 So, we are coming toward the end of this course actually, so there are just two more classes including today, today and then one more. And we have come a long way in Fano's analysis of the interaction between discrete bound to bound transitions and bound to continuum transitions. So, we have a situation in which a bound to bound discrete state is embedded in the continuum and we constructed the energy matrix in our previous class which has these elements right. And then we set up the configuration interaction which has got both the components the discrete state as well as the continuum because of the C i this is a result of the correlation which is ignored in the independent particle approximation. And then we used the Dirac Fano trick of using this Dirac delta function integration whenever the coefficient B would appear in the integrand. So, using that we agreed that the function z or z would be would correspond to appropriate boundary conditions and which is what gave us this form and we found the coefficient A e according to this. And I had pointed out that V e has the dimensions of a root of energy, so V e square would be the energy and it would correspond to the energy width of the resonance. So, now you have the complete configuration interaction wave function. So, it has got a discrete part then it has got this principle value integration and it has got this continuum state with scale by the factor z which is coming from the boundary condition. Now this is the width the gamma this is the width of the resonance and as you always have you know it is reciprocal scale by the factor h cross would give you the lifetime of the authorization state. Now our interest as we always emphasize in scattering theory is in seeking asymptotic solutions what happens as r tends to infinity right that is where you carry out your measurements. Now this discrete state function this is the bound state function, so it would vanish in the asymptotic limit, so there are 3 terms over here 1, 2 and 3 out of which this term goes to 0 as r tends to infinity ok, so we can drop that off ok. Now we are left with these 2 terms out of which this particular one is something that we have some familiarity with from scattering theory it is the principle value this is the term that we will have to determine. So now the r tending to infinity solution will be given by the remaining 2 terms, so these are the 2 terms which we are now left with for our analysis and now I have these 2 terms which describe the asymptotic wave function, so this is the de Broglie Schrodinger wave function the coordinate representation of the state vector and this one coming from scattering theory is the usual sinusoidal function by r plus the usual scattering phase shift which is in this case is the background phase shift. So this is known and what we have over here is pretty much the same except for the fact that this is weighted by this interaction term V e, now V e if you remember is let me go back to the previous slide if I have it on this slide or no I do not have it on this slide do I have it on the one prior to this here V e is nothing but the energy contribution to the energy matrix coming from the interaction between the discrete and the continuum right it is this interaction which is resulting in the V e, so that is the configuration interaction term which is contributing to that. So let me go back to this slide over here and here this wave function here is pretty much of this kind except that it is weighted by this configuration interaction V e right, so this has this usual sinusoidal behavior divided by r and then there is a phase shift coming from the background scattering phase shift then there is an energy dependent normalization this is the bound state wave function which is normalized independently it is the square integrable function. So this is well understood and this is the background phase shift as I mentioned earlier so we can write these two terms and look at the second term here the second term is just what I have written over here, so what is in this red box comes here with this explicit form inclusive of this energy dependent normalization the sinusoidal function by r and the background phase shift which is here. And now we look at this term out of which A e V e are here right the chi 0 r 2 coming from here comes here and then you have got this term which is similar to this except for the fact that now you have this A e V e prime popping up. So this is what takes care of the configuration interaction but the only thing is that now you have to deal with only the principal value of this integral. So one can work out this principal value integration in some details and all of you would have done this in your math physics courses and instead of going through all those detailed mathematical steps I will comment on the essential physics which goes into this analysis and here notice that this term is over here what you should remember that this eta e is a function this eta is energy dependent normalization it is a function of energy and this eta is the same function of energy. So if this is known this is also known so that is coming from usual scattering theory. So these two are essentially the same functions of energy except that the argument is different. So now our question is how to determine this principal value integral and that is what I will now discuss and I really like the treatment in this paper which came about ten years ago by A R P Rao who was I believe Fanot's first student and