including me, who is older than everybody else. So this is really not really modern in some sense. Okay, it has been there for a long time and it is important that our students who will be competing with others in their age group are familiar with these techniques because otherwise they will not be able to compete in international times. So these need to be introduced and once again I will like to thank the NPTEL administration, the leadership and the technology support which is absolutely superb and we had some introduction to topics and atomic physics. Dr. Arvind told you how to do spectroscopy with quantum mechanics and how quantum mechanics actually takes you, enables you to carry out observations and understand the physical universe around us because that's what spectroscopy is about. So one thing I would like to point out is Relativistic Quantum Mechanics because there is often a belief that quantum mechanics is important for tiny objects, microscopic objects. Relativity is important for those objects which are moving very fast, not at the speed of light, nothing goes at the speed of light except light itself but that objects which go at what we call as Relativistic speeds like 0.6 times the speed of light or 0.8 times the speed of light and so on. Now both of these notions are incorrect because quantum mechanics is applicable to everything no matter what the size is. When we introduced quantum mechanics we said that okay it is because you cannot measure position and momentum you know these are not compatible. Did we say that what is the size of the object? What is the volume of the object? What is the mass of the object? We never refer to these parameters and therefore quantum mechanics is applicable to everything to this object to large sized objects like you and me including me which is larger than anybody else and also the Earth itself the galaxies and you can write the Schrodinger equation for a galaxy and do astronomy using quantum theory because it is not at all a requirement that quantum mechanics should be used only for microscopic objects for all the microscopic phenomena for everything that happens over your quantum mechanics is applicable not only applicable it is necessary it is mandatory it is required the reason you are sitting on your chairs and not falling through the floor for example is because of certain interactions at the floor and these interactions for their completeness require a quantum description okay there are electrons and charges in the floor material in your feet and so on and unless all the interactions between them are correctly taken into account which includes the Coulomb interaction they include the exchange interaction the Fermi direct statistics okay so unless all of these interactions are taken into account correctly you will not be able to account for the fact that this object is not falling through the table okay so quantum mechanics is applicable to everything likewise relativity is applicable to everything not just to objects which are moving at relativistic speeds if an object is at rest it doesn't mean that it is free from relativistic effects I mentioned the sodium atom spectrum the spin orbit splitting between the 3p 3 half under 3p 1 half level that splitting will hold good for a sodium atom even if you bring it to a halt okay in laser cooling that's what you do you bring an alkali atom to some sort of a rest or as goes to rest as you can get and it will have the spin of its splitting and there is no way you can account for the spin of an electron without relativistic effects so these are some of the things that I would like to highlight very quickly so you have the Schrodinger equation over here and the Schrodinger equation as you can see has got the distance term okay V is a function of R and distance undergoes Lawrence contraction okay so the Schrodinger equation is not Lawrence invariant and that is the reason you need a relativistic formulation of quantum mechanics so you already realize that you need state vectors to describe the state of a system but then you also need the evolution of the state of the system to be described by an equation which is relativistically invariant which is Lawrence invariant which is what for electrons is the Dirac equation and events then take place in a four-dimensional world which we often call as the Minkowski world it all of the space-time continuum so I will not be going through all of this in detail because we are toward the end of the day but all of these points are discussed at considerable length in the two courses on atomic physics so in in different modules you know so some part is in module number three some and some other module but if you integrate you will find all of these ideas discussed at great length so quantization is not about discreteness it is about you know treating these quantum dynamical variables as operators representing the state of the system by vectors in a Hilbert space and then developing the algebra further to extract physical properties and when you do this you have to also take into account the Lawrence invariant so when you quantize momentum and instead of momentum we introduce operator gradient right minus i cross gradient was our operator for momentum and what is momentum momentum is mass and velocity dr by dt so it is a ratio of space over time and neither space nor time is Lawrence invariant okay so you have to have a more refined idea about these parameters before you quantize because quantization will involve having a appropriate operators for the momentum and what would be be these operators has to has to be introduced correctly and precisely so it takes a