he has this paper the title of this paper is perspectives on Fanot resonance formula which he wrote in physical script quite recently just ten years ago considering that Fanot's paper is so old it is relatively recent and what Professor Rao said yes is that we focus attention on the essential dynamics which is going into it rather than get lost in notation symbols and other factors which are not contributing to the main terms which are the focus of our discussion. So when you write the complete wave function you will of course have lots of factors you will have to worry about the logarithmic coulomb phase shift you will have to worry about the phase shift coming from the orbital angular momentum there is always this l pi by 2 phase shift which we all know right. So all of these terms will be there and he says that we will remove them from our discussion to simplify our notation and to simplify our discussion so that we focus on the essential aspects. So I am following this prescription by Professor Rao and what Rao does is to point out that the asymptotic structure of the pure continuum is given by this regular function which is a sinusoidal function okay the radial function of course has got that 1 over r so this is the radial function times r okay now this is the pure continuum. Now we do not have the pure continuum what we have is a configuration interaction between the discrete and the continuum so now when you have a superposition of a discrete and a continuum then the regular part alone will not be able to give you the full wave function you will need an admixture of the regular part so the solution will be some sort of a linear combination of a sinusoidal and a cosine function and whenever you have such possibilities then you invariably have a phase shift resulting from this okay which we have seen n number of times in our analysis of collision physics. So this will generate an additional phase shift and this is coming from the interaction between the discrete and the continuum and this is the configuration interaction so this results in an additional phase shift so that is what we are going to focus our discussion on today. You can also have additional terms in the potential for example in more complex forms of the potential when you take more realistic potentials many electron system then you know that the SO4 symmetry of the hydrogen atom is broken inside you know close to the core you have got a very complex form of the potential the potential is no longer 1 over r over the entire region of space and what one often does in other formalisms like quantum defect theory which is to partition the space in a short range region and a long range region when you do that you can separate the total phase shift in a short range part and a long range part when you do that again you will have a such a separation which is possible and you will have additional phase shifts which result from that. So our interest is in handling this particular integral out of which this part will go into the normalization as you can see this is scaling this wave function and the essential problem is in the evaluation of this integral other than the normalization part which goes into the integration of this function sign over the energy difference that is the main integral that we have to solve and we already know that these integrals can be solved using the Dirac-Fano procedure. So you insert this prescription which we have used in our discussion in our previous class and here this C e prime has been appended so that it will take care of this normalization A e v which is coming here that is this additional normalization this one is already there in the other normalization of the scattering solution. So this is contained in C e prime. So now this normalization has got v e which is nothing but the coupling matrix between the discrete and the continuum as I pointed out. Now this function over here has got 2 terms odd and even. So you can take 2 other terms like if you can write your sinusoidal function this one in terms of cosine k prime minus k and assign k prime minus k and this is just a trigonometric identity that you exploit. So you have got a correspondence from which you can determine the phase shift in terms of the z and c. So that is the trick that is the essential trick then rest of it is just plugging in and evaluating the integrals which is of straightforward procedure. So using this you now get so this is the expression for the sinusoidal function in terms of this trigonometric identity and now the solution is now just a superposition of this sine and cosine and this is this integral is easily solved the solutions will give you z e and pi factor. And I have just moved this C e inside this so that I have got C e z e and sine k r plus C e pi times cosine k r and you can see an additional phase shift popping out of this and you can get it if you introduce this angle delta this triangular delta as I have used. And if you introduce this delta as tan inverse as negative tan inverse of pi over z you can write this function as a sinusoidal function with a phase shift. So this is the additional phase shift which is resulting from the configuration interaction. Now this is what we have been expecting that the configuration interaction will essentially generate an additional phase shift. So that is this delta and the actual function that we wanted to work with was this so we used the simple analysis from Raou's paper. So we use that result and put it over here get the solution in