little while to do that so I won't get into these details but invite you to the full lectures which are integrated in this course so it has got some important consequences so I let I'm going to skip several slides but I would like to highlight one important aspect over here that if you consider this metric this p mu p mu and this is manifestly Lawrence invariant and this is what you would use to quantize the system so the momentum that you will be referring to would be coming from a relationship which is Lawrence invariant and this has got some important upshots you deal with the mass m and what is this mass m okay so there is a very famous equation e equal to mc square which is not correct okay it is not correct the correct equation is gamma mc square and you can see this very easily because if you put if you apply this equation for a photon which whose rest mass is zero the e equal to mc square would give zero energy for a photon and you would wonder what happened to that h mu which we have been using for the energy of the photon right so e equal to mc square is not correct and one really has to use equal to gamma mc square where gamma is root of one over root of one minus v square by c square and then you can expand this as a power series and then get various terms you get the rest mass and then rest mass energy and then you get the relativistic kinetic energy and so on so I won't go into these details but you will find that this equal to mc square will not give you the correct expression for the photon but if you use gamma mc square what does it give you for a photon for a photon m is zero so the numerator goes to zero but v is equal to c so the denominator also goes to zero so it doesn't give you a wrong answer it gives you an indeterminate quantity and it tells you that this is not the way to get the energy of the photon okay whereas equal to mc square would give you a wrong answer and an indeterminate answer is a better answer than a wrong answer because when it is indeterminate we learn that okay this is not the way to get the expression for the energy of the photon what you should do to get the energy of the photon is to use the relativistically invariant relationship and that will give you for the photon e equal to pc and that is no surprise because energy being associated with my initials is no coincidence at all and that is the correct expression for energy it will give you h nu as you should get right so so what has to do these things very carefully very rigorously and some of these things are often skipped out in many courses at the undergraduate level or even in the msc courses so i will not both of these details but then we have to we require a correct equation which is relativistically invariant to describe the state of the system and how it evolves with time which is the fundamental question in mechanics and that is done for an electron using the dirac equation so let me skip all of this but let me flash the dirac equation here it is at the very top of the slide what is in this red box over here is the dirac equation and what we call as the standard form and all along we have been saying that the electron has got an angular momentum which we call as orbital angular momentum we said that it has nothing to do with an orbit okay what is it it is an angular momentum which satisfies certain commutation rules and from that angular momentum we can get a magnetic moment and you see that magnetic moment in the z-man effect right and that's not the only magnetic moment you see in the z-man effect you see an additional source of the magnetic moment which must come from an additional source of angular momentum which is the spin angular momentum which we insisted has nothing to do with the kind of spin that you talk about when you talk about the earth's rotation or the spinning of a top right so if electron spin is not spin in this sense if orbital angular momentum has nothing to do with orbits of the sense then where does this spin come from so it comes from the dirac equation it comes from the relativistic dirac equation and here you have the dirac equation in front of you and do you see the spin over here do you see the do you see the spin orbit interaction the spin orbit interaction which we used in explaining the z-man effect and the anomalous z-man effect there was an s dot l term where is this s dot l in the dirac equation yet we argue that it is there and you have to dig into the dirac equation to find it okay and the way to do this is to carry out certain transformations of the dirac Hamiltonian and these transformations are known as fully Wodai's transformations which were worked out not so recently 1950 so there is nothing modern about this okay and this is the quantum mechanics that students really need to learn there is nothing new about it there is nothing advanced about it because it is quite old quantum mechanics it's more than 60 years old and for any student who is doing a masters in physics he has to know this so that he can really do some competitive physics which is at the very frontier of research in physics so one has to be introduced to what are known as the fully Wodai's transformations and what the fully Wodai's and transformations do is to carry out the dirac equation which is here so you write the dirac equation in a form which looks so similar to that of the Schrodinger equation but you carry out a transformation which are known as fully Wodai's and transformation and you carry out a transformation once and twice and three times and then rewrite the dirac equation in a form which in which after three fully Wodai's and transformations the H triple prime which is the third Hamiltonian in this sense has got this form and now you see the sigma dot L or the S dot L spin orbit coupling and you see the electron spin coming nicely out of the dirac equation but not unless you subject it to all of these transformations so these are some of the things that we deal with in these courses on atomic physics and the subsequent follow-up course which is called as the theory of atomic collisions and spectroscopy so it is abbreviated as TACS for theory of atomic collisions and spectroscopy but it rhymes as stacks and students normally say that okay this is a course which acts as them so it is again not a coincidence that it is called a stack so anyhow we also deal with how an atom is to be probed now can you if you have a target whatever how would you probe it you will probe it with something right so what will you probe it with you can probe it with light by electromagnetic radiation or you can fire some particles like electrons positrons or anything so is there anything else that you can think of either elementary particles or composite elementary particles like atoms or alpha particles or even molecules you can fire molecules and so on and when you do light you will say that you're doing spectroscopy when you do it with material particles you say that you're doing quantum collisions and these two are actually two facets of essentially the same physics so spectroscopy and quantum collisions are really not really two separate things you can see from this picture so if you look at the final state over here you have got an ion and an electron and you can get this final state through electron ions scattering experiment like this but you can also take a neutral atom and photo ionize it okay an electromagnetic shine electromagnetic energy on that and a photon is absorbed leaves an ion and an electron is kicked out so you'll you'll get a final state in which you have got an ion and an electron but you the initial ingredients are completely different in one case it is an electron and an ion in the other case it is a neutral atom and a photon so if you come this way you're doing spectroscopy if you come this way you're doing quantum collisions and it is not surprising that you should have the same final state in this case because these two processes are related to each other through a symmetry and this symmetry is what is called as the time reversal symmetry and it is not the same thing as t going to minus t as we saw in classical mechanics but it was all right for classical mechanics but when you do it in quantum theory you have to do more than that that t going to minus t is usually accompanied by the wave function undergoing complex conjugation so these are some details that I will not discuss at length we are toward the end of the day but you have the quantum collision and for organization processes related to each other through time reversal symmetry so we deal with quantum collisions the essential idea is how are you what kind of boundary conditions are you using over left and there are boundary conditions which are known as outgoing wave boundary conditions and in going wave boundary conditions so you you often say that e to the ikr by r is a spherically outgoing wave and it is meaningless to say that it is an outgoing wave without specifying what is the time dependence of this wave function and for a stationary state it is e to the minus i omega t so when you plug in the time dependence you have got the phase which goes as kr minus omega t and the surface of constant phase will have dr by dt to be a positive quantity so that the radial distance will increase which is what makes it a spherically outgoing wave so if you impose the in going wave boundary conditions you can get the solution for quantum collisions so these are determined they also determine the normalization of the wave function so these are several aspects that we deal with in these courses you can write the solutions not just as a differential equation like we do using the Schrodinger equation you can use the integral equation using the green's functions with appropriate boundary conditions and when you do that you get the Lippman Schrodinger equation for potential scattering you can develop sub approximation schemes like the bond approximation which is not just one approximation but it is a series of approximation so it is called as the bond series or bond approximations plural as I like to call it which comes from this multiple scattering process so these are some of the details that we discussed we also get the solution for the Coulomb scattering for which the usual methods of scattering really break down although it is a simple potential the one-over-all potential is the most common one and the simplest one as one might think but it does require some special techniques using solving the problem using parabolic quantum coordinates and some techniques from contour integration so these have also been dealt with in this course and then we deal with a very fascinating topic which is that of resonances and this I believe is an extremely important topic there are different kinds of resonances there are the shape resonances there are the final fresh bag resonances and the techniques are important not just in atomic physics but in all branches of physics in fact some of these techniques were developed by Wigner in the context of nuclear physics there are resonances and then there is shape analysis of these resonances Fano's paper has more citations than most other papers in literature so the bright Wigner relationship for resonances on the Fano analysis and so on so we have dealt with some of these topics then