terms of the cosine and the sine functions get the corresponding phase shift which is nothing but the same exactly the same expression which is negative of tan inverse pi over z ok z we have determined earlier from the boundary condition. So this is what the configuration interaction does which is to generate an additional phase shift. So the configuration interaction results in an additional phase shift which is negative of tan inverse pi over z. So this is the additional phase shift and your complete wave function now has got this one. So we have got the complete solution now. So the background phase shift was already there the kr term is always there right. We are dealing in this case the we dealt with l equal to 0 state so there is no l pi by 2 phase shift but otherwise that would also be there. So in more general forms of the solution you will see some additional terms and then you have got this resonance part this is coming from the resonant auto ionization resonant part of the interaction. So this is the additional phase shift z we have already introduced earlier. So this is just to summarize the essential results ok and notice that you have got the difference between the energy and E phi where E phi is the contribution to the energy matrix from the discrete states ok phi d is the discrete state. So the farther E is from E phi ok the lesser the contribution you would expect as is natural to C ok because if the discrete state is not close enough to the actual resonance process if the continuum energy is not close enough to the resonant process then the resonant part of the phase shift will become small ok. So if E minus E phi is large the resonant part of the phase shift will become small alright. So what happens is that over this width gamma which is which goes as V square over this width the resonance part of the phase shift however changes rapidly and it goes through pi as we have discussed in our previous classes. So the resonance energy as you see is when this denominator would vanish so this is when E r would become E phi plus F E so that becomes the resonant energy. So the resonance energy is not necessarily the energy at which the cross section is maximum or minimum the criterion for the resonance energy is this it is it relates to the phase shift and not to the cross section cross section and everything else comes as the consequence of that. So I would like to highlight some important references in this context in addition to Fano's original 1961 paper in which many things are worked out in the appendix of the paper. Macy and Burhip is an excellent source which is chapter 9 in their book which is volume 1 of this then Carvin's book I have referred to I have included that reference and this paper by Professor Rao which I cited just a little while ago. So these are excellent sources for a discussion on this point. Professor Rao also points out some very nice things which are nice to visit that if you look at the usual expression for the scattering cross section in terms of the sine square delta this is the usual expression you can use a simple trick to write this sine square delta in terms of these two cotangents of two different angles delta minus delta a and cotangent of delta a. So this is just a trigonometric identity and Macy and Burhip also do a similar thing but in a slightly different way and using this you can immediately see them relate to the Fano parameters q and epsilon. So they come so neatly out of this. So there are some very nice things which you find in Rao's literature and also in Macy and Burhip. Macy and Burhip do it in a slightly different way I will show you how they do it. So they go it is essentially the same idea so they go back to this expression in terms of the modulus square of the scattering amplitude and then break up the phase shift into the background part and the resonant part as we are quite familiar with already. And then they use certain trigonometric identities in terms of which they write this scattering cross section this is after doing a little bit of trigonometric manipulation and then what they find is that if you ignore the background contribution you get essentially the Breitwigner formula which we have already seen how it relates to the Fano parameters q and epsilon. So these are certain profiles that you get for this scattering cross section and the scattering cross section actually comes in all kinds of shapes. So these are different profiles naturalized shapes for different values of the q parameter of the Fano q parameter you can get very many different kinds of shapes and the resonance energy is given by this criterion E phi plus phi E as I mentioned earlier. This is the resonance width this is the width over which the phase shift changes rapidly through pi. The width is determined by the energy matrix contribution coming from the interaction between the discrete and the continuum. So that is what determines the width of the resonance. Lifetime is just the inverse of this width and one must remember that the idea of lifetime is quite different from the idea of what is called as time delay in scattering. So I will spend some time discussing the time delay in scattering. So before I do that I would like to show that some of these ideas are of immediate interest for the analysis of authorization resonances in every such group. So these are Moody's results and what there are other additional complexities because we have just dealt with the lifetimes and so on. But then what about the character of