these resonance profiles because they have very many different kinds of shapes so these can be analyzed using the Fano parameterization of the resonances we also deal with second quantization because these are powerful techniques and when you deal with a many electron atom you will need to go beyond the Hartree fog to take into account the electron correlations and when it becomes necessary to do that it becomes important to use methods of second quantization and then going for approximations is something that you cannot escape from okay because if you're dealing in classical mechanics even with a three-party problem you do not have analytical solutions which is why Poincaré and so on they ran into this theory of chaos okay the non-linear dynamics and the whole theory of chaos emerged from the fact that you could not get stability of the three-body system it became chaotic so when you deal with quantum phenomena forget the three bodies even for one single particle you need the uncertain de-principle and you cannot avoid statistical mechanics for that okay so like I said you need quantum mechanics not just for microscopic phenomena but they are applicable to objects of all sizes including earth sun galaxies you need relativity not just to deal with objects which are moving at speed of light but even for objects which are at rest because an electron at rest will have an intrinsic angular momentum which is a spin a sodium atom at rest will have the spin orbit interaction which will give you the d1 d2 lines okay and likewise you need statistical mechanics not just because you have a large number of objects and you have to do averaging to get the average properties like temperature and so on as we do in classical thermodynamics but you need statistical mechanics even for single particles for which you have uncertainty principle but not just for a single particle even for vacuum okay and brown said it in his book that if you are looking for exact solutions having no body at all is already too many so you need powerful methods and you need methods of second quantization because to get electron correlation which is a many electron problem for any atom and atom will have got n number of electrons a molecule has those many a bulk matter you talk about the electron properties of materials say that okay here is a metal this is a dielectric this is an insulator and so on and you try to explain it in terms of the chronic penny model a single particle Schrodinger equation so there are serious limitations because even if you take a small piece of any solid the number of electrons over there may just think of the Avogadro number and then every atom will have got so many electrons so you really have a large number of electrons and you cannot solve this problem exactly you need approximation methods and to develop these approximation methods you need methods of second quantization so you work with electron creation and destruction operators write the Hamiltonian in the second quantized formulation and then develop these approximations a very famous approximation is the random phase approximation which was developed by Bowman pines and people in condensed matter physics may be familiar with it this is the one which gives rise to these plasma on excitations okay so these are the collective oscillations of an electron gas and these are extremely important in modern technology in nano science and so on one has to understand these processes one also makes use of diagrammatic perturbation theory there are other ways of getting developing these approximations using Feynman diagrams and so on which also we provide a brief introduction to so we deal with this boom pines method of canonical transformations then we introduce the Dyson chronology operator and so on so doing the second quantization and using adiabatic switches to control these correlations so these are fairly sophisticated techniques but then they are not really modern they have been there for a long time and anybody who is graduating with a degree in physics with a master of science in physics needs to have a very good acquaintance with these techniques so these are some of the methods which we introduce for its applications and atomic physics but they have applications not just in atomic physics but also in all domains of physics including nuclear physics and this matter physics and so on so these are the get to be quite complicated but then it is not very difficult to handle these terms if you introduce some tricks and these tricks appear in terms of the Feynman diagrams and they make it very easy to deal with what looks like very complicated terms so we provide a little bit of introduction to that so I think I will stop here I will be happy to take some questions and then we will have an open session in which everybody can not just ask questions but also comment on how some of these things could have been done better your suggestions are always welcome and like Andrew and Patap said that we could you know develop collaborative courses for your students in your colleges so thank you all very much we offer physics courses to engineering students and then we also have programs which train scientists so the physics department for example offers what is called as a dual degree program which includes a bachelor of science and the master of science so this is an integrated bsms program so the intake is from the jee and they do not join a b-tech program but instead they join a bsms program and then we also offer a degree in engineering physics which is a b-tech in engineering physics and all the physics courses are offered to them but a good number of physics courses are taken by engineering students there are some which are mandatory there are some courses which are mandatory and there are some courses which are elective but a combination of these courses does provide an opportunity to all students whether he's a student of the dual degree program or a student of engineering to take all kinds of physics courses which are not just the basic core courses like ph 101 and so on which is taken by everybody but also electives and atomic physics condensed matter physics particle physics cosmology strain theory so all of these courses are available to both bsms students to b-tech and engineering physics and also to any student who is doing any engineering degree including civil engineering ocean engineering electrical engineering no matter what branch he goes for she or he can take these courses any other question as a matter of fact the undergraduate teaching is a very large part of the teaching program at it madras and one of the programs which we all enjoy very much yes yes yes so we have a very strong component of undergraduate teaching it's a lot of work and a lot of fun any other question or comment was the lunch all right well you need to give us some feedback so please please feel free to comment on anything whether it was the lunch or the coffee or the lectures or anything so actually yes the NPTEL has already prepared these certificates and I think you just have to collect them from the administrative staff these certificates are ready I believe and before you leave the premises do pick up your participation certificate and I want to personally thank each one of you for spending your entire day over here I realize that it may have been tiring but I certainly enjoyed it and I hope that you have something nice to take back home I'm sure that NPTEL is interested in reaching out to the physics faculty or not just to physics but also to other disciplines because there are courses in other branches of science and engineering and also fine arts like English literature even music you have seen that there are some courses so yes NPTEL will be making some of the some of the other attempt and Pratap and Andrew have already given some idea about it and they will be the best person to contact for details but I'm sure they do have a very positive initiative in this direction yes we have the NPTEL staff over here first of all every exam is an objective exam okay because there is nothing subjective that you can write in science okay so every exam is essentially an objective exam it doesn't necessarily mean that the questions will be fill in the blanks or fake true or false kind of statement so it will be a combination of those and whenever students ask me this question I always tell them that till I set the question paper I will not know what the question paper will be like and I think the student has to be ready for any kind of question it will basically test what you have learned so I don't think that that is the question which is which one should even worry about it is best to discourage students from raising such questions because they are actually asking for the question paper so so let's not get into that it will usually it is a mix of you know different formats and depending on you know whether you have one hour for the exam or two hours for the exam or three hours for the exam we usually design a test which will be a mix of various formats but that's just a matter of formatting a question because almost any question you can you know flip and format it in some other form so that that's an issue which is just of formatting a question and seeking an answer but all the questions will require an objective answer there's absolutely no doubt about it yes any other question or comment yes yes well right to them right to them you're to them because mean they look after the NPTEL administration and if you write to me I will forward your letter to them so you're quite free to write to me as well but essentially yes sure sure so so any college which is interested in this should write to NPTEL if you write to me I will get the answers from the NPTEL and they will be in the position to take a decision and then work out the logistics but in principle yes it is certainly possible for teachers to come down to your college and interact with the students directly it's certainly possible details will depend on what exactly the proposal is and so on so I think the devil is in the detail so let's let it come but yes I'm sure there will be a very positive response from the NPTEL NPTEL is all about reaching out so I'm sure that the NPTEL administration will do everything that is possible to meet these courses and the instruction and the material which is generated become easily accessible to whoever wants it so that's the whole philosophy of the program absolutely yeah the way we should put the quantum mechanics is very nice it needs a CG coefficient and a quantum mechanics yeah and the next lecture the next problem will be taking spectroscopy he won't explain that how the quantum are coming yeah absolutely they're just starts with jim and the spectroscopy because the nice connection begins with quantum and spectroscopy very cool he said you have done it will be useful so that you're actually the teachers should not say I don't know yeah you have to just go to the store absolutely yes really very properly yes sure absolutely yeah thank you for arranging on making knowledgeable that some workshop is useful for the teacher so we should take it right thanks a lot for that well thanks to NPTEL for providing the infrastructure and the facilities for that so very thankful to all of you