the resonances and so on there are many additional complex questions which come up and those are left for those of you who will reach original literature and move ahead but these have immediate interest in several research problems. Here is a figure of a funnel resonance, Shiva's result. So thank you very much Shiva for lending me this picture. Shiva also told me that the quality of the signal is considered as a measure of how good the setup is. So the next generation of the synchrotron is identified by the quality of the spectrum. So the helium spectrum is still used in characterizing and in calibrating in particular all the calibration procedures. So these are very exciting things and we have people over here who work with these things so it is nice to have you all in the audience. And I will now come to this time delay in scattering which is an idea which is different from lifetimes as I mentioned. And this comes from the analysis of Eisenwood and Wigner and the original analysis came in the context of phase shift analysis of S wave scattering at low energies. So typically if you have a free electron wave packet it is a superposition of a number of plane waves and this wave packet travels at a certain group velocity that group velocity is d omega by dk as we know from you know our studies of wave packets. There is a certain mean momentum okay which I represent by p0 corresponding to which I have got a wave vector k0 and what happens after the scattering if this wave packet meets an interaction potential it continues to propagate it continues to propagate at the same group velocity. The group velocity does not change there is it is transmitted through the barrier so let us think of a simple one dimensional case it goes through that but then it spends a little bit of time in the interaction region so that the transmission coefficient will be complex and there will be a phase delay coming as a result of that. So phi is a phase factor and the transmitted wave packet because of this phase factor it would appear whenever there is a phase lag okay what is the phase lag represent it says that okay if a wave front was to arrive at a certain time it is arriving either sooner or later. So there is a phase shift right and that means that as if it has originated from a different point of space that if it were to have originated from a certain source point if it is arriving earlier it would appear as if it has arrived from a point whose source is not here but closer if it is later it is like coming from a farther distance. So there is a spatial phase shift associated with this and this phase shift it would appear as if it is coming from a different point which is d phi by dk okay rather than from x0. So d phi by dk then is the spatial phase shift and if you divide this by the group velocity you will get the time delay right. So the time delay is nothing but this spatial phase shift divided by the group velocity and you can immediately see that this is nothing but the energy derivative of the phase shift. So this is the time delay in scattering theory our interest of course is in scattering in biospherical potentials and we are in a position to discuss this at some length because we have dealt with this in an earlier unit in considerable detail and in particular I will like to draw your attention to the lecture number 8 in unit 1 of this course okay and there is a discussion on slide 147 through 150 where you will find these details and you have already been through this but let me remind you some of the immediate results which are of interest to us. What we discussed at that point is that the radial function if there is a 0 potential so it is like a free particle which is propagating its solution will go as a sinusoidal function apart from the angular momentum phase shift l pi by 2 whereas when you have a potential there is a phase shift resulting from the scattering potential right. What it does is it changes the positions of the nodes because the node will come wherever the sinusoidal function goes to 0 and the argument of the sinusoidal function being different the position of the nodes is shifted okay. So we had drawn these pictures from Josh and his book okay and we discussed these solutions that the nodes are either pushed or pulled depending on the potential being either repulsive or attractive and accordingly you get a negative phase shift or a positive phase shift okay. So these details are there in Josh's book we discussed them in unit 1 and this is what happens that the positions of the nodes are given by what is the value of this right argument of the sinus function and in this case when it is pushed when the radial function is pushed the scattered wave emerges ahead of the unscattered wave and the unscattered wave is a free particle wave right. So the scattered wave will come ahead of that whereas in this case it is delayed it will come behind the unscattered wave and that is the time delay. So time delay will be either positive or negative depending on whether it is pushed or pulled okay. So there is an advancement of time or a retardation of time which goes according to this. Now we were very careful about it because we need to treat this as a wave packet okay because the electron is ejected as a wave packet and we conceded that this particular form of the scattering solution with the outgoing wave boundary conditions okay inclusive of the time factor this is written for a strictly monologetic beam which is an idealization whereas the actual particle will be described by a wave packet. So this is a realistic incident packet which is a superposition over a number of different momenta okay. So this may have a certain narrow range but whatever but it is not sharply mono energetic. So you have a realistic incident wave packet and we had asked the question if this relation d sigma by d omega being given by the modulus square of the scattering amplitude does it work also for a realistic incident wave packet when it is described by a superposition of a number of mono energetic waves and not just a mono energetic wave. So some of you will remember this discussion from unit 1 okay but otherwise you can refer back to those lectures and this point was discussed over a number of lectures in that unit lecture number 4, 5 and 6 of unit 1. So this is just to remind you of way to look back for the details and here if you look at this incident beam it has got this complex amplitude for each monochromatic wave. So this e to the i alpha k phase gets added up to this phase so you get a net phase of this function which is given by e to the i beta k and if beta were to change rapidly with k okay there would be so many oscillations that the incident packet would disappear. So the condition that it would not happen so that you have scattering at all is that the gradient of beta would be 0 right. So that is the condition that we discussed and when we put this condition we found that the wave packet moves at a certain group velocity which is given by this gradient of omega with respect to k as we always find right. So this is the relation that we get and you can write it in terms of you know like a kinematic equation r equal to vt plus r0 like in undergraduate mechanics courses right. So where the velocity is given by this gradient of omega with respect to k so you see that similar relations as we dealt with in the one dimensional case are present over here as well so we get essentially the same kind of consequence. We also considered the incident wave packet to come not head on at the target but if it is slightly displaced through an impact parameter and when we considered this we found that we can still write the scattering solution in the same form except that we now have an additional function which described the shape of the wave packet right. And we found that the scattering amplitude is given by its modulus and a phase factor and this phase is no longer just the phase of the mean momentum wave packet with the mean momentum but you have to expand it around it in a Taylor series. So you have got the leading term plus the first derivative times the difference of the momentum vectors. So this is what this gradient of the phase with respect to k is what is written by the by Rho this is I am following the notation from Jochen's book quantum collision theory and now you have got this phase which has been written which has been expanded about this mean momentum the phase for the mean momentum. The net phase is this this is the total phase right so this is the total phase which I call as zeta and this is the net phase angle of which these this part is given by these two terms are come they come over here and over here I simply separate the magnitude of Ki from the direction of Ki. So the last equation is nothing but the one prior to that except for the difference that I have separated the magnitude from the direction of this Ki and I do it for a reason because now you have this as the phase angle and you can ask the question how would you describe the surface of the scattered wave which is propagating along this particular unit vector and on this surface you would expect that d zeta by dk must vanish because that is the criterion for having a surface of constant phase. So this must be the criterion so you can put d zeta by dk equal to 0 what does it give you you just take the derivative of this term with respect to k put it equal to 0 you get an equation again the same kind of kinematic equation and you find that there is a spatial phase shift you divide the spatial phase shift by the group velocity you get the time delay. So this is the time delay it can also be a time advance of course depending on whether the potential is attractive or repulsive so this is the time delay in scattering. So this is essentially coming from the gradient of the phase with respect to k and what happens is that the wave front appears to have originated from a different point rather than what would seem like the origin of the wave front if it were to be traveling only at vi velocity vi over a time t starting from the origin so the origin is displaced so that is the spatial phase shift and corresponding to this phase shift if you divide it by the group velocity you will get the time delay because any change in phase will then result in the time delay. So this is the derivative of the phase shift with respect to k d delta by dk and now if you recognize the relation between energy and momentum which is h cross square k square by 2m and write this as d delta by dk so this dk now you write in terms of de rather than dk then you immediately find that the time delay is nothing but the energy derivative of the phase shift in units of h cross so that is the time delay and this is the analysis originates in the work of Eisenwood and Wigner. Smith has some nice work on this and with that I conclude today's class in the next class I will sum up show some applications of the various topics that we have done and that will pretty much conclude the course. Any question?